research article on interval-valued hesitant fuzzy soft...

Research ArticleOn Interval-Valued Hesitant Fuzzy Soft Sets

Haidong Zhang1 Lianglin Xiong2 and Weiyuan Ma1

1School of Mathematics and Computer Science Northwest University for Nationalities Lanzhou Gansu 730030 China2School of Mathematics and Computer Science Yunnan Minzu University Kunming Yunnan 650500 China

Correspondence should be addressed to Haidong Zhang lingdianstar163com and Lianglin Xiong lianglin 5318126com

Received 16 October 2014 Revised 13 January 2015 Accepted 19 January 2015

Academic Editor Shuming Wang

Copyright copy 2015 Haidong Zhang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

By combining the interval-valued hesitant fuzzy set and soft set models the purpose of this paper is to introduce the concept ofinterval-valued hesitant fuzzy soft sets Further some operations on the interval-valued hesitant fuzzy soft sets are investigated suchas complement ldquoANDrdquo ldquoORrdquo ring sum and ring product operations Then by means of reduct interval-valued fuzzy soft sets andlevel hesitant fuzzy soft sets we present an adjustable approach to interval-valued hesitant fuzzy soft sets based on decision makingand some numerical examples are provided to illustrate the developed approach Finally the weighted interval-valued hesitant fuzzysoft set is also introduced and its application in decision making problem is shown

1 Introduction

Soft sets initiated by Molodtsov [1] are a new mathematicaltool for dealing with uncertainties which are free frommany difficulties that have troubled the usual theoreticalapproaches It has been found that fuzzy sets rough sets andsoft sets are closely related concepts [2] Soft set theory haspotential applications in many different fields including thesmoothness of functions game theory operational researchPerron integration probability theory and measurementtheory [1 3]

Research works on soft sets develop very rapidly andnow are one of hotspots in the uncertainty research Forexample Maji et al [4] defined several operations on softsets and made a theoretical study on the theory of soft setsJun [5] introduced the concept of soft BCKBCI-algebrasSubsequently they also discussed the applications of softsets in ideal theory of BCKBCI-algebras [6] Feng et al [7]applied soft set theory to the study of semirings and initiatedthe notion of soft semirings Furthermore based on [4] Aliet al [8] introduced some new operations on soft sets andimproved the notion of complement of soft set They provedthat certain De-Morganrsquos laws hold in soft set theory Qinand Hong [9] introduced the concept of soft equality andestablished lattice structures and soft quotient algebras of softsets Meanwhile the study of hybrid models combining softsets with other mathematical structures is also emerging as

an active research topic of soft set theory Yang et al [10]introduced the interval-valued fuzzy soft sets by combininginterval-valued fuzzy set with soft set models and analyzed adecision problem by the model By using the multifuzzy setand soft set models Yang et al [11] presented the conceptof the multifuzzy soft sets and provided its application indecision making under an imprecise environment Maji et al[12] initiated the study on hybrid structures involving fuzzysets and soft sets They introduced the notion of fuzzysoft sets which can be seen as a fuzzy generalization ofsoft sets Furthermore based on [12] Roy and Maji [13]presented a novel method concerning object recognitionfrom an imprecise multiobserver data so as to cope withdecisionmaking based on fuzzy soft setsThenKong et al [14]revised the Roy-Maji method by considering ldquofuzzy choicevaluesrdquo Subsequently Feng et al [15 16] further discussedthe application of fuzzy soft sets and interval-valued fuzzysoft sets to decision making in an imprecise environmentThey proposed an adjustable approach to fuzzy soft setsand interval-valued fuzzy soft sets based decision makingin [15 16] By introducing the interval-valued intuitionisticfuzzy sets into soft sets Jiang et al [17] defined the conceptof interval-valued intuitionistic fuzzy soft set Moreoverthey also defined some operations on the interval-valuedintuitionistic fuzzy soft sets and investigated some basicproperties On the basis of [17] Zhang et al [18] developed

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 254764 17 pageshttpdxdoiorg1011552015254764

2 Mathematical Problems in Engineering

an adjustable approach to decision making problems basedon interval-valued intuitionistic fuzzy soft sets Very recentlyintegrating trapezoidal fuzzy sets with soft sets Xiao et al[19] initiated the trapezoidal fuzzy soft sets to deal withcertain linguistic assessments Further in order to capturethe vagueness of the attribute with linguistic assessmentsinformation Zhang et al [20] generalized trapezoidal fuzzysoft sets introduced by Xiao et al [19] and defined the conceptof generalized trapezoidal fuzzy soft sets

As a mathematical method to deal with vagueness ineveryday life fuzzy set was introduced by Zadeh in [21] Upto present several extensions have been developed such asintuitionistic fuzzy set [22] interval valued fuzzy sets [23ndash25] type-2 fuzzy set [26 27] and type-119899 fuzzy set [26]Recently Torra and Narukawa [28 29] extended fuzzy setsto hesitant fuzzy environment and initiated the notion ofhesitant fuzzy sets (HFSs) because they found that becauseof the doubts between a few different values it is very difficultto determine the membership of an element to a set under agroup setting [29] For example two decision makers discussthe membership degree of 119909 into 119860 One wants to assign 07but the other wants to assign 09 They cannot persuade eachother To avoid an argument the membership degrees of 119909into 119860 can be described as 07 09 After it was introducedby Torra the hesitant fuzzy set has attracted more and morescholarsrsquo attention [30ndash32]

As mentioned above there are close relationships amongthe uncertainty theories such as fuzzy sets rough sets andsoft sets So many hybrid models among them are proposedby the researchers such as fuzzy rough sets [33] rough fuzzysets [33] and fuzzy soft sets [12] Since the appearance ofhesitant fuzzy set the study on it has never been stoppedThe combination of hesitant fuzzy set with other uncertaintytheories is a hot spot of the current research recently Thereare several hybrid models in present such as hesitant fuzzyrough sets [34] and hesitant fuzzy soft sets [35 36] Infact Babitha and John [35] defined a hybrid model calledhesitant fuzzy soft sets and investigated some of their basicproperties Meanwhile Wang et al [36] also initiated theconcept of hesitant fuzzy soft sets by integrating hesitantfuzzy set with soft set model and presented an algorithm tosolve decision making problems based on hesitant fuzzy softsets By combining hesitant fuzzy set and rough set modelsYang et al [34] introduced the concept of the hesitant fuzzyrough sets and proposed an axiomatic approach to themodel

However Chen et al [37 38] pointed out that it is verydifficult for decision makers to exactly quantify their ideas byusing several crisp numbers because of the lack of availableinformation in many decision making events ThereforeChen et al [37 38] extended hesitant fuzzy sets into interval-valued hesitant fuzzy environment and introduced the con-cept of interval-valued hesitant fuzzy sets (IVHFSs) whichpermits the membership degrees of an element to a givenset to have a few different interval values It should benoted that when the upper and lower limits of the intervalvalues are identical IVHFS degenerates into HFS indicatingthat the latter is a special case of the former Recentlysimilarity distance and entropy measures for IVHFSs havebeen investigated by Farhadinia [39]Wei et al [40] discussed

some interval-valued hesitant fuzzy aggregation operatorsand gave their applications to multiple attribute decisionmaking based on interval-valued hesitant fuzzy sets

As a novel mathematical method to handle impreciseinformation the study of hybrid models combining IVHFSswith other uncertainty theories is emerging as an activeresearch topic of IVHFS theory Meanwhile we know thatthere are close relationships among the uncertainty theoriessuch as interval-valued hesitant fuzzy sets rough sets andsoft sets Therefore many scholars have been starting toresearch on the area For example Zhang et al [41] general-ized the hesitant fuzzy rough sets to interval-valued hesitantfuzzy environment and presented an interval-valued hesitantfuzzy rough set model by integrating interval-valued hesitantfuzzy set with rough set theory On the one hand it is unrea-sonable to use hesitant fuzzy soft sets to handle some deci-sion making problems because of insufficiency in availableinformation Instead adopting several interval numbers mayovercome the difficulty In that case it is necessary to extendhesitant fuzzy soft sets [36] into interval-valued hesitant fuzzyenvironment On the other hand by referring to a great dealof literature and expertise we find that the discussions aboutfusions of interval-valued hesitant fuzzy set and soft sets donot also exist in the related literatures Considering the abovefacts it is necessary for us to investigate the combinationof IVHFS and soft set The purpose of this paper is toinitiate the concept of interval-valued hesitant fuzzy soft setby combining interval-valued hesitant fuzzy set and soft settheory In order to illustrate the efficiency of the model anadjustable approach to interval-valued hesitant fuzzy soft setsbased on decision making is also presented Finally somenumerical examples are provided to illustrate the adjustableapproach

To facilitate our discussion we first review some back-grounds on soft sets fuzzy soft sets hesitant fuzzy sets andinterval numbers in Section 2 In Section 3 the concept ofinterval-valued hesitant fuzzy soft set with its operation rulesis presented In Section 4 an adjustable approach to interval-valued hesitant fuzzy soft sets based on decision makingis proposed In Section 5 the concept of weighted interval-valued hesitant fuzzy soft sets is defined and applied todecision making problems in which all the decision criteriamay not be of equal importance Finally we conclude thepaper with a summary and outlook for further research inSection 6

2 Preliminaries

In this section we briefly review the concepts of soft setsfuzzy soft sets hesitant fuzzy sets and interval numbers Thepair (119880 119864) will be called a soft universe Throughout thispaper unless otherwise stated 119880 refers to an initial universe119864 is a set of parameters119875(119880) is the power set of 119880 and119860 sube 119864

21 Soft Sets Fuzzy Soft Sets and Hesitant Fuzzy SetsAccording to [1] the concept of soft sets is defined as follows

Definition 1 (see [1]) A pair (119865 119860) is called a soft set over 119880where 119865 is a mapping given by 119865 119860 rarr 119875(119880)

Mathematical Problems in Engineering 3

Combining fuzzy sets and soft sets Maji et al [12]initiated the following hybrid model called fuzzy soft setswhich can be seen as an extension of both fuzzy sets and crispsoft sets

Definition 2 (see [12]) A pair (119865 119860) is called a fuzzy soft setover 119880 if 119860 sube 119864 and 119865 119860 rarr 119865(119880) where 119865(119880) is the set ofall fuzzy subsets of 119880

In the following we review some basic concepts relatedto hesitant fuzzy sets introduced by Torra [28 29]

Definition 3 (see [28 29]) Let119883be a fixed set a hesitant fuzzyset (HFS for short)119860 on119883 is in terms of a function ℎ

119860

(119909) thatwhen applied to119883 returns a subset of [0 1] that is

119860 = ⟨119909 ℎ

119860

(119909)⟩ | 119909 isin 119883 (1)

where ℎ

119860

(119909) is a set of some different values in [0 1]representing the possible membership degrees of the element119909 isin 119883 to 119860

For convenience we call ℎ

119860

(119909) a hesitant fuzzy element(HFE for short)

Example 4 Let 119883 = 1199091

1199092

1199093

be a reference set and letℎ

119860

(1199091

) = 07 04 05 ℎ

119860

(1199092

) = 02 04 and ℎ

119860

(1199093

) =

03 01 07 06 be the HFEs of 119909119894

(119894 = 1 2 3) to a set 119860respectively Then 119860 can be considered as a HFS that is

119860 = ⟨1199091

07 04 05⟩ ⟨1199092

02 04⟩

⟨1199093

03 01 07 06⟩

(2)

22 Interval Numbers In [42] Xu and Da gave the conceptof interval numbers and further investigated some of theirproperties

Definition 5 (see [42]) Let 119886 = [119886119871

119886119880

] = 119909 | 119886119871

le 119909 le 119886119880

then 119886 is called an interval number In particular 119886 is a realnumber if 119886119871 = 119886

119880


119886119880

] 119887 = [119887119871

119887119880

] and120582 ge 0 then one has the following

(1) 119886 = 119887 if 119886119871 = 119887119871 and 119886

119880

= 119887119880

(2) 119886 + 119887 = [119886119871

+ 119887119871

119886119880

+ 119887119880

](3) 120582119886 = [120582119886

119871

120582119886119880

] In particular 120582119886 = 0 if 120582 = 0


119886119880

] and 119887 = [119887119871

119887119880

] andlet 119897119886

= 119886119880

minus119886119871 and 119897

119887

= 119887119880

minus119887119871 then the degree of possibility

of 119886 ge 119887 is defined as

119901 (119886 ge 119887) = max1 minusmax(119887119880

minus 119886119871

119897119886

+ 119897119887

0) 0 (3)

Similarly the degree of possibility of 119887 ge 119886 is defined as

119901 (119887 ge 119886) = max1 minusmax(119886119880

minus 119887119871

119897119886

+ 119897119887

0) 0 (4)

Equations (3) and (4) are proposed in order to comparetwo interval numbers and to rank all the input argumentsFurther details could be found in [42]

3 Interval-Valued Hesitant Fuzzy Soft Sets

31 Concept of Interval-Valued Hesitant Fuzzy Sets In thesubsection we review some basic concepts related to interval-valued hesitant fuzzy sets introduced by Chen et al [37]

Definition 8 (see [37]) Let119883 be a fixed set and let Int[0 1] bethe set of all closed subintervals of [0 1] An interval-valuedhesitant fuzzy set (IVHFS for short) 119860 on119883 is defined as

119860 = ⟨119909 ℎ

119860

(119909)⟩ | 119909 isin 119883 (5)

where ℎ

119860

(119909) 119883 rarr Int[0 1] denotes all possible interval-valued membership degrees of the element 119909 isin 119883 to 119860


119860

(119909) an interval-valued hesi-tant fuzzy element (IVHFE for short) The set of all interval-valued hesitant fuzzy sets on119880 is denoted by IVHF(119880) FromDefinition 8 we can note that an IVHFS 119860 can be seen as aninterval-valued fuzzy set if there is only one element in ℎ

119860

(119909)which indicates that interval-valued fuzzy sets are a specialtype of IVHFSs

Example 9 Let 119883 = 1199091

1199092


119860

(1199091

) = [02 03] [04 06] [05 06] and ℎ

119860

(1199092

) = [03

05] [04 07] be the IVHFEs of 119909119894

(119894 = 1 2 3) to a set 119860respectively Then 119860 can be considered as an IVHFS that is

119860 = ⟨1199091

[02 03] [04 06] [05 06]⟩

⟨1199092

[03 05] [04 07]⟩

(6)

It is noted that the number of interval values in differentIVHFEs may be different and the interval values are usuallyout of order Suppose that 119897(ℎ) stands for the number ofinterval values in the IVHFE ℎ To operate correctly Chenet al [37] gave the following assumptions

(A1) All the elements in each IVHFE ℎ are arranged inincreasing order by (3) Let ℎ120590(119896) stand for the 119896th largestinterval numbers in the IVHFE ℎ where

ℎ120590(119896)

= [ℎ120590(119896)119871

ℎ120590(119896)119880

] (7)

is an interval number and ℎ120590(119896)119871

= inf ℎ120590(119896) ℎ120590(119896)119880 =

sup ℎ120590(119896) represent the lower and upper limits of ℎ120590(119896)

respectively(A2) If two IVHFEs ℎ

1

ℎ2

119897(ℎ1

) = 119897(ℎ2

) then 119897 =

max119897(ℎ1

) 119897(ℎ2

) To have a correct comparison the twoIVHFEs ℎ

1

and ℎ2

should have the same length 119897 If there arefewer elements in ℎ

1

than in ℎ2

an extension of ℎ1

should beconsidered optimistically by repeating its maximum elementuntil it has the same length with ℎ

2

Given three IVHFEs represented by ℎ ℎ

1

and ℎ2

Chenet al [37] defined some operations on them as follows

Definition 10 Let ℎ ℎ1

and ℎ2

be three IVHFEs then one hasthe following


(1) ℎ119888 = [1 minus 120574+

1 minus 120574minus

] | 120574 isin ℎ

(2) ℎ1

cup ℎ2

= [120574minus

1

or 120574minus

2

120574+

1

or 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

(3) ℎ1

cap ℎ2

= [120574minus

1

and 120574minus

2

120574+

1

and 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

(4) ℎ120582 = [(120574minus

1

)120582

(120574+

1

)120582

] | 120574 isin ℎ 120582 gt 0

(5) 120582ℎ = [1 minus (1 minus 120574minus

)120582

1 minus (1 minus 120574+

)120582

] | 120574 isin ℎ 120582 gt 0

(6) ℎ1

oplus ℎ2

= [120574minus

1

+ 120574minus

2

minus 120574minus

1

120574minus

2

120574+

1

+ 120574+

2

minus 120574+

1

120574+

2

] | 1205741

isin

ℎ1

1205742

isin ℎ2

(7) ℎ1

otimes ℎ2

= [120574minus

1

120574minus

2

120574+

1

120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

Further Chen et al [37] established some relationships for theabove operations on IVHFEs

Theorem 11 Let ℎ ℎ1

and ℎ2

be three IVHFEs one has thefollowing

(1) ℎ1

oplus ℎ2

= ℎ2

oplus ℎ1

(2) ℎ1

otimes ℎ2

= ℎ2

otimes ℎ1

(3) (ℎ1

cup ℎ2

)119888

= ℎ119888

1

cap ℎ119888

2

(4) (ℎ1

cap ℎ2

)119888

= ℎ119888

1

cup ℎ119888

2

(5) (120582ℎ)119888 = (ℎ119888

)120582

(6) (ℎ120582)119888 = 120582(ℎ119888

)

(7) (ℎ1

oplus ℎ2

)119888

= ℎ119888

1

otimes ℎ119888

2

(8) (ℎ1

otimes ℎ2

)119888

= ℎ119888

1

oplus ℎ119888

2

Example 12 Let ℎ1

= [05 06] [03 08] [03 06] and ℎ2

=

[04 05] [04 07] be two IVHFEs then by the operationallaws of IVHFEs given in Definition 10 we have

ℎ1

cup ℎ2

= [120574minus

1

or 120574minus

2

120574+

1

or 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

= [05 or 04 06 or 05] [05 or 04 06 or 07]

[03 or 04 08 or 05] [03 or 04 08 or 07]

[03 or 04 06 or 05] [03 or 04 06 or 07]

= [05 06] [05 07] [04 08] [04 08]

[04 06] [04 07]

ℎ1

cap ℎ2

= [120574minus

1

and 120574minus

2

120574+

1

and 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

= [05 and 04 06 and 05] [05 and 04 06 and 07]

[03 and 04 08 and 05] [03 and 04 08 and 07]

[03 and 04 06 and 05] [03 and 04 06 and 07]

= [04 05] [04 06] [03 05] [03 07]

[03 05] [03 06]

ℎ1

oplus ℎ2

= [120574minus

1

+ 120574minus

2

minus 120574minus

1

120574minus

2

120574+

1

+ 120574+

2

minus 120574+

1

120574+

2

] |

1205741

isin ℎ1

1205742

isin ℎ2

= [05 + 04 minus 05 sdot 04 06 + 05 minus 06 sdot 05]

[05 + 04 minus 05 sdot 04 06 + 07 minus 06 sdot 07]





= [07 08] [07 088] [058 09] [058 094]

[058 08] [058 088]

ℎ1

otimes ℎ2

= [120574minus

1

120574minus

2

120574+

1

120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

= [05 sdot 04 06 sdot 05] [05 sdot 04 06 sdot 07]

[03 sdot 04 08 sdot 05] [03 sdot 04 08 sdot 07]


= [02 03] [02 042] [012 04] [012 056]

[012 03] [012 042]

(8)

From Example 12 we can see that the dimension of thederived IVHFEmay increase as the addition ormultiplicativeoperations are done which may increase the complexityof the calculations To overcome the difficulty we developsome new methods to decrease the dimension of the derivedIVHFE when operating the IVHFEs on the premise ofthe assumptions given by Chen et al [37] The adjustedoperational laws are defined as follows


and ℎ2

be three IVHFEs and let 120582 bea positive real number then one has the following

(1) ℎ119888 = [1 minus ℎ120590(119896)119880

1 minus ℎ120590(119896)119871

] | 119896 = 1 2 119897

(2) ℎ1

cup ℎ2

= [ℎ120590(119896)119871

1

or ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

or ℎ120590(119896)119880

2

] | 119896 =

1 2 119897

(3) ℎ1

cap ℎ2

= [ℎ120590(119896)119871

1

and ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

and ℎ120590(119896)119880

2

] | 119896 =

1 2 119897

(4) ℎ120582 = [(ℎ120590(119896)119871

)120582

(ℎ120590(119896)119880

)120582

] | 119896 = 1 2 119897

(5) 120582ℎ = [1 minus (1 minus ℎ120590(119896)119871

)120582

1 minus (1 minus ℎ120590(119896)119880

)120582

] | 119896 =

1 2 119897

(6) ℎ1

oplusℎ2

= [ℎ120590(119896)119871

1

+ℎ120590(119896)119871

2

minusℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

+ℎ120590(119896)119880

2

minus

ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 119897

(7) ℎ1

otimes ℎ2

= [ℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 119897

where ℎ120590(119896)

119895

= [ℎ120590(119896)119871

119895

ℎ120590(119896)119880

119895

] is the 119896th largest intervalnumber in ℎ

119895



and ℎ2

be three IVHFEs For the newoperations in Definition 13 one has the following

(1) ℎ1

oplus ℎ2

= ℎ2

oplus ℎ1

(2) ℎ1

otimes ℎ2

= ℎ2

otimes ℎ1

(3) (ℎ

1

cup ℎ2

)119888

= ℎ119888

1

cap ℎ119888

2

(4) (ℎ

1

cap ℎ2

)119888

= ℎ119888

1

cup ℎ119888

2

(5) (120582ℎ)119888 = (ℎ119888

)120582

(6) (ℎ120582)119888 = 120582(ℎ119888

)(7) (ℎ

1

oplus ℎ2

)119888

= ℎ119888

1

otimes ℎ119888

2

(8) (ℎ

1

otimes ℎ2

)119888

= ℎ119888

1

oplus ℎ119888

2

Proof Theproofs are similar toTheorems 1 and 2 in [37]

Theorem 14 shows that Theorem 11 is still valid for thenew operations in Definition 13

Example 15 Reconsider Example 12 By (3) and assumptionsgiven by Chen et al [37] then ℎ

1

= [03 06] [03 08]

[05 06] and ℎ2

= [04 05] [04 07] [04 07] By virtueof Definition 13 we have

ℎ1

cup ℎ2

= [ℎ120590(119896)119871

1

or ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

or ℎ120590(119896)119880

2

] | 119896 = 1 2 3

= [03 or 04 06 or 05] [03 or 04 08 or 07]

[05 or 04 06 or 07]

= [04 06] [04 08] [05 07]

ℎ1

cap ℎ2

= [ℎ120590(119896)119871

1

and ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

and ℎ120590(119896)119880

2

] | 119896 = 1 2 3

= [03 and 04 06 and 05] [03 and 04 08 and 07]

[05 and 04 06 and 07]

= [03 05] [03 07] [04 06]

ℎ1

oplus ℎ2

= [ℎ120590(119896)119871

1

+ ℎ120590(119896)119871

2

minus ℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

+ ℎ120590(119896)119880

2

minus ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 3




= [058 08] [058 094] [07 088]

ℎ1

otimes ℎ2

= [ℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 3


[05 sdot 04 06 sdot 07]

= [012 03] [012 056] [02 042]

(9)

Comparing Example 12 with Example 15 we note thatthe adjusted operational laws given in Definition 13 indeed

decrease the dimension of the derived IVHFE when oper-ating the IVHFEs which brings grievous advantage for thepracticing application

32 Concept of Interval-Valued Hesitant Fuzzy Soft Sets In[35 36 43] researchers have introduced the concept ofhesitant fuzzy soft sets and developed some approaches tohesitant fuzzy soft sets based on decision making Howeverincompleteness and inaccuracy of information in the processof making decision are very important problems we haveto resolve Due to insufficiency in available information itis unreasonable for us to adopt hesitant fuzzy soft sets todeal with some decision making problems in which deci-sion makers only quantify their opinions with several crispnumbers Instead the basic characteristics of the decision-making problems described by several interval numbers mayovercome the difficulty Based on the above fact we extendhesitant fuzzy soft sets into the interval-valued hesitant fuzzyenvironment and introduce the concept of interval-valuedhesitant fuzzy sets

In this subsection we first introduce the notion ofinterval-valued hesitant fuzzy soft sets which is a hybridmodel combining interval-valued hesitant fuzzy sets and softsets

Definition 16 Let (119880 119864) be a soft universe and 119860 sube 119864 A pairS = (119865 119860) is called an interval-valued hesitant fuzzy soft setover 119880 where 119865 is a mapping given by 119865 119860 rarr IVHF(119880)

An interval-valued hesitant fuzzy soft set is a parameter-ized family of interval-valuedhesitant fuzzy subsets of119880Thatis to say 119865(119890) is an interval-valued hesitant fuzzy subset in119880forall119890 isin 119860 Following the standard notations119865(119890) can be writtenas

119865 (119890) = ⟨119909 119865 (119890) (119909)⟩ 119909 isin 119880 (10)

Sometimeswewrite119865 as (119865 119864) If119860 sube 119864 we can also havean interval-valued hesitant fuzzy soft set (119865 119860)

Example 17 Let 119880 be a set of four participants performingdance programme which is denoted by 119880 = 119909

1

1199092

1199093

1199094

Let 119864 be a parameter set where 119864 = 119890

1

1198902

1198903

= confidentcreative graceful Suppose that three judges think the precisemembership degrees of a candidate 119909

119895

to a parameter 119890119894

arehard to be specified To overcome this barrier they representthe membership degrees of a candidate 119909

119895

to a parameter119890119894

with several possible interval values Then interval-valuedhesitant fuzzy soft setS = (119865 119860) defined as follows gives theevaluation of the performance of candidates by three judges

119865 (1198901

) = ⟨1199091

[06 08] [06 07] [08 09]⟩

⟨1199092

[04 05] [07 08] [06 08]⟩

⟨1199093

[07 08] [06 08] [07 08]⟩

⟨1199094

[08 09] [07 09] [08 10]⟩


Table 1 Interval-valued hesitant fuzzy soft setS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[06 08] [06 07] [08 09] [05 06] [04 06] [05 07] [03 05] [04 07] [05 07]

1199092

[04 05] [07 08] [06 08] [07 08] [08 09] [07 08] [06 07] [05 08] [06 08]

1199093

[07 08] [06 08] [07 08] [06 09] [07 08] [07 09] [05 07] [06 08] [04 06]

1199094

[08 09] [07 09] [08 10] [06 07] [05 08] [06 08] [07 08] [08 09] [06 08]

119865 (1198902

) = ⟨1199091

[05 06] [04 06] [05 07]⟩

⟨1199092

[07 08] [08 09] [07 08]⟩

⟨1199093

[06 09] [07 08] [07 09]⟩

⟨1199094

[06 07] [05 08] [06 08]⟩

119865 (1198903

) = ⟨1199091

[03 05] [04 07] [05 07]⟩

⟨1199092

[06 07] [05 08] [06 08]⟩

⟨1199093

[05 07] [06 08] [04 06]⟩

⟨1199094

[07 08] [08 09] [06 08]⟩

(11)

The tabular representation ofS = (119865 119860) is shown in Table 1

All the available information on these participants per-forming dance programme can be characterized by aninterval-valued hesitant fuzzy soft setS = (119865 119860) In Table 1we can see that the precise evaluation for an alternative tosatisfy a criterion is unknown while possible interval valuesof such an evaluation are given For example we cannotpresent the precise degree of how confident the candidate1199091

performing dance programme is however the degreeto which the candidate 119909

1

performing dance programme isconfident can be represented by three possible interval values[06 08] [06 07] and [08 09]

In what follows we will compare some existing soft setsmodel with the newly proposed interval-valued hesitantfuzzy soft sets by using several examples Finally we illustratethe rationality of the newly proposed interval-valued hesitantfuzzy soft sets

Remark 18 InDefinition 16 if there is only one element in theIVHFE 119865(119890)(119909) we can note that an interval-valued hesitantfuzzy soft set degenerates into an interval-valued fuzzy softset [10] That is to say interval-valued hesitant fuzzy soft setsin Definition 16 are an extension of interval-valued fuzzy softsets proposed by Yang et al [10]


1

1199092

1199093

1199094


1

1198902

1198903

= confidentcreative graceful Now assume that there is only a judgewho is invited to evaluate the possible membership degreesof a candidate 119909

119895


with an interval valuewithin [0 1] In that case the evaluation of the performance

of candidates can be presented by an interval-valued fuzzysoft set which is defined as follows

119865 (1198901

) = ⟨1199091

[06 07]⟩ ⟨1199092

[06 08]⟩

⟨1199093

[07 08]⟩ ⟨1199094

[07 09]⟩

119865 (1198902

) = ⟨1199091

[05 06]⟩ ⟨1199092

[07 08]⟩

⟨1199093

[06 09]⟩ ⟨1199094

[05 08]⟩

119865 (1198903

) = ⟨1199091

[03 05]⟩ ⟨1199092

[06 07]⟩

⟨1199093

[04 06]⟩ ⟨1199094

[06 08]⟩

(12)

However we point out that it is unreasonable to inviteonly an expert to develop the policy with an interval numberbecause of the consideration of comprehension and ratio-nality in the process of decision making Therefore in manydecisionmaking problems it is necessary for decisionmakersto need several experts participating in developing the policyThus the decision results may be more comprehensive andreasonable In this case the evaluation of the performanceof candidates can be described as an interval-valued hesitantfuzzy soft set which is defined in Example 17

Comparing with the results of two models we observethat interval-valued hesitant fuzzy soft sets contain moreinformation than interval-valued fuzzy soft sets Hence wesay that the available information in interval-valued hesitantfuzzy soft sets is more comprehensive and reasonable thaninterval-valued fuzzy soft sets and interval-valued hesitantfuzzy soft sets are indeed an extension of interval-valuedfuzzy soft sets proposed by Yang et al [10]

Remark 20 When the upper and lower limits of all theinterval values in the IVHFE 119865(119890)(119909) are identical it shouldbe noted that an interval-valued hesitant fuzzy soft setdegenerates into a hesitant fuzzy soft set in [35 36] whichindicates that hesitant fuzzy soft sets are a special type ofinterval-valued hesitant fuzzy soft sets


1

1199092

1199093

1199094


1

1198902

1198903

= confidentcreative graceful Now we suppose that there are threejudges who are invited to evaluate the possible membershipdegrees of a candidate 119909

