algebraic hyperstructures of vague soft sets associated ...€¦ · researcharticle algebraic...

Research ArticleAlgebraic Hyperstructures of Vague Soft Sets Associated withHyperrings and Hyperideals

Ganeshsree Selvachandran1 and Abdul Razak Salleh2

1Department of Actuarial Science and Applied Statistics Faculty of Business and Information Science UCSI UniversityJalan Menara Gading Cheras 56000 Kuala Lumpur Malaysia2School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia (UKM)43600 Bangi Selangor Malaysia

Correspondence should be addressed to Ganeshsree Selvachandran ganeshsree86yahoocom

Received 28 September 2014 Accepted 25 May 2015

Academic Editor Xiaolong Xin

Copyright copy 2015 G Selvachandran and A R Salleh This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We apply the classical theory of hyperrings to vague soft sets to derive the concepts of vague soft hyperrings vague soft hyperidealsand vague soft hyperring homomorphism The properties and structural characteristics of these concepts are also studied anddiscussed Furthermore the relationship between the concepts introduced here and the corresponding concepts in classicalhyperring theory and soft hyperring theory is studied and investigated

1 Introduction

Hyperstructure theory which is a generalization of the notionof classical algebraic structures was introduced by Marty in1934 (see [1]) Since its inception hyperstructure theory hasseen tremendous development Corsini (see [2]) introducedthe concept of a hyperring and gave the definition of themost general form of hyperring besides being responsiblefor introducing the notion of hyperring homomorphismIn 1994 Vougiouklis (see [3]) introduced another type ofhyperrings called the119867]-ring which is a generalization of thenotion of hyperring introduced by [2] Vougiouklis (see [3])also introduced the notion of 119867]-subring and left and right119867]-ideal of a119867]-ring

The theory of hyperstructures has also been activelyapplied to variousmathematical theories such as the theory offuzzy sets introduced by Zadeh in 1965 (see [4]) intuitionisticfuzzy theory introduced by Atanassov in 1986 (see [5]) andsoft set theory introduced by Molodtsov in 1999 (see [6])to produce several important concepts and theories Thestudy of fuzzy algebra began with the introduction of thenotion of a fuzzy subgroup of a group by Rosenfeld (see[7]) Consequently this led to the study of fuzzy hyperalgebrabeginning with the introduction of the notion of a fuzzy

subhyperring of a hyperring (resp fuzzy 119867]-subrings) of ahyperring (resp119867]-ring) and fuzzy hyperideals (resp fuzzy119867]-ideals) of hyperrings (resp119867]-rings)Thenotion of fuzzyhyperideals introduced byDavvaz (see [8]) is a generalizationof the concept of Liursquos (see [9]) definition of a fuzzy ideal of aring

Soft set theory is a general mathematical tool whichwas designed to deal with uncertainties and is widely ac-knowledged as an effective alternative to the classical mathe-matical tools that were used to deal with uncertainties andimprecision Prior to the introduction of soft set theorymathematical theories such as the theory of fuzzy setsrough sets and probability theory were used as tools to dealwith uncertainties imprecision and vagueness Howeversoft set theory has been proven to be a superior alternativeto these classical theories as it is free from the inherentdifficulties that affect these classical mathematical tools Softset theory has been successfully applied in various areas ofmathematics such as in classical algebra and hyperalgebrafuzzy algebra and fuzzy hyperalgebra to develop variousalgebraic structures However an inherent weakness of softset theory is that it is difficult to be used to represent thevagueness of problem parameters particularly in problem-solving and decision making contexts To overcome this

Hindawi Publishing Corporatione Scientific World JournalVolume 2015 Article ID 780121 12 pageshttpdxdoiorg1011552015780121

2 The Scientific World Journal

disadvantage various generalizations of soft sets such as fuzzysoft sets vague soft sets and intuitionistic fuzzy soft sets wereintroduced as better alternatives to the concept of soft setsIn the context of this paper the notion of vague soft setsis used as a base to develop the initial theory of vague softhyperrings

In this paper we develop the initial theory of vaguesoft hyperrings in Rosenfeldrsquos sense as a continuation to thetheory of vague soft hypergroups introduced in [10] Subse-quently some of the fundamental properties and structuralcharacteristics of these concepts are studied and discussedLastly we prove that there exists a one-to-one correspon-dence between some of these concepts and their correspond-ing concepts in soft hyperring theory as well as classicalhyperring theory

2 Preliminaries

In this section some well-known and useful definitionspertaining to the development of the theory of vaguesoft hyperrings introduced here will be presented Thesedefinitions and concepts will be used throughout thischapter

Definition 1 (see [6]) A pair (119865 119860) is called a soft set over 119880where 119865 is a mapping given by 119865 119860 rarr 119875(119880) In otherwords a soft set over 119880 is a parameterized family of subsetsof the universe 119880 For 120576 isin 119860 119865(120576) may be considered as theset of 120576-elements of the soft set (119865 119860) or as the 120576-approximateelements of the soft set

Definition 2 (see [6]) For a soft set (119865 119860) the setSupp(119865 119860) = 119909 isin 119860 119865(119909) = 0 is called the supportof the soft set (119865 119860) Thus a null soft set is a soft set with anempty support and a soft set (119865 119860) is said to be nonnull ifSupp(119865 119860) = 0

Definition 3 (see [11]) Let 119883 be a space of points (objects)with a generic element of119883 denoted by 119909 A vague set 119881 in119883is characterized by a truth membership function 119905

119881 119883 rarr

[0 1] and a false membership function 119891119881 119883 rarr [0 1] The

value 119905119881(119909) is a lower bound on the grade of membership of

119909 derived from the evidence for 119909 and 119891119881(119909) is a lower bound

on the negation of 119909 derived from the evidence against 119909Thevalues 119905

119881(119909) and 119891

119881(119909) both associate a real number in the

interval [0 1] with each point in119883 where 119905119881(119909) + 119891

119881(119909) le 1

This approach bounds the grade of membership of 119909 to asubinterval [119905

119881(119909) 1 minus 119891

119881(119909)] of [0 1] Hence a vague set

is a form of fuzzy set albeit a more accurate form of fuzzyset

Definition 4 (see [12]) A pair (119865 119860) is called a vague soft setover 119880 where 119865 is a mapping given by 119865 119860 rarr 119881(119880) and119881(119880) is the power set of vague sets over 119880 In other words avague soft set over119880 is a parameterized family of vague sets ofthe universe 119880 Every set 119865(119890) for all 119890 isin 119860 from this familymay be considered as the set of 119890-approximate elements ofthe vague soft set (119865 119860) Hence the vague soft set (119865 119860) can

be viewed as consisting of a collection of approximations ofthe following form

(119865 119860)

= 119865119886(119909119894) 119894 = 1 2 3

=

[119905119865(119890119894)

(119909119894) 1 minus 119891

119865(119890119894)(119909119894)]

119909119894

119894 = 1 2 3

(1)

where 119865119886(119909119894) is a subset of (119865 119860) for all 119890 isin 119860 and for all

119909 isin 119880

Example 5 (see [12]) Consider a vague soft set (119865 119864)where 119880 is a set of six houses under consideration of adecision maker to purchase which is denoted by 119880 =

ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6 and 119864 is a parameter set where

119864 = 1198901 1198902 1198903 1198904 1198905 = expensive beautiful wooden

cheap in the green surroundings The vague soft set (119865 119864)describes the ldquoattractiveness of the housesrdquo to this decisionmaker

Suppose that

119865 (1198901) = ([01 02]

ℎ1[09 1]ℎ2

[03 05]

ℎ3[08 09]

ℎ4

[02 04]ℎ5

[04 06]

ℎ6)

119865 (1198902) = ([09 1]ℎ1

[02 07]

ℎ2[06 09]

ℎ3[02 04]

ℎ4

[03 04]ℎ5

[01 06]

ℎ6)

119865 (1198903) = ([0 0]ℎ1

[0 0]ℎ2

[1 1]ℎ3

[1 1]ℎ4

[1 1]ℎ5

[0 0]ℎ6

)

119865 (1198904) = ([08 09]

ℎ1[0 01]ℎ2

[05 07]

ℎ3[01 02]

ℎ4

[06 08]ℎ5

[04 06]

ℎ6)

119865 (1198905) = ([09 1]ℎ1

[02 03]

ℎ2[01 04]

ℎ3[01 02]

ℎ4

[02 04]ℎ5

[07 09]

ℎ6)

(2)

The vague soft set (119865 119864) is a parameterized family 119865(119890119894) 119894 =

1 2 3 4 5 of vague sets on 119880 and

(119865 119864) = Expensive Houses=([01 02]ℎ1

[09 1]ℎ2

[03 05]ℎ3

[08 09]

ℎ4[02 04]

ℎ5[04 06]

ℎ6)

The Scientific World Journal 3

Beautiful Houses=([09 1]ℎ1

[02 07]

ℎ2

[06 09]ℎ3

[02 04]

ℎ4[03 04]

ℎ5[01 06]

ℎ6)

(3)

Definition 6 (see [10]) Let (119865 119860) be a vague soft set over 119883The support of (119865 119860) denoted by Supp(119865 119860) is defined as

Supp (119865 119860) = 119886 isin119860 119865119886 (119909) = 0 ie 119905

119865119886(119909)

= 0 1 minus 119891119865119886(119909) = 0

(4)

for all 119909 isin 119883It is to be noted that a null vague soft set is a vague soft

set where both the truth and false membership functions areequal to zero Therefore a vague soft set (119865 119860) is said to benonnull if Supp(119865 119860) = 0

Definition 7 (see [12]) Let (119865 119860) and (119866 119861) be vague soft setsover 119880The set (119865 119860) is called a vague soft subset of (119866 119861) if119860 sube 119861 and for every 119886 isin 119860 119865(119886) and 119866(119886) are the sameapproximation In this case (119866 119861) is called the vague softsuperset of (119865 119860) and this relationship is denoted by (119865 119860) sube(119866 119861)

Definition 8 (see [12]) If (119865 119860) and (119866 119861) are two vague softsets over 119880 the intersection of (119865 119860) and (119866 119861) denoted asldquo(119865 119860) cap (119866 119861)rdquo is defined by (119865 119860) cap (119866 119861) = ( 119860 times 119861)where

119905119888(119909) =

119905119865119888(119909) 119888 isin 119860 minus 119861

119905119866119888(119909) 119888 isin 119861 minus 119860

min (119905119865119888(119909) 119905119866119888(119909)) 119888 isin 119861 cap 119860

1 minus 119891119888(119909)

=

1 minus 119891119865119888(119909) 119888 isin 119860 minus 119861

1 minus 119891119866119888(119909) 119888 isin 119861 minus 119860

min (1 minus 119891119865119888(119909) 1 minus 119891

119866119888(119909)) 119888 isin 119861 cap 119860

(5)

for all 119888 isin Supp( 119862) and for all 119909 isin 119880

Definition 9 (see [12]) Let (119865 119860) and (119866 119861) be vague soft setsover119880Then ldquo(119865 119860) AND (119866 119861)rdquo denoted by (119865 119860) and (119866 119861)is defined as (119865 119860) and (119866 119861) = ( 119860 times 119861) where

119905(120572120573)

(119909) = min 119905119865120572(119909) 119905119866120573

(119909)

1minus119891(120572120573)

(119909) = min 1minus119891119865(120572)

(119909) 1minus119891119866(120573)

(119909)

(6)

for every (120572 120573) isin 119860 times 119861 and 119909 isin 119880 where 119905(120572120573)

and 119891(120572120573)

are the truth membership function and false membershipfunction of

(120572120573) respectively This relationship can be

written as (120572120573)