119895


with crispnumbers In that case the evaluation of the performance of


candidates can be described as a hesitant fuzzy soft set definedas follows

119865 (1198901

) = ⟨1199091

06 07 08⟩ ⟨1199092

05 07 08⟩

⟨1199093

07 08 08⟩ ⟨1199094

08 09 09⟩

119865 (1198902

) = ⟨1199091

05 06 07⟩ ⟨1199092

07 08 08⟩

⟨1199093

06 07 09⟩ ⟨1199094

06 08 08⟩

119865 (1198903

) = ⟨1199091

04 05 07⟩ ⟨1199092

06 07 07⟩

⟨1199093

04 06 07⟩ ⟨1199094

07 08 08⟩

(13)

In this example if the available information and the expe-rience of experts are both short it is unreasonable to exactlyquantify their opinions by using several crisp numbers Thusthe decision makers are apt to lose information and maysupply incorrect policies through using hesitant fuzzy softset theory But decision makers can overcome the difficultyby adopting several interval numbers Thus the evaluation ofthe performance of candidates can be presented by interval-valued hesitant fuzzy soft sets defined in Example 17

Comparing with the results of twomodels we see that theavailable information in interval-valued hesitant fuzzy softsets is more comprehensive and scientific than hesitant fuzzysoft sets and hesitant fuzzy soft sets are indeed a special caseof interval-valued hesitant fuzzy soft sets

Remark 22 If there is only one interval value in the IVHFE119865(119890)(119909) whose upper and lower limits are identical interval-valued hesitant fuzzy soft sets inDefinition 16 degenerate intothe fuzzy soft set presented by Maji et al in [12] That is thefuzzy soft sets presented byMaji et al in [12] are a special caseof interval-valued hesitant fuzzy soft sets defined by us


1

1199092

1199093

1199094


1

1198902

1198903

= confidentcreative graceful Assume that there is a judge who is invitedto evaluate the possible membership degrees of a candidate119909119895


with a crisp number In that case theevaluation of the performance of candidates can be describedas fuzzy soft sets defined as follows

119865 (1198901

) =

06

1199091

+

05

1199092

+

08

1199093

+

09

1199094

119865 (1198902

) =

05

1199091

+

07

1199092

+

07

1199093

+

08

1199094

119865 (1198903

) =

04

1199091

+

06

1199092

+

07

1199093

+

08

1199094

(14)

Now we reconsider the example On the one hand inmany decision making events it is unreasonable to inviteonly an expert to develop the policy with a crisp numberSeveral experts participating in developing the policy canmake the decision results more comprehensive and objectiveOn the other hand if the expertsrsquo experience is short it is

very difficult for the experts to exactly quantify their opinionsby using several crisp numbers Instead adopting intervalnumbers may overcome the difficulty Considering the abovetwo facts the evaluation of the performance of candidates canbe described as interval-valuedhesitant fuzzy soft sets definedin Example 17

Based on the above discussions we can note that theavailable information in interval-valued hesitant fuzzy softsets is more comprehensive and objective than fuzzy soft setsand fuzzy soft sets are indeed a special type of interval-valuedhesitant fuzzy soft sets

From Remark 18 we can note that an interval-valuedfuzzy soft set can be induced by an interval-valued hesitantfuzzy soft set So we introduce reduct interval-valued fuzzysoft sets of interval-valued hesitant fuzzy soft sets

Definition 24 The optimistic reduct interval-valued fuzzysoft set (ORIVFS) of an interval-valued hesitant fuzzy soft set(119865 119860) is defined as an interval-valued fuzzy soft set (119865

+

119860)

over 119880 such that for all 120574120590(119896) isin 119865(119890)(119909)

119865+

(119890) = (119909 119865+

(119890) (119909)) 119909 isin 119880

= (119909

119897

⋁

119896=1

120574120590(119896)

) 119909 isin 119880 forall119890 isin 119860

(15)

where 120574120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880

] is the 119896th largest intervalnumber in the IVHFE 119865(119890)(119909) and 119897 stands for the numberof interval numbers in the IVHFE 119865(119890)(119909)

Definition 25 The neutral reduct interval-valued fuzzy softset (NRIVFS) of an interval-valued hesitant fuzzy soft set(119865 119860) is defined as an interval-valued fuzzy soft set (119865

119873

119860)


119865119873

(119890) = (119909 119865119873

(119890) (119909)) 119909 isin 119880

= (119909

1

119897

119897

sum

119896=1

120574120590(119896)

) 119909 isin 119880 forall119890 isin 119860

(16)

where 120574120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880


To illustrate the notions presented above we introducethe following example

Example 26 Reconsider Example 17 By (3) and Definitions24 and 25 we can compute the ORIVFS (119865

+

119860) and NRIVFS(119865119873

119860) of the interval-valued hesitant fuzzy soft set (119865 119860)shown in Tables 2 and 3 respectively

33 Operations on Interval-ValuedHesitant Fuzzy Soft Sets Inthe above subsection we have extended soft sets model intointerval-valued hesitant fuzzy environment and presentedinterval-valued hesitant fuzzy soft sets As the above sub-section mentioned interval-valued hesitant fuzzy soft sets


Table 2 ORIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[08 09] [05 07] [05 07]

1199092

[07 08] [08 09] [06 08]

1199093

[07 08] [07 09] [06 08]

1199094

[08 10] [06 08] [08 09]

Table 3 NRIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[067 070] [047 063] [040 063]

1199092

[057 070] [073 083] [057 077]

1199093

[067 080] [067 087] [050 070]

1199094

[077 093] [057 077] [070 083]

are an extension of several soft sets model such as interval-valued fuzzy soft sets hesitant fuzzy soft sets and fuzzy softsets In these existing soft sets models authors defined someoperations on their own model respectively For exampleWang et al [36] defined the complement ldquoANDrdquo and ldquoORrdquooperations on hesitant fuzzy soft sets In [43] some new oper-ations such as ring sum and ring product are also definedon hesitant fuzzy soft sets Meanwhile they also discussedsome of the interesting properties Along the lines of theseworks we will further generalize those operations defined inthese existing soft setsmodel to interval-valued hesitant fuzzyenvironment and present some new operations on interval-valued hesitant fuzzy soft sets Then some properties willbe further established for such operations on interval-valuedhesitant fuzzy soft sets

In the subsection unless otherwise stated the operationson IVHFEs are carried out by the assumptions given by Chenet al [37] and Definition 13 developed by us

First we give the definition of interval-valued hesitantfuzzy soft subsets

Definition 27 Let 119880 be an initial universe and let 119864 be a setof parameters Supposing that 119860 119861 sube 119864 (119865 119860) and (119866 119861)

are two interval-valued hesitant fuzzy soft sets one says that(119865 119860) is an interval-valued hesitant fuzzy soft subset of (119866 119861)if and only if

(1) 119860 sube 119861

(2) 120574120590(119896)1

le 120574120590(119896)

2

where for all 119890 isin 119860 119909 isin 119880 120574120590(119896)1

and 120574120590(119896)

2

stand for the 119896thlargest interval number in the IVHFEs 119865(119890)(119909) and 119866(119890)(119909)respectively

In this case wewrite (119865 119860) ⊑ (119866 119861) (119865 119860) is said to be aninterval-valued hesitant fuzzy soft super set of (119866 119861) if (119866 119861)is an interval-valued hesitant fuzzy soft subset of (119865 119860) Wedenote it by (119865 119860) ⊒ (119866 119861)

Example 28 Suppose that 119880 = 1199091

1199092

1199093

1199094

is an ini-tial universe and 119864 = 119890

1

1198902

1198903

is a set of parameters

Let 119860 = 1198901

1198902

119861 = 119864 = 1198901

1198902

1198903

Two interval-valuedhesitant fuzzy soft sets (119865 119860) and (119866 119860) are given as follows

119865 (1198901

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 09] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

119865 (1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119866 (1198901

) = ⟨1199091

[04 06] [02 09]⟩

⟨1199092

[02 07] [03 09] [07 09]⟩

⟨1199093

[06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119866 (1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119866 (1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

1199094

⟨[07 09] [08 10]⟩

(17)

By (3) and Definition 27 we have (119865 119860) ⊑ (119866 119861)

Definition 29 Let (119865 119860) and (119866 119861) be two interval-valuedhesitant fuzzy soft sets Now (119865 119860) and (119866 119861) are said to beinterval-valued hesitant fuzzy soft equal if and only if

(1) (119865 119860) ⊑ (119866 119861)

(2) (119866 119861) ⊑ (119865 119860)

which can be denoted by (119865 119860) = (119866 119861)

Definition 30 The complement of (119865 119860) denoted by (119865 119860)119888is defined by (119865 119860)119888 = (119865

119888

119860) where 119865119888 119860 rarr IVHF(119880) isa mapping given by 119865119888(119890) for all 119890 isin 119860 such that 119865119888(119890) is thecomplement of interval-valued hesitant fuzzy set 119865(119890) on 119880

Clearly we have ((119865 119860)119888)119888 = (119865 119860)


Example 31 Consider the interval-valued hesitant fuzzysoft set (119866 119861) over 119880 defined in Example 28 Thus byDefinition 30 we have

119866119888

(1198901

) = ⟨1199091

[01 08] [04 06]⟩

⟨1199092

[01 03] [01 07] [03 08]⟩

⟨1199093

[02 04]⟩

⟨1199094

[02 03] [02 04] [05 07]⟩

119866119888

(1198902

) = ⟨1199091

[02 06] [05 07]⟩

⟨1199092

[02 06] [03 07]⟩

⟨1199093

[02 05] [03 07] [05 07]⟩

⟨1199094

[04 05] [05 07]⟩

119866119888

(1198903

) = ⟨1199091

[00 02] [04 05] [05 07]⟩

⟨1199092

[01 03] [04 045]⟩

⟨1199093

[00 01] [015 02] [03 07]⟩

⟨1199094

[00 02] [01 03]⟩

(18)

By the suggestions given by Molodtsov in [1] we presentthe notion of ANDandORoperations on two interval-valuedhesitant fuzzy soft sets as follows

Definition 32 Let (119865 119860) and (119866 119861) be two interval-valuedhesitant fuzzy soft sets over 119880 The ldquo(119865 119860) AND (119866 119861)rdquodenoted by (119865 119860) and (119866 119861) is defined by

(119865 119860) and (119866 119861) = ( 119860 times 119861) (19)

where for all (120572 120573) isin 119860 times 119861

(120572 120573) = ⟨119909 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cap 119866 (120573) (119909)⟩ 119909 isin 119880

(20)

Definition 33 Let (119865 119860) and (119866 119861) be two interval-valuedhesitant fuzzy soft sets over 119880 The ldquo(119865 119860) OR (119866 119861)rdquodenoted by (119865 119860) or (119866 119861) is defined by

(119865 119860) or (119866 119861) = (119868 119860 times 119861) (21)


119868 (120572 120573) = ⟨119909 119868 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cup 119866 (120573) (119909)⟩ 119909 isin 119880

(22)

Example 34 Reconsider Example 28 Then we have (119865 119860) and(119866 119861) = ( 119860times119861) and (119865 119860)or(119866 119861) = (119868 119860times119861) as follows

(1198901

1198901

) = ⟨1199091

[02 05] [02 06]⟩

⟨1199092

[02 05] [03 07] [06 08]⟩

⟨1199093

[03 08] [05 08]⟩

⟨1199094

[02 04] [05 07] [05 07]⟩

(1198901

1198902

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[02 04] [05 06]⟩

(1198901

1198903

) = ⟨1199091

[02 05] [03 06] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 07] [05 08] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

(1198902

1198901

) = ⟨1199091

[01 04] [02 07] [02 08]⟩

⟨1199092

[02 05] [03 06] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06] [04 06]⟩

(1198902

1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[03 05] [04 06]⟩

⟨1199093

[02 04] [03 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

(1198902

1198903

) = ⟨1199091

[01 04] [03 06] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119868 (1198901

1198901

) = ⟨1199091

[04 06] [03 09]⟩

⟨1199092

[03 07] [04 09] [07 09]⟩

⟨1199093

[06 09] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198901

1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08] [06 08]⟩

⟨1199093

[03 09] [05 08] [05 08]⟩

⟨1199094

[03 05] [05 07]⟩


119868 (1198901

1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09] [07 09]⟩

⟨1199093

[03 09] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

119868 (1198902

1198901

) = ⟨1199091

[04 06] [03 09] [03 09]⟩

⟨1199092

[04 07] [05 09] [07 09]⟩

⟨1199093

[06 08] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198902

1198902

) = ⟨1199091

[03 05] [04 08] [04 08]⟩

⟨1199092

[04 07] [05 08]⟩

⟨1199093

[03 05] [04 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119868 (1198902

1198903

) = ⟨1199091

[03 05] [05 07] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

(23)

Theorem 35 Let (119865 119860) and (119866 119861) be two interval-valuedhesitant fuzzy soft sets over 119880 Then one has the following

(1) ((119865 119860) and (119866 119861))119888

= (119865 119860)119888

or (119866 119861)119888

(2) ((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888

Proof (1) Suppose that (119865 119860)and(119866 119861) = ( 119860times119861)Thereforeby Definitions 30 and 32 we have ((119865 119860)and(119866 119861))119888 = ( 119860times

119861)119888

= (119888

119860 times 119861) where for all 119909 isin 119880 and (120572 120573) isin 119860 times

119861 119888(120572 120573)(119909) = (119865(120572)(119909) cap 119866(120573)(119909))119888 From Theorem 14 it

follows that (119865(120572)(119909) cap 119866(120573)(119909))119888

= 119865119888

(120572)(119909) cup 119866119888

(120573)(119909)On the other hand by Definitions 30 and 33 we have

(119865 119860)119888

or(119866 119861)119888

= (119865119888

119860)or (119866119888

119861) = (119868 119860times119861) where for all119909 isin 119880 and (120572 120573) isin 119860 times 119861 119868(120572 120573)(119909) = 119865

119888

(120572)(119909) cup 119866119888

(120573)(119909)Hence (119888 119860 times 119861) = (119868 119860 times 119861)

(2)The result can be proved in a similar way

Theorem 36 Let (119865 119860) (119866 119861) and ( 119862) be three interval-valued hesitant fuzzy soft sets over 119880 Then one has thefollowing

(1) (119865 119860) and ((119866 119861) and ( 119862)) = ((119865 119860) and (119866 119861)) and ( 119862)(2) (119865 119860) or ((119866 119861) or ( 119862)) = ((119865 119860) or (119866 119861)) or ( 119862)(3) (119865 119860)and((119866 119861)or( 119862)) = ((119865 119860)and(119866 119861))or((119865 119860)and

( 119862))(4) (119865 119860)or((119866 119861)and( 119862)) = ((119865 119860)or(119866 119861))and((119865 119860)or

( 119862))

Proof (1) Consider forall119909 isin 119880 120572 isin 119860 120573 isin 119861 and 120574 isin 119862 wehave 119865(120572)(119909) cap (119866(120573)(119909) cap (120574)(119909)) = (119865(120572)(119909) cap119866(120573)(119909)) cap

(120574)(119909) from which we can conclude that (119865 119860) and ((119866 119861) and

( 119862)) = ((119865 119860) and (119866 119861)) and ( 119862) holdsSimilar to the above progress the proofs of (2) (3) and

(4) can be made

Remark 37 Suppose that (119865 119860) and (119866 119861) are two interval-valued hesitant fuzzy soft sets over 119880 It is noted that for all(120572 120573) isin 119860times119861 if 120572 = 120573 then (119866 119861)and (119865 119860) = (119865 119860)and (119866 119861)and (119866 119861) or (119865 119860) = (119865 119860) or (119866 119861)

Next on the basis of the operations in Definition 13we first present ring sum and ring product operations oninterval-valued hesitant fuzzy soft sets

Definition 38 The ring sum operation on the two interval-valued hesitant fuzzy soft sets 119865 and 119866 over 119880 denoted by119865 oplus 119866 = is a mapping given by

119864 997888rarr IVHF (119880) (24)

such that for all 119890 isin 119864

(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) oplus 119866 (119890) (119909)⟩ 119909 isin 119880

(25)

Definition 39 The ring product operation on the twointerval-valued hesitant fuzzy soft sets 119865 and 119866 over 119880denoted by 119865 otimes 119866 = is a mapping given by

119864 997888rarr IVHF (119880) (26)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) otimes 119866 (119890) (119909)⟩ 119909 isin 119880

(27)

Example 40 Let us consider the interval-valued hesitantfuzzy soft set 119866 in Example 28 Let 119865 be another interval-valued hesitant fuzzy soft set over 119880 defined as follows

119865 (1198901

) = ⟨1199091

[03 05] [03 08]⟩

⟨1199092

[01 05] [02 05] [04 07]⟩

⟨1199093

[02 05] [03 06] [05 08]⟩

⟨1199094

[03 04] [05 07]⟩


119865 (1198902

) = ⟨1199091

[04 05] [03 07]⟩

⟨1199092

[02 06] [03 06]⟩

⟨1199093

[02 03] [03 04] [05 08]⟩

⟨1199094

[02 04] [04 06]⟩

119865 (1198903

) = ⟨1199091

[02 05] [04 07]⟩

⟨1199092

[05 06] [05 08]⟩

⟨1199093

[03 05] [02 07] [03 08]⟩

⟨1199094

[03 05] [02 07]⟩

(28)

Then by Definition 38 we have

(119865 oplus 119866) (1198901

)

= ⟨1199091

[058 08] [044 098]⟩

⟨1199092

[028 085] [044 095] [082 097]⟩

⟨1199093

[068 09] [072 092] [08 096]⟩

⟨1199094

[051 07] [08 094] [085 094]⟩

(119865 oplus 119866) (1198902

)

= ⟨1199091

[058 075] [058 094]⟩

⟨1199092

[044 088] [058 092]⟩

⟨1199093

[044 065] [051 082] [075 096]⟩

⟨1199094

[044 07] [07 084]⟩

(119865 oplus 119866) (1198903

)

= ⟨1199091

[044 075] [07 088] [088 10]⟩

⟨1199092

[0775 084] [085 098]⟩

⟨1199093

[051 085] [084 0955] [093 10]⟩

⟨1199094

[079 095] [084 10]⟩

(29)

By Definition 39 then

(119865 otimes 119866) (1198901

)

= ⟨1199091

[012 03] [006 072]⟩

⟨1199092

[002 035] [006 045] [028 063]⟩

⟨1199093

[012 04] [018 048] [03 064]⟩

⟨1199094

[009 02] [03 056] [035 056]⟩

(119865 otimes 119866) (1198902

)

= ⟨1199091

[012 025] [012 056]⟩

⟨1199092

[006 042] [012 048]⟩

⟨1199093

[006 015] [009 028] [025 064]⟩

⟨1199094

[006 02] [02 036]⟩

(119865 otimes 119866) (1198903

)

= ⟨1199091

[006 025] [02 042] [032 07]⟩

⟨1199092

[0275 036] [035 072]⟩

⟨1199093

[009 035] [016 0595] [027 08]⟩

⟨1199094

[021 045] [016 07]⟩

(30)

Theorem 41 Let 119865 and 119866 be two interval-valued hesitantfuzzy soft sets over 119880 Then the following laws are valid

(1) 119865 oplus 119866 = 119866 oplus 119865(2) 119865 otimes 119866 = 119866 otimes 119865(3) (119865 oplus 119866)

119888

= 119865119888

otimes 119866119888

(4) (119865 otimes 119866)119888

= 119865119888

oplus 119866119888

Proof The proof directly follows from Definitions 38 and 39andTheorem 14

Theorem 42 Let 119865 119866 and be any three interval-valuedhesitant fuzzy soft sets over 119880 then the following hold

(1) (119865 oplus 119866) oplus = 119865 oplus (119866 oplus )(2) (119865 otimes 119866) otimes = 119865 otimes (119866 otimes )

Proof The properties follow from Definitions 13 38 and 39

Definition 43 An interval-valued hesitant fuzzy soft set issaid to be an empty interval-valued hesitant fuzzy soft setdenoted by if 119865 119864 rarr IVHF(119880) such that 119865(119890) =

⟨119909 119865(119890)(119909)⟩ 119909 isin 119880 = ⟨119909 [0 0]⟩ 119909 isin 119880 forall119890 isin 119864

Definition 44 An interval-valued hesitant fuzzy soft set issaid to be a full interval-valued hesitant fuzzy soft set denotedby if 119865 119864 rarr IVHF(119880) such that 119865(119890) = ⟨119909 119865(119890)(119909)⟩

119909 isin 119880 = ⟨119909 [1 1]⟩ 119909 isin 119880 forall119890 isin 119864

From Definitions 43 and 44 obviously we have

(1) ⊑ 119865 ⊑ (2) 119888 = (3) 119888 =

Theorem 45 Let 119865 be an interval-valued hesitant fuzzy softset over 119880 then the following hold


(1) 119865 or = 119865 119865 and = (2) 119865 or = 119865 and = 119865(3) 119865 oplus = 119865 119865 otimes = (4) 119865 oplus = 119865 otimes = 119865

Proof The proof is straightforward

Remark 46 Let119865 be an interval-valued hesitant fuzzy soft setover 119880 If 119865 = or 119865 = then 119865 or 119865

119888

= and 119865 and 119865119888

=

Remark 47 Let 119865 be an interval-valued hesitant fuzzy soft setover 119880 If 119865 = or 119865 = then 119865 oplus 119865

119888

= and 119865 otimes 119865119888

=

4 An Adjustable Approach toInterval-Valued Hesitant Fuzzy Soft SetsBased on Decision Making

After interval-valued hesitant fuzzy sets were introduced byChen et al [37] interval-valued hesitant fuzzy sets havebeen used in handlingmany fuzzy decisionmaking problems[38ndash40] With the development of soft sets the applicationof fuzzy soft sets in solving decision making problems hasbeen investigated by many researchers [13ndash15] In the currentsection we will present an adjustable approach to interval-valued hesitant fuzzy soft sets based on decision making Inthe following we will introduce the concept of level hesitantfuzzy soft sets of interval-valued hesitant fuzzy soft sets

Definition 48 LetS = (119865 119860) be the interval-valued hesitantfuzzy soft set over 119880 where 119860 sube 119864 and 119864 is the parameterset For 119905 isin Int[0 1] and 120574

120590(119896)

isin 119865(119890)(119909) (119896 = 1 2 119897) wedefine

120572120590(119896)

=

1 120574120590(119896)

ge 119905

0 120574120590(119896)

lt 119905

(31)

where Int[0 1] stands for the set of all closed subintervals of[0 1] and 120574

120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880

] is the 119896th largest intervalnumber in the IVHFE119865(119890)(119909)Then the hesitant fuzzy soft set(119865119905

119860) called a 119905-level hesitant fuzzy soft set119871(S 119905) = (119865119905

119860)is defined by

119865119905

(119890) = 119871 (119865 (119890) 119905)

= ⟨119909 119865119905

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(32)

for all 119890 isin 119860

Remark 49 In Definition 48 it is worth noting that 119905 whichis viewed as a given threshold is not a real number but aninterval value belonging to Int[0 1] on membership valuesIn practical applications these interval value thresholds arein advance given by decision makers according to theirrequirements on ldquomembership levelsrdquo It is easy to see thatHFEs of level hesitant fuzzy soft sets consist of 1 and 0 whichis very important in real-life applications of interval-valuedhesitant fuzzy soft sets based on decision making

Table 4 Tabular representation of 119871(S [06 08])

119880 1198901

1198902

1198903

1199091

1 0 1 0 0 0 0 0 0

1199092

0 1 1 1 1 1 0 0 1

1199093

1 1 1 1 1 1 0 1 0

1199094

1 1 1 0 0 1 1 1 1

To illustrate Definition 48 let us consider the interval-valued hesitant fuzzy soft set (119865 119860) shown in Example 17

Example 50 (the [06 08]-level hesitant fuzzy soft set ofthe interval-valued hesitant fuzzy soft set S) ReconsiderExample 17 Now taking 119905 = [06 08] then by (3) andDefinition 48 we have the following [06 08]-level hesitantfuzzy soft sets

119871 (119865 (1198901

) [06 08]) = ⟨1199091

1 0 1⟩ ⟨1199092

0 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

1 1 1⟩

119871 (119865 (1198902

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

1 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

0 0 1⟩

119871 (119865 (1198903

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

0 0 1⟩

⟨1199093

0 1 0⟩ ⟨1199094

1 1 1⟩

(33)

Then the [06 08]-level hesitant fuzzy soft set of S =

(119865 119860) is a special hesitant fuzzy soft set 119871(S [06 08]) =

(119865[0608]

119860) where for 119909119895

isin 119880 HFEs of the hesitant fuzzysets 119865

[0608]

(119890119894

) = 119871(119865(119890119894

) [06 08]) consist of 1 and 0For example 119865

[0608]

(1198901

) (1199092

) = 119871(119865(1198901

)(1199092

) [06 08]) =

0 1 1 which means that the second judge and the thirdjudge think the performance of candidate 119909

2

is confident onthe membership degree [06 08]

Table 4 gives the tabular representation of the levelhesitant fuzzy soft set 119871(S [06 08])

In Definition 48 we can note that the level threshold isalways a constant interval value 119905 isin Int[0 1] However inthe process of making decision decision makers may imposedifferent thresholds on different parameters So we substitutea function for the constant interval value as the thresholds onthe membership value to address this issue

Definition 51 Let (119865 119860) be the interval-valued hesitant fuzzysoft set over 119880 where 119860 sube 119864 and 119864 is the parameter set Let120582 119860 rarr Int[0 1] be an interval-valued fuzzy set in 119860 whichis called a threshold interval-valued fuzzy set For 119909 isin 119880 and120574120590(119896)

isin 119865(119890)(119909) (119896 = 1 2 119897) one defines

120572120590(119896)

=

1 120574120590(119896)

ge 120582 (119890)

0 120574120590(119896)

lt 120582 (119890)

(34)


Then the hesitant fuzzy soft set (119865120582

119860) called a level hesitantfuzzy soft set 119871(S 120582(119890)) = (119865

120582

119860) with respect to the fuzzyset 120582 is defined by

119865120582

(119890) = 119871 (119865 (119890) 120582 (119890))

= ⟨119909 119865120582

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(35)

for all 119890 isin 119860

In order to better understand the above definition let usconsider the following examples

Example 52 (the mid-level of interval-valued hesitant fuzzysoft set) Let S = (119865 119860) be an interval-valued hesitantfuzzy soft sets over 119880 where 119860 sube 119864 and 119864 is the parameterset Suppose that (119865

119873

119860) is a neutral reduct interval-valuedfuzzy soft set of the interval-valued hesitant fuzzy soft setS = (119865 119860) defined by Definition 25 Based on the neutralreduct interval-valued fuzzy soft set (119865

119873

119860) we can definean interval-valued fuzzy set midS 119860 rarr Int[0 1] by

midS (119890) =

1

|119880|

sum

119909isin119880

119865119873

(119890) (119909) (36)

for all 119890 isin 119860 The interval-valued fuzzy set midS is calledthe mid-threshold of the interval-valued hesitant fuzzy softset S Moreover the level hesitant fuzzy soft set of theinterval-valued hesitant fuzzy soft set S with respect tothe mid-threshold interval-valued fuzzy set midS namely119871(SmidS) is called the mid-level hesitant fuzzy soft set ofS which is simply denoted as 119871(Smid) In the discussionsbelow the mid-level decision rule means that we will adoptthe mid-threshold and consider the mid-level hesitant fuzzysoft set in interval-valued hesitant fuzzy soft sets based ondecision making

Reconsider the interval-valued hesitant fuzzy soft setS =

(119865 119860)with its tabular representation shown inTable 1 Table 3gives the tabular representation of the neutral reduct interval-valued fuzzy soft set (119865

119873

119860) of S From Table 3 the mid-threshold midS ofS can be given as follows

midS = (1198901

[067 07825]) (1198902

[061 0775])

(1198903

[05425 07325])

(37)

and the mid-level hesitant fuzzy soft set ofS is a special hes-itant fuzzy soft set 119871(Smid) with its tabular representationgiven by Table 5

Example 53 (the top-level of interval-valued hesitant fuzzysoft set) Let S = (119865 119860) be interval-valued hesitant fuzzysoft sets over 119880 where 119860 sube 119864 and 119864 is the parameterset Suppose that (119865

+

119860) is an optimistic reduct interval-valued fuzzy soft set defined by Definition 24 Based on theoptimistic reduct interval-valued fuzzy soft set (119865

+

119860) wecan define an interval-valued fuzzy setmaxS 119860 rarr Int[0 1]by

maxS (119890) = max119909isin119880

119865+

(119890) (119909) (38)

Table 5 Tabular representation of 119871(Smid)

119880 1198901

1198902

1198903

Choice value (119888119895

)

1199091

0 0 1 0 0 0 0 0 0 1198881

= 1

1199092

0 1 0 1 1 1 1 1 1 1198882

= 7

1199093

1 0 1 1 1 1 0 1 0 1198883

= 6

1199094

1 1 1 0 0 1 1 1 1 1198884

= 7

Table 6 Tabular representation of 119871(Smax)

119880 1198901

1198902

1198903


)

1199091

0 0 0 0 0 0 0 0 0 1198881

= 0

1199092

0 0 0 0 1 0 0 0 0 1198881

= 1

1199093

0 0 0 0 0 0 0 0 0 1198881

= 0

1199094

0 0 1 0 0 0 0 1 0 1198881

= 2

for all 119890 isin 119860 The interval-valued fuzzy set maxS is calledthe max-threshold of the interval-valued hesitant fuzzy softset S Moreover the level hesitant fuzzy soft set of theinterval-valued hesitant fuzzy soft set S with respect tothe max-threshold interval-valued fuzzy set maxS namely119871(SmaxS) is called the top-level hesitant fuzzy soft set ofS which is simply denoted as 119871(Smax) In the discussionsbelow the top-level decision rule means that we will usethe max-threshold and consider the top-level hesitant fuzzysoft set in interval-valued hesitant fuzzy soft sets based ondecision making

We also reconsider the interval-valued hesitant fuzzysoft set S = (119865 119860) with its tabular representation shownin Table 1 Table 2 gives the tabular representation of theoptimistic reduct interval-valued fuzzy soft set (119865

+

119860) of SFrom Table 2 the max-threshold maxS ofS can be given asfollows

maxS = (1198901

[08 10]) (1198902

[08 09]) (1198903

[08 09])

(39)

and the top-level hesitant fuzzy soft set ofS is a special hes-itant fuzzy soft set 119871(Smax) with its tabular representationgiven by Table 6

In the following we will present an adjustable approachto interval-valued hesitant fuzzy soft sets based on decisionmaking

Algorithm 54

Step 1 Input an interval-valued hesitant fuzzy soft set S =

(119865 119860)

Step 2 Input a threshold interval-valued fuzzy set 120582 119860 rarr

Int[0 1] (or give a threshold value 119905 isin Int[0 1] or choose themid-level decision rule or choose the top-level decision rule)for decision making

Step 3 Compute the level hesitant fuzzy soft set 119871(S 120582) ofS with respect to the threshold interval-valued fuzzy set 120582(or the 119905-level hesitant fuzzy soft set 119871(S 119905) or the mid-level


hesitant fuzzy soft set 119871(Smid) or the top-level hesitantfuzzy soft set 119871(Smax))

Step 4 Present the level hesitant fuzzy soft set 119871(S 120582) (or119871(S 119905) or 119871(Smid) or 119871(Smax)) in tabular form For any119909119895

isin 119880 compute the choice value 119888119895

of 119909119895

Step 5 The optimal decision is to select 119909119896

if 119888119896

= max119895

119888119895

Step 6 If 119896 has more than one value then any one of 119909119896

maybe chosen

Remark 55 Thebasic motivation for designing Algorithm 54is to solve interval-valued hesitant fuzzy soft set based ondecision making problem by using level hesitant fuzzy softsets initiated in this study By choosing certain thresholdsor decision strategies such as the mid-level or the top-leveldecision rules we can convert a complex interval-valuedhesitant fuzzy soft set into a simple hesitant fuzzy soft setcalled the level hesitant fuzzy soft set Thus we need not treatinterval-valued hesitant fuzzy soft sets directly in decisionmaking but only deal with the level hesitant fuzzy soft setsderived from them

To illustrate the basic idea of Algorithm 54 we give thefollowing example

Example 56 Let us reconsider the decision making problemthat involves the interval-valued hesitant fuzzy soft set S =

(119865 119860) with tabular representation as in Table 1 Assume thatall the opinions of three judges must be considered by us Sowe intend to select the object fulfilling the different standardsof three judges in most aspects as the optimal person Henceit is reasonable to use the mid-level decision rule for decisionmaking in such cases In Example 52 we have obtained themid-level hesitant fuzzy soft set 119871(Smid) ofS with respectto the threshold interval-valued fuzzy set midS with itstabular representation given by Table 5

In Table 5 we can calculate choice values as follows

1198881

= 1 + 0 + 0 = 1

1198882

= 1 + 1 times 3 + 1 times 3 = 7

1198883

= 1 times 2 + 1 times 3 + 1 = 6

1198884

= 1 times 3 + 1 + 1 times 3 = 7

(40)

From the choice value listed in Table 5 we can concludethat the ranking of the four candidates is 119909

4

= 1199092

gt 1199093

gt 1199091

and the best choice is 1199094

and 1199092

Next we use the top-level decision rule to handle the

decision making problem Table 6 gives the tabular represen-tation of the level hesitant fuzzy soft set119871(Smax) ofS FromTable 6 it follows that the ranking of the four candidates is1199094

gt 1199092

gt 1199093

= 1199091

and the optimal decision is to select1199094

5 Weighted Interval-Valued Hesitant FuzzySoft Set Based on Decision Making

Section 4 has investigated the application of interval-valuedhesitant fuzzy soft sets based on decision making It is worthnoting that the importance of the parameters is not reflectedin the process of the above application about interval-valued hesitant fuzzy soft set Therefore the weights of theparameters should be considered by decisionmakers inmanydecision making events Suppose that the parameters that weconsider have different weights In that case we should takeinto account the weighted interval-valued hesitant fuzzy softset based on decision making

In the present section inspired by the idea of Feng et al in[15 16] we first introduce the concept of weighted interval-valued hesitant fuzzy soft sets and discuss its applications todecision making problems

Definition 57 Let (119865 119860) be the interval-valued hesitant fuzzysoft set over 119880 where 119860 sube 119864 and 119864 is the parameter setA weighted interval-valued hesitant fuzzy soft set is a tripleweierp = (119865 119860 120596) where 120596 119860 rarr [0 1] is a weight functionspecifying the weight 120596

119894

= 120596(119890119894

) for each attribute 119890119894

isin 119860

Remark 58 In Definition 57 if there is only one elementin the IVHFE 119865(119890)(119909) we can note that weighted interval-valued hesitant fuzzy soft sets degenerate into the weightedinterval-valued fuzzy soft sets presented by Feng et al in [16]

Remark 59 When there is only one interval value in theIVHFE 119865(119890)(119909) whose upper and lower limits are identical itshould be noted that weighted interval-valued hesitant fuzzysoft sets inDefinition 57 degenerate to theweighted fuzzy softset introduced by Feng et al in [15] In general the weightedfuzzy soft sets and weighted interval-valued fuzzy soft setspresented by Feng et al in [15 16] are two special cases ofweighted interval-valued hesitant fuzzy soft sets defined byus

In reality all the parameters may not be of equal impor-tance in many decision making problemsThe importance ofdifferent parameters can be described as the weight functionin a weighted interval-valued hesitant fuzzy soft set So theweighted interval-valued hesitant fuzzy soft set can be appliedto some decision making problems in which the parametersare not of equal importance

In order to deal with the weighted interval-valued hesi-tant fuzzy soft sets based on decision making problems wedevelop a revised version of Algorithm 54 by using weightedchoice values 119888

119895

as substitutes for choice values 119888119895

(seeAlgorithm 60)

Algorithm 60

Step 1 Input a weighted interval-valued hesitant fuzzy soft setweierp = (119865 119860 120596)


Int[0 1] (or give a threshold value 119905 isin Int[0 1] or choose


Table 7 Tabular representation ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]

Table 8 Tabular representation of weierp = (119865 119860 120596)

119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]

the mid-level decision rule or choose the top-level decisionrule) for decision making

Step 3 Compute the level hesitant fuzzy soft set 119871((119865 119860) 120582) ofweierpwith respect to the threshold interval-valued fuzzy set 120582 (orthe 119905-level hesitant fuzzy soft set 119871((119865 119860) 119905) or the mid-levelhesitant fuzzy soft set 119871((119865 119860)mid) or the top-level hesitantfuzzy soft set 119871((119865 119860)max))

Step 4 Present the level hesitant fuzzy soft set 119871((119865 119860) 120582) (or119871((119865 119860) 119905) or 119871((119865 119860)mid) or 119871((119865 119860)max)) in tabularform For any 119909

119895

isin 119880 compute the weighted choice value119888119895

of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen

To illustrate the above idea we consider the followingexample

Example 61 A car company is eager to select the optimumrobot for its manufacturing process Suppose that 119880 =

ℎ1

ℎ2

ℎ3

ℎ4

ℎ5

is a set of five robots and 119860 is a set of fourcriteria where 119860 = 119890

1

1198902

1198903

1198904

= load capacity speedmemory capacity degree of freedom

Now assume that two evaluation experts impose theweights on the parameters in 119860 as follows for the parameterldquoload capacityrdquo 120596

1

= 04 for the parameter ldquospeedrdquo 1205962

=

02 for the parameter ldquomemory capacityrdquo 1205963

= 015 forthe parameter ldquodegree of freedomrdquo 120596

4

= 025 The twoexperts evaluate the performances of the alternatives ℎ

119895

(119895 =

1 2 3 4 5) with respect to criteria 119890119894

(119894 = 1 2 3 4) andconstruct an interval-valued hesitant fuzzy soft setG = (119865 119860)

shown in Table 7Then the weight function 120596 119860 rarr [0 1] and the

interval-valued hesitant fuzzy soft set G = (119865 119860) aretransformed into a weighted interval-valued hesitant fuzzysoft set weierp = (119865 119860 120596) The tabular representation of weierp =

(119865 119860 120596) is shown in Table 8

Assume that all the evaluations of two experts for fiverobots must be taken into account by us In that case weshould deal with this problem by using mid-level decisionrule That is we will use the mid-threshold midG In orderto obtain the mid-threshold midG ofG by Definition 25 wecan calculate the neutral reduct interval-valued fuzzy soft setofG given in Table 9 So themid-thresholdmidG ofG can beobtained as follows

midG = (1198901

[055 071]) (1198902

[063 078])

(1198903

[061 075]) (1198904

[066 074])

(41)

By Definition 51 we obtain the mid-level hesitant fuzzy softset of S with respect to the mid-threshold midG Table 10gives the tabular representation of the mid-level hesitantfuzzy soft set 119871(Gmid) with weighted choice values InTable 10 we can calculate choice values as follows

1198881

= 04 times 2 + 0 + 0 + 0 = 08

1198882

= 0 + 02 times 1 + 015 times 1 + 025 times 2 = 085

1198883

= 04 times 1 + 02 times 1 + 015 times 1 + 025 times 2 = 125

1198884


1198885

= 04 times 2 + 02 times 1 + 0 + 025 times 1 = 125

(42)

Therefore according to Table 10 we can conclude that theranking of the five alternatives is ℎ

4

gt ℎ5

= ℎ3

gt ℎ2

gt ℎ1

andthe best choice is ℎ

4

As an adjustable approach one can also use other rules

in the above decision making problem and in general thefinal decision result will change accordingly For instance weuse the top-level decision rule to handle the decision makingproblem Similar to Example 53 Table 11 gives the tabularrepresentation of the level hesitant fuzzy soft set 119871(Gmax)of G From Table 11 it follows that the ranking of the fivealternatives is ℎ

4

gt ℎ3

gt ℎ1

= ℎ2

= ℎ5

and the optimaldecision is to select ℎ

4


Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]

Table 10 Tabular representation of 119871(Gmid) with weighted choice values

119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025 Weighted choice value (119888119895

)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125

Table 11 Tabular representation of 119871(Gmax) with weighted choice values

119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion

In this paper we propose the concept of interval-valuedhesitant fuzzy soft sets and their properties are also investi-gated Meanwhile an adjustable approach to interval-valuedhesitant fuzzy soft sets based on decision making is proposedby using level hesitant fuzzy soft sets Also we illustrate theefficiency of the novel method by some examples More-over the weighted interval-valued hesitant fuzzy soft set isintroduced and an example problem is provided to show theeffectiveness of the proposedmodelThis work can be viewedas the extension of Feng et al [15 16]

In the future it is important and interesting to furtherinvestigate level hesitant fuzzy soft set approach to decisionmaking based on other extensions of hesitant fuzzy soft settheory such as dual hesitant fuzzy soft set theory and dualinterval-valued fuzzy soft set theory

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous refereesfor their valuable comments and suggestions This work issupported by the National Natural Science Foundation ofChina (no 11461082)

References

[1] D Molodtsov ldquoSoft set theory-first resultsrdquo Computers andMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[2] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[3] D Molodtsov The Theory of Soft Sets URSS PublishersMoscow Russia 2004 (Russian)

[4] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[5] Y B Jun ldquoSoft BCKBCI-algebrasrdquo Computers amp Mathematicswith Applications vol 56 no 5 pp 1408ndash1413 2008

[6] Y B Jun andCH Park ldquoApplications of soft sets in ideal theoryof BCKBCI-algebrasrdquo Information Sciences vol 178 no 11 pp2466ndash2475 2008

[7] F Feng Y B Jun and X Z Zhao ldquoSoft semiringsrdquo ComputersampMathematics with Applications vol 56 no 10 pp 2621ndash26282008

[8] M I Ali F Feng X Liu W K Min and M Shabir ldquoOn somenew operations in soft set theoryrdquo Computers and Mathematicswith Applications vol 57 no 9 pp 1547ndash1553 2009

[9] K Qin and Z Hong ldquoOn soft equalityrdquo Journal of Computa-tional and Applied Mathematics vol 234 no 5 pp 1347ndash13552010

[10] X Yang T Y Lin J Yang Y Li and D Yu ldquoCombination ofinterval-valued fuzzy set and soft setrdquo Computers ampMathemat-ics with Applications vol 58 no 3 pp 521ndash527 2009

[11] Y Yang X Tan and C C Meng ldquoThe multi-fuzzy soft setand its application in decision makingrdquo Applied MathematicalModelling vol 37 no 7 pp 4915ndash4923 2013


[12] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[13] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[14] Z Kong L Q Gao and L F Wang ldquoComment on lsquoA fuzzy softset theoretic approach to decision making problemsrsquordquo Journalof Computational and Applied Mathematics vol 223 no 2 pp540ndash542 2009

[15] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[16] F Feng Y M Li and V Leoreanu-Fotea ldquoApplication of levelsoft sets in decision making based on interval-valued fuzzy softsetsrdquo Computers amp Mathematics with Applications vol 60 no6 pp 1756ndash1767 2010

[17] Y Jiang Y Tang Q Chen H Liu and J Tang ldquoInterval-valuedintuitionistic fuzzy soft sets and their propertiesrdquo Computers ampMathematics with Applications vol 60 no 3 pp 906ndash918 2010

[18] Z Zhang C Wang D Tian and K Li ldquoA novel approachto interval-valued intuitionistic fuzzy soft set based decisionmakingrdquo Applied Mathematical Modelling vol 38 no 4 pp1255ndash1270 2014

[19] Z Xiao S Xia K Gong and D Li ldquoThe trapezoidal fuzzysoft set and its application in MCDMrdquo Applied MathematicalModelling vol 36 no 12 pp 5844ndash5855 2012

[20] H D Zhang L Shu and S L Liao ldquoGeneralized trapezoidalfuzzy soft set and its application in medical diagnosisrdquo Journalof Applied Mathematics vol 2014 Article ID 312069 12 pages2014

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[23] M B Gorzałczany ldquoA method of inference in approximatereasoning based on interval-valued fuzzy setsrdquo Fuzzy Sets andSystems vol 21 no 1 pp 1ndash17 1987

[24] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003

[25] G Deschrijver and E E Kerre ldquoImplicators based on binaryaggregation operators in interval-valued fuzzy set theoryrdquoFuzzySets and Systems vol 153 no 2 pp 229ndash248 2005

[26] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications vol 144 ofMathematics in Science and EngineeringAcademic Press New York NY USA 1980

[27] S Miyamoto ldquoRemarks on basics of fuzzy sets and fuzzymultisetsrdquo Fuzzy Sets and Systems vol 156 no 3 pp 427ndash4312005

[28] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009

[29] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[30] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[31] M Xia and Z Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011

[32] Z S Xu and M M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[33] D Dubois and H Prade ldquoRough fuzzy sets and fuzzy roughsetsrdquo International Journal of General Systems vol 17 pp 191ndash209 1990

[34] X B Yang X N Song Y S Qi and J Y Yang ldquoConstructiveand axiomatic approaches to hesitant fuzzy rough setrdquo SoftComputing vol 18 no 6 pp 1067ndash1077 2014

[35] K V Babitha and S J John ldquoHesitant fuzzy soft setsrdquo Journal ofNew Results in Science vol 3 pp 98ndash107 2013

[36] F Q Wang X H Li and X H Chen ldquoHesitant fuzzy soft setand its applications in multicriteria decision makingrdquo Journalof Applied Mathematics vol 2014 Article ID 643785 10 pages2014

[37] N Chen Z S Xu and M M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013

[38] N Chen Z S Xu and M M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013

[39] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013

[40] G W Wei X F Zhao and R Lin ldquoSome hesitant interval-valued fuzzy aggregation operators and their applications tomultiple attribute decision makingrdquo Knowledge-Based Systemsvol 46 pp 43ndash53 2013

[41] H D Zhang L Shu and S L Liao ldquoOn interval-valued hesitantfuzzy rough approximation operatorsrdquo Soft Computing 2014

[42] Z S Xu and Q L Da ldquoThe uncertain OWA operatorrdquo Interna-tional Journal of Intelligent Systems vol 17 no 6 pp 569ndash5752002

[43] J Q Wang X E Li and X H Chen ldquoHesitant fuzzy softsets with application in multicriteria group decision makingproblemsrdquo The Scientific World Journal vol 2015 Article ID806983 14 pages 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


an adjustable approach to decision making problems basedon interval-valued intuitionistic fuzzy soft sets Very recentlyintegrating trapezoidal fuzzy sets with soft sets Xiao et al[19] initiated the trapezoidal fuzzy soft sets to deal withcertain linguistic assessments Further in order to capturethe vagueness of the attribute with linguistic assessmentsinformation Zhang et al [20] generalized trapezoidal fuzzysoft sets introduced by Xiao et al [19] and defined the conceptof generalized trapezoidal fuzzy soft sets

As a mathematical method to deal with vagueness ineveryday life fuzzy set was introduced by Zadeh in [21] Upto present several extensions have been developed such asintuitionistic fuzzy set [22] interval valued fuzzy sets [23ndash25] type-2 fuzzy set [26 27] and type-119899 fuzzy set [26]Recently Torra and Narukawa [28 29] extended fuzzy setsto hesitant fuzzy environment and initiated the notion ofhesitant fuzzy sets (HFSs) because they found that becauseof the doubts between a few different values it is very difficultto determine the membership of an element to a set under agroup setting [29] For example two decision makers discussthe membership degree of 119909 into 119860 One wants to assign 07but the other wants to assign 09 They cannot persuade eachother To avoid an argument the membership degrees of 119909into 119860 can be described as 07 09 After it was introducedby Torra the hesitant fuzzy set has attracted more and morescholarsrsquo attention [30ndash32]

As mentioned above there are close relationships amongthe uncertainty theories such as fuzzy sets rough sets andsoft sets So many hybrid models among them are proposedby the researchers such as fuzzy rough sets [33] rough fuzzysets [33] and fuzzy soft sets [12] Since the appearance ofhesitant fuzzy set the study on it has never been stoppedThe combination of hesitant fuzzy set with other uncertaintytheories is a hot spot of the current research recently Thereare several hybrid models in present such as hesitant fuzzyrough sets [34] and hesitant fuzzy soft sets [35 36] Infact Babitha and John [35] defined a hybrid model calledhesitant fuzzy soft sets and investigated some of their basicproperties Meanwhile Wang et al [36] also initiated theconcept of hesitant fuzzy soft sets by integrating hesitantfuzzy set with soft set model and presented an algorithm tosolve decision making problems based on hesitant fuzzy softsets By combining hesitant fuzzy set and rough set modelsYang et al [34] introduced the concept of the hesitant fuzzyrough sets and proposed an axiomatic approach to themodel

However Chen et al [37 38] pointed out that it is verydifficult for decision makers to exactly quantify their ideas byusing several crisp numbers because of the lack of availableinformation in many decision making events ThereforeChen et al [37 38] extended hesitant fuzzy sets into interval-valued hesitant fuzzy environment and introduced the con-cept of interval-valued hesitant fuzzy sets (IVHFSs) whichpermits the membership degrees of an element to a givenset to have a few different interval values It should benoted that when the upper and lower limits of the intervalvalues are identical IVHFS degenerates into HFS indicatingthat the latter is a special case of the former Recentlysimilarity distance and entropy measures for IVHFSs havebeen investigated by Farhadinia [39]Wei et al [40] discussed

some interval-valued hesitant fuzzy aggregation operatorsand gave their applications to multiple attribute decisionmaking based on interval-valued hesitant fuzzy sets

As a novel mathematical method to handle impreciseinformation the study of hybrid models combining IVHFSswith other uncertainty theories is emerging as an activeresearch topic of IVHFS theory Meanwhile we know thatthere are close relationships among the uncertainty theoriessuch as interval-valued hesitant fuzzy sets rough sets andsoft sets Therefore many scholars have been starting toresearch on the area For example Zhang et al [41] general-ized the hesitant fuzzy rough sets to interval-valued hesitantfuzzy environment and presented an interval-valued hesitantfuzzy rough set model by integrating interval-valued hesitantfuzzy set with rough set theory On the one hand it is unrea-sonable to use hesitant fuzzy soft sets to handle some deci-sion making problems because of insufficiency in availableinformation Instead adopting several interval numbers mayovercome the difficulty In that case it is necessary to extendhesitant fuzzy soft sets [36] into interval-valued hesitant fuzzyenvironment On the other hand by referring to a great dealof literature and expertise we find that the discussions aboutfusions of interval-valued hesitant fuzzy set and soft sets donot also exist in the related literatures Considering the abovefacts it is necessary for us to investigate the combinationof IVHFS and soft set The purpose of this paper is toinitiate the concept of interval-valued hesitant fuzzy soft setby combining interval-valued hesitant fuzzy set and soft settheory In order to illustrate the efficiency of the model anadjustable approach to interval-valued hesitant fuzzy soft setsbased on decision making is also presented Finally somenumerical examples are provided to illustrate the adjustableapproach

To facilitate our discussion we first review some back-grounds on soft sets fuzzy soft sets hesitant fuzzy sets andinterval numbers in Section 2 In Section 3 the concept ofinterval-valued hesitant fuzzy soft set with its operation rulesis presented In Section 4 an adjustable approach to interval-valued hesitant fuzzy soft sets based on decision makingis proposed In Section 5 the concept of weighted interval-valued hesitant fuzzy soft sets is defined and applied todecision making problems in which all the decision criteriamay not be of equal importance Finally we conclude thepaper with a summary and outlook for further research inSection 6

2 Preliminaries

In this section we briefly review the concepts of soft setsfuzzy soft sets hesitant fuzzy sets and interval numbers Thepair (119880 119864) will be called a soft universe Throughout thispaper unless otherwise stated 119880 refers to an initial universe119864 is a set of parameters119875(119880) is the power set of 119880 and119860 sube 119864

21 Soft Sets Fuzzy Soft Sets and Hesitant Fuzzy SetsAccording to [1] the concept of soft sets is defined as follows

Definition 1 (see [1]) A pair (119865 119860) is called a soft set over 119880where 119865 is a mapping given by 119865 119860 rarr 119875(119880)






119860


119860 = ⟨119909 ℎ

119860

(119909)⟩ | 119909 isin 119883 (1)

where ℎ

119860



119860


Example 4 Let 119883 = 1199091

1199092

1199093


119860

(1199091

) = 07 04 05 ℎ

119860

(1199092

) = 02 04 and ℎ

119860

(1199093

) =

03 01 07 06 be the HFEs of 119909119894


119860 = ⟨1199091

07 04 05⟩ ⟨1199092

02 04⟩

⟨1199093

03 01 07 06⟩

(2)



119886119880

] = 119909 | 119886119871

le 119909 le 119886119880


119880


119886119880

] 119887 = [119887119871

119887119880


(1) 119886 = 119887 if 119886119871 = 119887119871 and 119886

119880

= 119887119880

(2) 119886 + 119887 = [119886119871

+ 119887119871

119886119880

+ 119887119880

](3) 120582119886 = [120582119886

119871

120582119886119880

] In particular 120582119886 = 0 if 120582 = 0


119886119880

] and 119887 = [119887119871

119887119880

] andlet 119897119886

= 119886119880

minus119886119871 and 119897

119887

= 119887119880



119901 (119886 ge 119887) = max1 minusmax(119887119880

minus 119886119871

119897119886

+ 119897119887

0) 0 (3)


119901 (119887 ge 119886) = max1 minusmax(119886119880

minus 119887119871

119897119886

+ 119897119887

0) 0 (4)





119860 = ⟨119909 ℎ

119860

(119909)⟩ | 119909 isin 119883 (5)

where ℎ

119860



119860


119860


Example 9 Let 119883 = 1199091

1199092


119860

(1199091

) = [02 03] [04 06] [05 06] and ℎ

119860

(1199092

) = [03

05] [04 07] be the IVHFEs of 119909119894


119860 = ⟨1199091

[02 03] [04 06] [05 06]⟩

⟨1199092

[03 05] [04 07]⟩

(6)



ℎ120590(119896)

= [ℎ120590(119896)119871

ℎ120590(119896)119880

] (7)


= inf ℎ120590(119896) ℎ120590(119896)119880 =



1

ℎ2

119897(ℎ1

) = 119897(ℎ2

) then 119897 =

max119897(ℎ1

) 119897(ℎ2


1

and ℎ2


1

than in ℎ2



2


1

and ℎ2



and ℎ2



(1) ℎ119888 = [1 minus 120574+

1 minus 120574minus

] | 120574 isin ℎ

(2) ℎ1

cup ℎ2

= [120574minus

1

or 120574minus

2

120574+

1

or 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

(3) ℎ1

cap ℎ2

= [120574minus

1

and 120574minus

2

120574+

1

and 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

(4) ℎ120582 = [(120574minus

1

)120582

(120574+

1

)120582

] | 120574 isin ℎ 120582 gt 0


)120582


)120582

] | 120574 isin ℎ 120582 gt 0

(6) ℎ1

oplus ℎ2

= [120574minus

1

+ 120574minus

2

minus 120574minus

1

120574minus

2

120574+

1

+ 120574+

2

minus 120574+

1

120574+

2

] | 1205741

isin

ℎ1

1205742

isin ℎ2

(7) ℎ1

otimes ℎ2

= [120574minus

1

120574minus

2

120574+

1

120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2



and ℎ2


(1) ℎ1

oplus ℎ2

= ℎ2

oplus ℎ1

(2) ℎ1

otimes ℎ2

= ℎ2

otimes ℎ1

(3) (ℎ1

cup ℎ2

)119888

= ℎ119888

1

cap ℎ119888

2

(4) (ℎ1

cap ℎ2

)119888

= ℎ119888

1

cup ℎ119888

2

(5) (120582ℎ)119888 = (ℎ119888

)120582

(6) (ℎ120582)119888 = 120582(ℎ119888

)

(7) (ℎ1

oplus ℎ2

)119888

= ℎ119888

1

otimes ℎ119888

2

(8) (ℎ1

otimes ℎ2

)119888

= ℎ119888

1

oplus ℎ119888

2

Example 12 Let ℎ1

= [05 06] [03 08] [03 06] and ℎ2

=


ℎ1

cup ℎ2

= [120574minus

1

or 120574minus

2

120574+

1

or 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

= [05 or 04 06 or 05] [05 or 04 06 or 07]

[03 or 04 08 or 05] [03 or 04 08 or 07]

[03 or 04 06 or 05] [03 or 04 06 or 07]

= [05 06] [05 07] [04 08] [04 08]

[04 06] [04 07]

ℎ1

cap ℎ2

= [120574minus

1

and 120574minus

2

120574+

1

and 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

= [05 and 04 06 and 05] [05 and 04 06 and 07]

[03 and 04 08 and 05] [03 and 04 08 and 07]

[03 and 04 06 and 05] [03 and 04 06 and 07]

= [04 05] [04 06] [03 05] [03 07]

[03 05] [03 06]

ℎ1

oplus ℎ2

= [120574minus

1

+ 120574minus

2

minus 120574minus

1

120574minus

2

120574+

1

+ 120574+

2

minus 120574+

1

120574+

2

] |

1205741

isin ℎ1

1205742

isin ℎ2







= [07 08] [07 088] [058 09] [058 094]

[058 08] [058 088]

ℎ1

otimes ℎ2

= [120574minus

1

120574minus

2

120574+

1

120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2




= [02 03] [02 042] [012 04] [012 056]

[012 03] [012 042]

(8)



and ℎ2


(1) ℎ119888 = [1 minus ℎ120590(119896)119880

1 minus ℎ120590(119896)119871

] | 119896 = 1 2 119897

(2) ℎ1

cup ℎ2

= [ℎ120590(119896)119871

1

or ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

or ℎ120590(119896)119880

2

] | 119896 =

1 2 119897

(3) ℎ1

cap ℎ2

= [ℎ120590(119896)119871

1

and ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

and ℎ120590(119896)119880

2

] | 119896 =

1 2 119897

(4) ℎ120582 = [(ℎ120590(119896)119871

)120582

(ℎ120590(119896)119880

)120582

] | 119896 = 1 2 119897

(5) 120582ℎ = [1 minus (1 minus ℎ120590(119896)119871

)120582

1 minus (1 minus ℎ120590(119896)119880

)120582

] | 119896 =

1 2 119897

(6) ℎ1

oplusℎ2

= [ℎ120590(119896)119871

1

+ℎ120590(119896)119871

2

minusℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

+ℎ120590(119896)119880

2

minus

ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 119897

(7) ℎ1

otimes ℎ2

= [ℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 119897

where ℎ120590(119896)

119895

= [ℎ120590(119896)119871

119895

ℎ120590(119896)119880

119895


119895



and ℎ2


(1) ℎ1

oplus ℎ2

= ℎ2

oplus ℎ1

(2) ℎ1

otimes ℎ2

= ℎ2

otimes ℎ1

(3) (ℎ

1

cup ℎ2

)119888

= ℎ119888

1

cap ℎ119888

2

(4) (ℎ

1

cap ℎ2

)119888

= ℎ119888

1

cup ℎ119888

2

(5) (120582ℎ)119888 = (ℎ119888

)120582

(6) (ℎ120582)119888 = 120582(ℎ119888

)(7) (ℎ

1

oplus ℎ2

)119888

= ℎ119888

1

otimes ℎ119888

2

(8) (ℎ

1

otimes ℎ2

)119888

= ℎ119888

1

oplus ℎ119888

2




1

= [03 06] [03 08]

[05 06] and ℎ2


ℎ1

cup ℎ2

= [ℎ120590(119896)119871

1

or ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

or ℎ120590(119896)119880

2

] | 119896 = 1 2 3

= [03 or 04 06 or 05] [03 or 04 08 or 07]

[05 or 04 06 or 07]

= [04 06] [04 08] [05 07]

ℎ1

cap ℎ2

= [ℎ120590(119896)119871

1

and ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

and ℎ120590(119896)119880

2

] | 119896 = 1 2 3

= [03 and 04 06 and 05] [03 and 04 08 and 07]

[05 and 04 06 and 07]

= [03 05] [03 07] [04 06]

ℎ1

oplus ℎ2

= [ℎ120590(119896)119871

1

+ ℎ120590(119896)119871

2

minus ℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

+ ℎ120590(119896)119880

2

minus ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 3




= [058 08] [058 094] [07 088]

ℎ1

otimes ℎ2

= [ℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 3


[05 sdot 04 06 sdot 07]

= [012 03] [012 056] [02 042]

(9)







119865 (119890) = ⟨119909 119865 (119890) (119909)⟩ 119909 isin 119880 (10)



1

1199092

1199093

1199094


1

1198902

1198903


119895



119895



119865 (1198901

) = ⟨1199091

[06 08] [06 07] [08 09]⟩

⟨1199092

[04 05] [07 08] [06 08]⟩

⟨1199093

[07 08] [06 08] [07 08]⟩

⟨1199094

[08 09] [07 09] [08 10]⟩



119880 1198901

1198902

1198903

1199091

[06 08] [06 07] [08 09] [05 06] [04 06] [05 07] [03 05] [04 07] [05 07]