(119909) = 119865120572(119909) cap 119866

120573(119909) where cap represents the

intersection operation between two vague soft sets as definedin the case of 119888 isin 119861 cap 119860 in Definition 8

Definition 10 (see [10]) Let (119865 119860) be a vague soft set over 119880Then for all 120572 120573 isin [0 1] where 120572 le 120573 the (120572 120573)-cut orthe vague soft (120572 120573)-cut of (119865 119860) is a subset of 119880 which isas defined below

(119865 119860)(120572120573)

= 119909 isin119880 119905119865119886(119909) ge 120572 1 minus 119891

119865119886(119909)

ge 120573 ie 119865119886 (119909) ge [120572 120573]

(7)

for all 119886 isin 119860If 120572 = 120573 then it is called the vague soft (120572 120572)-cut of (119865 119860)

or the 120572-level set of (119865 119860) denoted by (119865 119860)(120572120572)

is a subsetof 119880 which is as defined below

(119865 119860)(120572120572)

= 119909 isin119880 119905119865119886(119909) ge 120572 1 minus 119891

119865119886(119909)

ge 120572 ie 119865119886 (119909) ge [120572 120572]

(8)

for all 119886 isin 119860

Definition 11 (see [10]) Let (119865 119860) be a vague soft set over 119883and let 119866 be a nonnull subset of 119883 Then (119865 119860)

119866is called

a vague soft characteristic set over 119866 in [0 1] and the lowerbound and upper bound of (119865

119886)119866are as defined below

119905(119865119886)119866

(119909) = 1minus119891(119865119886)119866

(119909) =

119904 if 119909 isin 119866

119908 otherwise(9)

where (119865119886)119866is a subset of (119865 119860)

119866 119909 isin 119883 119904 119908 isin [0 1] and

119904 gt 119908

This paper pertains to the application of classical hyper-ring theory to the concept of vague soft sets Furthermore therelationships that exist between the concepts introduced inthis paper and the corresponding concepts in classical hyper-ring theory are also studied and discussed The definitionsof several important concepts pertaining to the theory ofhyperstructures are presented below

Definition 12 (see [1]) A hypergroup ⟨119867 ∘⟩ is a set 119867

equipped with an associative hyperoperation (∘) 119867 times 119867 rarr

119875(119867)which satisfies the reproduction axiom given by 119909∘119867 =

119867 ∘ 119909 = 119867 for all 119909 in119867

Definition 13 (see [13]) A hyperstructure ⟨119867 ∘⟩ is called an119867V-group if the following axioms hold

(i) 119909 ∘ (119910 ∘ 119911) cap (119909 ∘ 119910) ∘ 119911 = 0 for all 119909 119910 119911 isin 119867(ii) 119909 ∘ 119867 = 119867 ∘ 119909 = 119867 for all 119909 in119867

If ⟨119867 ∘⟩ only satisfies the first axiom then it is called a 119867V-semigroup


Definition 14 (see [1]) A subset 119870 of 119867 is called a subhyper-group if ⟨119870 ∘⟩ is a hypergroup

Definition 15 (see [3]) A 119867]-ring is a multivalued system(119877 + ∘) which satisfies the following axioms

(i) (119877 +) is a119867]-group(ii) (119877 ∘) is a119867]-semigroup(iii) the hyperoperation ldquo∘rdquo is weak distributive over the

hyperoperation ldquo+rdquo that is for each 119909 119910 119911 isin 119877 thefollowing holds true

119909 ∘ (119910 + 119911) cap ((119909 ∘ 119910) + (119909 ∘ 119911)) = 120601

(119909 + 119910) ∘ 119911 cap ((119909 ∘ 119911) + (119910 ∘ 119911)) = 120601

(10)

Definition 16 (see [3]) A nonempty subset 1198771015840 of 119877 is called asubhyperring of (119877 + ∘) if (1198771015840 +) is a subhypergroup of (119877 +)and for all 119909 119910 119911 isin 119877

1015840 119909 ∘119910 isin 119875lowast(1198771015840) where 119875lowast(1198771015840) is the set

of all nonempty subsets of 1198771015840

Definition 17 (see [3]) Let119877 be a119867V-ring A nonempty subset119868 of 119877 is called a left (resp right) 119867V-ideal if the followingaxioms hold

(i) (119868 +) is a119867V-subgroup of (119877 +)(ii) 119877 ∘ 119868 sube 119868 (resp 119868 ∘ 119877 sube 119868)

If 119868 is both left and right119867V-ideals of 119877 then 119868 is said to be a119867V-ideal of 119877

3 Vague Soft Hyperrings

In this section the concept of vague soft hyperrings inRosenfeldrsquos sense is introduced as a continuation to the theoryof vague soft hypergroups introduced by [10] We derive thedefinition of vague soft hyperrings by applying the theory ofhyperrings to vague soft sets The properties of this conceptare studied and discussed

From now on let (119877 + ∘) be a hyperring (or 119867]-ring)let 119864 be a set of parameters and let 119860 sube 119864 For the sake ofsimplicity we will denote (119877 + ∘) by 119877

Definition 18 (see [14]) Let (119865 119860) be a nonnull soft set over119877 Then (119865 119860) is called a soft hyperring over 119877 if 119865(119909) is asubhyperring of 119877 for all 119909 isin Supp(119865 119860)

Definition 19 (see [10]) Let (119865 119860) be a nonnull vague softset over a hypergroup ⟨119867 ∘⟩ Then (119865 119860) is called a vaguesoft semihypergroup over ⟨119867 ∘⟩ if the following conditions aresatisfied for all 119886 isin Supp(119865 119860)

For all 119909 119910 isin ⟨119867 ∘⟩ min119905119865119886(119909) 119905119865119886(119910) le inf119905

119865119886(119911) 119911 isin

119909 ∘ 119910 and min1 minus 119891119865119886(119909) 1 minus 119891

119865119886(119910) le inf1 minus 119891

119865119886(119911) 119911 isin

119909 ∘ 119910 that is min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910

Definition 20 (see [10]) Let (119865 119860) be a nonnull vague softset over 119867 Then (119865 119860) is called a vague soft hypergroup

over 119867 if the following conditions are satisfied for all119886 isin Supp(119865 119860)

(i) For all 119909 119910 isin 119867min119905119865119886(119909) 119905119865119886(119910) le inf119905

119865119886(119911)

119911 isin 119909∘119910 andmin1 minus 119891119865119886(119909) 1 minus 119891

119865119886(119910) le inf1minus

119891119865119886(119911) 119911 isin 119909 ∘ 119910 that is min119865

119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910

(ii) For all 119908 119909 isin 119867 there exists 119910 isin 119867 such that119909 isin 119908 ∘ 119910 and min119905

119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)

(iii) For all 119908 119909 isin 119867 there exists 119911 isin 119867 such that119909 isin 119911 ∘ 119908 and min119905

119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)

Conditions (ii) and (iii) represent the left and right repro-duction axioms respectively Then 119865

119886is a nonnull vague

subhypergroup of ⟨119867 ∘⟩ (in Rosenfeldrsquos sense) for all 119886 isin

Supp(119865 119860)

Definition 21 Let (119865 119860) be a nonnull vague soft set over 119877Then (119865 119860) is called a vague soft hyperring over 119877 if for all119886 isin Supp(119865 119860) the following conditions are satisfied

(i) For all 119909 119910 isin 119877 min119905119865119886(119909) 119905119865119886(119910) le inf119905

119865119886(119911)

119911 isin 119909 + 119910 and min1 minus 119891119865119886(119909) 1 minus 119891

119865119886(119910) le

inf1 minus 119891119865119886(119911) 119911 isin 119909 + 119910 which means

min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910

(ii) For all 119908 119909 isin 119877 there exists 119910 isin 119877 such that119909 isin 119908 + 119910 and min119905

119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)

(iii) For all 119908 119909 isin 119877 there exists 119911 isin 119877 such that119909 isin 119911 + 119908 and min119905

119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)

(iv) For all 119909 119910 isin 119877 min119905119865119886(119909) 119905119865119886(119910) le inf119905

119865119886(119911)

119911 isin 119909 ∘ 119910 and min1 minus 119891119865119886(119909) 1 minus 119891

119865119886(119910) le

inf1 minus 119891119865119886(119911) 119911 isin 119909∘119910 that is min119865

119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910

That is 119865119886is a nonnull vague subhyperring (in Rosenfeldrsquos

sense) of 119877 for each 119886 isin Supp(119865 119860)Conditions (i) (ii) and (iii) indicate that 119865

119886is a vague

subhypergroup of (119877 +) while condition (iv) indicates that119865119886is a vague subsemihypergroup of (119877 ∘) Therefore it can

be concluded that (119865 119860) is a vague soft hyperring over 119877 if(119865 119860) is a vague soft hypergroup over (119877 +) and a vague softsemihypergroup over (119877 ∘)

Theorem 22 Let (119865 119860) be a nonnull vague soft set over 119877Then the necessary and sufficient condition for (119865 119860) to be


a vague soft hyperring over 119877 is for (119865 119860) to be a vague softsemihypergroup over (119877 ∘) and also a vague soft hypergroupover (119877 +)

Proof The proof is straightforward by Definition 21

Example 23 A vague soft hyperring (119865 119860) for which 119860 is asingleton is a vague subhyperring Hence vague subhyper-rings and classical hyperrings are a particular type of vaguesoft hyperrings

Example 24 Let (119865 119860) be a nonnull vague soft set over ahyperring 119877 which is as defined below

(119865 119860)+

= (119865119886)+

= 119909 isin119877 119905(119865119886)+ (119909) = 119905

119865119886(119909) + 1

minus 119905119865119886(119890) 1 minus 119891

(119865119886)+ (119909) = 1 minus 119891

119865119886(119909)

+119891119865119886(119890) ie (119865119886)

+

(119909) = 119865119886 (119909) + 1minus119865119886 (119890)

(11)

Then (119865 119860)+ is a vague soft hyperring over 119877

Example 25 Let119865119886be a vague subhyperring of a hyperring119877

Thismeans that119865119886satisfies the axioms stated inDefinition 21

Now consider the family of 120572-level sets for 119865119886

givenby

(119865 119860)120572= 119909 isin119867 119865

119886 (119909) ge 120572 (12)

for all 119886 isin 119860 and 120572 isin [0 1] Then for all 120572 isin [0 1] 119865119886is a

subhyperring of 119877 Hence (119865 [0 1]) is a soft hyperring over119877

Example 26 Any soft hyperring is a vague subhyperringsince any characteristic function of a subhyperring is a vaguesubhyperring

Proposition 27 Let (119865 119860) and (119866 119861) be vague soft hyperringsover 119877Then (119865 119860) and (119866 119861) is a vague soft hyperring over 119877 ifit is nonnull

Proof Suppose that (119865 119860) and (119866 119861) are vague soft hyper-rings over 119877 Now let (119865 119860) and (119866 119861) = ( 119860 times 119861) Then forall (120572 120573) isin Supp( 119860 times 119861) and for all 119909 isin 119877 we have

119905(120572120573)

(119909) = min 119905119865120572(119909) 119905119866120573

(119909)

1minus119891(120572120573)

(119909) = min 1minus119891119865120572(119909) 1minus119891

119866120573(119909)

(13)

that is

(120572120573) (119909) = 119865

120572 (119909) cap119866120573 (119909) (14)

Since (119865 119860) and (119866 119861) are both nonnull and ( 119860 times 119861)

represents the basic intersection between (119865 119860) and (119866 119861)

it follows that ( 119860 times 119861) must be nonnull For the sake of

similarity we only show the proof for the truth membershipfunction of

(120572120573)given by 119905

(120572120573) The proof for 1 minus 119891

(120572120573)can

be derived in the same mannerHence for all 119909 119910 isin 119877 and (120572 120573) isin Supp( 119860 times 119861) we

obtain

min 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= min 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le min 119905119865120572(119909) cap 119905