1199092

[04 05] [07 08] [06 08] [07 08] [08 09] [07 08] [06 07] [05 08] [06 08]

1199093

[07 08] [06 08] [07 08] [06 09] [07 08] [07 09] [05 07] [06 08] [04 06]

1199094

[08 09] [07 09] [08 10] [06 07] [05 08] [06 08] [07 08] [08 09] [06 08]

119865 (1198902

) = ⟨1199091

[05 06] [04 06] [05 07]⟩

⟨1199092

[07 08] [08 09] [07 08]⟩

⟨1199093

[06 09] [07 08] [07 09]⟩

⟨1199094

[06 07] [05 08] [06 08]⟩

119865 (1198903

) = ⟨1199091

[03 05] [04 07] [05 07]⟩

⟨1199092

[06 07] [05 08] [06 08]⟩

⟨1199093

[05 07] [06 08] [04 06]⟩

⟨1199094

[07 08] [08 09] [06 08]⟩

(11)




1





1

1199092

1199093

1199094


1

1198902

1198903


119895




119865 (1198901

) = ⟨1199091

[06 07]⟩ ⟨1199092

[06 08]⟩

⟨1199093

[07 08]⟩ ⟨1199094

[07 09]⟩

119865 (1198902

) = ⟨1199091

[05 06]⟩ ⟨1199092

[07 08]⟩

⟨1199093

[06 09]⟩ ⟨1199094

[05 08]⟩

119865 (1198903

) = ⟨1199091

[03 05]⟩ ⟨1199092

[06 07]⟩

⟨1199093

[04 06]⟩ ⟨1199094

[06 08]⟩

(12)





1

1199092

1199093

1199094


1

1198902

1198903


119895





119865 (1198901

) = ⟨1199091

06 07 08⟩ ⟨1199092

05 07 08⟩

⟨1199093

07 08 08⟩ ⟨1199094

08 09 09⟩

119865 (1198902

) = ⟨1199091

05 06 07⟩ ⟨1199092

07 08 08⟩

⟨1199093

06 07 09⟩ ⟨1199094

06 08 08⟩

119865 (1198903

) = ⟨1199091

04 05 07⟩ ⟨1199092

06 07 07⟩

⟨1199093

04 06 07⟩ ⟨1199094

07 08 08⟩

(13)





1

1199092

1199093

1199094


1

1198902

1198903




119865 (1198901

) =

06

1199091

+

05

1199092

+

08

1199093

+

09

1199094

119865 (1198902

) =

05

1199091

+

07

1199092

+

07

1199093

+

08

1199094

119865 (1198903

) =

04

1199091

+

06

1199092

+

07

1199093

+

08

1199094

(14)






+

119860)


119865+

(119890) = (119909 119865+

(119890) (119909)) 119909 isin 119880

= (119909

119897

⋁

119896=1

120574120590(119896)

) 119909 isin 119880 forall119890 isin 119860

(15)

where 120574120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880



119873

119860)


119865119873

(119890) = (119909 119865119873

(119890) (119909)) 119909 isin 119880

= (119909

1

119897

119897

sum

119896=1

120574120590(119896)

) 119909 isin 119880 forall119890 isin 119860

(16)

where 120574120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




+

119860) and NRIVFS(119865119873




Table 2 ORIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[08 09] [05 07] [05 07]

1199092

[07 08] [08 09] [06 08]

1199093

[07 08] [07 09] [06 08]

1199094

[08 10] [06 08] [08 09]

Table 3 NRIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[067 070] [047 063] [040 063]

1199092

[057 070] [073 083] [057 077]

1199093

[067 080] [067 087] [050 070]

1199094

[077 093] [057 077] [070 083]






(1) 119860 sube 119861

(2) 120574120590(119896)1

le 120574120590(119896)

2


and 120574120590(119896)

2




1199092

1199093

1199094


1

1198902

1198903


Let 119860 = 1198901

1198902

119861 = 119864 = 1198901

1198902

1198903


119865 (1198901

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 09] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

119865 (1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119866 (1198901

) = ⟨1199091

[04 06] [02 09]⟩

⟨1199092

[02 07] [03 09] [07 09]⟩

⟨1199093

[06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119866 (1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119866 (1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

1199094

⟨[07 09] [08 10]⟩

(17)



(1) (119865 119860) ⊑ (119866 119861)

(2) (119866 119861) ⊑ (119865 119860)



119888


Clearly we have ((119865 119860)119888)119888 = (119865 119860)



119866119888

(1198901

) = ⟨1199091

[01 08] [04 06]⟩

⟨1199092

[01 03] [01 07] [03 08]⟩

⟨1199093

[02 04]⟩

⟨1199094

[02 03] [02 04] [05 07]⟩

119866119888

(1198902

) = ⟨1199091

[02 06] [05 07]⟩

⟨1199092

[02 06] [03 07]⟩

⟨1199093

[02 05] [03 07] [05 07]⟩

⟨1199094

[04 05] [05 07]⟩

119866119888

(1198903

) = ⟨1199091

[00 02] [04 05] [05 07]⟩

⟨1199092

[01 03] [04 045]⟩

⟨1199093

[00 01] [015 02] [03 07]⟩

⟨1199094

[00 02] [01 03]⟩

(18)



(119865 119860) and (119866 119861) = ( 119860 times 119861) (19)


(120572 120573) = ⟨119909 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cap 119866 (120573) (119909)⟩ 119909 isin 119880

(20)


(119865 119860) or (119866 119861) = (119868 119860 times 119861) (21)


119868 (120572 120573) = ⟨119909 119868 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cup 119866 (120573) (119909)⟩ 119909 isin 119880

(22)


(1198901

1198901

) = ⟨1199091

[02 05] [02 06]⟩

⟨1199092

[02 05] [03 07] [06 08]⟩

⟨1199093

[03 08] [05 08]⟩

⟨1199094

[02 04] [05 07] [05 07]⟩

(1198901

1198902

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[02 04] [05 06]⟩

(1198901

1198903

) = ⟨1199091

[02 05] [03 06] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 07] [05 08] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

(1198902

1198901

) = ⟨1199091

[01 04] [02 07] [02 08]⟩

⟨1199092

[02 05] [03 06] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06] [04 06]⟩

(1198902

1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[03 05] [04 06]⟩

⟨1199093

[02 04] [03 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

(1198902

1198903

) = ⟨1199091

[01 04] [03 06] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119868 (1198901

1198901

) = ⟨1199091

[04 06] [03 09]⟩

⟨1199092

[03 07] [04 09] [07 09]⟩

⟨1199093

[06 09] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198901

1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08] [06 08]⟩

⟨1199093

[03 09] [05 08] [05 08]⟩

⟨1199094

[03 05] [05 07]⟩


119868 (1198901

1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09] [07 09]⟩

⟨1199093

[03 09] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

119868 (1198902

1198901

) = ⟨1199091

[04 06] [03 09] [03 09]⟩

⟨1199092

[04 07] [05 09] [07 09]⟩

⟨1199093

[06 08] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198902

1198902

) = ⟨1199091

[03 05] [04 08] [04 08]⟩

⟨1199092

[04 07] [05 08]⟩

⟨1199093

[03 05] [04 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119868 (1198902

1198903

) = ⟨1199091

[03 05] [05 07] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

(23)


(1) ((119865 119860) and (119866 119861))119888

= (119865 119860)119888

or (119866 119861)119888

(2) ((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


119861)119888

= (119888



follows that (119865(120572)(119909) cap 119866(120573)(119909))119888

= 119865119888

(120572)(119909) cup 119866119888


(119865 119860)119888

or(119866 119861)119888

= (119865119888

119860)or (119866119888


119888

(120572)(119909) cup 119866119888





( 119862))(4) (119865 119860)or((119866 119861)and( 119862)) = ((119865 119860)or(119866 119861))and((119865 119860)or

( 119862))




(4) can be made




119864 997888rarr IVHF (119880) (24)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) oplus 119866 (119890) (119909)⟩ 119909 isin 119880

(25)


119864 997888rarr IVHF (119880) (26)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) otimes 119866 (119890) (119909)⟩ 119909 isin 119880

(27)


119865 (1198901

) = ⟨1199091

[03 05] [03 08]⟩

⟨1199092

[01 05] [02 05] [04 07]⟩

⟨1199093

[02 05] [03 06] [05 08]⟩

⟨1199094

[03 04] [05 07]⟩


119865 (1198902

) = ⟨1199091

[04 05] [03 07]⟩

⟨1199092

[02 06] [03 06]⟩

⟨1199093

[02 03] [03 04] [05 08]⟩

⟨1199094

[02 04] [04 06]⟩

119865 (1198903

) = ⟨1199091

[02 05] [04 07]⟩

⟨1199092

[05 06] [05 08]⟩

⟨1199093

[03 05] [02 07] [03 08]⟩

⟨1199094

[03 05] [02 07]⟩

(28)


(119865 oplus 119866) (1198901

)

= ⟨1199091

[058 08] [044 098]⟩

⟨1199092

[028 085] [044 095] [082 097]⟩

⟨1199093

[068 09] [072 092] [08 096]⟩

⟨1199094

[051 07] [08 094] [085 094]⟩

(119865 oplus 119866) (1198902

)

= ⟨1199091

[058 075] [058 094]⟩

⟨1199092

[044 088] [058 092]⟩

⟨1199093

[044 065] [051 082] [075 096]⟩

⟨1199094

[044 07] [07 084]⟩

(119865 oplus 119866) (1198903

)

= ⟨1199091

[044 075] [07 088] [088 10]⟩

⟨1199092

[0775 084] [085 098]⟩

⟨1199093

[051 085] [084 0955] [093 10]⟩

⟨1199094

[079 095] [084 10]⟩

(29)


(119865 otimes 119866) (1198901

)

= ⟨1199091

[012 03] [006 072]⟩

⟨1199092

[002 035] [006 045] [028 063]⟩

⟨1199093

[012 04] [018 048] [03 064]⟩

⟨1199094

[009 02] [03 056] [035 056]⟩

(119865 otimes 119866) (1198902

)

= ⟨1199091

[012 025] [012 056]⟩

⟨1199092

[006 042] [012 048]⟩

⟨1199093

[006 015] [009 028] [025 064]⟩

⟨1199094

[006 02] [02 036]⟩

(119865 otimes 119866) (1198903

)

= ⟨1199091

[006 025] [02 042] [032 07]⟩

⟨1199092

[0275 036] [035 072]⟩

⟨1199093

[009 035] [016 0595] [027 08]⟩

⟨1199094

[021 045] [016 07]⟩

(30)



119888

= 119865119888

otimes 119866119888

(4) (119865 otimes 119866)119888

= 119865119888

oplus 119866119888










(1) ⊑ 119865 ⊑ (2) 119888 = (3) 119888 =






119888

= and 119865 and 119865119888

=


119888

= and 119865 otimes 119865119888

=




120590(119896)

isin 119865(119890)(119909) (119896 = 1 2 119897) wedefine

120572120590(119896)

=

1 120574120590(119896)

ge 119905

0 120574120590(119896)

lt 119905

(31)


120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




119865119905

(119890) = 119871 (119865 (119890) 119905)

= ⟨119909 119865119905

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(32)

for all 119890 isin 119860



119880 1198901

1198902

1198903

1199091

1 0 1 0 0 0 0 0 0

1199092

0 1 1 1 1 1 0 0 1

1199093

1 1 1 1 1 1 0 1 0

1199094

1 1 1 0 0 1 1 1 1



119871 (119865 (1198901

) [06 08]) = ⟨1199091

1 0 1⟩ ⟨1199092

0 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

1 1 1⟩

119871 (119865 (1198902

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

1 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

0 0 1⟩

119871 (119865 (1198903

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

0 0 1⟩

⟨1199093

0 1 0⟩ ⟨1199094

1 1 1⟩

(33)



(119865[0608]

119860) where for 119909119895


[0608]

(119890119894

) = 119871(119865(119890119894


[0608]

(1198901

) (1199092

) = 119871(119865(1198901

)(1199092

) [06 08]) =


2





isin 119865(119890)(119909) (119896 = 1 2 119897) one defines

120572120590(119896)

=

1 120574120590(119896)

ge 120582 (119890)

0 120574120590(119896)

lt 120582 (119890)

(34)




120582


119865120582

(119890) = 119871 (119865 (119890) 120582 (119890))

= ⟨119909 119865120582

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(35)

for all 119890 isin 119860



119873


119873


midS (119890) =

1

|119880|

sum

119909isin119880

119865119873

(119890) (119909) (36)




119873


midS = (1198901

[067 07825]) (1198902

[061 0775])

(1198903

[05425 07325])

(37)



+


+


maxS (119890) = max119909isin119880

119865+

(119890) (119909) (38)


119880 1198901

1198902

1198903


)

1199091

0 0 1 0 0 0 0 0 0 1198881

= 1

1199092

0 1 0 1 1 1 1 1 1 1198882

= 7

1199093

1 0 1 1 1 1 0 1 0 1198883

= 6

1199094

1 1 1 0 0 1 1 1 1 1198884

= 7


119880 1198901

1198902

1198903


)

1199091

0 0 0 0 0 0 0 0 0 1198881

= 0

1199092

0 0 0 0 1 0 0 0 0 1198881

= 1

1199093

0 0 0 0 0 0 0 0 0 1198881

= 0

1199094

0 0 1 0 0 0 0 1 0 1198881

= 2



+


maxS = (1198901

[08 10]) (1198902

[08 09]) (1198903

[08 09])

(39)



Algorithm 54


(119865 119860)








of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen






1198881

= 1 + 0 + 0 = 1

1198882

= 1 + 1 times 3 + 1 times 3 = 7

1198883

= 1 times 2 + 1 times 3 + 1 = 6

1198884

= 1 times 3 + 1 + 1 times 3 = 7

(40)


4

= 1199092

gt 1199093

gt 1199091


and 1199092



gt 1199092

gt 1199093

= 1199091






119894

= 120596(119890119894


isin 119860





119895


(seeAlgorithm 60)

Algorithm 60






119880 1198901

1198902

1198903

1198904

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]




119895


of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen



ℎ1

ℎ2

ℎ3

ℎ4

ℎ5


1

1198902

1198903

1198904



1


=



4


119895

(119895 =





(119865 119860 120596) is shown in Table 8


midG = (1198901

[055 071]) (1198902

[063 078])

(1198903

[061 075]) (1198904

[066 074])

(41)


1198881

= 04 times 2 + 0 + 0 + 0 = 08

1198882


1198883


1198884


1198885


(42)


4

gt ℎ5

= ℎ3

gt ℎ2

gt ℎ1


4



4

gt ℎ3

gt ℎ1

= ℎ2

= ℎ5


4


Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion





Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119860


119860 = ⟨119909 ℎ

119860

(119909)⟩ | 119909 isin 119883 (1)

where ℎ

119860



119860


Example 4 Let 119883 = 1199091

1199092

1199093


119860

(1199091

) = 07 04 05 ℎ

119860

(1199092

) = 02 04 and ℎ

119860

(1199093

) =

03 01 07 06 be the HFEs of 119909119894


119860 = ⟨1199091

07 04 05⟩ ⟨1199092

02 04⟩

⟨1199093

03 01 07 06⟩

(2)



119886119880

] = 119909 | 119886119871

le 119909 le 119886119880


119880


119886119880

] 119887 = [119887119871

119887119880


(1) 119886 = 119887 if 119886119871 = 119887119871 and 119886

119880

= 119887119880

(2) 119886 + 119887 = [119886119871

+ 119887119871

119886119880

+ 119887119880

](3) 120582119886 = [120582119886

119871

120582119886119880

] In particular 120582119886 = 0 if 120582 = 0


119886119880

] and 119887 = [119887119871

119887119880

] andlet 119897119886

= 119886119880

minus119886119871 and 119897

119887

= 119887119880



119901 (119886 ge 119887) = max1 minusmax(119887119880

minus 119886119871

119897119886

+ 119897119887

0) 0 (3)


119901 (119887 ge 119886) = max1 minusmax(119886119880

minus 119887119871

119897119886

+ 119897119887

0) 0 (4)





119860 = ⟨119909 ℎ

119860

(119909)⟩ | 119909 isin 119883 (5)

where ℎ

119860



119860


119860


Example 9 Let 119883 = 1199091

1199092


119860

(1199091

) = [02 03] [04 06] [05 06] and ℎ

119860

(1199092

) = [03

05] [04 07] be the IVHFEs of 119909119894


119860 = ⟨1199091

[02 03] [04 06] [05 06]⟩

⟨1199092

[03 05] [04 07]⟩

(6)



ℎ120590(119896)

= [ℎ120590(119896)119871

ℎ120590(119896)119880

] (7)


= inf ℎ120590(119896) ℎ120590(119896)119880 =



1

ℎ2

119897(ℎ1

) = 119897(ℎ2

) then 119897 =

max119897(ℎ1

) 119897(ℎ2


1

and ℎ2


1

than in ℎ2



2


1

and ℎ2



and ℎ2



(1) ℎ119888 = [1 minus 120574+

1 minus 120574minus

] | 120574 isin ℎ

(2) ℎ1

cup ℎ2

= [120574minus

1

or 120574minus

2

120574+

1

or 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

(3) ℎ1

cap ℎ2

= [120574minus

1

and 120574minus

2

120574+

1

and 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

(4) ℎ120582 = [(120574minus

1

)120582

(120574+

1

)120582

] | 120574 isin ℎ 120582 gt 0


)120582


)120582

] | 120574 isin ℎ 120582 gt 0

(6) ℎ1

oplus ℎ2

= [120574minus

1

+ 120574minus

2

minus 120574minus

1

120574minus

2

120574+

1

+ 120574+

2

minus 120574+

1

120574+

2

] | 1205741

isin

ℎ1

1205742

isin ℎ2

(7) ℎ1

otimes ℎ2

= [120574minus

1

120574minus

2

120574+

1

120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2



and ℎ2


(1) ℎ1

oplus ℎ2

= ℎ2

oplus ℎ1

(2) ℎ1

otimes ℎ2

= ℎ2

otimes ℎ1

(3) (ℎ1

cup ℎ2

)119888

= ℎ119888

1

cap ℎ119888

2

(4) (ℎ1

cap ℎ2

)119888

= ℎ119888

1

cup ℎ119888

2

(5) (120582ℎ)119888 = (ℎ119888

)120582

(6) (ℎ120582)119888 = 120582(ℎ119888

)

(7) (ℎ1

oplus ℎ2

)119888

= ℎ119888

1

otimes ℎ119888

2

(8) (ℎ1

otimes ℎ2

)119888

= ℎ119888

1

oplus ℎ119888

2

Example 12 Let ℎ1

= [05 06] [03 08] [03 06] and ℎ2

=


ℎ1

cup ℎ2

= [120574minus

1

or 120574minus

2

120574+

1

or 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

= [05 or 04 06 or 05] [05 or 04 06 or 07]

[03 or 04 08 or 05] [03 or 04 08 or 07]

[03 or 04 06 or 05] [03 or 04 06 or 07]

= [05 06] [05 07] [04 08] [04 08]

[04 06] [04 07]

ℎ1

cap ℎ2

= [120574minus

1

and 120574minus

2

120574+

1

and 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

= [05 and 04 06 and 05] [05 and 04 06 and 07]

[03 and 04 08 and 05] [03 and 04 08 and 07]

[03 and 04 06 and 05] [03 and 04 06 and 07]

= [04 05] [04 06] [03 05] [03 07]

[03 05] [03 06]

ℎ1

oplus ℎ2

= [120574minus

1

+ 120574minus

2

minus 120574minus

1

120574minus

2

120574+

1

+ 120574+

2

minus 120574+

1

120574+

2

] |

1205741

isin ℎ1

1205742

isin ℎ2







= [07 08] [07 088] [058 09] [058 094]

[058 08] [058 088]

ℎ1

otimes ℎ2

= [120574minus

1

120574minus

2

120574+

1

120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2




= [02 03] [02 042] [012 04] [012 056]

[012 03] [012 042]

(8)



and ℎ2


(1) ℎ119888 = [1 minus ℎ120590(119896)119880

1 minus ℎ120590(119896)119871

] | 119896 = 1 2 119897

(2) ℎ1

cup ℎ2

= [ℎ120590(119896)119871

1

or ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

or ℎ120590(119896)119880

2

] | 119896 =

1 2 119897

(3) ℎ1

cap ℎ2

= [ℎ120590(119896)119871

1

and ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

and ℎ120590(119896)119880

2

] | 119896 =

1 2 119897

(4) ℎ120582 = [(ℎ120590(119896)119871

)120582

(ℎ120590(119896)119880

)120582

] | 119896 = 1 2 119897

(5) 120582ℎ = [1 minus (1 minus ℎ120590(119896)119871

)120582

1 minus (1 minus ℎ120590(119896)119880

)120582

] | 119896 =

1 2 119897

(6) ℎ1

oplusℎ2

= [ℎ120590(119896)119871

1

+ℎ120590(119896)119871

2

minusℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

+ℎ120590(119896)119880

2

minus

ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 119897

(7) ℎ1

otimes ℎ2

= [ℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 119897

where ℎ120590(119896)

119895

= [ℎ120590(119896)119871

119895

ℎ120590(119896)119880

119895


119895



and ℎ2


(1) ℎ1

oplus ℎ2

= ℎ2

oplus ℎ1

(2) ℎ1

otimes ℎ2

= ℎ2

otimes ℎ1

(3) (ℎ

1

cup ℎ2

)119888

= ℎ119888

1

cap ℎ119888

2

(4) (ℎ

1

cap ℎ2

)119888

= ℎ119888

1

cup ℎ119888

2

(5) (120582ℎ)119888 = (ℎ119888

)120582

(6) (ℎ120582)119888 = 120582(ℎ119888

)(7) (ℎ

1

oplus ℎ2

)119888

= ℎ119888

1

otimes ℎ119888

2

(8) (ℎ

1

otimes ℎ2

)119888

= ℎ119888

1

oplus ℎ119888

2




1

= [03 06] [03 08]

[05 06] and ℎ2


ℎ1

cup ℎ2

= [ℎ120590(119896)119871

1

or ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

or ℎ120590(119896)119880

2

] | 119896 = 1 2 3

= [03 or 04 06 or 05] [03 or 04 08 or 07]

[05 or 04 06 or 07]

= [04 06] [04 08] [05 07]

ℎ1

cap ℎ2

= [ℎ120590(119896)119871

1

and ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

and ℎ120590(119896)119880

2

] | 119896 = 1 2 3

= [03 and 04 06 and 05] [03 and 04 08 and 07]

[05 and 04 06 and 07]

= [03 05] [03 07] [04 06]

ℎ1

oplus ℎ2

= [ℎ120590(119896)119871

1

+ ℎ120590(119896)119871

2

minus ℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

+ ℎ120590(119896)119880

2

minus ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 3




= [058 08] [058 094] [07 088]

ℎ1

otimes ℎ2

= [ℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 3


[05 sdot 04 06 sdot 07]

= [012 03] [012 056] [02 042]

(9)







119865 (119890) = ⟨119909 119865 (119890) (119909)⟩ 119909 isin 119880 (10)



1

1199092

1199093

1199094


1

1198902

1198903


119895



119895



119865 (1198901

) = ⟨1199091

[06 08] [06 07] [08 09]⟩

⟨1199092

[04 05] [07 08] [06 08]⟩

⟨1199093

[07 08] [06 08] [07 08]⟩

⟨1199094

[08 09] [07 09] [08 10]⟩



119880 1198901

1198902

1198903

1199091

[06 08] [06 07] [08 09] [05 06] [04 06] [05 07] [03 05] [04 07] [05 07]

1199092

[04 05] [07 08] [06 08] [07 08] [08 09] [07 08] [06 07] [05 08] [06 08]

1199093

[07 08] [06 08] [07 08] [06 09] [07 08] [07 09] [05 07] [06 08] [04 06]

1199094

[08 09] [07 09] [08 10] [06 07] [05 08] [06 08] [07 08] [08 09] [06 08]

119865 (1198902

) = ⟨1199091

[05 06] [04 06] [05 07]⟩

⟨1199092

[07 08] [08 09] [07 08]⟩

⟨1199093

[06 09] [07 08] [07 09]⟩

⟨1199094

[06 07] [05 08] [06 08]⟩

119865 (1198903

) = ⟨1199091

[03 05] [04 07] [05 07]⟩

⟨1199092

[06 07] [05 08] [06 08]⟩

⟨1199093

[05 07] [06 08] [04 06]⟩

⟨1199094

[07 08] [08 09] [06 08]⟩

(11)




1





1

1199092

1199093

1199094


1

1198902

1198903


119895




119865 (1198901

) = ⟨1199091

[06 07]⟩ ⟨1199092

[06 08]⟩

⟨1199093

[07 08]⟩ ⟨1199094

[07 09]⟩

119865 (1198902

) = ⟨1199091

[05 06]⟩ ⟨1199092

[07 08]⟩

⟨1199093

[06 09]⟩ ⟨1199094

[05 08]⟩

119865 (1198903

) = ⟨1199091

[03 05]⟩ ⟨1199092

[06 07]⟩

⟨1199093

[04 06]⟩ ⟨1199094

[06 08]⟩

(12)





1

1199092

1199093

1199094


1

1198902

1198903


119895





119865 (1198901

) = ⟨1199091

06 07 08⟩ ⟨1199092

05 07 08⟩

⟨1199093

07 08 08⟩ ⟨1199094

08 09 09⟩

119865 (1198902

) = ⟨1199091

05 06 07⟩ ⟨1199092

07 08 08⟩

⟨1199093

06 07 09⟩ ⟨1199094

06 08 08⟩

119865 (1198903

) = ⟨1199091

04 05 07⟩ ⟨1199092

06 07 07⟩

⟨1199093

04 06 07⟩ ⟨1199094

07 08 08⟩

(13)





1

1199092

1199093

1199094


1

1198902

1198903




119865 (1198901

) =

06

1199091

+

05

1199092

+

08

1199093

+

09

1199094

119865 (1198902

) =

05

1199091

+

07

1199092

+

07

1199093

+

08

1199094

119865 (1198903

) =

04

1199091

+

06

1199092

+

07

1199093

+

08

1199094

(14)






+

119860)


119865+

(119890) = (119909 119865+

(119890) (119909)) 119909 isin 119880

= (119909

119897

⋁

119896=1

120574120590(119896)

) 119909 isin 119880 forall119890 isin 119860

(15)

where 120574120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880



119873

119860)


119865119873

(119890) = (119909 119865119873

(119890) (119909)) 119909 isin 119880

= (119909

1

119897

119897

sum

119896=1

120574120590(119896)

) 119909 isin 119880 forall119890 isin 119860

(16)

where 120574120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




+

119860) and NRIVFS(119865119873




Table 2 ORIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[08 09] [05 07] [05 07]

1199092

[07 08] [08 09] [06 08]

1199093

[07 08] [07 09] [06 08]

1199094

[08 10] [06 08] [08 09]

Table 3 NRIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[067 070] [047 063] [040 063]

1199092

[057 070] [073 083] [057 077]

1199093

[067 080] [067 087] [050 070]

1199094

[077 093] [057 077] [070 083]






(1) 119860 sube 119861

(2) 120574120590(119896)1

le 120574120590(119896)

2


and 120574120590(119896)

2




1199092

1199093

1199094


1

1198902

1198903


Let 119860 = 1198901

1198902

119861 = 119864 = 1198901

1198902

1198903


119865 (1198901

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 09] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

119865 (1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119866 (1198901

) = ⟨1199091

[04 06] [02 09]⟩

⟨1199092

[02 07] [03 09] [07 09]⟩

⟨1199093

[06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119866 (1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119866 (1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

1199094

⟨[07 09] [08 10]⟩

(17)



(1) (119865 119860) ⊑ (119866 119861)

(2) (119866 119861) ⊑ (119865 119860)



119888


Clearly we have ((119865 119860)119888)119888 = (119865 119860)



119866119888

(1198901

) = ⟨1199091

[01 08] [04 06]⟩

⟨1199092

[01 03] [01 07] [03 08]⟩

⟨1199093

[02 04]⟩

⟨1199094

[02 03] [02 04] [05 07]⟩

119866119888

(1198902

) = ⟨1199091

[02 06] [05 07]⟩

⟨1199092

[02 06] [03 07]⟩

⟨1199093

[02 05] [03 07] [05 07]⟩

⟨1199094

[04 05] [05 07]⟩

119866119888

(1198903

) = ⟨1199091

[00 02] [04 05] [05 07]⟩

⟨1199092

[01 03] [04 045]⟩

⟨1199093

[00 01] [015 02] [03 07]⟩

⟨1199094

[00 02] [01 03]⟩

(18)



(119865 119860) and (119866 119861) = ( 119860 times 119861) (19)


(120572 120573) = ⟨119909 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cap 119866 (120573) (119909)⟩ 119909 isin 119880

(20)


(119865 119860) or (119866 119861) = (119868 119860 times 119861) (21)


119868 (120572 120573) = ⟨119909 119868 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cup 119866 (120573) (119909)⟩ 119909 isin 119880

(22)


(1198901

1198901

) = ⟨1199091

[02 05] [02 06]⟩

⟨1199092

[02 05] [03 07] [06 08]⟩

⟨1199093

[03 08] [05 08]⟩

⟨1199094

[02 04] [05 07] [05 07]⟩

(1198901

1198902

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[02 04] [05 06]⟩

(1198901

1198903

) = ⟨1199091

[02 05] [03 06] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 07] [05 08] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

(1198902

1198901

) = ⟨1199091

[01 04] [02 07] [02 08]⟩

⟨1199092

[02 05] [03 06] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06] [04 06]⟩

(1198902

1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[03 05] [04 06]⟩

⟨1199093

[02 04] [03 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

(1198902

1198903

) = ⟨1199091

[01 04] [03 06] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119868 (1198901

1198901

) = ⟨1199091

[04 06] [03 09]⟩

⟨1199092

[03 07] [04 09] [07 09]⟩

⟨1199093

[06 09] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198901

1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08] [06 08]⟩

⟨1199093

[03 09] [05 08] [05 08]⟩

⟨1199094

[03 05] [05 07]⟩


119868 (1198901

1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09] [07 09]⟩

⟨1199093

[03 09] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

119868 (1198902

1198901

) = ⟨1199091

[04 06] [03 09] [03 09]⟩

⟨1199092

[04 07] [05 09] [07 09]⟩

⟨1199093

[06 08] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198902

1198902

) = ⟨1199091

[03 05] [04 08] [04 08]⟩

⟨1199092

[04 07] [05 08]⟩

⟨1199093

[03 05] [04 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119868 (1198902

1198903

) = ⟨1199091

[03 05] [05 07] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

(23)


(1) ((119865 119860) and (119866 119861))119888

= (119865 119860)119888

or (119866 119861)119888

(2) ((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


119861)119888

= (119888



follows that (119865(120572)(119909) cap 119866(120573)(119909))119888

= 119865119888

(120572)(119909) cup 119866119888


(119865 119860)119888

or(119866 119861)119888

= (119865119888

119860)or (119866119888


119888

(120572)(119909) cup 119866119888





( 119862))(4) (119865 119860)or((119866 119861)and( 119862)) = ((119865 119860)or(119866 119861))and((119865 119860)or

( 119862))




(4) can be made




119864 997888rarr IVHF (119880) (24)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) oplus 119866 (119890) (119909)⟩ 119909 isin 119880

(25)


119864 997888rarr IVHF (119880) (26)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) otimes 119866 (119890) (119909)⟩ 119909 isin 119880

(27)


119865 (1198901

) = ⟨1199091

[03 05] [03 08]⟩

⟨1199092

[01 05] [02 05] [04 07]⟩

⟨1199093

[02 05] [03 06] [05 08]⟩

⟨1199094

[03 04] [05 07]⟩


119865 (1198902

) = ⟨1199091

[04 05] [03 07]⟩

⟨1199092

[02 06] [03 06]⟩

⟨1199093

[02 03] [03 04] [05 08]⟩

⟨1199094

[02 04] [04 06]⟩

119865 (1198903

) = ⟨1199091

[02 05] [04 07]⟩

⟨1199092

[05 06] [05 08]⟩

⟨1199093

[03 05] [02 07] [03 08]⟩

⟨1199094

[03 05] [02 07]⟩

(28)


(119865 oplus 119866) (1198901

)

= ⟨1199091

[058 08] [044 098]⟩

⟨1199092

[028 085] [044 095] [082 097]⟩

⟨1199093

[068 09] [072 092] [08 096]⟩

⟨1199094

[051 07] [08 094] [085 094]⟩

(119865 oplus 119866) (1198902

)

= ⟨1199091

[058 075] [058 094]⟩

⟨1199092

[044 088] [058 092]⟩

⟨1199093

[044 065] [051 082] [075 096]⟩

⟨1199094

[044 07] [07 084]⟩

(119865 oplus 119866) (1198903

)

= ⟨1199091

[044 075] [07 088] [088 10]⟩

⟨1199092

[0775 084] [085 098]⟩

⟨1199093

[051 085] [084 0955] [093 10]⟩

⟨1199094

[079 095] [084 10]⟩

(29)


(119865 otimes 119866) (1198901

)

= ⟨1199091

[012 03] [006 072]⟩

⟨1199092

[002 035] [006 045] [028 063]⟩

⟨1199093

[012 04] [018 048] [03 064]⟩

⟨1199094

[009 02] [03 056] [035 056]⟩

(119865 otimes 119866) (1198902

)

= ⟨1199091

[012 025] [012 056]⟩

⟨1199092

[006 042] [012 048]⟩

⟨1199093

[006 015] [009 028] [025 064]⟩

⟨1199094

[006 02] [02 036]⟩

(119865 otimes 119866) (1198903

)

= ⟨1199091

[006 025] [02 042] [032 07]⟩

⟨1199092

[0275 036] [035 072]⟩

⟨1199093

[009 035] [016 0595] [027 08]⟩

⟨1199094

[021 045] [016 07]⟩

(30)



119888

= 119865119888

otimes 119866119888

(4) (119865 otimes 119866)119888

= 119865119888

oplus 119866119888










(1) ⊑ 119865 ⊑ (2) 119888 = (3) 119888 =






119888

= and 119865 and 119865119888

=


119888

= and 119865 otimes 119865119888

=




120590(119896)

isin 119865(119890)(119909) (119896 = 1 2 119897) wedefine

120572120590(119896)

=

1 120574120590(119896)

ge 119905

0 120574120590(119896)

lt 119905

(31)


120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




119865119905

(119890) = 119871 (119865 (119890) 119905)

= ⟨119909 119865119905

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(32)

for all 119890 isin 119860



119880 1198901

1198902

1198903

1199091

1 0 1 0 0 0 0 0 0

1199092

0 1 1 1 1 1 0 0 1

1199093

1 1 1 1 1 1 0 1 0

1199094

1 1 1 0 0 1 1 1 1



119871 (119865 (1198901

) [06 08]) = ⟨1199091

1 0 1⟩ ⟨1199092

0 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

1 1 1⟩

119871 (119865 (1198902

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

1 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

0 0 1⟩

119871 (119865 (1198903

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

0 0 1⟩

⟨1199093

0 1 0⟩ ⟨1199094

1 1 1⟩

(33)



(119865[0608]

119860) where for 119909119895


[0608]

(119890119894

) = 119871(119865(119890119894


[0608]

(1198901

) (1199092

) = 119871(119865(1198901

)(1199092

) [06 08]) =


2





isin 119865(119890)(119909) (119896 = 1 2 119897) one defines

120572120590(119896)

=

1 120574120590(119896)

ge 120582 (119890)

0 120574120590(119896)

lt 120582 (119890)

(34)




120582


119865120582

(119890) = 119871 (119865 (119890) 120582 (119890))

= ⟨119909 119865120582

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(35)

for all 119890 isin 119860



119873


119873


midS (119890) =

1

|119880|

sum

119909isin119880

119865119873

(119890) (119909) (36)




119873


midS = (1198901

[067 07825]) (1198902

[061 0775])

(1198903

[05425 07325])

(37)



+


+


maxS (119890) = max119909isin119880

119865+

(119890) (119909) (38)


119880 1198901

1198902

1198903


)

1199091

0 0 1 0 0 0 0 0 0 1198881

= 1

1199092

0 1 0 1 1 1 1 1 1 1198882

= 7

1199093

1 0 1 1 1 1 0 1 0 1198883

= 6

1199094

1 1 1 0 0 1 1 1 1 1198884

= 7


119880 1198901

1198902

1198903


)

1199091

0 0 0 0 0 0 0 0 0 1198881

= 0

1199092

0 0 0 0 1 0 0 0 0 1198881

= 1

1199093

0 0 0 0 0 0 0 0 0 1198881

= 0

1199094

0 0 1 0 0 0 0 1 0 1198881

= 2



+


maxS = (1198901

[08 10]) (1198902

[08 09]) (1198903

[08 09])

(39)



Algorithm 54


(119865 119860)








of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen






1198881

= 1 + 0 + 0 = 1

1198882

= 1 + 1 times 3 + 1 times 3 = 7

1198883

= 1 times 2 + 1 times 3 + 1 = 6

1198884

= 1 times 3 + 1 + 1 times 3 = 7

(40)


4

= 1199092

gt 1199093

gt 1199091


and 1199092



gt 1199092

gt 1199093

= 1199091






119894

= 120596(119890119894


isin 119860





119895


(seeAlgorithm 60)

Algorithm 60






119880 1198901

1198902

1198903

1198904

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]




119895


of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen



ℎ1

ℎ2

ℎ3

ℎ4

ℎ5


1

1198902

1198903

1198904



1


=



4


119895

(119895 =





(119865 119860 120596) is shown in Table 8


midG = (1198901

[055 071]) (1198902

[063 078])

(1198903

[061 075]) (1198904

[066 074])

(41)


1198881

= 04 times 2 + 0 + 0 + 0 = 08

1198882


1198883


1198884


1198885


(42)


4

gt ℎ5

= ℎ3

gt ℎ2

gt ℎ1


4



4

gt ℎ3

gt ℎ1

= ℎ2

= ℎ5


4


Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion





Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(1) ℎ119888 = [1 minus 120574+

1 minus 120574minus

] | 120574 isin ℎ

(2) ℎ1

cup ℎ2

= [120574minus

1

or 120574minus

2

120574+

1

or 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

(3) ℎ1

cap ℎ2

= [120574minus

1

and 120574minus

2

120574+

1

and 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

(4) ℎ120582 = [(120574minus

1

)120582

(120574+

1

)120582

] | 120574 isin ℎ 120582 gt 0


)120582


)120582

] | 120574 isin ℎ 120582 gt 0

(6) ℎ1

oplus ℎ2

= [120574minus

1

+ 120574minus

2

minus 120574minus

1

120574minus

2

120574+

1

+ 120574+

2

minus 120574+

1

120574+

2

] | 1205741

isin

ℎ1

1205742

isin ℎ2

(7) ℎ1

otimes ℎ2

= [120574minus

1

120574minus

2

120574+

1

120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2



and ℎ2


(1) ℎ1

oplus ℎ2

= ℎ2

oplus ℎ1

(2) ℎ1

otimes ℎ2

= ℎ2

otimes ℎ1

(3) (ℎ1

cup ℎ2

)119888

= ℎ119888

1

cap ℎ119888

2

(4) (ℎ1

cap ℎ2

)119888

= ℎ119888

1

cup ℎ119888

2

(5) (120582ℎ)119888 = (ℎ119888

)120582

(6) (ℎ120582)119888 = 120582(ℎ119888

)

(7) (ℎ1

oplus ℎ2

)119888

= ℎ119888

1

otimes ℎ119888

2

(8) (ℎ1

otimes ℎ2

)119888

= ℎ119888

1

oplus ℎ119888

2

Example 12 Let ℎ1

= [05 06] [03 08] [03 06] and ℎ2

=


ℎ1

cup ℎ2

= [120574minus

1

or 120574minus

2

120574+

1

or 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

= [05 or 04 06 or 05] [05 or 04 06 or 07]

[03 or 04 08 or 05] [03 or 04 08 or 07]

[03 or 04 06 or 05] [03 or 04 06 or 07]

= [05 06] [05 07] [04 08] [04 08]

[04 06] [04 07]

ℎ1

cap ℎ2

= [120574minus

1

and 120574minus

2

120574+

1

and 120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2

= [05 and 04 06 and 05] [05 and 04 06 and 07]

[03 and 04 08 and 05] [03 and 04 08 and 07]

[03 and 04 06 and 05] [03 and 04 06 and 07]

= [04 05] [04 06] [03 05] [03 07]

[03 05] [03 06]

ℎ1

oplus ℎ2

= [120574minus

1

+ 120574minus

2

minus 120574minus

1

120574minus

2

120574+

1

+ 120574+

2

minus 120574+

1

120574+

2

] |

1205741

isin ℎ1

1205742

isin ℎ2







= [07 08] [07 088] [058 09] [058 094]

[058 08] [058 088]

ℎ1

otimes ℎ2

= [120574minus

1

120574minus

2

120574+

1

120574+

2

] | 1205741

isin ℎ1

1205742

isin ℎ2




= [02 03] [02 042] [012 04] [012 056]

[012 03] [012 042]

(8)



and ℎ2


(1) ℎ119888 = [1 minus ℎ120590(119896)119880

1 minus ℎ120590(119896)119871

] | 119896 = 1 2 119897

(2) ℎ1

cup ℎ2

= [ℎ120590(119896)119871

1

or ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

or ℎ120590(119896)119880

2

] | 119896 =

1 2 119897

(3) ℎ1

cap ℎ2

= [ℎ120590(119896)119871

1

and ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

and ℎ120590(119896)119880

2

] | 119896 =

1 2 119897

(4) ℎ120582 = [(ℎ120590(119896)119871

)120582

(ℎ120590(119896)119880

)120582

] | 119896 = 1 2 119897

(5) 120582ℎ = [1 minus (1 minus ℎ120590(119896)119871

)120582

1 minus (1 minus ℎ120590(119896)119880

)120582

] | 119896 =

1 2 119897

(6) ℎ1

oplusℎ2

= [ℎ120590(119896)119871

1

+ℎ120590(119896)119871

2

minusℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

+ℎ120590(119896)119880

2

minus

ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 119897

(7) ℎ1

otimes ℎ2

= [ℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 119897

where ℎ120590(119896)

119895

= [ℎ120590(119896)119871

119895

ℎ120590(119896)119880

119895


119895



and ℎ2


(1) ℎ1

oplus ℎ2

= ℎ2

oplus ℎ1

(2) ℎ1

otimes ℎ2

= ℎ2

otimes ℎ1

(3) (ℎ

1

cup ℎ2

)119888

= ℎ119888

1

cap ℎ119888

2

(4) (ℎ

1

cap ℎ2

)119888

= ℎ119888

1

cup ℎ119888

2

(5) (120582ℎ)119888 = (ℎ119888

)120582

(6) (ℎ120582)119888 = 120582(ℎ119888

)(7) (ℎ

1

oplus ℎ2

)119888

= ℎ119888

1

otimes ℎ119888

2

(8) (ℎ

1

otimes ℎ2

)119888

= ℎ119888

1

oplus ℎ119888

2




1

= [03 06] [03 08]

[05 06] and ℎ2


ℎ1

cup ℎ2

= [ℎ120590(119896)119871

1

or ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

or ℎ120590(119896)119880

2

] | 119896 = 1 2 3

= [03 or 04 06 or 05] [03 or 04 08 or 07]

[05 or 04 06 or 07]

= [04 06] [04 08] [05 07]

ℎ1

cap ℎ2

= [ℎ120590(119896)119871

1

and ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

and ℎ120590(119896)119880

2

] | 119896 = 1 2 3

= [03 and 04 06 and 05] [03 and 04 08 and 07]

[05 and 04 06 and 07]

= [03 05] [03 07] [04 06]

ℎ1

oplus ℎ2

= [ℎ120590(119896)119871

1

+ ℎ120590(119896)119871

2

minus ℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

+ ℎ120590(119896)119880

2

minus ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 3




= [058 08] [058 094] [07 088]

ℎ1

otimes ℎ2

= [ℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 3


[05 sdot 04 06 sdot 07]

= [012 03] [012 056] [02 042]

(9)







119865 (119890) = ⟨119909 119865 (119890) (119909)⟩ 119909 isin 119880 (10)



1

1199092

1199093

1199094


1

1198902

1198903


119895



119895



119865 (1198901

) = ⟨1199091

[06 08] [06 07] [08 09]⟩

⟨1199092

[04 05] [07 08] [06 08]⟩

⟨1199093

[07 08] [06 08] [07 08]⟩

⟨1199094

[08 09] [07 09] [08 10]⟩



119880 1198901

1198902

1198903

1199091

[06 08] [06 07] [08 09] [05 06] [04 06] [05 07] [03 05] [04 07] [05 07]

1199092

[04 05] [07 08] [06 08] [07 08] [08 09] [07 08] [06 07] [05 08] [06 08]

1199093

[07 08] [06 08] [07 08] [06 09] [07 08] [07 09] [05 07] [06 08] [04 06]

1199094

[08 09] [07 09] [08 10] [06 07] [05 08] [06 08] [07 08] [08 09] [06 08]

119865 (1198902

) = ⟨1199091

[05 06] [04 06] [05 07]⟩

⟨1199092

[07 08] [08 09] [07 08]⟩

⟨1199093

[06 09] [07 08] [07 09]⟩

⟨1199094

[06 07] [05 08] [06 08]⟩

119865 (1198903

) = ⟨1199091

[03 05] [04 07] [05 07]⟩

⟨1199092

[06 07] [05 08] [06 08]⟩

⟨1199093

[05 07] [06 08] [04 06]⟩

⟨1199094

[07 08] [08 09] [06 08]⟩

(11)




1





1

1199092

1199093

1199094


1

1198902

1198903


119895




119865 (1198901

) = ⟨1199091

[06 07]⟩ ⟨1199092

[06 08]⟩

⟨1199093

[07 08]⟩ ⟨1199094

[07 09]⟩

119865 (1198902

) = ⟨1199091

[05 06]⟩ ⟨1199092

[07 08]⟩

⟨1199093

[06 09]⟩ ⟨1199094

[05 08]⟩

119865 (1198903

) = ⟨1199091

[03 05]⟩ ⟨1199092

[06 07]⟩

⟨1199093

[04 06]⟩ ⟨1199094

[06 08]⟩

(12)





1

1199092

1199093

1199094


1

1198902

1198903


119895





119865 (1198901

) = ⟨1199091

06 07 08⟩ ⟨1199092

05 07 08⟩

⟨1199093

07 08 08⟩ ⟨1199094

08 09 09⟩

119865 (1198902

) = ⟨1199091

05 06 07⟩ ⟨1199092

07 08 08⟩

⟨1199093

06 07 09⟩ ⟨1199094

06 08 08⟩

119865 (1198903

) = ⟨1199091

04 05 07⟩ ⟨1199092

06 07 07⟩

⟨1199093

04 06 07⟩ ⟨1199094

07 08 08⟩

(13)





1

1199092

1199093

1199094


1

1198902

1198903




119865 (1198901

) =

06

1199091

+

05

1199092

+

08

1199093

+

09

1199094

119865 (1198902

) =

05

1199091

+

07

1199092

+

07

1199093

+

08

1199094

119865 (1198903

) =

04

1199091

+

06

1199092

+

07

1199093

+

08

1199094

(14)






+

119860)


119865+

(119890) = (119909 119865+

(119890) (119909)) 119909 isin 119880

= (119909

119897

⋁

119896=1

120574120590(119896)

) 119909 isin 119880 forall119890 isin 119860

(15)

where 120574120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880



119873

119860)


119865119873

(119890) = (119909 119865119873

(119890) (119909)) 119909 isin 119880

= (119909

1

119897

119897

sum

119896=1

120574120590(119896)

) 119909 isin 119880 forall119890 isin 119860

(16)

where 120574120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




+

119860) and NRIVFS(119865119873




Table 2 ORIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[08 09] [05 07] [05 07]

1199092

[07 08] [08 09] [06 08]

1199093

[07 08] [07 09] [06 08]

1199094

[08 10] [06 08] [08 09]

Table 3 NRIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[067 070] [047 063] [040 063]

1199092

[057 070] [073 083] [057 077]

1199093

[067 080] [067 087] [050 070]

1199094

[077 093] [057 077] [070 083]






(1) 119860 sube 119861

(2) 120574120590(119896)1

le 120574120590(119896)

2


and 120574120590(119896)

2




1199092

1199093

1199094


1

1198902

1198903


Let 119860 = 1198901

1198902

119861 = 119864 = 1198901

1198902

1198903


119865 (1198901

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 09] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

119865 (1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119866 (1198901

) = ⟨1199091

[04 06] [02 09]⟩

⟨1199092

[02 07] [03 09] [07 09]⟩

⟨1199093

[06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119866 (1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119866 (1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

1199094

⟨[07 09] [08 10]⟩

(17)



(1) (119865 119860) ⊑ (119866 119861)

(2) (119866 119861) ⊑ (119865 119860)



119888


Clearly we have ((119865 119860)119888)119888 = (119865 119860)



119866119888

(1198901

) = ⟨1199091

[01 08] [04 06]⟩

⟨1199092

[01 03] [01 07] [03 08]⟩

⟨1199093

[02 04]⟩

⟨1199094

[02 03] [02 04] [05 07]⟩

119866119888

(1198902

) = ⟨1199091

[02 06] [05 07]⟩

⟨1199092

[02 06] [03 07]⟩

⟨1199093

[02 05] [03 07] [05 07]⟩

⟨1199094

[04 05] [05 07]⟩

119866119888

(1198903

) = ⟨1199091

[00 02] [04 05] [05 07]⟩

⟨1199092

[01 03] [04 045]⟩

⟨1199093

[00 01] [015 02] [03 07]⟩

⟨1199094

[00 02] [01 03]⟩

(18)



(119865 119860) and (119866 119861) = ( 119860 times 119861) (19)


(120572 120573) = ⟨119909 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cap 119866 (120573) (119909)⟩ 119909 isin 119880

(20)


(119865 119860) or (119866 119861) = (119868 119860 times 119861) (21)


119868 (120572 120573) = ⟨119909 119868 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cup 119866 (120573) (119909)⟩ 119909 isin 119880

(22)


(1198901

1198901

) = ⟨1199091

[02 05] [02 06]⟩

⟨1199092

[02 05] [03 07] [06 08]⟩

⟨1199093

[03 08] [05 08]⟩

⟨1199094

[02 04] [05 07] [05 07]⟩

(1198901

1198902

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[02 04] [05 06]⟩

(1198901

1198903

) = ⟨1199091

[02 05] [03 06] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 07] [05 08] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

(1198902

1198901

) = ⟨1199091

[01 04] [02 07] [02 08]⟩

⟨1199092

[02 05] [03 06] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06] [04 06]⟩

(1198902

1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[03 05] [04 06]⟩

⟨1199093

[02 04] [03 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

(1198902

1198903

) = ⟨1199091

[01 04] [03 06] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119868 (1198901

1198901

) = ⟨1199091

[04 06] [03 09]⟩

⟨1199092

[03 07] [04 09] [07 09]⟩

⟨1199093

[06 09] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198901

1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08] [06 08]⟩

⟨1199093

[03 09] [05 08] [05 08]⟩

⟨1199094

[03 05] [05 07]⟩


119868 (1198901

1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09] [07 09]⟩

⟨1199093

[03 09] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

119868 (1198902

1198901

) = ⟨1199091

[04 06] [03 09] [03 09]⟩

⟨1199092

[04 07] [05 09] [07 09]⟩

⟨1199093

[06 08] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198902

1198902

) = ⟨1199091

[03 05] [04 08] [04 08]⟩

⟨1199092

[04 07] [05 08]⟩

⟨1199093

[03 05] [04 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119868 (1198902

1198903

) = ⟨1199091

[03 05] [05 07] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

(23)


(1) ((119865 119860) and (119866 119861))119888

= (119865 119860)119888

or (119866 119861)119888

(2) ((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


119861)119888

= (119888



follows that (119865(120572)(119909) cap 119866(120573)(119909))119888

= 119865119888

(120572)(119909) cup 119866119888


(119865 119860)119888

or(119866 119861)119888

= (119865119888

119860)or (119866119888


119888

(120572)(119909) cup 119866119888





( 119862))(4) (119865 119860)or((119866 119861)and( 119862)) = ((119865 119860)or(119866 119861))and((119865 119860)or

( 119862))




(4) can be made




119864 997888rarr IVHF (119880) (24)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) oplus 119866 (119890) (119909)⟩ 119909 isin 119880

(25)


119864 997888rarr IVHF (119880) (26)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) otimes 119866 (119890) (119909)⟩ 119909 isin 119880

(27)


119865 (1198901

) = ⟨1199091

[03 05] [03 08]⟩

⟨1199092

[01 05] [02 05] [04 07]⟩

⟨1199093

[02 05] [03 06] [05 08]⟩

⟨1199094

[03 04] [05 07]⟩


119865 (1198902

) = ⟨1199091

[04 05] [03 07]⟩

⟨1199092

[02 06] [03 06]⟩

⟨1199093

[02 03] [03 04] [05 08]⟩

⟨1199094

[02 04] [04 06]⟩

119865 (1198903

) = ⟨1199091

[02 05] [04 07]⟩

⟨1199092

[05 06] [05 08]⟩

⟨1199093

[03 05] [02 07] [03 08]⟩

⟨1199094

[03 05] [02 07]⟩

(28)


(119865 oplus 119866) (1198901

)

= ⟨1199091

[058 08] [044 098]⟩

⟨1199092

[028 085] [044 095] [082 097]⟩

⟨1199093

[068 09] [072 092] [08 096]⟩

⟨1199094

[051 07] [08 094] [085 094]⟩

(119865 oplus 119866) (1198902

)

= ⟨1199091

[058 075] [058 094]⟩

⟨1199092

[044 088] [058 092]⟩

⟨1199093

[044 065] [051 082] [075 096]⟩

⟨1199094

[044 07] [07 084]⟩

(119865 oplus 119866) (1198903

)

= ⟨1199091

[044 075] [07 088] [088 10]⟩

⟨1199092

[0775 084] [085 098]⟩

⟨1199093

[051 085] [084 0955] [093 10]⟩

⟨1199094

[079 095] [084 10]⟩

(29)


(119865 otimes 119866) (1198901

)

= ⟨1199091

[012 03] [006 072]⟩

⟨1199092

[002 035] [006 045] [028 063]⟩

⟨1199093

[012 04] [018 048] [03 064]⟩

⟨1199094

[009 02] [03 056] [035 056]⟩

(119865 otimes 119866) (1198902

)

= ⟨1199091

[012 025] [012 056]⟩

⟨1199092

[006 042] [012 048]⟩

⟨1199093

[006 015] [009 028] [025 064]⟩

⟨1199094

[006 02] [02 036]⟩

(119865 otimes 119866) (1198903

)

= ⟨1199091

[006 025] [02 042] [032 07]⟩

⟨1199092

[0275 036] [035 072]⟩

⟨1199093

[009 035] [016 0595] [027 08]⟩

⟨1199094

[021 045] [016 07]⟩

(30)



119888

= 119865119888

otimes 119866119888

(4) (119865 otimes 119866)119888

= 119865119888

oplus 119866119888










(1) ⊑ 119865 ⊑ (2) 119888 = (3) 119888 =






119888

= and 119865 and 119865119888

=


119888

= and 119865 otimes 119865119888

=




120590(119896)

isin 119865(119890)(119909) (119896 = 1 2 119897) wedefine

120572120590(119896)

=

1 120574120590(119896)

ge 119905

0 120574120590(119896)

lt 119905

(31)


120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




119865119905

(119890) = 119871 (119865 (119890) 119905)

= ⟨119909 119865119905

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(32)

for all 119890 isin 119860



119880 1198901

1198902

1198903

1199091

1 0 1 0 0 0 0 0 0

1199092

0 1 1 1 1 1 0 0 1

1199093

1 1 1 1 1 1 0 1 0

1199094

1 1 1 0 0 1 1 1 1



119871 (119865 (1198901

) [06 08]) = ⟨1199091

1 0 1⟩ ⟨1199092

0 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

1 1 1⟩

119871 (119865 (1198902

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

1 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

0 0 1⟩

119871 (119865 (1198903

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

0 0 1⟩

⟨1199093

0 1 0⟩ ⟨1199094

1 1 1⟩

(33)



(119865[0608]

119860) where for 119909119895


[0608]

(119890119894

) = 119871(119865(119890119894


[0608]

(1198901

) (1199092

) = 119871(119865(1198901

)(1199092

) [06 08]) =


2





isin 119865(119890)(119909) (119896 = 1 2 119897) one defines

120572120590(119896)

=

1 120574120590(119896)

ge 120582 (119890)

0 120574120590(119896)

lt 120582 (119890)

(34)




120582


119865120582

(119890) = 119871 (119865 (119890) 120582 (119890))

= ⟨119909 119865120582

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(35)

for all 119890 isin 119860



119873


119873


midS (119890) =

1

|119880|

sum

119909isin119880

119865119873

(119890) (119909) (36)




119873


midS = (1198901

[067 07825]) (1198902

[061 0775])

(1198903

[05425 07325])

(37)



+


+


maxS (119890) = max119909isin119880

119865+

(119890) (119909) (38)


119880 1198901

1198902

1198903


)

1199091

0 0 1 0 0 0 0 0 0 1198881

= 1

1199092

0 1 0 1 1 1 1 1 1 1198882

= 7

1199093

1 0 1 1 1 1 0 1 0 1198883

= 6

1199094

1 1 1 0 0 1 1 1 1 1198884

= 7


119880 1198901

1198902

1198903


)

1199091

0 0 0 0 0 0 0 0 0 1198881

= 0

1199092

0 0 0 0 1 0 0 0 0 1198881

= 1

1199093

0 0 0 0 0 0 0 0 0 1198881

= 0

1199094

0 0 1 0 0 0 0 1 0 1198881

= 2



+


maxS = (1198901

[08 10]) (1198902

[08 09]) (1198903

[08 09])

(39)



Algorithm 54


(119865 119860)








of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen






1198881

= 1 + 0 + 0 = 1

1198882

= 1 + 1 times 3 + 1 times 3 = 7

1198883

= 1 times 2 + 1 times 3 + 1 = 6

1198884

= 1 times 3 + 1 + 1 times 3 = 7

(40)


4

= 1199092

gt 1199093

gt 1199091


and 1199092



gt 1199092

gt 1199093

= 1199091






119894

= 120596(119890119894


isin 119860





119895


(seeAlgorithm 60)

Algorithm 60






119880 1198901

1198902

1198903

1198904

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]




119895


of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen



ℎ1

ℎ2

ℎ3

ℎ4

ℎ5


1

1198902

1198903

1198904



1


=



4


119895

(119895 =





(119865 119860 120596) is shown in Table 8


midG = (1198901

[055 071]) (1198902

[063 078])

(1198903

[061 075]) (1198904

[066 074])

(41)


1198881

= 04 times 2 + 0 + 0 + 0 = 08

1198882


1198883


1198884


1198885


(42)


4

gt ℎ5

= ℎ3

gt ℎ2

gt ℎ1


4



4

gt ℎ3

gt ℎ1

= ℎ2

= ℎ5


4


Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion





Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











and ℎ2


(1) ℎ1

oplus ℎ2

= ℎ2

oplus ℎ1

(2) ℎ1

otimes ℎ2

= ℎ2

otimes ℎ1

(3) (ℎ

1

cup ℎ2

)119888

= ℎ119888

1

cap ℎ119888

2

(4) (ℎ

1

cap ℎ2

)119888

= ℎ119888

1

cup ℎ119888

2

(5) (120582ℎ)119888 = (ℎ119888

)120582

(6) (ℎ120582)119888 = 120582(ℎ119888

)(7) (ℎ

1

oplus ℎ2

)119888

= ℎ119888

1

otimes ℎ119888

2

(8) (ℎ

1

otimes ℎ2

)119888

= ℎ119888

1

oplus ℎ119888

2




1

= [03 06] [03 08]

[05 06] and ℎ2


ℎ1

cup ℎ2

= [ℎ120590(119896)119871

1

or ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

or ℎ120590(119896)119880

2

] | 119896 = 1 2 3

= [03 or 04 06 or 05] [03 or 04 08 or 07]

[05 or 04 06 or 07]

= [04 06] [04 08] [05 07]

ℎ1

cap ℎ2

= [ℎ120590(119896)119871

1

and ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

and ℎ120590(119896)119880

2

] | 119896 = 1 2 3

= [03 and 04 06 and 05] [03 and 04 08 and 07]

[05 and 04 06 and 07]

= [03 05] [03 07] [04 06]