119865120572(119910) 119905

119866120573(119909) cap 119905

119866120573(119910)

le min 119905119865120572(119909) 119905119865120572(119910) capmin 119905

119866120573(119909) 119905119866120573

(119910)

le inf 119905119865120572(119911) 119911 isin 119909 +119910

cap inf 119905119866120573

(119911) 119911 isin 119909 +119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 +119910

= inf 119905(120572120573)

119911 isin 119909 +119910

(15)

Furthermore for all 119908 119909 isin 119877 we have

min 119905(120572120573)

(119908) 119905(120572120573)

(119909)

= min 119905119865120572(119908) cap 119905

119866120573(119908) 119905

119865120572(119909) cap 119905

119866120573(119909)

le min 119905119865120572(119908) cap 119905

119865120572(119909) 119905119866120573

(119908) cap 119905119866120573

(119909)

le min 119905119865120572(119908) 119905

119865120572(119909) capmin 119905

119866120573(119908) 119905

119866120573(119909)

le 119905119865120572(119910) cap 119905

119866120573(119910) = 119905

(120572120573)(119910)

(16)

where 119910 isin 119877 such that 119909 isin 119908 + 119910 Similarly it can also beproved that min119905

(120572120573)(119908) 119905(120572120573)

(119909) le 119905(120572120573)

(119911) where 119911 isin 119877

such that 119909 isin 119911+119908Therefore it has been proved that (120572120573)

isa vague subhypergroup of (119877 +) whichmeans that ( 119860times119861)

is a vague soft hypergroup over (119877 +)Lastly for all 119909 119910 isin 119877 we have

min 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= min 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le min 119905119865120572(119909) cap 119905

119865120572(119910) 119905

119866120573(119909) cap 119905

119866120573(119910)

le min 119905119865120572(119909) 119905119865120572(119910) capmin 119905

119866120573(119909) 119905119866120573

(119910)


le inf 119905119865120572(119911) 119911 isin 119909 ∘ 119910

cap inf 119905119866120573

(119911) 119911 isin 119909 ∘ 119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 ∘ 119910

= inf 119905(120572120573)

119911 isin 119909 ∘ 119910

(17)

This proves that (120572120573)

is a vague subhyperring of 119877 for all(120572 120573) isin Supp( 119860 times 119861) Hence ( 119860 times 119861) = (119865 119860) and (119866 119861)

is a vague soft hyperring over 119877

Theorem 28 Let (119865 119860) be a vague soft set over119877Then (119865 119860)is a vague soft hyperring over 119877 if and only if for every 119905 isin

[0 1] (119865 119860)(119905119905)

is a soft hyperring over 119877

Proof (rArr) Since 119877 is a hyperring (119877 +) is a commutativehypergroup and (119877 ∘) is a semihypergroup Let (119865 119860) be avague soft hyperring over (119877 + ∘) and 119905 isin [0 1]Then for all119886 isin Supp(119865 119860) the corresponding119865

119886is a vague subhyperring

of 119877 Now let 119909 119910 isin (119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and

1 minus 119891119865119886(119909) 1 minus 119891

119865119886(119910) ge 119905 which means that 119865

119886(119909) ge 119905 and

119865119886(119910) ge 119905 Furthermore since 119865

119886is a vague subhypergroup of

(119877 +) we have inf119865119886(119911) 119911 isin 119909+119910 ge min119865

119886(119909) 119865119886(119910) ge 119905

This implies that 119911 isin (119865119886)(119905119905)

and therefore for every 119911 isin 119909+119910we obtain 119909 + 119910 sube (119865

119886)(119905119905)

As such for every 119911 isin (119865119886)(119905119905)

we obtain 119911 + (119865

119886)(119905119905)

sube (119865119886)(119905119905)

Now let 119909 119911 isin (119865119886)(119905119905)

Then 119905

119865119886(119909) 119905119865119886(119911) ge 119905 and 1 minus 119891

119865119886(119909) 1 minus 119891

119865119886(119911) ge 119905 which

means that 119865119886(119909) ge 119905 and 119865

119886(119911) ge 119905 Due to the fact that

119865119886is a vague subhypergroup of (119877 +) there exists 119910 isin 119877

such that 119909 isin 119911 + 119910 and min119865119886(119911) 119865119886(119909) le 119865

119886(119910) Since

119909 119911 isin (119865119886)(119905119905)

we obtain min119865119886(119911) 119865119886(119909) ge min119905 119905 = 119905

and this implies that 119865119886(119910) ge 119905 and as a result 119910 isin (119865

119886)(119905119905)

Therefore we obtain (119865119886)(119905119905)

sube 119911+(119865119886)(119905119905)

As such we obtain119911 + (119865

119886)(119905119905)

= (119865119886)(119905119905)

As a result (119865119886)(119905119905)

is a subhypergroupof (119877 +) Hence (119865 119860)

(119905119905)is a soft hypergroup over (119877 +)

Furthermore since (119877 ∘) is a semihypergroup119865119886is a nonnull

vague subsemihypergroup of (119877 ∘) for all 119886 isin Supp(119865 119860) andmin119865

119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 Now let 119909 119910 isin

(119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and 1 minus 119891

119865119886(119909) 1 minus 119891

119865119886(119910) ge

119905 which means that 119865119886(119909) ge 119905 and 119865

119886(119910) ge 119905 Therefore

it follows that min119865119886(119909) 119865119886(119910) ge min119905 119905 = 119905 and this

implies that inf119865119886(119911) 119911 isin 119909∘119910 ge 119905This in turn implies that

119911 isin (119865119886)(119905119905)

and consequently 119909 ∘ 119910 sube (119865119886)(119905119905)

Therefore forevery 119909 119910 isin (119865

119886)(119905119905)

we obtain 119909∘119910 isin 119875lowast(119877) As such (119865

119886)(119905119905)

is a subhyperring of 119877 Hence (119865 119860)(119905119905)

is a soft hyperringover 119877

(lArr)Assume that (119865 119860)(119905119905)

is a soft hyperring over119877Thusfor every 119905 isin [0 1] (119865

119886)(119905119905)

is a nonnull subhyperring of 119877Now for every 119909 119910 isin 119877 we have min119865

119886(119909) 119865119886(119910) le 119865

119886(119909)

ormin119865119886(119909) 119865119886(119910) le 119865

119886(119910) So if we letmin119865

119886(119909) 119865119886(119910) =

1199050 then 119909 119910 isin (119865119886)(1199050 1199050)

and therefore 119909 + 119910 sube (119865119886)(1199050 1199050)

Therefore for every 119911 isin 119909 + 119910 we have 119865

119886(119911) ge 1199050 which

implies that min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 +

119910 As such condition (i) of Definition 21 is verified Nextlet 119908 119909 isin 119877 and min119865

119886(119908) 119865

119886(119909) = 1199051 This implies

that 119908 119909 isin (119865119886)(1199051 1199051)

which means that there exists 119910 isin

(119865119886)(1199051 1199051)

such that 119909 isin 119908 ∘ 119910 Since 119910 isin (119865119886)(1199051 1199051)

wehave 119905

119865119886(119910) ge 1199051 and 1 minus 119905

119865119886(119910) ge 1199051 which means that

119865119886(119910) ge 1199051 and thus min119865

119886(119908) 119865

119886(119909) = 1199051 le 119865

119886(119910)

which means that min119865119886(119908) 119865

119886(119909) le 119865

119886(119910) Therefore

condition (ii) of Definition 21 has been verified Condition(iii) of Definition 21 can be verified in a similar mannerThusit has been proven that 119865

119886is a vague subhypergroup of (119877 +)

Now (119865119886)(119905119905)

is a subsemihypergroup of the semihypergroup(119877 ∘) Let min119865

119886(119909) 119865119886(119910) = 1199052 for every 119909 119910 isin 119877 Thus

119909 119910 isin (119865119886)(1199052 1199052)

and therefore we have 119909 ∘ 119910 sube (119865119886)(1199052 1199052)

Thus for every 119911 isin 119909 ∘ 119910 we obtain 119865

119886(119911) ge 1199052 As a result

we obtain min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 which

proves that condition (iv) of Definition 21 has been verifiedAs such 119865

119886is a vague subhyperring of 119877 and therefore (119865 119860)


Theorem 28 proves that there exists a one-to-one corre-spondence between vague soft hyperrings and softhyperringsover a hyperring

Theorem 29 Let (119865 119860) be a vague soft hyperring over 119877Then for each 120572 120573 isin [0 1] (119865 119860)

(120572120573)is a subhyperring of 119877

Proof Let (119865 119860) be a vague soft hyperring over (119877 + ∘)Thenfor each 119886 isin Supp(119865 119860) the corresponding 119865

119886is a vague

subhyperring of119877 Now let119909 119910 isin (119865 119860)(120572120573)

Then 119905119865119886(119909) ge 120572

1 minus 119891119865119886(119909) ge 120573 and 119905

119865119886(119910) ge 120572 1 minus 119891

119865119886(119910) ge 120573 Thus the

following is obtained

min 119905119865119886(119909) 119905119865119886(119910) ge min 120572 120572 = 120572

min 1 minus 119891119865119886(119909) 1minus119891

119865119886(119910) ge min 120573 120573 = 120573

(18)

However since119865119886is a vague subhypergroup of (119877 +) we have

inf119905119865119886(119911) 119911 isin 119909+119910 ge 120572 and inf1 minus 119891

119865119886(119911) 119911 isin 119909+119910 ge 120573

Therefore for every 119911 isin 119909 + 119910 we obtain 119911 isin (119865 119860)(120572120573)

andthus 119909 + 119910 sube (119865 119860)

(120572120573) As such it can be concluded that

119911 + (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

Furthermore since 119865119886is a vague

subhypergroup of (119877 +) there exists119910 isin 119877 such that 119909 isin 119911+119910

and min119865119886(119909) 119865119886(119911) le 119865

119886(119910) Now let 119909 119911 isin (119865 119860)

(120572120573)

Then the following is obtained

min 119905119865119886(119909) 119905119865119886(119911) ge min 120572 120572 = 120572

min 1 minus 119891119865119886(119909) 1minus119891

119865119886(119910) ge min 120573 120573 = 120573

(19)

This means that 119905119865119886(119910) ge 120572 and 1 minus 119891

119865119886(119910) ge 120573 which implies

that119910 isin (119865 119860)(120572120573)

Therefore for all119909 119911 isin (119865 119860)(120572120573)

we have


119909 isin 119911 + 119910 and 119910 isin (119865 119860)(120572120573)

As such it can be concludedthat (119865 119860)

(120572120573)sube 119911 + (119865 119860)

(120572120573)and this implies that for

any 119911 isin (119865 119860)(120572120573)

119911 + (119865 119860)(120572120573)

= (119865 119860)(120572120573)

Hence ithas been proven that (119865 119860)

(120572120573)is a subhypergroup of (119877 +)

Furthermore (119877 ∘) is a semihypergroup and 119865119886is also a vague

subsemihypergroup of (119877 ∘) for all 119886 isin Supp(119865 119860) Now let119909 119910 isin (119865 119860)

(120572120573) Then min119905

119865119886(119909) 119905119865119886(119910) ge min120572 120572 = 120572

and min1 minus 119891119865119886(119909) 1 minus 119891

119865119886(119910) ge min120573 120573 = 120573 and this

results in

inf 119905119865119886(119911) 119911 isin 119909 ∘ 119910 ge min 119905

119865119886(119909) 119905119865119886(119910) ge 120572

inf 1 minus 119891119865119886(119911) 119911 isin 119909 ∘ 119910

ge min 1 minus 119891119865119886(119909) 1minus119891

119865119886(119910) ge 120573

(20)