ℎ1

oplus ℎ2

= [ℎ120590(119896)119871

1

+ ℎ120590(119896)119871

2

minus ℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

+ ℎ120590(119896)119880

2

minus ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 3




= [058 08] [058 094] [07 088]

ℎ1

otimes ℎ2

= [ℎ120590(119896)119871

1

ℎ120590(119896)119871

2

ℎ120590(119896)119880

1

ℎ120590(119896)119880

2

] | 119896 = 1 2 3


[05 sdot 04 06 sdot 07]

= [012 03] [012 056] [02 042]

(9)







119865 (119890) = ⟨119909 119865 (119890) (119909)⟩ 119909 isin 119880 (10)



1

1199092

1199093

1199094


1

1198902

1198903


119895



119895



119865 (1198901

) = ⟨1199091

[06 08] [06 07] [08 09]⟩

⟨1199092

[04 05] [07 08] [06 08]⟩

⟨1199093

[07 08] [06 08] [07 08]⟩

⟨1199094

[08 09] [07 09] [08 10]⟩



119880 1198901

1198902

1198903

1199091

[06 08] [06 07] [08 09] [05 06] [04 06] [05 07] [03 05] [04 07] [05 07]

1199092

[04 05] [07 08] [06 08] [07 08] [08 09] [07 08] [06 07] [05 08] [06 08]

1199093

[07 08] [06 08] [07 08] [06 09] [07 08] [07 09] [05 07] [06 08] [04 06]

1199094

[08 09] [07 09] [08 10] [06 07] [05 08] [06 08] [07 08] [08 09] [06 08]

119865 (1198902

) = ⟨1199091

[05 06] [04 06] [05 07]⟩

⟨1199092

[07 08] [08 09] [07 08]⟩

⟨1199093

[06 09] [07 08] [07 09]⟩

⟨1199094

[06 07] [05 08] [06 08]⟩

119865 (1198903

) = ⟨1199091

[03 05] [04 07] [05 07]⟩

⟨1199092

[06 07] [05 08] [06 08]⟩

⟨1199093

[05 07] [06 08] [04 06]⟩

⟨1199094

[07 08] [08 09] [06 08]⟩

(11)




1





1

1199092

1199093

1199094


1

1198902

1198903


119895




119865 (1198901

) = ⟨1199091

[06 07]⟩ ⟨1199092

[06 08]⟩

⟨1199093

[07 08]⟩ ⟨1199094

[07 09]⟩

119865 (1198902

) = ⟨1199091

[05 06]⟩ ⟨1199092

[07 08]⟩

⟨1199093

[06 09]⟩ ⟨1199094

[05 08]⟩

119865 (1198903

) = ⟨1199091

[03 05]⟩ ⟨1199092

[06 07]⟩

⟨1199093

[04 06]⟩ ⟨1199094

[06 08]⟩

(12)





1

1199092

1199093

1199094


1

1198902

1198903


119895





119865 (1198901

) = ⟨1199091

06 07 08⟩ ⟨1199092

05 07 08⟩

⟨1199093

07 08 08⟩ ⟨1199094

08 09 09⟩

119865 (1198902

) = ⟨1199091

05 06 07⟩ ⟨1199092

07 08 08⟩

⟨1199093

06 07 09⟩ ⟨1199094

06 08 08⟩

119865 (1198903

) = ⟨1199091

04 05 07⟩ ⟨1199092

06 07 07⟩

⟨1199093

04 06 07⟩ ⟨1199094

07 08 08⟩

(13)





1

1199092

1199093

1199094


1

1198902

1198903




119865 (1198901

) =

06

1199091

+

05

1199092

+

08

1199093

+

09

1199094

119865 (1198902

) =

05

1199091

+

07

1199092

+

07

1199093

+

08

1199094

119865 (1198903

) =

04

1199091

+

06

1199092

+

07

1199093

+

08

1199094

(14)






+

119860)


119865+

(119890) = (119909 119865+

(119890) (119909)) 119909 isin 119880

= (119909

119897

⋁

119896=1

120574120590(119896)

) 119909 isin 119880 forall119890 isin 119860

(15)

where 120574120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880



119873

119860)


119865119873

(119890) = (119909 119865119873

(119890) (119909)) 119909 isin 119880

= (119909

1

119897

119897

sum

119896=1

120574120590(119896)

) 119909 isin 119880 forall119890 isin 119860

(16)

where 120574120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




+

119860) and NRIVFS(119865119873




Table 2 ORIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[08 09] [05 07] [05 07]

1199092

[07 08] [08 09] [06 08]

1199093

[07 08] [07 09] [06 08]

1199094

[08 10] [06 08] [08 09]

Table 3 NRIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[067 070] [047 063] [040 063]

1199092

[057 070] [073 083] [057 077]

1199093

[067 080] [067 087] [050 070]

1199094

[077 093] [057 077] [070 083]






(1) 119860 sube 119861

(2) 120574120590(119896)1

le 120574120590(119896)

2


and 120574120590(119896)

2




1199092

1199093

1199094


1

1198902

1198903


Let 119860 = 1198901

1198902

119861 = 119864 = 1198901

1198902

1198903


119865 (1198901

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 09] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

119865 (1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119866 (1198901

) = ⟨1199091

[04 06] [02 09]⟩

⟨1199092

[02 07] [03 09] [07 09]⟩

⟨1199093

[06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119866 (1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119866 (1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

1199094

⟨[07 09] [08 10]⟩

(17)



(1) (119865 119860) ⊑ (119866 119861)

(2) (119866 119861) ⊑ (119865 119860)



119888


Clearly we have ((119865 119860)119888)119888 = (119865 119860)



119866119888

(1198901

) = ⟨1199091

[01 08] [04 06]⟩

⟨1199092

[01 03] [01 07] [03 08]⟩

⟨1199093

[02 04]⟩

⟨1199094

[02 03] [02 04] [05 07]⟩

119866119888

(1198902

) = ⟨1199091

[02 06] [05 07]⟩

⟨1199092

[02 06] [03 07]⟩

⟨1199093

[02 05] [03 07] [05 07]⟩

⟨1199094

[04 05] [05 07]⟩

119866119888

(1198903

) = ⟨1199091

[00 02] [04 05] [05 07]⟩

⟨1199092

[01 03] [04 045]⟩

⟨1199093

[00 01] [015 02] [03 07]⟩

⟨1199094

[00 02] [01 03]⟩

(18)



(119865 119860) and (119866 119861) = ( 119860 times 119861) (19)


(120572 120573) = ⟨119909 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cap 119866 (120573) (119909)⟩ 119909 isin 119880

(20)


(119865 119860) or (119866 119861) = (119868 119860 times 119861) (21)


119868 (120572 120573) = ⟨119909 119868 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cup 119866 (120573) (119909)⟩ 119909 isin 119880

(22)


(1198901

1198901

) = ⟨1199091

[02 05] [02 06]⟩

⟨1199092

[02 05] [03 07] [06 08]⟩

⟨1199093

[03 08] [05 08]⟩

⟨1199094

[02 04] [05 07] [05 07]⟩

(1198901

1198902

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[02 04] [05 06]⟩

(1198901

1198903

) = ⟨1199091

[02 05] [03 06] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 07] [05 08] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

(1198902

1198901

) = ⟨1199091

[01 04] [02 07] [02 08]⟩

⟨1199092

[02 05] [03 06] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06] [04 06]⟩

(1198902

1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[03 05] [04 06]⟩

⟨1199093

[02 04] [03 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

(1198902

1198903

) = ⟨1199091

[01 04] [03 06] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119868 (1198901

1198901

) = ⟨1199091

[04 06] [03 09]⟩

⟨1199092

[03 07] [04 09] [07 09]⟩

⟨1199093

[06 09] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198901

1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08] [06 08]⟩

⟨1199093

[03 09] [05 08] [05 08]⟩

⟨1199094

[03 05] [05 07]⟩


119868 (1198901

1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09] [07 09]⟩

⟨1199093

[03 09] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

119868 (1198902

1198901

) = ⟨1199091

[04 06] [03 09] [03 09]⟩

⟨1199092

[04 07] [05 09] [07 09]⟩

⟨1199093

[06 08] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198902

1198902

) = ⟨1199091

[03 05] [04 08] [04 08]⟩

⟨1199092

[04 07] [05 08]⟩

⟨1199093

[03 05] [04 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119868 (1198902

1198903

) = ⟨1199091

[03 05] [05 07] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

(23)


(1) ((119865 119860) and (119866 119861))119888

= (119865 119860)119888

or (119866 119861)119888

(2) ((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


119861)119888

= (119888



follows that (119865(120572)(119909) cap 119866(120573)(119909))119888

= 119865119888

(120572)(119909) cup 119866119888


(119865 119860)119888

or(119866 119861)119888

= (119865119888

119860)or (119866119888


119888

(120572)(119909) cup 119866119888





( 119862))(4) (119865 119860)or((119866 119861)and( 119862)) = ((119865 119860)or(119866 119861))and((119865 119860)or

( 119862))




(4) can be made




119864 997888rarr IVHF (119880) (24)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) oplus 119866 (119890) (119909)⟩ 119909 isin 119880

(25)


119864 997888rarr IVHF (119880) (26)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) otimes 119866 (119890) (119909)⟩ 119909 isin 119880

(27)


119865 (1198901

) = ⟨1199091

[03 05] [03 08]⟩

⟨1199092

[01 05] [02 05] [04 07]⟩

⟨1199093

[02 05] [03 06] [05 08]⟩

⟨1199094

[03 04] [05 07]⟩


119865 (1198902

) = ⟨1199091

[04 05] [03 07]⟩

⟨1199092

[02 06] [03 06]⟩

⟨1199093

[02 03] [03 04] [05 08]⟩

⟨1199094

[02 04] [04 06]⟩

119865 (1198903

) = ⟨1199091

[02 05] [04 07]⟩

⟨1199092

[05 06] [05 08]⟩

⟨1199093

[03 05] [02 07] [03 08]⟩

⟨1199094

[03 05] [02 07]⟩

(28)


(119865 oplus 119866) (1198901

)

= ⟨1199091

[058 08] [044 098]⟩

⟨1199092

[028 085] [044 095] [082 097]⟩

⟨1199093

[068 09] [072 092] [08 096]⟩

⟨1199094

[051 07] [08 094] [085 094]⟩

(119865 oplus 119866) (1198902

)

= ⟨1199091

[058 075] [058 094]⟩

⟨1199092

[044 088] [058 092]⟩

⟨1199093

[044 065] [051 082] [075 096]⟩

⟨1199094

[044 07] [07 084]⟩

(119865 oplus 119866) (1198903

)

= ⟨1199091

[044 075] [07 088] [088 10]⟩

⟨1199092

[0775 084] [085 098]⟩

⟨1199093

[051 085] [084 0955] [093 10]⟩

⟨1199094

[079 095] [084 10]⟩

(29)


(119865 otimes 119866) (1198901

)

= ⟨1199091

[012 03] [006 072]⟩

⟨1199092

[002 035] [006 045] [028 063]⟩

⟨1199093

[012 04] [018 048] [03 064]⟩

⟨1199094

[009 02] [03 056] [035 056]⟩

(119865 otimes 119866) (1198902

)

= ⟨1199091

[012 025] [012 056]⟩

⟨1199092

[006 042] [012 048]⟩

⟨1199093

[006 015] [009 028] [025 064]⟩

⟨1199094

[006 02] [02 036]⟩

(119865 otimes 119866) (1198903

)

= ⟨1199091

[006 025] [02 042] [032 07]⟩

⟨1199092

[0275 036] [035 072]⟩

⟨1199093

[009 035] [016 0595] [027 08]⟩

⟨1199094

[021 045] [016 07]⟩

(30)



119888

= 119865119888

otimes 119866119888

(4) (119865 otimes 119866)119888

= 119865119888

oplus 119866119888










(1) ⊑ 119865 ⊑ (2) 119888 = (3) 119888 =






119888

= and 119865 and 119865119888

=


119888

= and 119865 otimes 119865119888

=




120590(119896)

isin 119865(119890)(119909) (119896 = 1 2 119897) wedefine

120572120590(119896)

=

1 120574120590(119896)

ge 119905

0 120574120590(119896)

lt 119905

(31)


120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




119865119905

(119890) = 119871 (119865 (119890) 119905)

= ⟨119909 119865119905

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(32)

for all 119890 isin 119860



119880 1198901

1198902

1198903

1199091

1 0 1 0 0 0 0 0 0

1199092

0 1 1 1 1 1 0 0 1

1199093

1 1 1 1 1 1 0 1 0

1199094

1 1 1 0 0 1 1 1 1



119871 (119865 (1198901

) [06 08]) = ⟨1199091

1 0 1⟩ ⟨1199092

0 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

1 1 1⟩

119871 (119865 (1198902

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

1 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

0 0 1⟩

119871 (119865 (1198903

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

0 0 1⟩

⟨1199093

0 1 0⟩ ⟨1199094

1 1 1⟩

(33)



(119865[0608]

119860) where for 119909119895


[0608]

(119890119894

) = 119871(119865(119890119894


[0608]

(1198901

) (1199092

) = 119871(119865(1198901

)(1199092

) [06 08]) =


2





isin 119865(119890)(119909) (119896 = 1 2 119897) one defines

120572120590(119896)

=

1 120574120590(119896)

ge 120582 (119890)

0 120574120590(119896)

lt 120582 (119890)

(34)




120582


119865120582

(119890) = 119871 (119865 (119890) 120582 (119890))

= ⟨119909 119865120582

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(35)

for all 119890 isin 119860



119873


119873


midS (119890) =

1

|119880|

sum

119909isin119880

119865119873

(119890) (119909) (36)




119873


midS = (1198901

[067 07825]) (1198902

[061 0775])

(1198903

[05425 07325])

(37)



+


+


maxS (119890) = max119909isin119880

119865+

(119890) (119909) (38)


119880 1198901

1198902

1198903


)

1199091

0 0 1 0 0 0 0 0 0 1198881

= 1

1199092

0 1 0 1 1 1 1 1 1 1198882

= 7

1199093

1 0 1 1 1 1 0 1 0 1198883

= 6

1199094

1 1 1 0 0 1 1 1 1 1198884

= 7


119880 1198901

1198902

1198903


)

1199091

0 0 0 0 0 0 0 0 0 1198881

= 0

1199092

0 0 0 0 1 0 0 0 0 1198881

= 1

1199093

0 0 0 0 0 0 0 0 0 1198881

= 0

1199094

0 0 1 0 0 0 0 1 0 1198881

= 2



+


maxS = (1198901

[08 10]) (1198902

[08 09]) (1198903

[08 09])

(39)



Algorithm 54


(119865 119860)








of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen






1198881

= 1 + 0 + 0 = 1

1198882

= 1 + 1 times 3 + 1 times 3 = 7

1198883

= 1 times 2 + 1 times 3 + 1 = 6

1198884

= 1 times 3 + 1 + 1 times 3 = 7

(40)


4

= 1199092

gt 1199093

gt 1199091


and 1199092



gt 1199092

gt 1199093

= 1199091






119894

= 120596(119890119894


isin 119860





119895


(seeAlgorithm 60)

Algorithm 60






119880 1198901

1198902

1198903

1198904

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]




119895


of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen



ℎ1

ℎ2

ℎ3

ℎ4

ℎ5


1

1198902

1198903

1198904



1


=



4


119895

(119895 =





(119865 119860 120596) is shown in Table 8


midG = (1198901

[055 071]) (1198902

[063 078])

(1198903

[061 075]) (1198904

[066 074])

(41)


1198881

= 04 times 2 + 0 + 0 + 0 = 08

1198882


1198883


1198884


1198885


(42)


4

gt ℎ5

= ℎ3

gt ℎ2

gt ℎ1


4



4

gt ℎ3

gt ℎ1

= ℎ2

= ℎ5


4


Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion





Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119880 1198901

1198902

1198903

1199091

[06 08] [06 07] [08 09] [05 06] [04 06] [05 07] [03 05] [04 07] [05 07]

1199092

[04 05] [07 08] [06 08] [07 08] [08 09] [07 08] [06 07] [05 08] [06 08]

1199093

[07 08] [06 08] [07 08] [06 09] [07 08] [07 09] [05 07] [06 08] [04 06]

1199094

[08 09] [07 09] [08 10] [06 07] [05 08] [06 08] [07 08] [08 09] [06 08]

119865 (1198902

) = ⟨1199091

[05 06] [04 06] [05 07]⟩

⟨1199092

[07 08] [08 09] [07 08]⟩

⟨1199093

[06 09] [07 08] [07 09]⟩

⟨1199094

[06 07] [05 08] [06 08]⟩

119865 (1198903

) = ⟨1199091

[03 05] [04 07] [05 07]⟩

⟨1199092

[06 07] [05 08] [06 08]⟩

⟨1199093

[05 07] [06 08] [04 06]⟩

⟨1199094

[07 08] [08 09] [06 08]⟩

(11)




1





1

1199092

1199093

1199094


1

1198902

1198903


119895




119865 (1198901

) = ⟨1199091

[06 07]⟩ ⟨1199092

[06 08]⟩

⟨1199093

[07 08]⟩ ⟨1199094

[07 09]⟩

119865 (1198902

) = ⟨1199091

[05 06]⟩ ⟨1199092

[07 08]⟩

⟨1199093

[06 09]⟩ ⟨1199094

[05 08]⟩

119865 (1198903

) = ⟨1199091

[03 05]⟩ ⟨1199092

[06 07]⟩

⟨1199093

[04 06]⟩ ⟨1199094

[06 08]⟩

(12)





1

1199092

1199093

1199094


1

1198902

1198903


119895





119865 (1198901

) = ⟨1199091

06 07 08⟩ ⟨1199092

05 07 08⟩

⟨1199093

07 08 08⟩ ⟨1199094

08 09 09⟩

119865 (1198902

) = ⟨1199091

05 06 07⟩ ⟨1199092

07 08 08⟩

⟨1199093

06 07 09⟩ ⟨1199094

06 08 08⟩

119865 (1198903

) = ⟨1199091

04 05 07⟩ ⟨1199092

06 07 07⟩

⟨1199093

04 06 07⟩ ⟨1199094

07 08 08⟩

(13)





1

1199092

1199093

1199094


1

1198902

1198903




119865 (1198901

) =

06

1199091

+

05

1199092

+

08

1199093

+

09

1199094

119865 (1198902

) =

05

1199091

+

07

1199092

+

07

1199093

+

08

1199094

119865 (1198903

) =

04

1199091

+

06

1199092

+

07

1199093

+

08

1199094

(14)






+

119860)


119865+

(119890) = (119909 119865+

(119890) (119909)) 119909 isin 119880

= (119909

119897

⋁

119896=1

120574120590(119896)

) 119909 isin 119880 forall119890 isin 119860

(15)

where 120574120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880



119873

119860)


119865119873

(119890) = (119909 119865119873

(119890) (119909)) 119909 isin 119880

= (119909

1

119897

119897

sum

119896=1

120574120590(119896)

) 119909 isin 119880 forall119890 isin 119860

(16)

where 120574120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




+

119860) and NRIVFS(119865119873




Table 2 ORIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[08 09] [05 07] [05 07]

1199092

[07 08] [08 09] [06 08]

1199093

[07 08] [07 09] [06 08]

1199094

[08 10] [06 08] [08 09]

Table 3 NRIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[067 070] [047 063] [040 063]

1199092

[057 070] [073 083] [057 077]

1199093

[067 080] [067 087] [050 070]

1199094

[077 093] [057 077] [070 083]






(1) 119860 sube 119861

(2) 120574120590(119896)1

le 120574120590(119896)

2


and 120574120590(119896)

2




1199092

1199093

1199094


1

1198902

1198903


Let 119860 = 1198901

1198902

119861 = 119864 = 1198901

1198902

1198903


119865 (1198901

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 09] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

119865 (1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119866 (1198901

) = ⟨1199091

[04 06] [02 09]⟩

⟨1199092

[02 07] [03 09] [07 09]⟩

⟨1199093

[06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119866 (1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119866 (1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

1199094

⟨[07 09] [08 10]⟩

(17)



(1) (119865 119860) ⊑ (119866 119861)

(2) (119866 119861) ⊑ (119865 119860)



119888


Clearly we have ((119865 119860)119888)119888 = (119865 119860)



119866119888

(1198901

) = ⟨1199091

[01 08] [04 06]⟩

⟨1199092

[01 03] [01 07] [03 08]⟩

⟨1199093

[02 04]⟩

⟨1199094

[02 03] [02 04] [05 07]⟩

119866119888

(1198902

) = ⟨1199091

[02 06] [05 07]⟩

⟨1199092

[02 06] [03 07]⟩

⟨1199093

[02 05] [03 07] [05 07]⟩

⟨1199094

[04 05] [05 07]⟩

119866119888

(1198903

) = ⟨1199091

[00 02] [04 05] [05 07]⟩

⟨1199092

[01 03] [04 045]⟩

⟨1199093

[00 01] [015 02] [03 07]⟩

⟨1199094

[00 02] [01 03]⟩

(18)



(119865 119860) and (119866 119861) = ( 119860 times 119861) (19)


(120572 120573) = ⟨119909 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cap 119866 (120573) (119909)⟩ 119909 isin 119880

(20)


(119865 119860) or (119866 119861) = (119868 119860 times 119861) (21)


119868 (120572 120573) = ⟨119909 119868 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cup 119866 (120573) (119909)⟩ 119909 isin 119880

(22)


(1198901

1198901

) = ⟨1199091

[02 05] [02 06]⟩

⟨1199092

[02 05] [03 07] [06 08]⟩

⟨1199093

[03 08] [05 08]⟩

⟨1199094

[02 04] [05 07] [05 07]⟩

(1198901

1198902

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[02 04] [05 06]⟩

(1198901

1198903

) = ⟨1199091

[02 05] [03 06] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 07] [05 08] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

(1198902

1198901

) = ⟨1199091

[01 04] [02 07] [02 08]⟩

⟨1199092

[02 05] [03 06] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06] [04 06]⟩

(1198902

1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[03 05] [04 06]⟩

⟨1199093

[02 04] [03 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

(1198902

1198903

) = ⟨1199091

[01 04] [03 06] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119868 (1198901

1198901

) = ⟨1199091

[04 06] [03 09]⟩

⟨1199092

[03 07] [04 09] [07 09]⟩

⟨1199093

[06 09] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198901

1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08] [06 08]⟩

⟨1199093

[03 09] [05 08] [05 08]⟩

⟨1199094

[03 05] [05 07]⟩


119868 (1198901

1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09] [07 09]⟩

⟨1199093

[03 09] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

119868 (1198902

1198901

) = ⟨1199091

[04 06] [03 09] [03 09]⟩

⟨1199092

[04 07] [05 09] [07 09]⟩

⟨1199093

[06 08] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198902

1198902

) = ⟨1199091

[03 05] [04 08] [04 08]⟩

⟨1199092

[04 07] [05 08]⟩

⟨1199093

[03 05] [04 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119868 (1198902

1198903

) = ⟨1199091

[03 05] [05 07] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

(23)


(1) ((119865 119860) and (119866 119861))119888

= (119865 119860)119888

or (119866 119861)119888

(2) ((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


119861)119888

= (119888



follows that (119865(120572)(119909) cap 119866(120573)(119909))119888

= 119865119888

(120572)(119909) cup 119866119888


(119865 119860)119888

or(119866 119861)119888

= (119865119888

119860)or (119866119888


119888

(120572)(119909) cup 119866119888





( 119862))(4) (119865 119860)or((119866 119861)and( 119862)) = ((119865 119860)or(119866 119861))and((119865 119860)or

( 119862))




(4) can be made




119864 997888rarr IVHF (119880) (24)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) oplus 119866 (119890) (119909)⟩ 119909 isin 119880

(25)


119864 997888rarr IVHF (119880) (26)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) otimes 119866 (119890) (119909)⟩ 119909 isin 119880

(27)


119865 (1198901

) = ⟨1199091

[03 05] [03 08]⟩

⟨1199092

[01 05] [02 05] [04 07]⟩

⟨1199093

[02 05] [03 06] [05 08]⟩

⟨1199094

[03 04] [05 07]⟩


119865 (1198902

) = ⟨1199091

[04 05] [03 07]⟩

⟨1199092

[02 06] [03 06]⟩

⟨1199093

[02 03] [03 04] [05 08]⟩

⟨1199094

[02 04] [04 06]⟩

119865 (1198903

) = ⟨1199091

[02 05] [04 07]⟩

⟨1199092

[05 06] [05 08]⟩

⟨1199093

[03 05] [02 07] [03 08]⟩

⟨1199094

[03 05] [02 07]⟩

(28)


(119865 oplus 119866) (1198901

)

= ⟨1199091

[058 08] [044 098]⟩

⟨1199092

[028 085] [044 095] [082 097]⟩

⟨1199093

[068 09] [072 092] [08 096]⟩

⟨1199094

[051 07] [08 094] [085 094]⟩

(119865 oplus 119866) (1198902

)

= ⟨1199091

[058 075] [058 094]⟩

⟨1199092

[044 088] [058 092]⟩

⟨1199093

[044 065] [051 082] [075 096]⟩

⟨1199094

[044 07] [07 084]⟩

(119865 oplus 119866) (1198903

)

= ⟨1199091

[044 075] [07 088] [088 10]⟩

⟨1199092

[0775 084] [085 098]⟩

⟨1199093

[051 085] [084 0955] [093 10]⟩

⟨1199094

[079 095] [084 10]⟩

(29)


(119865 otimes 119866) (1198901

)

= ⟨1199091

[012 03] [006 072]⟩

⟨1199092

[002 035] [006 045] [028 063]⟩

⟨1199093

[012 04] [018 048] [03 064]⟩

⟨1199094

[009 02] [03 056] [035 056]⟩

(119865 otimes 119866) (1198902

)

= ⟨1199091

[012 025] [012 056]⟩

⟨1199092

[006 042] [012 048]⟩

⟨1199093

[006 015] [009 028] [025 064]⟩

⟨1199094

[006 02] [02 036]⟩

(119865 otimes 119866) (1198903

)

= ⟨1199091

[006 025] [02 042] [032 07]⟩

⟨1199092

[0275 036] [035 072]⟩

⟨1199093

[009 035] [016 0595] [027 08]⟩

⟨1199094

[021 045] [016 07]⟩

(30)



119888

= 119865119888

otimes 119866119888

(4) (119865 otimes 119866)119888

= 119865119888

oplus 119866119888










(1) ⊑ 119865 ⊑ (2) 119888 = (3) 119888 =






119888

= and 119865 and 119865119888

=


119888

= and 119865 otimes 119865119888

=




120590(119896)

isin 119865(119890)(119909) (119896 = 1 2 119897) wedefine

120572120590(119896)

=

1 120574120590(119896)

ge 119905

0 120574120590(119896)

lt 119905

(31)


120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




119865119905

(119890) = 119871 (119865 (119890) 119905)

= ⟨119909 119865119905

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(32)

for all 119890 isin 119860



119880 1198901

1198902

1198903

1199091

1 0 1 0 0 0 0 0 0

1199092

0 1 1 1 1 1 0 0 1

1199093

1 1 1 1 1 1 0 1 0

1199094

1 1 1 0 0 1 1 1 1



119871 (119865 (1198901

) [06 08]) = ⟨1199091

1 0 1⟩ ⟨1199092

0 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

1 1 1⟩

119871 (119865 (1198902

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

1 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

0 0 1⟩

119871 (119865 (1198903

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

0 0 1⟩

⟨1199093

0 1 0⟩ ⟨1199094

1 1 1⟩

(33)



(119865[0608]

119860) where for 119909119895


[0608]

(119890119894

) = 119871(119865(119890119894


[0608]

(1198901

) (1199092

) = 119871(119865(1198901

)(1199092

) [06 08]) =


2





isin 119865(119890)(119909) (119896 = 1 2 119897) one defines

120572120590(119896)

=

1 120574120590(119896)

ge 120582 (119890)

0 120574120590(119896)

lt 120582 (119890)

(34)




120582


119865120582

(119890) = 119871 (119865 (119890) 120582 (119890))

= ⟨119909 119865120582

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(35)

for all 119890 isin 119860



119873


119873


midS (119890) =

1

|119880|

sum

119909isin119880

119865119873

(119890) (119909) (36)




119873


midS = (1198901

[067 07825]) (1198902

[061 0775])

(1198903

[05425 07325])

(37)



+


+


maxS (119890) = max119909isin119880

119865+

(119890) (119909) (38)


119880 1198901

1198902

1198903


)

1199091

0 0 1 0 0 0 0 0 0 1198881

= 1

1199092

0 1 0 1 1 1 1 1 1 1198882

= 7

1199093

1 0 1 1 1 1 0 1 0 1198883

= 6

1199094

1 1 1 0 0 1 1 1 1 1198884

= 7


119880 1198901

1198902

1198903


)

1199091

0 0 0 0 0 0 0 0 0 1198881

= 0

1199092

0 0 0 0 1 0 0 0 0 1198881

= 1

1199093

0 0 0 0 0 0 0 0 0 1198881

= 0

1199094

0 0 1 0 0 0 0 1 0 1198881

= 2



+


maxS = (1198901

[08 10]) (1198902

[08 09]) (1198903

[08 09])

(39)



Algorithm 54


(119865 119860)








of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen






1198881

= 1 + 0 + 0 = 1

1198882

= 1 + 1 times 3 + 1 times 3 = 7

1198883

= 1 times 2 + 1 times 3 + 1 = 6

1198884

= 1 times 3 + 1 + 1 times 3 = 7

(40)


4

= 1199092

gt 1199093

gt 1199091


and 1199092



gt 1199092

gt 1199093

= 1199091






119894

= 120596(119890119894


isin 119860





119895


(seeAlgorithm 60)

Algorithm 60






119880 1198901

1198902

1198903

1198904

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]




119895


of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen



ℎ1

ℎ2

ℎ3

ℎ4

ℎ5


1

1198902

1198903

1198904



1


=



4


119895

(119895 =





(119865 119860 120596) is shown in Table 8


midG = (1198901

[055 071]) (1198902

[063 078])

(1198903

[061 075]) (1198904

[066 074])

(41)


1198881

= 04 times 2 + 0 + 0 + 0 = 08

1198882


1198883


1198884


1198885


(42)


4

gt ℎ5

= ℎ3

gt ℎ2

gt ℎ1


4



4

gt ℎ3

gt ℎ1

= ℎ2

= ℎ5


4


Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion





Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119865 (1198901

) = ⟨1199091

06 07 08⟩ ⟨1199092

05 07 08⟩

⟨1199093

07 08 08⟩ ⟨1199094

08 09 09⟩

119865 (1198902

) = ⟨1199091

05 06 07⟩ ⟨1199092

07 08 08⟩

⟨1199093

06 07 09⟩ ⟨1199094

06 08 08⟩

119865 (1198903

) = ⟨1199091

04 05 07⟩ ⟨1199092

06 07 07⟩

⟨1199093

04 06 07⟩ ⟨1199094

07 08 08⟩

(13)





1

1199092

1199093

1199094


1

1198902

1198903




119865 (1198901

) =

06

1199091

+

05

1199092

+

08

1199093

+

09

1199094

119865 (1198902

) =

05

1199091

+

07

1199092

+

07

1199093

+

08

1199094

119865 (1198903

) =

04

1199091

+

06

1199092

+

07

1199093

+

08

1199094

(14)






+

119860)


119865+

(119890) = (119909 119865+

(119890) (119909)) 119909 isin 119880

= (119909

119897

⋁

119896=1

120574120590(119896)

) 119909 isin 119880 forall119890 isin 119860

(15)

where 120574120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880



119873

119860)


119865119873

(119890) = (119909 119865119873

(119890) (119909)) 119909 isin 119880

= (119909

1

119897

119897

sum

119896=1

120574120590(119896)

) 119909 isin 119880 forall119890 isin 119860

(16)

where 120574120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




+

119860) and NRIVFS(119865119873




Table 2 ORIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[08 09] [05 07] [05 07]

1199092

[07 08] [08 09] [06 08]

1199093

[07 08] [07 09] [06 08]

1199094

[08 10] [06 08] [08 09]

Table 3 NRIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[067 070] [047 063] [040 063]

1199092

[057 070] [073 083] [057 077]

1199093

[067 080] [067 087] [050 070]

1199094

[077 093] [057 077] [070 083]






(1) 119860 sube 119861

(2) 120574120590(119896)1

le 120574120590(119896)

2


and 120574120590(119896)

2




1199092

1199093

1199094


1

1198902

1198903


Let 119860 = 1198901

1198902

119861 = 119864 = 1198901

1198902

1198903


119865 (1198901

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 09] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

119865 (1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119866 (1198901

) = ⟨1199091

[04 06] [02 09]⟩

⟨1199092

[02 07] [03 09] [07 09]⟩

⟨1199093

[06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119866 (1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119866 (1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

1199094

⟨[07 09] [08 10]⟩

(17)



(1) (119865 119860) ⊑ (119866 119861)

(2) (119866 119861) ⊑ (119865 119860)



119888


Clearly we have ((119865 119860)119888)119888 = (119865 119860)



119866119888

(1198901

) = ⟨1199091

[01 08] [04 06]⟩

⟨1199092

[01 03] [01 07] [03 08]⟩

⟨1199093

[02 04]⟩

⟨1199094

[02 03] [02 04] [05 07]⟩

119866119888

(1198902

) = ⟨1199091

[02 06] [05 07]⟩

⟨1199092

[02 06] [03 07]⟩

⟨1199093

[02 05] [03 07] [05 07]⟩

⟨1199094

[04 05] [05 07]⟩

119866119888

(1198903

) = ⟨1199091

[00 02] [04 05] [05 07]⟩

⟨1199092

[01 03] [04 045]⟩

⟨1199093

[00 01] [015 02] [03 07]⟩

⟨1199094

[00 02] [01 03]⟩

(18)



(119865 119860) and (119866 119861) = ( 119860 times 119861) (19)


(120572 120573) = ⟨119909 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cap 119866 (120573) (119909)⟩ 119909 isin 119880

(20)


(119865 119860) or (119866 119861) = (119868 119860 times 119861) (21)


119868 (120572 120573) = ⟨119909 119868 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cup 119866 (120573) (119909)⟩ 119909 isin 119880

(22)


(1198901

1198901

) = ⟨1199091

[02 05] [02 06]⟩

⟨1199092

[02 05] [03 07] [06 08]⟩

⟨1199093

[03 08] [05 08]⟩

⟨1199094

[02 04] [05 07] [05 07]⟩

(1198901

1198902

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[02 04] [05 06]⟩

(1198901

1198903

) = ⟨1199091

[02 05] [03 06] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 07] [05 08] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

(1198902

1198901

) = ⟨1199091

[01 04] [02 07] [02 08]⟩

⟨1199092

[02 05] [03 06] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06] [04 06]⟩

(1198902

1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[03 05] [04 06]⟩

⟨1199093

[02 04] [03 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

(1198902

1198903

) = ⟨1199091

[01 04] [03 06] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119868 (1198901

1198901

) = ⟨1199091

[04 06] [03 09]⟩

⟨1199092

[03 07] [04 09] [07 09]⟩

⟨1199093

[06 09] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198901

1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08] [06 08]⟩

⟨1199093

[03 09] [05 08] [05 08]⟩

⟨1199094

[03 05] [05 07]⟩


119868 (1198901

1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09] [07 09]⟩

⟨1199093

[03 09] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

119868 (1198902

1198901

) = ⟨1199091

[04 06] [03 09] [03 09]⟩

⟨1199092

[04 07] [05 09] [07 09]⟩

⟨1199093

[06 08] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198902

1198902

) = ⟨1199091

[03 05] [04 08] [04 08]⟩

⟨1199092

[04 07] [05 08]⟩

⟨1199093

[03 05] [04 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119868 (1198902

1198903

) = ⟨1199091

[03 05] [05 07] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

(23)


(1) ((119865 119860) and (119866 119861))119888

= (119865 119860)119888

or (119866 119861)119888

(2) ((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


119861)119888

= (119888



follows that (119865(120572)(119909) cap 119866(120573)(119909))119888

= 119865119888

(120572)(119909) cup 119866119888


(119865 119860)119888

or(119866 119861)119888

= (119865119888

119860)or (119866119888


119888

(120572)(119909) cup 119866119888





( 119862))(4) (119865 119860)or((119866 119861)and( 119862)) = ((119865 119860)or(119866 119861))and((119865 119860)or

( 119862))




(4) can be made




119864 997888rarr IVHF (119880) (24)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) oplus 119866 (119890) (119909)⟩ 119909 isin 119880

(25)


119864 997888rarr IVHF (119880) (26)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) otimes 119866 (119890) (119909)⟩ 119909 isin 119880

(27)


119865 (1198901

) = ⟨1199091

[03 05] [03 08]⟩

⟨1199092

[01 05] [02 05] [04 07]⟩

⟨1199093

[02 05] [03 06] [05 08]⟩

⟨1199094

[03 04] [05 07]⟩


119865 (1198902

) = ⟨1199091

[04 05] [03 07]⟩

⟨1199092

[02 06] [03 06]⟩

⟨1199093

[02 03] [03 04] [05 08]⟩

⟨1199094

[02 04] [04 06]⟩

119865 (1198903

) = ⟨1199091

[02 05] [04 07]⟩

⟨1199092

[05 06] [05 08]⟩

⟨1199093

[03 05] [02 07] [03 08]⟩

⟨1199094

[03 05] [02 07]⟩

(28)


(119865 oplus 119866) (1198901

)

= ⟨1199091

[058 08] [044 098]⟩

⟨1199092

[028 085] [044 095] [082 097]⟩

⟨1199093

[068 09] [072 092] [08 096]⟩

⟨1199094

[051 07] [08 094] [085 094]⟩

(119865 oplus 119866) (1198902

)

= ⟨1199091

[058 075] [058 094]⟩

⟨1199092

[044 088] [058 092]⟩

⟨1199093

[044 065] [051 082] [075 096]⟩

⟨1199094

[044 07] [07 084]⟩

(119865 oplus 119866) (1198903

)

= ⟨1199091

[044 075] [07 088] [088 10]⟩

⟨1199092

[0775 084] [085 098]⟩

⟨1199093

[051 085] [084 0955] [093 10]⟩

⟨1199094

[079 095] [084 10]⟩

(29)


(119865 otimes 119866) (1198901

)

= ⟨1199091

[012 03] [006 072]⟩

⟨1199092

[002 035] [006 045] [028 063]⟩

⟨1199093

[012 04] [018 048] [03 064]⟩

⟨1199094

[009 02] [03 056] [035 056]⟩

(119865 otimes 119866) (1198902

)

= ⟨1199091

[012 025] [012 056]⟩

⟨1199092

[006 042] [012 048]⟩

⟨1199093

[006 015] [009 028] [025 064]⟩

⟨1199094

[006 02] [02 036]⟩

(119865 otimes 119866) (1198903

)

= ⟨1199091

[006 025] [02 042] [032 07]⟩

⟨1199092

[0275 036] [035 072]⟩

⟨1199093

[009 035] [016 0595] [027 08]⟩

⟨1199094

[021 045] [016 07]⟩

(30)



119888

= 119865119888

otimes 119866119888

(4) (119865 otimes 119866)119888

= 119865119888

oplus 119866119888










(1) ⊑ 119865 ⊑ (2) 119888 = (3) 119888 =






119888

= and 119865 and 119865119888

=


119888

= and 119865 otimes 119865119888

=




120590(119896)

isin 119865(119890)(119909) (119896 = 1 2 119897) wedefine

120572120590(119896)

=

1 120574120590(119896)

ge 119905

0 120574120590(119896)

lt 119905

(31)


120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




119865119905

(119890) = 119871 (119865 (119890) 119905)

= ⟨119909 119865119905

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(32)

for all 119890 isin 119860



119880 1198901

1198902

1198903

1199091

1 0 1 0 0 0 0 0 0

1199092

0 1 1 1 1 1 0 0 1

1199093

1 1 1 1 1 1 0 1 0

1199094

1 1 1 0 0 1 1 1 1



119871 (119865 (1198901

) [06 08]) = ⟨1199091

1 0 1⟩ ⟨1199092

0 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

1 1 1⟩

119871 (119865 (1198902

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

1 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

0 0 1⟩

119871 (119865 (1198903

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

0 0 1⟩

⟨1199093

0 1 0⟩ ⟨1199094

1 1 1⟩

(33)



(119865[0608]

119860) where for 119909119895


[0608]

(119890119894

) = 119871(119865(119890119894


[0608]

(1198901

) (1199092

) = 119871(119865(1198901

)(1199092

) [06 08]) =


2





isin 119865(119890)(119909) (119896 = 1 2 119897) one defines

120572120590(119896)

=

1 120574120590(119896)

ge 120582 (119890)

0 120574120590(119896)

lt 120582 (119890)

(34)




120582


119865120582

(119890) = 119871 (119865 (119890) 120582 (119890))

= ⟨119909 119865120582

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(35)

for all 119890 isin 119860



119873


119873


midS (119890) =

1

|119880|

sum

119909isin119880

119865119873

(119890) (119909) (36)




119873


midS = (1198901

[067 07825]) (1198902

[061 0775])

(1198903

[05425 07325])

(37)



+


+


maxS (119890) = max119909isin119880

119865+

(119890) (119909) (38)


119880 1198901

1198902

1198903


)

1199091

0 0 1 0 0 0 0 0 0 1198881

= 1

1199092

0 1 0 1 1 1 1 1 1 1198882

= 7

1199093

1 0 1 1 1 1 0 1 0 1198883

= 6

1199094

1 1 1 0 0 1 1 1 1 1198884

= 7


119880 1198901

1198902

1198903


)

1199091

0 0 0 0 0 0 0 0 0 1198881

= 0

1199092

0 0 0 0 1 0 0 0 0 1198881

= 1

1199093

0 0 0 0 0 0 0 0 0 1198881

= 0

1199094

0 0 1 0 0 0 0 1 0 1198881

= 2



+


maxS = (1198901

[08 10]) (1198902

[08 09]) (1198903

[08 09])

(39)



Algorithm 54


(119865 119860)








of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen






1198881

= 1 + 0 + 0 = 1

1198882

= 1 + 1 times 3 + 1 times 3 = 7

1198883

= 1 times 2 + 1 times 3 + 1 = 6

1198884

= 1 times 3 + 1 + 1 times 3 = 7

(40)


4

= 1199092

gt 1199093

gt 1199091


and 1199092



gt 1199092

gt 1199093

= 1199091






119894

= 120596(119890119894


isin 119860





119895


(seeAlgorithm 60)

Algorithm 60






119880 1198901

1198902

1198903

1198904

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]




119895


of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen



ℎ1

ℎ2

ℎ3

ℎ4

ℎ5


1

1198902

1198903

1198904



1


=



4


119895

(119895 =





(119865 119860 120596) is shown in Table 8


midG = (1198901

[055 071]) (1198902

[063 078])

(1198903

[061 075]) (1198904

[066 074])

(41)


1198881

= 04 times 2 + 0 + 0 + 0 = 08

1198882


1198883


1198884


1198885


(42)


4

gt ℎ5

= ℎ3

gt ℎ2

gt ℎ1


4



4

gt ℎ3

gt ℎ1

= ℎ2

= ℎ5


4


Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion





Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 2 ORIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[08 09] [05 07] [05 07]

1199092

[07 08] [08 09] [06 08]

1199093

[07 08] [07 09] [06 08]

1199094

[08 10] [06 08] [08 09]

Table 3 NRIVFS ofS = (119865 119860)

119880 1198901

1198902

1198903

1199091

[067 070] [047 063] [040 063]

1199092

[057 070] [073 083] [057 077]

1199093

[067 080] [067 087] [050 070]

1199094

[077 093] [057 077] [070 083]






(1) 119860 sube 119861

(2) 120574120590(119896)1

le 120574120590(119896)

2


and 120574120590(119896)

2




1199092

1199093

1199094


1

1198902

1198903


Let 119860 = 1198901

1198902

119861 = 119864 = 1198901

1198902

1198903


119865 (1198901

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 09] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

119865 (1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119866 (1198901

) = ⟨1199091

[04 06] [02 09]⟩

⟨1199092

[02 07] [03 09] [07 09]⟩

⟨1199093

[06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119866 (1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119866 (1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

1199094

⟨[07 09] [08 10]⟩

(17)



(1) (119865 119860) ⊑ (119866 119861)

(2) (119866 119861) ⊑ (119865 119860)



119888


Clearly we have ((119865 119860)119888)119888 = (119865 119860)



119866119888

(1198901

) = ⟨1199091

[01 08] [04 06]⟩

⟨1199092

[01 03] [01 07] [03 08]⟩

⟨1199093

[02 04]⟩

⟨1199094

[02 03] [02 04] [05 07]⟩

119866119888

(1198902

) = ⟨1199091

[02 06] [05 07]⟩

⟨1199092

[02 06] [03 07]⟩

⟨1199093

[02 05] [03 07] [05 07]⟩

⟨1199094

[04 05] [05 07]⟩

119866119888

(1198903

) = ⟨1199091

[00 02] [04 05] [05 07]⟩

⟨1199092

[01 03] [04 045]⟩

⟨1199093

[00 01] [015 02] [03 07]⟩

⟨1199094

[00 02] [01 03]⟩

(18)



(119865 119860) and (119866 119861) = ( 119860 times 119861) (19)


(120572 120573) = ⟨119909 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cap 119866 (120573) (119909)⟩ 119909 isin 119880

(20)


(119865 119860) or (119866 119861) = (119868 119860 times 119861) (21)


119868 (120572 120573) = ⟨119909 119868 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cup 119866 (120573) (119909)⟩ 119909 isin 119880

(22)


(1198901

1198901

) = ⟨1199091

[02 05] [02 06]⟩

⟨1199092

[02 05] [03 07] [06 08]⟩

⟨1199093

[03 08] [05 08]⟩

⟨1199094

[02 04] [05 07] [05 07]⟩

(1198901

1198902

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[02 04] [05 06]⟩

(1198901

1198903

) = ⟨1199091

[02 05] [03 06] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 07] [05 08] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

(1198902

1198901

) = ⟨1199091

[01 04] [02 07] [02 08]⟩

⟨1199092

[02 05] [03 06] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06] [04 06]⟩

(1198902

1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[03 05] [04 06]⟩

⟨1199093

[02 04] [03 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

(1198902

1198903

) = ⟨1199091

[01 04] [03 06] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119868 (1198901

1198901

) = ⟨1199091

[04 06] [03 09]⟩

⟨1199092

[03 07] [04 09] [07 09]⟩

⟨1199093

[06 09] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198901

1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08] [06 08]⟩

⟨1199093

[03 09] [05 08] [05 08]⟩

⟨1199094

[03 05] [05 07]⟩


119868 (1198901

1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09] [07 09]⟩

⟨1199093

[03 09] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

119868 (1198902

1198901

) = ⟨1199091

[04 06] [03 09] [03 09]⟩

⟨1199092

[04 07] [05 09] [07 09]⟩

⟨1199093

[06 08] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198902

1198902

) = ⟨1199091

[03 05] [04 08] [04 08]⟩

⟨1199092

[04 07] [05 08]⟩

⟨1199093

[03 05] [04 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119868 (1198902

1198903

) = ⟨1199091

[03 05] [05 07] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

(23)


(1) ((119865 119860) and (119866 119861))119888

= (119865 119860)119888

or (119866 119861)119888

(2) ((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


119861)119888

= (119888



follows that (119865(120572)(119909) cap 119866(120573)(119909))119888

= 119865119888

(120572)(119909) cup 119866119888


(119865 119860)119888

or(119866 119861)119888

= (119865119888

119860)or (119866119888


119888

(120572)(119909) cup 119866119888





( 119862))(4) (119865 119860)or((119866 119861)and( 119862)) = ((119865 119860)or(119866 119861))and((119865 119860)or

( 119862))




(4) can be made




119864 997888rarr IVHF (119880) (24)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) oplus 119866 (119890) (119909)⟩ 119909 isin 119880

(25)


119864 997888rarr IVHF (119880) (26)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) otimes 119866 (119890) (119909)⟩ 119909 isin 119880

(27)


119865 (1198901

) = ⟨1199091

[03 05] [03 08]⟩

⟨1199092

[01 05] [02 05] [04 07]⟩

⟨1199093

[02 05] [03 06] [05 08]⟩

⟨1199094

[03 04] [05 07]⟩


119865 (1198902

) = ⟨1199091

[04 05] [03 07]⟩

⟨1199092

[02 06] [03 06]⟩

⟨1199093

[02 03] [03 04] [05 08]⟩

⟨1199094

[02 04] [04 06]⟩

119865 (1198903

) = ⟨1199091

[02 05] [04 07]⟩

⟨1199092

[05 06] [05 08]⟩

⟨1199093

[03 05] [02 07] [03 08]⟩

⟨1199094

[03 05] [02 07]⟩

(28)


(119865 oplus 119866) (1198901

)

= ⟨1199091

[058 08] [044 098]⟩

⟨1199092

[028 085] [044 095] [082 097]⟩

⟨1199093

[068 09] [072 092] [08 096]⟩

⟨1199094

[051 07] [08 094] [085 094]⟩

(119865 oplus 119866) (1198902

)

= ⟨1199091

[058 075] [058 094]⟩

⟨1199092

[044 088] [058 092]⟩

⟨1199093

[044 065] [051 082] [075 096]⟩

⟨1199094

[044 07] [07 084]⟩

(119865 oplus 119866) (1198903

)

= ⟨1199091

[044 075] [07 088] [088 10]⟩

⟨1199092

[0775 084] [085 098]⟩

⟨1199093

[051 085] [084 0955] [093 10]⟩

⟨1199094

[079 095] [084 10]⟩

(29)


(119865 otimes 119866) (1198901

)

= ⟨1199091

[012 03] [006 072]⟩

⟨1199092

[002 035] [006 045] [028 063]⟩

⟨1199093

[012 04] [018 048] [03 064]⟩

⟨1199094

[009 02] [03 056] [035 056]⟩

(119865 otimes 119866) (1198902

)

= ⟨1199091

[012 025] [012 056]⟩

⟨1199092

[006 042] [012 048]⟩

⟨1199093

[006 015] [009 028] [025 064]⟩

⟨1199094

[006 02] [02 036]⟩

(119865 otimes 119866) (1198903

)

= ⟨1199091

[006 025] [02 042] [032 07]⟩

⟨1199092

[0275 036] [035 072]⟩

⟨1199093

[009 035] [016 0595] [027 08]⟩

⟨1199094

[021 045] [016 07]⟩

(30)



119888

= 119865119888

otimes 119866119888

(4) (119865 otimes 119866)119888

= 119865119888

oplus 119866119888










(1) ⊑ 119865 ⊑ (2) 119888 = (3) 119888 =






119888

= and 119865 and 119865119888

=


119888

= and 119865 otimes 119865119888

=




120590(119896)

isin 119865(119890)(119909) (119896 = 1 2 119897) wedefine

120572120590(119896)

=

1 120574120590(119896)

ge 119905

0 120574120590(119896)

lt 119905

(31)


120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




119865119905

(119890) = 119871 (119865 (119890) 119905)

= ⟨119909 119865119905

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(32)

for all 119890 isin 119860



119880 1198901

1198902

1198903

1199091

1 0 1 0 0 0 0 0 0

1199092

0 1 1 1 1 1 0 0 1

1199093

1 1 1 1 1 1 0 1 0

1199094

1 1 1 0 0 1 1 1 1



119871 (119865 (1198901

) [06 08]) = ⟨1199091

1 0 1⟩ ⟨1199092

0 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

1 1 1⟩

119871 (119865 (1198902

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

1 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

0 0 1⟩

119871 (119865 (1198903

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

0 0 1⟩

⟨1199093

0 1 0⟩ ⟨1199094

1 1 1⟩

(33)



(119865[0608]

119860) where for 119909119895


[0608]

(119890119894

) = 119871(119865(119890119894


[0608]

(1198901

) (1199092

) = 119871(119865(1198901

)(1199092

) [06 08]) =


2





isin 119865(119890)(119909) (119896 = 1 2 119897) one defines

120572120590(119896)

=

1 120574120590(119896)

ge 120582 (119890)

0 120574120590(119896)

lt 120582 (119890)

(34)




120582


119865120582

(119890) = 119871 (119865 (119890) 120582 (119890))

= ⟨119909 119865120582

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(35)

for all 119890 isin 119860



119873


119873


midS (119890) =

1

|119880|

sum

119909isin119880

119865119873

(119890) (119909) (36)




119873


midS = (1198901

[067 07825]) (1198902

[061 0775])

(1198903

[05425 07325])

(37)



+


+


maxS (119890) = max119909isin119880

119865+

(119890) (119909) (38)


119880 1198901

1198902

1198903


)

1199091

0 0 1 0 0 0 0 0 0 1198881

= 1

1199092

0 1 0 1 1 1 1 1 1 1198882

= 7

1199093

1 0 1 1 1 1 0 1 0 1198883

= 6

1199094

1 1 1 0 0 1 1 1 1 1198884

= 7


119880 1198901

1198902

1198903


)

1199091

0 0 0 0 0 0 0 0 0 1198881

= 0

1199092

0 0 0 0 1 0 0 0 0 1198881

= 1

1199093

0 0 0 0 0 0 0 0 0 1198881

= 0

1199094

0 0 1 0 0 0 0 1 0 1198881

= 2



+


maxS = (1198901

[08 10]) (1198902

[08 09]) (1198903

[08 09])

(39)



Algorithm 54


(119865 119860)








of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen






1198881

= 1 + 0 + 0 = 1

1198882

= 1 + 1 times 3 + 1 times 3 = 7

1198883

= 1 times 2 + 1 times 3 + 1 = 6

1198884

= 1 times 3 + 1 + 1 times 3 = 7

(40)


4

= 1199092

gt 1199093

gt 1199091


and 1199092



gt 1199092

gt 1199093

= 1199091






119894

= 120596(119890119894


isin 119860





119895


(seeAlgorithm 60)

Algorithm 60






119880 1198901

1198902

1198903

1198904

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]




119895


of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen



ℎ1

ℎ2

ℎ3

ℎ4

ℎ5


1

1198902

1198903

1198904



1


=



4


119895

(119895 =





(119865 119860 120596) is shown in Table 8


midG = (1198901

[055 071]) (1198902

[063 078])

(1198903

[061 075]) (1198904

[066 074])

(41)


1198881

= 04 times 2 + 0 + 0 + 0 = 08

1198882


1198883


1198884


1198885


(42)


4

gt ℎ5

= ℎ3

gt ℎ2

gt ℎ1


4



4

gt ℎ3

gt ℎ1

= ℎ2

= ℎ5


4


Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion





Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119866119888

(1198901

) = ⟨1199091

[01 08] [04 06]⟩

⟨1199092

[01 03] [01 07] [03 08]⟩

⟨1199093

[02 04]⟩

⟨1199094

[02 03] [02 04] [05 07]⟩

119866119888

(1198902

) = ⟨1199091

[02 06] [05 07]⟩

⟨1199092

[02 06] [03 07]⟩

⟨1199093

[02 05] [03 07] [05 07]⟩

⟨1199094

[04 05] [05 07]⟩

119866119888

(1198903

) = ⟨1199091

[00 02] [04 05] [05 07]⟩

⟨1199092

[01 03] [04 045]⟩

⟨1199093

[00 01] [015 02] [03 07]⟩

⟨1199094

[00 02] [01 03]⟩

(18)



(119865 119860) and (119866 119861) = ( 119860 times 119861) (19)


(120572 120573) = ⟨119909 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cap 119866 (120573) (119909)⟩ 119909 isin 119880

(20)


(119865 119860) or (119866 119861) = (119868 119860 times 119861) (21)


119868 (120572 120573) = ⟨119909 119868 (120572 120573) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (120572) (119909) cup 119866 (120573) (119909)⟩ 119909 isin 119880

(22)


(1198901

1198901

) = ⟨1199091

[02 05] [02 06]⟩

⟨1199092

[02 05] [03 07] [06 08]⟩

⟨1199093

[03 08] [05 08]⟩

⟨1199094

[02 04] [05 07] [05 07]⟩

(1198901

1198902

) = ⟨1199091

[02 05] [03 06]⟩

⟨1199092

[03 05] [04 07] [04 08]⟩

⟨1199093

[03 05] [03 07] [05 08]⟩

⟨1199094

[02 04] [05 06]⟩

(1198901

1198903

) = ⟨1199091

[02 05] [03 06] [03 06]⟩

⟨1199092

[03 05] [04 07] [06 08]⟩

⟨1199093

[03 07] [05 08] [05 08]⟩

⟨1199094

[02 04] [05 07]⟩

(1198902

1198901

) = ⟨1199091

[01 04] [02 07] [02 08]⟩

⟨1199092

[02 05] [03 06] [05 06]⟩

⟨1199093

[02 04] [04 05]⟩

⟨1199094

[02 03] [04 06] [04 06]⟩

(1198902

1198902

) = ⟨1199091

[01 04] [03 07] [03 08]⟩

⟨1199092

[03 05] [04 06]⟩

⟨1199093

[02 04] [03 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

(1198902

1198903

) = ⟨1199091

[01 04] [03 06] [03 08]⟩

⟨1199092

[04 05] [05 06]⟩

⟨1199093

[02 04] [04 05] [04 05]⟩

⟨1199094

[02 03] [04 06]⟩

119868 (1198901

1198901

) = ⟨1199091

[04 06] [03 09]⟩

⟨1199092

[03 07] [04 09] [07 09]⟩

⟨1199093

[06 09] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198901

1198902

) = ⟨1199091

[03 05] [04 08]⟩

⟨1199092

[03 07] [04 08] [06 08]⟩

⟨1199093

[03 09] [05 08] [05 08]⟩

⟨1199094

[03 05] [05 07]⟩


119868 (1198901

1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09] [07 09]⟩

⟨1199093

[03 09] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

119868 (1198902

1198901

) = ⟨1199091

[04 06] [03 09] [03 09]⟩

⟨1199092

[04 07] [05 09] [07 09]⟩

⟨1199093

[06 08] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198902

1198902

) = ⟨1199091

[03 05] [04 08] [04 08]⟩

⟨1199092

[04 07] [05 08]⟩

⟨1199093

[03 05] [04 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119868 (1198902

1198903

) = ⟨1199091

[03 05] [05 07] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

(23)


(1) ((119865 119860) and (119866 119861))119888

= (119865 119860)119888

or (119866 119861)119888

(2) ((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


119861)119888

= (119888



follows that (119865(120572)(119909) cap 119866(120573)(119909))119888

= 119865119888

(120572)(119909) cup 119866119888


(119865 119860)119888

or(119866 119861)119888

= (119865119888

119860)or (119866119888


119888

(120572)(119909) cup 119866119888





( 119862))(4) (119865 119860)or((119866 119861)and( 119862)) = ((119865 119860)or(119866 119861))and((119865 119860)or

( 119862))




(4) can be made




119864 997888rarr IVHF (119880) (24)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) oplus 119866 (119890) (119909)⟩ 119909 isin 119880

(25)


119864 997888rarr IVHF (119880) (26)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) otimes 119866 (119890) (119909)⟩ 119909 isin 119880

(27)


119865 (1198901

) = ⟨1199091

[03 05] [03 08]⟩

⟨1199092

[01 05] [02 05] [04 07]⟩

⟨1199093

[02 05] [03 06] [05 08]⟩

⟨1199094

[03 04] [05 07]⟩


119865 (1198902

) = ⟨1199091

[04 05] [03 07]⟩

⟨1199092

[02 06] [03 06]⟩

⟨1199093

[02 03] [03 04] [05 08]⟩

⟨1199094

[02 04] [04 06]⟩

119865 (1198903

) = ⟨1199091

[02 05] [04 07]⟩

⟨1199092

[05 06] [05 08]⟩

⟨1199093

[03 05] [02 07] [03 08]⟩

⟨1199094

[03 05] [02 07]⟩

(28)


(119865 oplus 119866) (1198901

)

= ⟨1199091

[058 08] [044 098]⟩

⟨1199092

[028 085] [044 095] [082 097]⟩

⟨1199093

[068 09] [072 092] [08 096]⟩

⟨1199094

[051 07] [08 094] [085 094]⟩

(119865 oplus 119866) (1198902

)

= ⟨1199091

[058 075] [058 094]⟩

⟨1199092

[044 088] [058 092]⟩

⟨1199093

[044 065] [051 082] [075 096]⟩

⟨1199094

[044 07] [07 084]⟩

(119865 oplus 119866) (1198903

)

= ⟨1199091

[044 075] [07 088] [088 10]⟩

⟨1199092

[0775 084] [085 098]⟩

⟨1199093

[051 085] [084 0955] [093 10]⟩

⟨1199094

[079 095] [084 10]⟩

(29)