This implies that for all 119911 isin 119909∘119910 we obtain 119911 isin (119865 119860)(120572120573)

andtherefore 119909 ∘ 119910 sube (119865 119860)

(120572120573) Thus for every 119909 119910 isin (119865 119860)

(120572120573)

we obtain 119909 ∘ 119910 isin 119875lowast((119865 119860)

(120572120573)) As such (119865 119860)

(120572120573)is

a subsemihypergroup of the semihypergroup (119877 ∘) Hence(119865 119860)


As a result of Theorem 29 we obtain Corollary 30

Corollary 30 Let (119865 119860) be a vague soft hyperring over 119877Then for each 120572 isin [0 1] (119865 119860)


Theorem 31 Let 119879 be a nonempty subset of 119877 let (119865 119860) be anonnull vague soft set over 119877 and let (119865 119860)

119879be a vague soft

characteristic set over 119879 If (119865 119860) is a vague soft hyperring over119877 then 119879 is a subhyperring of 119877

Proof Let (119865 119860) be a vague soft hyperring over 119877 Thenby Definition 21 (119865 119860) is a vague soft hypergroup over(119877 +) and a vague soft semihypergroup over (119877 ∘) and foreach 119886 isin Supp(119865 119860) the corresponding vague subset 119865

119886

is a vague subhyperring of 119877 Furthermore since (119877 + ∘)

is a hyperring (119877 +) is a commutative hypergroup and(119877 ∘) is a semihypergroup Now let 119909 119910 isin 119879 Then119905(119865119886)119879

(119909) = 119905(119865119886)119879

(119910) = 119904 and 119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119910) = 119908which means that (119865

119886)119879(119909) = (119865

119886)119879(119910) = 119904 Therefore

min(119865119886)119879(119909) (119865

119886)119879(119910) = min119904 119904 = 119904 and so inf(119865

119886)119879(119911)

119911 isin 119909+119910 ge 119904 (∵ 119865119886is a vague subhyperring of 119877 and a vague

subhypergroup of (119877 +)) Thus for every 119911 isin 119909 + 119910 we have119911 isin 119879 and so it follows that 119909 + 119910 sube 119879 As such for every119911 isin 119879 we have 119911 + 119879 sube 119879

Now let 119909 119911 isin 119879 Then 119905(119865119886)119879

(119909) = 119905(119865119886)119879

(119911) = 119904 and119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119911) = 119908 and so (119865119886)119879(119909) = (119865

119886)119879(119911) =

119904 Since 119865119886is a vague subhypergroup of (119877 +) for all 119886 isin

Supp(119865 119860) there exists 119910 isin 119877 such that 119909 isin 119911 + 119910

and min(119865119886)119879(119909) (119865

119886)119879(119911) le (119865

119886)119879(119910) Thus we obtain

min(119865119886)119879(119909) (119865

119886)119879(119911) = min119904 119904 = 119904 and therefore

(119865119886)119879(119910) ge 119904 which implies that 119910 isin 119879 Since 119909 119910 isin 119879 this

implies that 119879 sube 119911+119879 and so we have 119911 +119879 = 119879 for all 119911 isin 119879

Therefore it has been proven that (119879 +) is a subhypergroupof (119877 +) Let 119909 119910 isin 119879 Then 119905

(119865119886)119879(119909) = 119905

(119865119886)119879(119910) =

119904 and 119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119910) = 119908 and thus (119865119886)119879(119909) =

(119865119886)119879(119910) = 119904 Now since (119877 ∘) is a semihypergroup 119865

119886is a

vague subhypergroup of (119877 ∘) for all 119886 isin Supp(119865 119860) Thusit follows that min(119865

119886)119879(119909) (119865

119886)119879(119910) = min119904 119904 = 119904 and

therefore inf(119865119886)119879(119911) 119911 isin 119909 ∘ 119910 ge 119904 too As a result for

every 119911 isin 119909 ∘ 119910 we have 119911 isin 119879 and therefore 119909 ∘ 119910 sube 119879 Assuch for every 119909 119910 isin 119879 we have 119909 ∘ 119910 isin 119875

lowast(119879) Hence 119879 is a

subhyperring of 119877



characteristic set over 119879 Then (119865 119860) is a vague soft hyperringover 119877 if and only if 119879 is a subhyperring of 119877

Proof The proof is similar to that of Theorem 31

Theorem 32 implies that there exists a one-to-one cor-respondence between vague soft hyperrings and classicalsubhyperrings of a hyperring

4 Vague Soft Hyperideals

The ideal of a ring is an important concept in the study ofrings The concept of a fuzzy ideal of a ring was introducedby Liu (see [9]) Davvaz (see [8]) introduced the notionof a fuzzy hyperideal (resp fuzzy 119867]-ideal) of a hyperring(resp 119867]-ring) In this section the notion of vague softhyperideals of a hyperring is defined using Rosenfeldrsquosapproach

Definition 33 Let (119865 119860) be a nonnull soft set over 119877 Then(119865 119860) is called a soft left (resp right) hyperideal of 119877 if 119865(119909)is a left (resp right) hyperideal of 119877 for all 119909 isin Supp(119865 119860)

Definition 34 Let (119865 119860) be a nonnull soft set over 119877 Then(119865 119860) is called a soft hyperideal of 119877 if 119865(119909) is a hyperideal of119877 that is 119865(119909) is both right and left hyperideals of 119877 for all119909 isin Supp(119865 119860)

Definition 35 Let (119865 119860) be a nonnull vague soft set over 119877Then (119865 119860) is called a vague soft left (resp right) hyperidealof 119877 if the following axioms hold for all 119886 isin Supp(119865 119860)

(i) for all 119909 119910 isin 119877 min119905119865119886(119909) 119905119865119886(119910) le inf119905

119865119886(119911)


119865119886(119910) le

inf1 minus 119891119865119886(119911) 119911 isin 119909 + 119910 which means that

min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910

(ii) for all 119908 119909 isin 119877 there exists 119910 isin 119877 such that119909 isin 119908 + 119910 and min119905

119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)

(iii) for all 119908 119909 isin 119877 there exists 119911 isin 119877 such that119909 isin 119911 + 119908 and min119905

119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and


min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)

(iv) for all 119909 119910 isin 119877 119905119865119886(119910) le inf119905

119865119886(119911) 119911 isin 119909 ∘ 119910 (resp

119905119865119886(119909) le inf119905

119865119886(119911) 119911 isin 119909 ∘ 119910) and 1 minus 119891

119865119886(119910) le

inf1 minus 119891119865119886(119911) 119911 isin 119909 ∘ 119910 (resp 1 minus 119891

119865119886(119909) le

inf1 minus 119891119865119886(119911) 119911 isin 119909∘119910) that is119865

119886(119910) le inf119865

119886(119911)

119911 isin 119909 ∘ 119910 (resp 119865119886(119909) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910)

That is 119865119886is a vague left (resp right) hyperideal of 119877 (in

Rosenfeldrsquos sense) for all 119886 isin Supp(119865 119860)Conditions (i) (ii) and (iii) prove that 119865

119886is a vague

subhypergroup of (119877 +)Thismeans that (119865 119860) is a vague softleft (resp right) hyperideal of 119877 if for all 119886 isin Supp(119865 119860) 119865

119886

is a vague subhypergroup of (119877 +) and 119865119886also satisfies the

condition 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 (resp 119865

119886(119909) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910)

Definition 36 Let (119865 119860) be a nonnull vague soft set over119877 Then (119865 119860) is called a vague soft hyperideal of 119877 if thefollowing axioms hold for all 119886 isin Supp(119865 119860)


119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)

(iv) for all 119909 119910 isin 119877 max119905119865119886(119909) 119905119865119886(119910) le inf119905

119865119886(119911)

119911 isin 119909 ∘ 119910 and max1 minus 119891119865119886(119909) 1 minus 119891

119865119886(119910) le

inf1 minus 119891119865119886(119911) 119911 isin 119909∘119910 that ismax119865

119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910

That is 119865119886is a vague hyperideal of 119877 (in Rosenfeldrsquos sense) for

all 119886 isin Supp(119865 119860)Similar to Definition 35 conditions (i) (ii) and (iii) of

Definition 36 indicate that 119865119886is a vague subhypergroup of

(119877 +) while condition (iv) is a combination of both theconditions given in Definition 35 (condition (iv))

Theorem 37 Let (119865 119860) be a nonnull vague soft set over 119877Then the necessary and sufficient condition for (119865 119860) to be avague soft hyperideal is for (119865 119860) to be a vague soft hypergroupof (119877 +) and (119865 119860) is both a vague soft left hyperideal and avague soft right hyperideal of 119877

Proof The proof is straightforward by Definitions 35 and 36

Proposition 38 Let (119865 119860) and (119866 119861) be two vague softhyperideals of119877Then (119865 119860) and (119866 119861) is a vague soft hyperidealof 119877 if it is nonnull

Proof Suppose that (119865 119860) and (119866 119861) are vague soft hyperide-als of119877Now to prove that ( 119860times119861) is a vague soft hyperidealof 119877 we have to prove that all the conditions of Definition 36have been satisfied However in Proposition 27 it has beenproven that conditions (i) (ii) and (iii) have been satisfiedTherefore it is only necessary to prove that ( 119860times119861) satisfiescondition (iv) of Definition 36

Thus for all 119909 119910 isin 119877 and (120572 120573) isin Supp( 119860times119861)we have

max 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= max 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le max 119905119865120572(119909) 119905119865120572(119910) capmax 119905

119866120573(119909) 119905119866120573

(119910)

le inf 119905119865120572(119911) 119911 isin 119909 ∘ 119910

cap inf 119905119866120573

(119911) 119911 isin 119909 ∘ 119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 ∘ 119910

= inf 119905(120572120573)

119911 isin 119909 ∘ 119910

(21)

Similarly it can be proven that the conditionmax1 minus 119891

(120572120573)(119909) 1 minus119891

(120572120573)(119910) le inf1 minus 119891

(120572120573) 119911 isin 119909 ∘ 119910

is satisfied This means that max(120572120573)

(119909) (120572120573)

(119910) le

inf(120572120573)

(119911) 119911 isin 119909 ∘ 119910 Thus it has been proven that (120572120573)

is a vague hyperideal of 119877 for all (120572 120573) isin Supp( 119860 times 119861) andas such ( 119860times119861) = (119865 119860) and (119866 119861) is a vague soft hyperidealof 119877

Theorem 39 Let (119865 119860) be a nonnull vague soft set over 119877Then (119865 119860) is a vague soft hyperideal of 119877 if and only if foreach 119905 isin [0 1] (119865 119860)

(119905119905)is a soft hyperideal of 119877

Proof (rArr) Let (119865 119860) be a vague soft hyperideal of 119877 and119905 isin [0 1] Then 119865


Now let 119909 119910 isin (119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and

1 minus 119891119865119886(119909) 1 minus 119891


119886(119909) ge 119905 and

119865119886(119910) ge 119905 In Theorem 28 it was proved that if 119865

119886is a vague

subhypergroup of (119877 +) then (119865119886)(119905119905)

is a subhypergroupof (119877 +) This means that it is only necessary to prove that(119865119886)(119905119905)

satisfies the following conditions

119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(22)


Now let 119909 isin 119877 and 119910 isin (119865119886)(119905119905)