(119865 otimes 119866) (1198901

)

= ⟨1199091

[012 03] [006 072]⟩

⟨1199092

[002 035] [006 045] [028 063]⟩

⟨1199093

[012 04] [018 048] [03 064]⟩

⟨1199094

[009 02] [03 056] [035 056]⟩

(119865 otimes 119866) (1198902

)

= ⟨1199091

[012 025] [012 056]⟩

⟨1199092

[006 042] [012 048]⟩

⟨1199093

[006 015] [009 028] [025 064]⟩

⟨1199094

[006 02] [02 036]⟩

(119865 otimes 119866) (1198903

)

= ⟨1199091

[006 025] [02 042] [032 07]⟩

⟨1199092

[0275 036] [035 072]⟩

⟨1199093

[009 035] [016 0595] [027 08]⟩

⟨1199094

[021 045] [016 07]⟩

(30)



119888

= 119865119888

otimes 119866119888

(4) (119865 otimes 119866)119888

= 119865119888

oplus 119866119888










(1) ⊑ 119865 ⊑ (2) 119888 = (3) 119888 =






119888

= and 119865 and 119865119888

=


119888

= and 119865 otimes 119865119888

=




120590(119896)

isin 119865(119890)(119909) (119896 = 1 2 119897) wedefine

120572120590(119896)

=

1 120574120590(119896)

ge 119905

0 120574120590(119896)

lt 119905

(31)


120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




119865119905

(119890) = 119871 (119865 (119890) 119905)

= ⟨119909 119865119905

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(32)

for all 119890 isin 119860



119880 1198901

1198902

1198903

1199091

1 0 1 0 0 0 0 0 0

1199092

0 1 1 1 1 1 0 0 1

1199093

1 1 1 1 1 1 0 1 0

1199094

1 1 1 0 0 1 1 1 1



119871 (119865 (1198901

) [06 08]) = ⟨1199091

1 0 1⟩ ⟨1199092

0 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

1 1 1⟩

119871 (119865 (1198902

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

1 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

0 0 1⟩

119871 (119865 (1198903

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

0 0 1⟩

⟨1199093

0 1 0⟩ ⟨1199094

1 1 1⟩

(33)



(119865[0608]

119860) where for 119909119895


[0608]

(119890119894

) = 119871(119865(119890119894


[0608]

(1198901

) (1199092

) = 119871(119865(1198901

)(1199092

) [06 08]) =


2





isin 119865(119890)(119909) (119896 = 1 2 119897) one defines

120572120590(119896)

=

1 120574120590(119896)

ge 120582 (119890)

0 120574120590(119896)

lt 120582 (119890)

(34)




120582


119865120582

(119890) = 119871 (119865 (119890) 120582 (119890))

= ⟨119909 119865120582

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(35)

for all 119890 isin 119860



119873


119873


midS (119890) =

1

|119880|

sum

119909isin119880

119865119873

(119890) (119909) (36)




119873


midS = (1198901

[067 07825]) (1198902

[061 0775])

(1198903

[05425 07325])

(37)



+


+


maxS (119890) = max119909isin119880

119865+

(119890) (119909) (38)


119880 1198901

1198902

1198903


)

1199091

0 0 1 0 0 0 0 0 0 1198881

= 1

1199092

0 1 0 1 1 1 1 1 1 1198882

= 7

1199093

1 0 1 1 1 1 0 1 0 1198883

= 6

1199094

1 1 1 0 0 1 1 1 1 1198884

= 7


119880 1198901

1198902

1198903


)

1199091

0 0 0 0 0 0 0 0 0 1198881

= 0

1199092

0 0 0 0 1 0 0 0 0 1198881

= 1

1199093

0 0 0 0 0 0 0 0 0 1198881

= 0

1199094

0 0 1 0 0 0 0 1 0 1198881

= 2



+


maxS = (1198901

[08 10]) (1198902

[08 09]) (1198903

[08 09])

(39)



Algorithm 54


(119865 119860)








of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen






1198881

= 1 + 0 + 0 = 1

1198882

= 1 + 1 times 3 + 1 times 3 = 7

1198883

= 1 times 2 + 1 times 3 + 1 = 6

1198884

= 1 times 3 + 1 + 1 times 3 = 7

(40)


4

= 1199092

gt 1199093

gt 1199091


and 1199092



gt 1199092

gt 1199093

= 1199091






119894

= 120596(119890119894


isin 119860





119895


(seeAlgorithm 60)

Algorithm 60






119880 1198901

1198902

1198903

1198904

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]




119895


of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen



ℎ1

ℎ2

ℎ3

ℎ4

ℎ5


1

1198902

1198903

1198904



1


=



4


119895

(119895 =





(119865 119860 120596) is shown in Table 8


midG = (1198901

[055 071]) (1198902

[063 078])

(1198903

[061 075]) (1198904

[066 074])

(41)


1198881

= 04 times 2 + 0 + 0 + 0 = 08

1198882


1198883


1198884


1198885


(42)


4

gt ℎ5

= ℎ3

gt ℎ2

gt ℎ1


4



4

gt ℎ3

gt ℎ1

= ℎ2

= ℎ5


4


Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion





Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119868 (1198901

1198903

) = ⟨1199091

[03 05] [05 06] [08 10]⟩

⟨1199092

[055 06] [07 09] [07 09]⟩

⟨1199093

[03 09] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

119868 (1198902

1198901

) = ⟨1199091

[04 06] [03 09] [03 09]⟩

⟨1199092

[04 07] [05 09] [07 09]⟩

⟨1199093

[06 08] [06 08]⟩

⟨1199094

[03 05] [06 08] [07 08]⟩

119868 (1198902

1198902

) = ⟨1199091

[03 05] [04 08] [04 08]⟩

⟨1199092

[04 07] [05 08]⟩

⟨1199093

[03 05] [04 07] [05 08]⟩

⟨1199094

[03 05] [05 06]⟩

119868 (1198902

1198903

) = ⟨1199091

[03 05] [05 07] [08 10]⟩

⟨1199092

[055 06] [07 09]⟩

⟨1199093

[03 07] [08 085] [09 10]⟩

⟨1199094

[07 09] [08 10]⟩

(23)


(1) ((119865 119860) and (119866 119861))119888

= (119865 119860)119888

or (119866 119861)119888

(2) ((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


119861)119888

= (119888



follows that (119865(120572)(119909) cap 119866(120573)(119909))119888

= 119865119888

(120572)(119909) cup 119866119888


(119865 119860)119888

or(119866 119861)119888

= (119865119888

119860)or (119866119888


119888

(120572)(119909) cup 119866119888





( 119862))(4) (119865 119860)or((119866 119861)and( 119862)) = ((119865 119860)or(119866 119861))and((119865 119860)or

( 119862))




(4) can be made




119864 997888rarr IVHF (119880) (24)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) oplus 119866 (119890) (119909)⟩ 119909 isin 119880

(25)


119864 997888rarr IVHF (119880) (26)


(119890) = ⟨119909 (119890) (119909)⟩ 119909 isin 119880

= ⟨119909 119865 (119890) (119909) otimes 119866 (119890) (119909)⟩ 119909 isin 119880

(27)


119865 (1198901

) = ⟨1199091

[03 05] [03 08]⟩

⟨1199092

[01 05] [02 05] [04 07]⟩

⟨1199093

[02 05] [03 06] [05 08]⟩

⟨1199094

[03 04] [05 07]⟩


119865 (1198902

) = ⟨1199091

[04 05] [03 07]⟩

⟨1199092

[02 06] [03 06]⟩

⟨1199093

[02 03] [03 04] [05 08]⟩

⟨1199094

[02 04] [04 06]⟩

119865 (1198903

) = ⟨1199091

[02 05] [04 07]⟩

⟨1199092

[05 06] [05 08]⟩

⟨1199093

[03 05] [02 07] [03 08]⟩

⟨1199094

[03 05] [02 07]⟩

(28)


(119865 oplus 119866) (1198901

)

= ⟨1199091

[058 08] [044 098]⟩

⟨1199092

[028 085] [044 095] [082 097]⟩

⟨1199093

[068 09] [072 092] [08 096]⟩

⟨1199094

[051 07] [08 094] [085 094]⟩

(119865 oplus 119866) (1198902

)

= ⟨1199091

[058 075] [058 094]⟩

⟨1199092

[044 088] [058 092]⟩

⟨1199093

[044 065] [051 082] [075 096]⟩

⟨1199094

[044 07] [07 084]⟩

(119865 oplus 119866) (1198903

)

= ⟨1199091

[044 075] [07 088] [088 10]⟩

⟨1199092

[0775 084] [085 098]⟩

⟨1199093

[051 085] [084 0955] [093 10]⟩

⟨1199094

[079 095] [084 10]⟩

(29)


(119865 otimes 119866) (1198901

)

= ⟨1199091

[012 03] [006 072]⟩

⟨1199092

[002 035] [006 045] [028 063]⟩

⟨1199093

[012 04] [018 048] [03 064]⟩

⟨1199094

[009 02] [03 056] [035 056]⟩

(119865 otimes 119866) (1198902

)

= ⟨1199091

[012 025] [012 056]⟩

⟨1199092

[006 042] [012 048]⟩

⟨1199093

[006 015] [009 028] [025 064]⟩

⟨1199094

[006 02] [02 036]⟩

(119865 otimes 119866) (1198903

)

= ⟨1199091

[006 025] [02 042] [032 07]⟩

⟨1199092

[0275 036] [035 072]⟩

⟨1199093

[009 035] [016 0595] [027 08]⟩

⟨1199094

[021 045] [016 07]⟩

(30)



119888

= 119865119888

otimes 119866119888

(4) (119865 otimes 119866)119888

= 119865119888

oplus 119866119888










(1) ⊑ 119865 ⊑ (2) 119888 = (3) 119888 =






119888

= and 119865 and 119865119888

=


119888

= and 119865 otimes 119865119888

=




120590(119896)

isin 119865(119890)(119909) (119896 = 1 2 119897) wedefine

120572120590(119896)

=

1 120574120590(119896)

ge 119905

0 120574120590(119896)

lt 119905

(31)


120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




119865119905

(119890) = 119871 (119865 (119890) 119905)

= ⟨119909 119865119905

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(32)

for all 119890 isin 119860



119880 1198901

1198902

1198903

1199091

1 0 1 0 0 0 0 0 0

1199092

0 1 1 1 1 1 0 0 1

1199093

1 1 1 1 1 1 0 1 0

1199094

1 1 1 0 0 1 1 1 1



119871 (119865 (1198901

) [06 08]) = ⟨1199091

1 0 1⟩ ⟨1199092

0 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

1 1 1⟩

119871 (119865 (1198902

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

1 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

0 0 1⟩

119871 (119865 (1198903

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

0 0 1⟩

⟨1199093

0 1 0⟩ ⟨1199094

1 1 1⟩

(33)



(119865[0608]

119860) where for 119909119895


[0608]

(119890119894

) = 119871(119865(119890119894


[0608]

(1198901

) (1199092

) = 119871(119865(1198901

)(1199092

) [06 08]) =


2





isin 119865(119890)(119909) (119896 = 1 2 119897) one defines

120572120590(119896)

=

1 120574120590(119896)

ge 120582 (119890)

0 120574120590(119896)

lt 120582 (119890)

(34)




120582


119865120582

(119890) = 119871 (119865 (119890) 120582 (119890))

= ⟨119909 119865120582

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(35)

for all 119890 isin 119860



119873


119873


midS (119890) =

1

|119880|

sum

119909isin119880

119865119873

(119890) (119909) (36)




119873


midS = (1198901

[067 07825]) (1198902

[061 0775])

(1198903

[05425 07325])

(37)



+


+


maxS (119890) = max119909isin119880

119865+

(119890) (119909) (38)


119880 1198901

1198902

1198903


)

1199091

0 0 1 0 0 0 0 0 0 1198881

= 1

1199092

0 1 0 1 1 1 1 1 1 1198882

= 7

1199093

1 0 1 1 1 1 0 1 0 1198883

= 6

1199094

1 1 1 0 0 1 1 1 1 1198884

= 7


119880 1198901

1198902

1198903


)

1199091

0 0 0 0 0 0 0 0 0 1198881

= 0

1199092

0 0 0 0 1 0 0 0 0 1198881

= 1

1199093

0 0 0 0 0 0 0 0 0 1198881

= 0

1199094

0 0 1 0 0 0 0 1 0 1198881

= 2



+


maxS = (1198901

[08 10]) (1198902

[08 09]) (1198903

[08 09])

(39)



Algorithm 54


(119865 119860)








of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen






1198881

= 1 + 0 + 0 = 1

1198882

= 1 + 1 times 3 + 1 times 3 = 7

1198883

= 1 times 2 + 1 times 3 + 1 = 6

1198884

= 1 times 3 + 1 + 1 times 3 = 7

(40)


4

= 1199092

gt 1199093

gt 1199091


and 1199092



gt 1199092

gt 1199093

= 1199091






119894

= 120596(119890119894


isin 119860





119895


(seeAlgorithm 60)

Algorithm 60






119880 1198901

1198902

1198903

1198904

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]




119895


of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen



ℎ1

ℎ2

ℎ3

ℎ4

ℎ5


1

1198902

1198903

1198904



1


=



4


119895

(119895 =





(119865 119860 120596) is shown in Table 8


midG = (1198901

[055 071]) (1198902

[063 078])

(1198903

[061 075]) (1198904

[066 074])

(41)


1198881

= 04 times 2 + 0 + 0 + 0 = 08

1198882


1198883


1198884


1198885


(42)


4

gt ℎ5

= ℎ3

gt ℎ2

gt ℎ1


4



4

gt ℎ3

gt ℎ1

= ℎ2

= ℎ5


4


Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion





Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119865 (1198902

) = ⟨1199091

[04 05] [03 07]⟩

⟨1199092

[02 06] [03 06]⟩

⟨1199093

[02 03] [03 04] [05 08]⟩

⟨1199094

[02 04] [04 06]⟩

119865 (1198903

) = ⟨1199091

[02 05] [04 07]⟩

⟨1199092

[05 06] [05 08]⟩

⟨1199093

[03 05] [02 07] [03 08]⟩

⟨1199094

[03 05] [02 07]⟩

(28)


(119865 oplus 119866) (1198901

)

= ⟨1199091

[058 08] [044 098]⟩

⟨1199092

[028 085] [044 095] [082 097]⟩

⟨1199093

[068 09] [072 092] [08 096]⟩

⟨1199094

[051 07] [08 094] [085 094]⟩

(119865 oplus 119866) (1198902

)

= ⟨1199091

[058 075] [058 094]⟩

⟨1199092

[044 088] [058 092]⟩

⟨1199093

[044 065] [051 082] [075 096]⟩

⟨1199094

[044 07] [07 084]⟩

(119865 oplus 119866) (1198903

)

= ⟨1199091

[044 075] [07 088] [088 10]⟩

⟨1199092

[0775 084] [085 098]⟩

⟨1199093

[051 085] [084 0955] [093 10]⟩

⟨1199094

[079 095] [084 10]⟩

(29)


(119865 otimes 119866) (1198901

)

= ⟨1199091

[012 03] [006 072]⟩

⟨1199092

[002 035] [006 045] [028 063]⟩

⟨1199093

[012 04] [018 048] [03 064]⟩

⟨1199094

[009 02] [03 056] [035 056]⟩

(119865 otimes 119866) (1198902

)

= ⟨1199091

[012 025] [012 056]⟩

⟨1199092

[006 042] [012 048]⟩

⟨1199093

[006 015] [009 028] [025 064]⟩

⟨1199094

[006 02] [02 036]⟩

(119865 otimes 119866) (1198903

)

= ⟨1199091

[006 025] [02 042] [032 07]⟩

⟨1199092

[0275 036] [035 072]⟩

⟨1199093

[009 035] [016 0595] [027 08]⟩

⟨1199094

[021 045] [016 07]⟩

(30)



119888

= 119865119888

otimes 119866119888

(4) (119865 otimes 119866)119888

= 119865119888

oplus 119866119888










(1) ⊑ 119865 ⊑ (2) 119888 = (3) 119888 =






119888

= and 119865 and 119865119888

=


119888

= and 119865 otimes 119865119888

=




120590(119896)

isin 119865(119890)(119909) (119896 = 1 2 119897) wedefine

120572120590(119896)

=

1 120574120590(119896)

ge 119905

0 120574120590(119896)

lt 119905

(31)


120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




119865119905

(119890) = 119871 (119865 (119890) 119905)

= ⟨119909 119865119905

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(32)

for all 119890 isin 119860



119880 1198901

1198902

1198903

1199091

1 0 1 0 0 0 0 0 0

1199092

0 1 1 1 1 1 0 0 1

1199093

1 1 1 1 1 1 0 1 0

1199094

1 1 1 0 0 1 1 1 1



119871 (119865 (1198901

) [06 08]) = ⟨1199091

1 0 1⟩ ⟨1199092

0 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

1 1 1⟩

119871 (119865 (1198902

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

1 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

0 0 1⟩

119871 (119865 (1198903

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

0 0 1⟩

⟨1199093

0 1 0⟩ ⟨1199094

1 1 1⟩

(33)



(119865[0608]

119860) where for 119909119895


[0608]

(119890119894

) = 119871(119865(119890119894


[0608]

(1198901

) (1199092

) = 119871(119865(1198901

)(1199092

) [06 08]) =


2





isin 119865(119890)(119909) (119896 = 1 2 119897) one defines

120572120590(119896)

=

1 120574120590(119896)

ge 120582 (119890)

0 120574120590(119896)

lt 120582 (119890)

(34)




120582


119865120582

(119890) = 119871 (119865 (119890) 120582 (119890))

= ⟨119909 119865120582

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(35)

for all 119890 isin 119860



119873


119873


midS (119890) =

1

|119880|

sum

119909isin119880

119865119873

(119890) (119909) (36)




119873


midS = (1198901

[067 07825]) (1198902

[061 0775])

(1198903

[05425 07325])

(37)



+


+


maxS (119890) = max119909isin119880

119865+

(119890) (119909) (38)


119880 1198901

1198902

1198903


)

1199091

0 0 1 0 0 0 0 0 0 1198881

= 1

1199092

0 1 0 1 1 1 1 1 1 1198882

= 7

1199093

1 0 1 1 1 1 0 1 0 1198883

= 6

1199094

1 1 1 0 0 1 1 1 1 1198884

= 7


119880 1198901

1198902

1198903


)

1199091

0 0 0 0 0 0 0 0 0 1198881

= 0

1199092

0 0 0 0 1 0 0 0 0 1198881

= 1

1199093

0 0 0 0 0 0 0 0 0 1198881

= 0

1199094

0 0 1 0 0 0 0 1 0 1198881

= 2



+


maxS = (1198901

[08 10]) (1198902

[08 09]) (1198903

[08 09])

(39)



Algorithm 54


(119865 119860)








of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen






1198881

= 1 + 0 + 0 = 1

1198882

= 1 + 1 times 3 + 1 times 3 = 7

1198883

= 1 times 2 + 1 times 3 + 1 = 6

1198884

= 1 times 3 + 1 + 1 times 3 = 7

(40)


4

= 1199092

gt 1199093

gt 1199091


and 1199092



gt 1199092

gt 1199093

= 1199091






119894

= 120596(119890119894


isin 119860





119895


(seeAlgorithm 60)

Algorithm 60






119880 1198901

1198902

1198903

1198904

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]




119895


of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen



ℎ1

ℎ2

ℎ3

ℎ4

ℎ5


1

1198902

1198903

1198904



1


=



4


119895

(119895 =





(119865 119860 120596) is shown in Table 8


midG = (1198901

[055 071]) (1198902

[063 078])

(1198903

[061 075]) (1198904

[066 074])

(41)


1198881

= 04 times 2 + 0 + 0 + 0 = 08

1198882


1198883


1198884


1198885


(42)


4

gt ℎ5

= ℎ3

gt ℎ2

gt ℎ1


4



4

gt ℎ3

gt ℎ1

= ℎ2

= ℎ5


4


Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion





Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119888

= and 119865 and 119865119888

=


119888

= and 119865 otimes 119865119888

=




120590(119896)

isin 119865(119890)(119909) (119896 = 1 2 119897) wedefine

120572120590(119896)

=

1 120574120590(119896)

ge 119905

0 120574120590(119896)

lt 119905

(31)


120590(119896)

= [120574120590(119896)119871

120574120590(119896)119880




119865119905

(119890) = 119871 (119865 (119890) 119905)

= ⟨119909 119865119905

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(32)

for all 119890 isin 119860



119880 1198901

1198902

1198903

1199091

1 0 1 0 0 0 0 0 0

1199092

0 1 1 1 1 1 0 0 1

1199093

1 1 1 1 1 1 0 1 0

1199094

1 1 1 0 0 1 1 1 1



119871 (119865 (1198901

) [06 08]) = ⟨1199091

1 0 1⟩ ⟨1199092

0 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

1 1 1⟩

119871 (119865 (1198902

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

1 1 1⟩

⟨1199093

1 1 1⟩ ⟨1199094

0 0 1⟩

119871 (119865 (1198903

) [06 08]) = ⟨1199091

0 0 0⟩ ⟨1199092

0 0 1⟩

⟨1199093

0 1 0⟩ ⟨1199094

1 1 1⟩

(33)



(119865[0608]

119860) where for 119909119895


[0608]

(119890119894

) = 119871(119865(119890119894


[0608]

(1198901

) (1199092

) = 119871(119865(1198901

)(1199092

) [06 08]) =


2





isin 119865(119890)(119909) (119896 = 1 2 119897) one defines

120572120590(119896)

=

1 120574120590(119896)

ge 120582 (119890)

0 120574120590(119896)

lt 120582 (119890)

(34)




120582


119865120582

(119890) = 119871 (119865 (119890) 120582 (119890))

= ⟨119909 119865120582

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(35)

for all 119890 isin 119860



119873


119873


midS (119890) =

1

|119880|

sum

119909isin119880

119865119873

(119890) (119909) (36)




119873


midS = (1198901

[067 07825]) (1198902

[061 0775])

(1198903

[05425 07325])

(37)



+


+


maxS (119890) = max119909isin119880

119865+

(119890) (119909) (38)


119880 1198901

1198902

1198903


)

1199091

0 0 1 0 0 0 0 0 0 1198881

= 1

1199092

0 1 0 1 1 1 1 1 1 1198882

= 7

1199093

1 0 1 1 1 1 0 1 0 1198883

= 6

1199094

1 1 1 0 0 1 1 1 1 1198884

= 7


119880 1198901

1198902

1198903


)

1199091

0 0 0 0 0 0 0 0 0 1198881

= 0

1199092

0 0 0 0 1 0 0 0 0 1198881

= 1

1199093

0 0 0 0 0 0 0 0 0 1198881

= 0

1199094

0 0 1 0 0 0 0 1 0 1198881

= 2



+


maxS = (1198901

[08 10]) (1198902

[08 09]) (1198903

[08 09])

(39)



Algorithm 54


(119865 119860)








of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen






1198881

= 1 + 0 + 0 = 1

1198882

= 1 + 1 times 3 + 1 times 3 = 7

1198883

= 1 times 2 + 1 times 3 + 1 = 6

1198884

= 1 times 3 + 1 + 1 times 3 = 7

(40)


4

= 1199092

gt 1199093

gt 1199091


and 1199092



gt 1199092

gt 1199093

= 1199091






119894

= 120596(119890119894


isin 119860





119895


(seeAlgorithm 60)

Algorithm 60






119880 1198901

1198902

1198903

1198904

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]




119895


of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen



ℎ1

ℎ2

ℎ3

ℎ4

ℎ5


1

1198902

1198903

1198904



1


=



4


119895

(119895 =





(119865 119860 120596) is shown in Table 8


midG = (1198901

[055 071]) (1198902

[063 078])

(1198903

[061 075]) (1198904

[066 074])

(41)


1198881

= 04 times 2 + 0 + 0 + 0 = 08

1198882


1198883


1198884


1198885


(42)


4

gt ℎ5

= ℎ3

gt ℎ2

gt ℎ1


4



4

gt ℎ3

gt ℎ1

= ℎ2

= ℎ5


4


Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion





Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120582


119865120582

(119890) = 119871 (119865 (119890) 120582 (119890))

= ⟨119909 119865120582

(119890) (119909)⟩ 119909 isin 119880 = ⟨119909 120572120590(119896)

⟩ 119909 isin 119880

(35)

for all 119890 isin 119860



119873


119873


midS (119890) =

1

|119880|

sum

119909isin119880

119865119873

(119890) (119909) (36)




119873


midS = (1198901

[067 07825]) (1198902

[061 0775])

(1198903

[05425 07325])

(37)



+


+


maxS (119890) = max119909isin119880

119865+

(119890) (119909) (38)


119880 1198901

1198902

1198903


)

1199091

0 0 1 0 0 0 0 0 0 1198881

= 1

1199092

0 1 0 1 1 1 1 1 1 1198882

= 7

1199093

1 0 1 1 1 1 0 1 0 1198883

= 6

1199094

1 1 1 0 0 1 1 1 1 1198884

= 7


119880 1198901

1198902

1198903


)

1199091

0 0 0 0 0 0 0 0 0 1198881

= 0

1199092

0 0 0 0 1 0 0 0 0 1198881

= 1

1199093

0 0 0 0 0 0 0 0 0 1198881

= 0

1199094

0 0 1 0 0 0 0 1 0 1198881

= 2



+


maxS = (1198901

[08 10]) (1198902

[08 09]) (1198903

[08 09])

(39)



Algorithm 54


(119865 119860)








of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen






1198881

= 1 + 0 + 0 = 1

1198882

= 1 + 1 times 3 + 1 times 3 = 7

1198883

= 1 times 2 + 1 times 3 + 1 = 6

1198884

= 1 times 3 + 1 + 1 times 3 = 7

(40)


4

= 1199092

gt 1199093

gt 1199091


and 1199092



gt 1199092

gt 1199093

= 1199091






119894

= 120596(119890119894


isin 119860





119895


(seeAlgorithm 60)

Algorithm 60






119880 1198901

1198902

1198903

1198904

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]




119895


of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen



ℎ1

ℎ2

ℎ3

ℎ4

ℎ5


1

1198902

1198903

1198904



1


=



4


119895

(119895 =





(119865 119860 120596) is shown in Table 8


midG = (1198901

[055 071]) (1198902

[063 078])

(1198903

[061 075]) (1198904

[066 074])

(41)


1198881

= 04 times 2 + 0 + 0 + 0 = 08

1198882


1198883


1198884


1198885


(42)


4

gt ℎ5

= ℎ3

gt ℎ2

gt ℎ1


4



4

gt ℎ3

gt ℎ1

= ℎ2

= ℎ5


4


Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion





Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen






1198881

= 1 + 0 + 0 = 1

1198882

= 1 + 1 times 3 + 1 times 3 = 7

1198883

= 1 times 2 + 1 times 3 + 1 = 6

1198884

= 1 times 3 + 1 + 1 times 3 = 7

(40)


4

= 1199092

gt 1199093

gt 1199091


and 1199092



gt 1199092

gt 1199093

= 1199091






119894

= 120596(119890119894


isin 119860





119895


(seeAlgorithm 60)

Algorithm 60






119880 1198901

1198902

1198903

1198904

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]




119895


of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen



ℎ1

ℎ2

ℎ3

ℎ4

ℎ5


1

1198902

1198903

1198904



1


=



4


119895

(119895 =





(119865 119860 120596) is shown in Table 8


midG = (1198901

[055 071]) (1198902

[063 078])

(1198903

[061 075]) (1198904

[066 074])

(41)


1198881

= 04 times 2 + 0 + 0 + 0 = 08

1198882


1198883


1198884


1198885


(42)


4

gt ℎ5

= ℎ3

gt ℎ2

gt ℎ1


4



4

gt ℎ3

gt ℎ1

= ℎ2

= ℎ5


4


Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion





Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119880 1198901

1198902

1198903

1198904

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964

= 025

ℎ1

[06 08] [06 07] [06 08] [06 06] [05 06] [06 07] [05 05] [04 05]

ℎ2

[02 03] [03 05] [06 08] [07 08] [07 08] [05 08] [07 07] [06 09]

ℎ3

[05 07] [06 08] [07 09] [06 08] [06 08] [07 07] [08 09] [09 09]

ℎ4

[06 07] [08 09] [05 07] [07 08] [08 09] [07 08] [06 06] [08 09]

ℎ5

[06 08] [07 09] [06 08] [07 08] [06 08] [04 06] [05 07] [08 08]




119895


of 119909119895


if 119888119896

= max119895

119888119895


maybe chosen



ℎ1

ℎ2

ℎ3

ℎ4

ℎ5


1

1198902

1198903

1198904



1


=



4


119895

(119895 =





(119865 119860 120596) is shown in Table 8


midG = (1198901

[055 071]) (1198902

[063 078])

(1198903

[061 075]) (1198904

[066 074])

(41)


1198881

= 04 times 2 + 0 + 0 + 0 = 08

1198882


1198883


1198884


1198885


(42)


4

gt ℎ5

= ℎ3

gt ℎ2

gt ℎ1


4



4

gt ℎ3

gt ℎ1

= ℎ2

= ℎ5


4


Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion





Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 9 NRIVFS ofG = (119865 119860)

119880 1198901

1198902

1198903

1198904

ℎ1

[060 075] [060 070] [055 065] [045 050]

ℎ2

[025 040] [065 080] [060 080] [065 080]

ℎ3

[055 075] [065 085] [065 075] [085 090]

ℎ4

[070 080] [060 075] [075 085] [070 075]

ℎ5

[065 085] [065 080] [050 070] [065 075]


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

1 1 0 0 0 0 0 0 1198881

= 08

ℎ2

0 0 0 1 1 0 1 1 1198882

= 085

ℎ3

0 1 1 0 0 1 1 1 1198883

= 125

ℎ4

1 1 0 1 1 1 0 1 1198884

= 155

ℎ5

1 1 0 1 0 0 0 1 1198885

= 125


119880 1198901

1205961

= 04 1198902

1205962

= 02 1198903

1205963

= 015 1198904

1205964


)

ℎ1

0 0 0 0 0 0 0 0 1198881

= 0

ℎ2

0 0 0 0 0 0 0 0 1198882

= 0

ℎ3

0 0 1 0 0 0 0 1 1198883

= 045

ℎ4

0 1 0 0 1 0 0 0 1198884

= 055

ℎ5

0 0 0 0 0 0 0 0 1198885

= 0

6 Conclusion





Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article on interval-valued hesitant fuzzy soft...

Documents