Then 119905119865119886(119910) ge 119905 and

1 minus 119891119865119886(119910) ge 119905 which means that 119865

119886(119910) ge 119905 Since 119865

119886is a

vague hyperideal of 119877 for all 119886 isin Supp(119865 119860) 119865119886must satisfy

the following condition

119905119865119886(119910) le inf 119905

119865119886(119911) 119911 isin 119909 ∘ 119910

1 minus 119891119865119886(119910) le inf 1 minus 119891

119865119886(119911) 119911 isin 119909 ∘ 119910

(23)

which means 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 Therefore we

obtain inf119865119886(119911) 119911 isin 119909 ∘ 119910 ge 119865

119886(119910) ge 119905 This proves

that 119911 isin (119865119886)(119905119905)

for all 119911 isin 119909 ∘ 119910 and as a result we obtain119909 ∘ 119910 sube (119865

119886)(119905119905)

Thus for every 119909 isin 119877 and 119910 isin (119865119886)(119905119905)

weobtain 119877 ∘ (119865

119886)(119905119905)

sube (119865119886)(119905119905)

and this proves that (119865119886)(119905119905)

isa left hyperideal of 119877 This implies that (119865 119860)

(119905119905)is a soft left

hyperideal of 119877 Similarly it can been proven that (119865 119860)(119905119905)

is a soft right hyperideal of 119877 Thus (119865 119860)(119905119905)

is a soft righthyperideal of 119877 Hence (119865 119860)

(119905119905)is a soft left hyperideal of 119877

and a soft right hyperideal of 119877which proves that (119865 119860)(119905119905)

isa soft hyperideal of 119877

(lArr) Let (119865 119860)(119905119905)

be a soft hyperideal of 119877 Then foreach 119886 isin Supp(119865 119860) (119865

119886)(119905119905)

is a nonnull hyperideal of119877 Therefore by Definition 16 (119865

119886)(119905119905)

is a subhypergroupof (119877 +) This implies that (119865 119860)

(119905119905)is a soft hypergroup

over (119877 +) In [10] it was proven that if (119865 119860)(119905119905)

is a softhypergroup then (119865 119860) is a vague soft hypergroup Thus itcan be concluded that (119865 119860) is a vague soft hypergroup over(119877 +) Furthermore since (119865

119886)(119905119905)

is both a left hyperideal anda right hyperideal of 119877 the following conditions are satisfied

119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(24)


or 119909 isin (119865119886)(119905119905)

and 119910 isin 119877Thus we obtain 119909 ∘ 119910 sube (119865

119886)(119905119905)

and 119910 ∘ 119909 sube (119865119886)(119905119905)

whichimplies that 119911 isin 119909 ∘ 119910 where 119911 isin (119865

119886)(119905119905)

As such we obtaininf119865119886(119911) 119911 isin 119909 ∘ 119910 ge 119905 Now let 119909 119910 isin (119865

119886)(119905119905)

Then119865119886(119909) ge 119905 and 119865

119886(119910) ge 119905 Therefore we obtain inf119865

119886(119911) 119911 isin

119909 ∘ 119910 ge 119865119886(119909) ge 119905 as well as inf119865

119886(119911) 119911 isin 119909 ∘ 119910 ge 119865

119886(119910) ge 119905

which means that 119865119886(119909) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 and 119865

119886(119910) le

inf119865119886(119911) 119911 isin 119909∘119910 Combining these conditions we obtain

max119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 Thus condition

(iv) of Definition 36 has also been verified This means that119865119886is a vague hyperideal of 119877 and hence it proves that (119865 119860)

is a vague soft hyperideal of 119877

Theorem 39 proves that there exists a one-to-one cor-respondence between vague soft hyperideal and the softhyperideal of a hyperring

Theorem 40 Let (119865 119860) be a nonnull vague soft set over 119877left (resp right) hyperideal of 119877 Then for all 120572 120573 isin [0 1](119865 119860)

(120572120573)is a left (resp right) hyperideal of 119877

Proof Let (119865 119860) be a vague soft left (resp right) hyperidealof119877Then by Definition 35 (119865 119860) is a vague soft hypergroupover (119877 +) Now in Theorem 29 it was proven that if (119865 119860)is a vague soft hypergroup over (119877 +) then (119865 119860)

(120572120573)is a

subhypergroup of (119877 +) Therefore in order to prove that(119865 119860)

(120572120573)is a left (resp right) hyperideal of 119877 we only have

to prove that 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

(resp (119865 119860)(120572120573)

∘

119877 sube (119865 119860)(120572120573)

) Now let 119909 isin 119877 and 119910 isin (119865 119860)(120572120573)

Then119905119865119886(119910) ge 120572 and 1 minus 119891

119865119886(119910) ge 120573 Since 119865

119886is a vague left ideal of

119877 for all 119886 isin Supp(119865 119860) 119865119886satisfies the conditions 119905

119865119886(119910) le

inf119905119865119886(119911) 119911 isin 119909 ∘ 119910 and 1 minus 119891

119865119886(119910) le inf1 minus 119891

119865119886(119911) 119911 isin

119909∘119910 Therefore we obtain inf119905119865119886(119911) 119911 isin 119909∘119910 ge 119905

119865119886(119910) ge 120572

and inf1 minus 119891119865119886(119911) 119911 isin 119909 ∘ 119910 ge 1 minus 119891

119865119886(119910) ge 120573 which

implies that for all 119911 isin 119909 ∘ 119910 119911 isin (119865 119860)(120572120573)

As suchwe obtain 119909 ∘ 119910 sube (119865 119860)

(120572120573) Thus for every 119909 isin 119877 and

119910 isin (119865 119860)(120572120573)

we obtain 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

Hence(119865 119860)

(120572120573)is a left hyperideal of 119877 Next let 119909 isin (119865 119860)

(120572120573)

and 119910 isin 119877 Then 119905119865119886(119909) ge 120572 and 1 minus 119891

119865119886(119909) ge 120573 Similarly

we obtain inf119905119865119886(119911) 119911 isin 119909 ∘ 119910 ge 119905

119865119886(119909) ge 120572 and

inf1 minus 119891119865119886(119911) 119911 isin 119909 ∘ 119910 ge 1 minus 119891

119865119886(119909) ge 120573 This implies

that 119911 isin (119865 119860)(120572120573)

for all 119911 isin 119909 ∘ 119910 and therefore we obtain119909∘119910 sube (119865 119860)

(120572120573) As such for every119909 isin (119865 119860)

(120572120573)and119910 isin 119877

we obtain (119865 119860)(120572120573)

∘ 119877 sube (119865 119860)(120572120573)

which means (119865 119860)(120572120573)

is a right hyperideal of 119877

Theorem 41 Let (119865 119860) be a nonnull vague soft set over 119877Then (119865 119860) is a vague soft left (resp right) hyperideal of 119877 ifand only if for all 120572 120573 isin [0 1] (119865 119860)

(120572120573)is a left (resp right)

hyperideal of 119877

Proof The proof is straightforward

Theorem 42 Let 119872 be a nonempty subset of 119877 and (119865 119860)119872

be a vague soft characteristic set over 119872 If (119865 119860) is a vaguesoft hyperideal of 119877 then119872 is a hyperideal of 119877

Proof Let (119865 119860) be a vague soft hyperideal of119877Therefore byDefinition 36 (119865 119860) is a vague soft hypergroup over (119877 +)and satisfies the condition max119865

119886(119909) 119865119886(119910) le inf119865

119886(119911)

119911 isin 119909 ∘ 119910 In order to prove that 119872 is a hyperideal of 119877 weneed to prove that (119872 +) is a subhypergroup of (119877 +) and119877 ∘ 119872 sube 119872 and 119872 ∘ 119877 sube 119872 In Theorem 31 it was proventhat (119872 +) is a subhypergroup of (119877 +) if (119865 119860) is a vaguesoft hypergroup over (119877 +) Now let 119909 isin 119877 and 119910 isin 119872 Then119905119865119886(119910) = 1 minus 119891

119865119886(119910) = 119904 which means that 119865

119886(119910) = 119904 Since

119865119886is a vague hyperideal of 119877 for all 119886 isin Supp(119865 119860) it must

satisfy the following conditions

119865119886(119910) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

119865119886 (119909) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

(25)

Therefore since 119865119886(119910) = 119904 for all 119886 isin Supp(119865 119860) and 119910 isin 119872


119886(119910) = 119904 which means


that inf119865119886(119911) 119911 isin 119909 ∘ 119910 ge 119904 This implies that 119911 isin 119872 for

all 119911 isin 119909 ∘ 119910 and this proves that 119909 ∘ 119910 sube 119872 for all 119909 isin 119877 and119910 isin 119872 Therefore we obtain 119877 ∘119872 sube 119872 and this implies that119872 is a left hyperideal of 119877 Now let 119909 isin 119872 and 119910 isin 119877 Then119905119865119886(119909) = 1 minus 119891

119865119886= 119904 which means that 119865

119886(119909) = 119904 Similarly


119886(119909) = 119904 which means


all 119911 isin 119909 ∘ 119910 and consequently we obtain 119909 ∘ 119910 sube 119872 for all119909 isin 119872 and 119910 isin 119877 This proves that119872∘119877 sube 119872 As such119872 isa right hyperideal of 119877 Hence it has been proven that119872 is ahyperideal of 119877

Theorem 43 Let 119872 be a nonempty subset of 119877 let (119865 119860) bea nonnull vague soft set over 119877 and let (119865 119860)


characteristic set over119872Then (119865 119860) is a vague soft hyperidealof 119877 if and only if119872 is a hyperideal of 119877


5 Vague Soft Hyperring Homomorphism

In this section we introduce the notion of vague soft hyper-ring homomorphism by applying the concept of vague softfunctions as well as the notion of the image and preimage of avague soft set under a vague soft function introduced by [10]in the context of vague soft hyperrings Lastly it is proven thatthe vague soft hyperring homomorphism preserves vaguesoft hyperrings

Definition 44 (see [2]) Let (1198771 +1 ∘1) and (1198772 +2 ∘2) be twohyperrings A map 119891 1198771 rarr 1198772 is called a hyperringhomomorphism if for all 119909 119910 isin 1198771 one has 119891(119909 +1 119910) sube

119891(119909) +2 119891(119910) and 119891(119909 ∘1 119910) sube 119891(119909) ∘2 119891(119910)

Definition 45 (see [10]) Let 120593 119883 rarr 119884 and 120595 119860 rarr 119861

be two functions where 119860 and 119861 are parameter sets for theclassical sets 119883 and 119884 respectively Let (119865 119860) and (119866 119861) bevague soft sets over 119883 and 119884 respectively Then the orderedpair (120593 120595) is called a vague soft function from (119865 119860) to (119866 119861)and is denoted by (120593 120595) (119865 119860) rarr (119866 119861)

Definition 46 (see [10]) Let (119865 119860) and (119866 119861) be vague softsets over 119883 and 119884 respectively Let (120593 120595) (119865 119860) rarr (119866 119861)

be a vague soft function

(i) The image of (119865 119860) under the vague soft function(120593 120595) which is denoted by (120593 120595)(119865 119860) is a vague softset over 119884 which is defined as

(120593 120595) (119865 119860) = (120593 (119865) 120595 (119860)) (26)

where

120593 (119865119886) (120593 (119909)) = (120593 (119865))

120595(119886)

(119910) (27)

for all 119886 isin 119860 119909 isin 119883 and 119910 isin 119884

(ii) The preimage of (119866 119861) under the vague soft function(120593 120595) which is denoted by (120593 120595)minus1(119866 119861) is a vaguesoft set over119883 which is defined as

(120593 120595)minus1(119866 119861) = (120593

minus1(119866) 120595

minus1(119861)) (28)

where

120593minus1(119866119887) (120593minus1(119910)) = (120593

minus1(119866))120595minus1(119887)

(119909) (29)


If 120593 and 120595 are injective (surjective) then the vague softfunction (120593 120595) is said to be injective (surjective)

Definition 47 Let (1198771 +1 ∘1) and (119877

2 +2 ∘2) be two hyper-

rings Also let (119865 119860) and (119866 119861) be two vague soft hyperringsover 119877

1and 119877

2 respectively and let (120593 120595) (119865 119860) rarr (119866 119861)

be a vague soft function Then (120593 120595) (119865 119860) rarr (119866 119861) iscalled a vague soft hyperring homomorphism if the followingconditions are satisfied

(i) 120593 is a hyperring homomorphism from 1198771to 1198772

(ii) 120595 is a function from 119860 to 119861

(iii) 120593(119865(119909)) = 119866(120595(119909)) for all 119909 isin Supp(119865 119860)

Theorem 48 Let (1198771 +1 ∘1) and (1198772 +2 ∘2) be two hyper-rings Let (119865 119860) and (119866 119861) be two vague soft hyperrings over1198771and 1198772 respectively Let (120593 120595) (119865 119860) rarr (119866 119861) be a vaguesoft hyperring homomorphism and let 120595 119860 rarr 119861 be aninjective mapping Then (120593(119865) 120595(119860)) is a vague soft hyperringover (1198772 +2 ∘2)

Proof Since (120593 120595) is a vague soft hyperring homomorphismit can be seen that

Supp (120593 (119865) 120595 (119860)) sube 120595 (Supp (119865 119860)) (30)

Now let 1199081 1199082 isin 1198772 Then there exists 119909 119910 isin 1198771 such that120593(119909) = 1199081 and 120593(119910) = 1199082 Therefore for all 119909 119910 isin 1198771 119886 isin

Supp(119865 119860) we obtain

min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119906) 119906 isin1199081 +2 1199082

= inf 120593 (119865119886) (120593 (119911))

(31)

where 119911 isin 1198771 such that 119911 isin 119909 +1 119910 and 120593(119911) = 119906


Furthermore for all 119908 119909 isin 1198771 there exists 1199061 1199062 isin 1198772such that 120593(119908) = 1199061 and 120593(119909) = 1199062 Therefore for all 119908 119909 isin

1198771 and for each 119886 isin Supp(119865 119860) we obtain

min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119887) 1199062 isin 1199061 +2 119887

= inf 120593 (119865119886) (120593 (119910))

(32)

where 119910 isin 1198771 such that 119909 isin 119908 +1 119910 and 120593(119910) = 119887 and also

min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119888) 1199062 isin 119888 +2 1199061

= inf 120593 (119865119886) (120593 (119911))

(33)

where 119911 isin 1198771 such that 119909 isin 119911 +1 119908 and 120593(119911) = 119888Thus it has been proven that 120593(119865

119886) is a nonnull subhyper-

group of (119877 +2) and consequently (120593(119865) 120595(119860)) is a vague soft

hypergroup over (119877 +2)

Lastly let 119909 119910 isin 1198771 Then there exists 1199081 1199082 isin 1198772 suchthat 120593(119909) = 1199081 and 120593(119910) = 1199082 Therefore for all 119909 119910 isin 1198771 and119886 isin Supp(119865 119860) we obtain

min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119901) 119901 isin1199081 ∘2 1199082

= inf 120593 (119865119886) (120593 (119911))

(34)

where 119911 isin 1198771 such that 119911 isin 119909∘1119910 and 120593(119911) = 119901Therefore it has been proven that 120593(119865

119886) is a nonnull

subsemihypergroup of (119877 ∘2) and consequently (120593(119865) 120595(119860))

is a vague soft semihypergroup over (119877 ∘2) As such 120593(119865

119886) is

a vague subhyperring of (1198772 +2 ∘2) Hence (120593(119865) 120595(119860)) is a

vague soft hyperring over (1198772 +2 ∘2)

Theorem 49 Let (1198771 +1 ∘1) and (1198772 +2 ∘2) be two hyper-rings Let (119865 119860) and (119866 119861) be two vague soft hypergroups over1198771 and 1198772 respectively Let (120593 120595) (119865 119860) rarr (119866 119861) be avague soft hyperring homomorphism Then (120593

minus1(119866) 120595

minus1(119861))

is a vague soft hyperring over (1198771 +1 ∘1)


Theorems 48 and 49 prove that the homomorphic imageand preimage of a vague soft hyperring are also a vague softhyperring This implies that the vague soft hyperring homo-morphism preserves vague soft hyperrings

Theorem 50 Let 1198771 1198772 and 1198773 be nonnull hyperrings and let(119865 119860) (119866 119861) and (119869 119862) be vague soft hyperrings over 1198771 1198772and1198773 respectively Let (120593 120595) (119865 119860) rarr (119866 119861) and (1205931015840 1205951015840) (119866 119861) rarr (119869 119862) be vague soft hyperring homomorphismsThen (1205931015840 ∘120593 1205951015840 ∘120595) (119865 119860) rarr (119869 119862) is a vague soft hyperringhomomorphism


6 Conclusion

In this paper we introduced and developed the initial theoryof vague soft hyperrings as a continuation to the theory ofvague soft hypergroups It was proved that there exists aone-to-one correspondence between the concepts introducedhere and the corresponding concepts in soft hyperring theoryand classical hyperring theory Lastly it was proved that thenotion of vague soft hyperring homomorphism introducedhere preserves vague soft hyperrings This is an importantstep in the formulation and advancement of knowledge in thearea of vague soft hyperalgebra

Conflict of Interests

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

Acknowledgments

The authors would like to gratefully acknowledge the finan-cial assistance received from the Ministry of EducationMalaysia and Universiti Kebangsaan Malaysia under Grantno FRGS22013SG04UKM023

References

[1] F Marty ldquoSur une Generalisation de la Notion de Grouperdquo inProceedings of the 8th Congress Mathematiciens Scandinaves pp45ndash49 Stockholm Sweden 1934

[2] P Corsini Prolegomena of Hypergroup Theory Aviani EditorTricesimo Italy 2nd edition 1993

[3] T Vougiouklis Hyperstructures and Their RepresentationsHadronic Press Palm Harbor Fla USA 1994

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

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

[6] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[7] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 pp 512ndash517 1971

[8] B Davvaz ldquoOn H]-rings and fuzzy H]-idealrdquo Journal of FuzzyMathematics vol 6 no 1 pp 33ndash42 1998


[9] W J Liu ldquoFuzzy invariant subgroups and fuzzy idealsrdquo FuzzySets and Systems vol 8 no 2 pp 133ndash139 1982

[10] G Selvachandran and A R Salleh ldquoVague soft hypergroupsand vague soft hypergroup homomorphismrdquoAdvances in FuzzySystems vol 2014 Article ID 758637 10 pages 2014

[11] W-L Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[12] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers and Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[13] T Vougiouklis ldquoA new class of hyperstructuresrdquo Journal ofCombination and Information Systems and Sciences vol 20 pp229ndash235 1995

[14] G Selvachandran ldquoIntroduction to the theory of soft hyper-rings and soft hyperring homomorphismrdquo JP Journal of AlgebraNumber Theory and Applications In press


disadvantage various generalizations of soft sets such as fuzzysoft sets vague soft sets and intuitionistic fuzzy soft sets wereintroduced as better alternatives to the concept of soft setsIn the context of this paper the notion of vague soft setsis used as a base to develop the initial theory of vague softhyperrings

In this paper we develop the initial theory of vaguesoft hyperrings in Rosenfeldrsquos sense as a continuation to thetheory of vague soft hypergroups introduced in [10] Subse-quently some of the fundamental properties and structuralcharacteristics of these concepts are studied and discussedLastly we prove that there exists a one-to-one correspon-dence between some of these concepts and their correspond-ing concepts in soft hyperring theory as well as classicalhyperring theory

2 Preliminaries

In this section some well-known and useful definitionspertaining to the development of the theory of vaguesoft hyperrings introduced here will be presented Thesedefinitions and concepts will be used throughout thischapter

Definition 1 (see [6]) A pair (119865 119860) is called a soft set over 119880where 119865 is a mapping given by 119865 119860 rarr 119875(119880) In otherwords a soft set over 119880 is a parameterized family of subsetsof the universe 119880 For 120576 isin 119860 119865(120576) may be considered as theset of 120576-elements of the soft set (119865 119860) or as the 120576-approximateelements of the soft set

Definition 2 (see [6]) For a soft set (119865 119860) the setSupp(119865 119860) = 119909 isin 119860 119865(119909) = 0 is called the supportof the soft set (119865 119860) Thus a null soft set is a soft set with anempty support and a soft set (119865 119860) is said to be nonnull ifSupp(119865 119860) = 0

Definition 3 (see [11]) Let 119883 be a space of points (objects)with a generic element of119883 denoted by 119909 A vague set 119881 in119883is characterized by a truth membership function 119905

119881 119883 rarr

[0 1] and a false membership function 119891119881 119883 rarr [0 1] The

value 119905119881(119909) is a lower bound on the grade of membership of

119909 derived from the evidence for 119909 and 119891119881(119909) is a lower bound

on the negation of 119909 derived from the evidence against 119909Thevalues 119905

119881(119909) and 119891

119881(119909) both associate a real number in the

interval [0 1] with each point in119883 where 119905119881(119909) + 119891

119881(119909) le 1

This approach bounds the grade of membership of 119909 to asubinterval [119905

119881(119909) 1 minus 119891

119881(119909)] of [0 1] Hence a vague set

is a form of fuzzy set albeit a more accurate form of fuzzyset

Definition 4 (see [12]) A pair (119865 119860) is called a vague soft setover 119880 where 119865 is a mapping given by 119865 119860 rarr 119881(119880) and119881(119880) is the power set of vague sets over 119880 In other words avague soft set over119880 is a parameterized family of vague sets ofthe universe 119880 Every set 119865(119890) for all 119890 isin 119860 from this familymay be considered as the set of 119890-approximate elements ofthe vague soft set (119865 119860) Hence the vague soft set (119865 119860) can

be viewed as consisting of a collection of approximations ofthe following form

(119865 119860)

= 119865119886(119909119894) 119894 = 1 2 3

=

[119905119865(119890119894)

(119909119894) 1 minus 119891

119865(119890119894)(119909119894)]

119909119894

119894 = 1 2 3

(1)

where 119865119886(119909119894) is a subset of (119865 119860) for all 119890 isin 119860 and for all

119909 isin 119880

Example 5 (see [12]) Consider a vague soft set (119865 119864)where 119880 is a set of six houses under consideration of adecision maker to purchase which is denoted by 119880 =

ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6 and 119864 is a parameter set where

119864 = 1198901 1198902 1198903 1198904 1198905 = expensive beautiful wooden

cheap in the green surroundings The vague soft set (119865 119864)describes the ldquoattractiveness of the housesrdquo to this decisionmaker

Suppose that

119865 (1198901) = ([01 02]

ℎ1[09 1]ℎ2

[03 05]

ℎ3[08 09]

ℎ4

[02 04]ℎ5

[04 06]

ℎ6)

119865 (1198902) = ([09 1]ℎ1

[02 07]

ℎ2[06 09]

ℎ3[02 04]

ℎ4

[03 04]ℎ5

[01 06]

ℎ6)

119865 (1198903) = ([0 0]ℎ1

[0 0]ℎ2

[1 1]ℎ3

[1 1]ℎ4

[1 1]ℎ5

[0 0]ℎ6

)

119865 (1198904) = ([08 09]

ℎ1[0 01]ℎ2

[05 07]

ℎ3[01 02]

ℎ4

[06 08]ℎ5

[04 06]

ℎ6)

119865 (1198905) = ([09 1]ℎ1

[02 03]

ℎ2[01 04]

ℎ3[01 02]

ℎ4

[02 04]ℎ5

[07 09]

ℎ6)

(2)

The vague soft set (119865 119864) is a parameterized family 119865(119890119894) 119894 =

1 2 3 4 5 of vague sets on 119880 and

(119865 119864) = Expensive Houses=([01 02]ℎ1

[09 1]ℎ2

[03 05]ℎ3

[08 09]

ℎ4[02 04]

ℎ5[04 06]

ℎ6)



[02 07]

ℎ2

[06 09]ℎ3

[02 04]

ℎ4[03 04]

ℎ5[01 06]

ℎ6)

(3)


Supp (119865 119860) = 119886 isin119860 119865119886 (119909) = 0 ie 119905

119865119886(119909)

= 0 1 minus 119891119865119886(119909) = 0

(4)





119905119888(119909) =

119905119865119888(119909) 119888 isin 119860 minus 119861

119905119866119888(119909) 119888 isin 119861 minus 119860

min (119905119865119888(119909) 119905119866119888(119909)) 119888 isin 119861 cap 119860

1 minus 119891119888(119909)

=

1 minus 119891119865119888(119909) 119888 isin 119860 minus 119861

1 minus 119891119866119888(119909) 119888 isin 119861 minus 119860

min (1 minus 119891119865119888(119909) 1 minus 119891

119866119888(119909)) 119888 isin 119861 cap 119860

(5)



119905(120572120573)

(119909) = min 119905119865120572(119909) 119905119866120573

(119909)

1minus119891(120572120573)

(119909) = min 1minus119891119865(120572)

(119909) 1minus119891119866(120573)

(119909)

(6)


and 119891(120572120573)



written as (120572120573)

(119909) = 119865120572(119909) cap 119866




(119865 119860)(120572120573)

= 119909 isin119880 119905119865119886(119909) ge 120572 1 minus 119891

119865119886(119909)

ge 120573 ie 119865119886 (119909) ge [120572 120573]

(7)




(119865 119860)(120572120572)

= 119909 isin119880 119905119865119886(119909) ge 120572 1 minus 119891

119865119886(119909)

ge 120572 ie 119865119886 (119909) ge [120572 120572]

(8)

for all 119886 isin 119860


119866is called



119905(119865119886)119866

(119909) = 1minus119891(119865119886)119866

(119909) =

119904 if 119909 isin 119866

119908 otherwise(9)


119866 119909 isin 119883 119904 119908 isin [0 1] and

119904 gt 119908





119867 ∘ 119909 = 119867 for all 119909 in119867









119909 ∘ (119910 + 119911) cap ((119909 ∘ 119910) + (119909 ∘ 119911)) = 120601

(119909 + 119910) ∘ 119911 cap ((119909 ∘ 119911) + (119910 ∘ 119911)) = 120601

(10)













119865119886(119911) 119911 isin


119865119886(119910) le inf1 minus 119891

119865119886(119911) 119911 isin

119909 ∘ 119910 that is min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910




119865119886(119911)


119865119886(119910) le inf1minus

119891119865119886(119911) 119911 isin 119909 ∘ 119910 that is min119865

119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)




Supp(119865 119860)



119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911)


119865119886(119910) le


119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910



119886is a vague









(119865 119860)+

= (119865119886)+

= 119909 isin119877 119905(119865119886)+ (119909) = 119905

119865119886(119909) + 1

minus 119905119865119886(119890) 1 minus 119891

(119865119886)+ (119909) = 1 minus 119891

119865119886(119909)

+119891119865119886(119890) ie (119865119886)

+

(119909) = 119865119886 (119909) + 1minus119865119886 (119890)

(11)





givenby

(119865 119860)120572= 119909 isin119867 119865

119886 (119909) ge 120572 (12)






119905(120572120573)

(119909) = min 119905119865120572(119909) 119905119866120573

(119909)

1minus119891(120572120573)

(119909) = min 1minus119891119865120572(119909) 1minus119891

119866120573(119909)

(13)

that is

(120572120573) (119909) = 119865

120572 (119909) cap119866120573 (119909) (14)





(120572120573)given by 119905


(120572120573)can


obtain

min 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= min 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le min 119905119865120572(119909) cap 119905

119865120572(119910) 119905

119866120573(119909) cap 119905

119866120573(119910)

le min 119905119865120572(119909) 119905119865120572(119910) capmin 119905

119866120573(119909) 119905119866120573

(119910)

le inf 119905119865120572(119911) 119911 isin 119909 +119910

cap inf 119905119866120573

(119911) 119911 isin 119909 +119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 +119910

= inf 119905(120572120573)

119911 isin 119909 +119910

(15)


min 119905(120572120573)

(119908) 119905(120572120573)

(119909)

= min 119905119865120572(119908) cap 119905

119866120573(119908) 119905

119865120572(119909) cap 119905

119866120573(119909)

le min 119905119865120572(119908) cap 119905

119865120572(119909) 119905119866120573

(119908) cap 119905119866120573

(119909)

le min 119905119865120572(119908) 119905

119865120572(119909) capmin 119905

119866120573(119908) 119905

119866120573(119909)

le 119905119865120572(119910) cap 119905

119866120573(119910) = 119905

(120572120573)(119910)

(16)


(120572120573)(119908) 119905(120572120573)

(119909) le 119905(120572120573)

(119911) where 119911 isin 119877




min 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= min 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le min 119905119865120572(119909) cap 119905

119865120572(119910) 119905

119866120573(119909) cap 119905

119866120573(119910)

le min 119905119865120572(119909) 119905119865120572(119910) capmin 119905

119866120573(119909) 119905119866120573

(119910)


le inf 119905119865120572(119911) 119911 isin 119909 ∘ 119910

cap inf 119905119866120573

(119911) 119911 isin 119909 ∘ 119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 ∘ 119910

= inf 119905(120572120573)

119911 isin 119909 ∘ 119910

(17)





[0 1] (119865 119860)(119905119905)




of 119877 Now let 119909 119910 isin (119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and

1 minus 119891119865119886(119909) 1 minus 119891


119886(119909) ge 119905 and




119886(119909) 119865119886(119910) ge 119905



119886)(119905119905)


we obtain 119911 + (119865

119886)(119905119905)

sube (119865119886)(119905119905)

Now let 119909 119911 isin (119865119886)(119905119905)

Then 119905

119865119886(119909) 119905119865119886(119911) ge 119905 and 1 minus 119891

119865119886(119909) 1 minus 119891

119865119886(119911) ge 119905 which

means that 119865119886(119909) ge 119905 and 119865




119886(119910) Since

119909 119911 isin (119865119886)(119905119905)



119886)(119905119905)


sube 119911+(119865119886)(119905119905)


119886)(119905119905)

= (119865119886)(119905119905)

As a result (119865119886)(119905119905)





119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 Now let 119909 119910 isin

(119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and 1 minus 119891

119865119886(119909) 1 minus 119891

119865119886(119910) ge


119886(119910) ge 119905 Therefore



119911 isin (119865119886)(119905119905)



119886)(119905119905)


119886)(119905119905)



(lArr)Assume that (119865 119860)(119905119905)


119886)(119905119905)


119886(119909) 119865119886(119910) le 119865

119886(119909)

ormin119865119886(119909) 119865119886(119910) le 119865

119886(119910) So if we letmin119865

119886(119909) 119865119886(119910) =

1199050 then 119909 119910 isin (119865119886)(1199050 1199050)



119886(119911) ge 1199050 which


119886(119911) 119911 isin 119909 +


119886(119908) 119865

119886(119909) = 1199051 This implies

that 119908 119909 isin (119865119886)(1199051 1199051)


(119865119886)(1199051 1199051)


wehave 119905

119865119886(119910) ge 1199051 and 1 minus 119905


119865119886(119910) ge 1199051 and thus min119865

119886(119908) 119865

119886(119909) = 1199051 le 119865

119886(119910)


119886(119909) le 119865

119886(119910) Therefore



Now (119865119886)(119905119905)



119909 119910 isin (119865119886)(1199052 1199052)



119886(119911) ge 1199052 As a result


119886(119911) 119911 isin 119909 ∘ 119910 which








119886is a vague


Then 119905119865119886(119909) ge 120572

1 minus 119891119865119886(119909) ge 120573 and 119905

119865119886(119910) ge 120572 1 minus 119891

119865119886(119910) ge 120573 Thus the


min 119905119865119886(119909) 119905119865119886(119910) ge min 120572 120572 = 120572

min 1 minus 119891119865119886(119909) 1minus119891

119865119886(119910) ge min 120573 120573 = 120573

(18)



119865119886(119911) 119911 isin 119909+119910 ge 120573


andthus 119909 + 119910 sube (119865 119860)


119911 + (119865 119860)(120572120573)

sube (119865 119860)(120572120573)



and min119865119886(119909) 119865119886(119911) le 119865

119886(119910) Now let 119909 119911 isin (119865 119860)

(120572120573)


min 119905119865119886(119909) 119905119865119886(119911) ge min 120572 120572 = 120572

min 1 minus 119891119865119886(119909) 1minus119891

119865119886(119910) ge min 120573 120573 = 120573

(19)


119865119886(119910) ge 120573 which implies

that119910 isin (119865 119860)(120572120573)


we have


119909 isin 119911 + 119910 and 119910 isin (119865 119860)(120572120573)


(120572120573)sube 119911 + (119865 119860)


any 119911 isin (119865 119860)(120572120573)

119911 + (119865 119860)(120572120573)

= (119865 119860)(120572120573)





(120572120573) Then min119905

119865119886(119909) 119905119865119886(119910) ge min120572 120572 = 120572


119865119886(119910) ge min120573 120573 = 120573 and this

results in

inf 119905119865119886(119911) 119911 isin 119909 ∘ 119910 ge min 119905

119865119886(119909) 119905119865119886(119910) ge 120572

inf 1 minus 119891119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119910) ge 120573

(20)




(120572120573)


(120572120573)) As such (119865 119860)

(120572120573)is










119886



(119909) = 119905(119865119886)119879

(119910) = 119904 and 119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119910) = 119908which means that (119865

119886)119879(119909) = (119865

119886)119879(119910) = 119904 Therefore

min(119865119886)119879(119909) (119865

119886)119879(119910) = min119904 119904 = 119904 and so inf(119865

119886)119879(119911)



Now let 119909 119911 isin 119879 Then 119905(119865119886)119879

(119909) = 119905(119865119886)119879

(119911) = 119904 and119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119911) = 119908 and so (119865119886)119879(119909) = (119865

119886)119879(119911) =



and min(119865119886)119879(119909) (119865

119886)119879(119911) le (119865

119886)119879(119910) Thus we obtain

min(119865119886)119879(119909) (119865





(119865119886)119879(119909) = 119905

(119865119886)119879(119910) =

119904 and 119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119910) = 119908 and thus (119865119886)119879(119909) =


119886is a


119886)119879(119909) (119865

119886)119879(119910) = min119904 119904 = 119904 and
















119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and


min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911) 119911 isin 119909 ∘ 119910 (resp

119905119865119886(119909) le inf119905

119865119886(119911) 119911 isin 119909 ∘ 119910) and 1 minus 119891

119865119886(119910) le


119865119886(119909) le


119886(119910) le inf119865

119886(119911)

119911 isin 119909 ∘ 119910 (resp 119865119886(119909) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910)



119886is a vague


119886


condition 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 (resp 119865

119886(119909) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910)



119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911)


119865119886(119910) le


119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910










max 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= max 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le max 119905119865120572(119909) 119905119865120572(119910) capmax 119905

119866120573(119909) 119905119866120573

(119910)

le inf 119905119865120572(119911) 119911 isin 119909 ∘ 119910

cap inf 119905119866120573

(119911) 119911 isin 119909 ∘ 119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 ∘ 119910

= inf 119905(120572120573)

119911 isin 119909 ∘ 119910

(21)


(120572120573)(119909) 1 minus119891

(120572120573)(119910) le inf1 minus 119891

(120572120573) 119911 isin 119909 ∘ 119910


(119909) (120572120573)

(119910) le

inf(120572120573)







Now let 119909 119910 isin (119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and

1 minus 119891119865119886(119909) 1 minus 119891


119886(119909) ge 119905 and


119886is a vague




119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(22)



Then 119905119865119886(119910) ge 119905 and


119886(119910) ge 119905 Since 119865

119886is a



119905119865119886(119910) le inf 119905

119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119911) 119911 isin 119909 ∘ 119910

(23)




119886(119910) ge 119905 This proves

that 119911 isin (119865119886)(119905119905)


119886)(119905119905)


weobtain 119877 ∘ (119865

119886)(119905119905)

sube (119865119886)(119905119905)



(119905119905)is a soft left







(lArr) Let (119865 119860)(119905119905)


119886)(119905119905)


119886)(119905119905)





119886)(119905119905)


119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(24)


or 119909 isin (119865119886)(119905119905)


119886)(119905119905)

and 119910 ∘ 119909 sube (119865119886)(119905119905)


119886)(119905119905)


119886)(119905119905)

Then119865119886(119909) ge 119905 and 119865


119886(119911) 119911 isin


119886(119911) 119911 isin 119909 ∘ 119910 ge 119865

119886(119910) ge 119905


119886(119911) 119911 isin 119909 ∘ 119910 and 119865

119886(119910) le


max119865119886(119909) 119865119886(119910) le inf119865








(120572120573)is a



to prove that 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

(resp (119865 119860)(120572120573)

∘

119877 sube (119865 119860)(120572120573)


Then119905119865119886(119910) ge 120572 and 1 minus 119891

119865119886(119910) ge 120573 Since 119865



119865119886(119910) le


119865119886(119910) le inf1 minus 119891

119865119886(119911) 119911 isin


119865119886(119910) ge 120572


119865119886(119910) ge 120573 which




119910 isin (119865 119860)(120572120573)

we obtain 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

Hence(119865 119860)


(120572120573)


119865119886(119909) ge 120573 Similarly


119865119886(119909) ge 120572 and


119865119886(119909) ge 120573 This implies

that 119911 isin (119865 119860)(120572120573)



(120572120573)and119910 isin 119877

we obtain (119865 119860)(120572120573)

∘ 119877 sube (119865 119860)(120572120573)

which means (119865 119860)(120572120573)









119886(119909) 119865119886(119910) le inf119865

119886(119911)


119865119886(119910) = 119904 which means that 119865

119886(119910) = 119904 Since



119865119886(119910) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

119865119886 (119909) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

(25)



119886(119910) = 119904 which means




119865119886= 119904 which means that 119865

119886(119909) = 119904 Similarly


119886(119909) = 119904 which means










119891(119909) +2 119891(119910) and 119891(119909 ∘1 119910) sube 119891(119909) ∘2 119891(119910)






(120593 120595) (119865 119860) = (120593 (119865) 120595 (119860)) (26)

where

120593 (119865119886) (120593 (119909)) = (120593 (119865))

120595(119886)

(119910) (27)



(120593 120595)minus1(119866 119861) = (120593

minus1(119866) 120595

minus1(119861)) (28)

where

120593minus1(119866119887) (120593minus1(119910)) = (120593

minus1(119866))120595minus1(119887)

(119909) (29)






1and 119877








Supp (120593 (119865) 120595 (119860)) sube 120595 (Supp (119865 119860)) (30)



min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119906) 119906 isin1199081 +2 1199082

= inf 120593 (119865119886) (120593 (119911))

(31)





min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119887) 1199062 isin 1199061 +2 119887

= inf 120593 (119865119886) (120593 (119910))

(32)


min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119888) 1199062 isin 119888 +2 1199061

= inf 120593 (119865119886) (120593 (119911))

(33)






min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119901) 119901 isin1199081 ∘2 1199082

= inf 120593 (119865119886) (120593 (119911))

(34)





119886) is




minus1(119866) 120595

minus1(119861))






6 Conclusion




Acknowledgments


References


















[02 07]

ℎ2

[06 09]ℎ3

[02 04]

ℎ4[03 04]

ℎ5[01 06]

ℎ6)

(3)


Supp (119865 119860) = 119886 isin119860 119865119886 (119909) = 0 ie 119905

119865119886(119909)

= 0 1 minus 119891119865119886(119909) = 0

(4)





119905119888(119909) =

119905119865119888(119909) 119888 isin 119860 minus 119861

119905119866119888(119909) 119888 isin 119861 minus 119860

min (119905119865119888(119909) 119905119866119888(119909)) 119888 isin 119861 cap 119860

1 minus 119891119888(119909)

=

1 minus 119891119865119888(119909) 119888 isin 119860 minus 119861

1 minus 119891119866119888(119909) 119888 isin 119861 minus 119860

min (1 minus 119891119865119888(119909) 1 minus 119891

119866119888(119909)) 119888 isin 119861 cap 119860

(5)



119905(120572120573)

(119909) = min 119905119865120572(119909) 119905119866120573

(119909)

1minus119891(120572120573)

(119909) = min 1minus119891119865(120572)

(119909) 1minus119891119866(120573)

(119909)

(6)


and 119891(120572120573)



written as (120572120573)

(119909) = 119865120572(119909) cap 119866




(119865 119860)(120572120573)

= 119909 isin119880 119905119865119886(119909) ge 120572 1 minus 119891

119865119886(119909)

ge 120573 ie 119865119886 (119909) ge [120572 120573]

(7)




(119865 119860)(120572120572)

= 119909 isin119880 119905119865119886(119909) ge 120572 1 minus 119891

119865119886(119909)

ge 120572 ie 119865119886 (119909) ge [120572 120572]

(8)

for all 119886 isin 119860


119866is called



119905(119865119886)119866

(119909) = 1minus119891(119865119886)119866

(119909) =

119904 if 119909 isin 119866

119908 otherwise(9)


119866 119909 isin 119883 119904 119908 isin [0 1] and

119904 gt 119908





119867 ∘ 119909 = 119867 for all 119909 in119867









119909 ∘ (119910 + 119911) cap ((119909 ∘ 119910) + (119909 ∘ 119911)) = 120601

(119909 + 119910) ∘ 119911 cap ((119909 ∘ 119911) + (119910 ∘ 119911)) = 120601

(10)













119865119886(119911) 119911 isin


119865119886(119910) le inf1 minus 119891

119865119886(119911) 119911 isin

119909 ∘ 119910 that is min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910




119865119886(119911)


119865119886(119910) le inf1minus

119891119865119886(119911) 119911 isin 119909 ∘ 119910 that is min119865

119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)




Supp(119865 119860)



119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911)


119865119886(119910) le


119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910



119886is a vague









(119865 119860)+

= (119865119886)+

= 119909 isin119877 119905(119865119886)+ (119909) = 119905

119865119886(119909) + 1

minus 119905119865119886(119890) 1 minus 119891

(119865119886)+ (119909) = 1 minus 119891

119865119886(119909)

+119891119865119886(119890) ie (119865119886)

+

(119909) = 119865119886 (119909) + 1minus119865119886 (119890)

(11)





givenby

(119865 119860)120572= 119909 isin119867 119865

119886 (119909) ge 120572 (12)






119905(120572120573)

(119909) = min 119905119865120572(119909) 119905119866120573

(119909)

1minus119891(120572120573)

(119909) = min 1minus119891119865120572(119909) 1minus119891

119866120573(119909)

(13)

that is

(120572120573) (119909) = 119865

120572 (119909) cap119866120573 (119909) (14)





(120572120573)given by 119905


(120572120573)can


obtain

min 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= min 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le min 119905119865120572(119909) cap 119905

119865120572(119910) 119905

119866120573(119909) cap 119905

119866120573(119910)

le min 119905119865120572(119909) 119905119865120572(119910) capmin 119905

119866120573(119909) 119905119866120573

(119910)

le inf 119905119865120572(119911) 119911 isin 119909 +119910

cap inf 119905119866120573

(119911) 119911 isin 119909 +119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 +119910

= inf 119905(120572120573)

119911 isin 119909 +119910

(15)


min 119905(120572120573)

(119908) 119905(120572120573)

(119909)

= min 119905119865120572(119908) cap 119905

119866120573(119908) 119905

119865120572(119909) cap 119905

119866120573(119909)

le min 119905119865120572(119908) cap 119905

119865120572(119909) 119905119866120573

(119908) cap 119905119866120573

(119909)

le min 119905119865120572(119908) 119905

119865120572(119909) capmin 119905

119866120573(119908) 119905

119866120573(119909)

le 119905119865120572(119910) cap 119905

119866120573(119910) = 119905

(120572120573)(119910)

(16)


(120572120573)(119908) 119905(120572120573)

(119909) le 119905(120572120573)

(119911) where 119911 isin 119877




min 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= min 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le min 119905119865120572(119909) cap 119905

119865120572(119910) 119905

119866120573(119909) cap 119905

119866120573(119910)

le min 119905119865120572(119909) 119905119865120572(119910) capmin 119905

119866120573(119909) 119905119866120573

(119910)


le inf 119905119865120572(119911) 119911 isin 119909 ∘ 119910

cap inf 119905119866120573

(119911) 119911 isin 119909 ∘ 119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 ∘ 119910

= inf 119905(120572120573)

119911 isin 119909 ∘ 119910

(17)





[0 1] (119865 119860)(119905119905)




of 119877 Now let 119909 119910 isin (119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and

1 minus 119891119865119886(119909) 1 minus 119891


119886(119909) ge 119905 and




119886(119909) 119865119886(119910) ge 119905



119886)(119905119905)


we obtain 119911 + (119865

119886)(119905119905)

sube (119865119886)(119905119905)

Now let 119909 119911 isin (119865119886)(119905119905)

Then 119905

119865119886(119909) 119905119865119886(119911) ge 119905 and 1 minus 119891

119865119886(119909) 1 minus 119891

119865119886(119911) ge 119905 which

means that 119865119886(119909) ge 119905 and 119865




119886(119910) Since

119909 119911 isin (119865119886)(119905119905)



119886)(119905119905)


sube 119911+(119865119886)(119905119905)


119886)(119905119905)

= (119865119886)(119905119905)

As a result (119865119886)(119905119905)





119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 Now let 119909 119910 isin

(119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and 1 minus 119891

119865119886(119909) 1 minus 119891

119865119886(119910) ge


119886(119910) ge 119905 Therefore



119911 isin (119865119886)(119905119905)



119886)(119905119905)


119886)(119905119905)



(lArr)Assume that (119865 119860)(119905119905)


119886)(119905119905)


119886(119909) 119865119886(119910) le 119865

119886(119909)

ormin119865119886(119909) 119865119886(119910) le 119865

119886(119910) So if we letmin119865

119886(119909) 119865119886(119910) =

1199050 then 119909 119910 isin (119865119886)(1199050 1199050)



119886(119911) ge 1199050 which


119886(119911) 119911 isin 119909 +


119886(119908) 119865

119886(119909) = 1199051 This implies

that 119908 119909 isin (119865119886)(1199051 1199051)


(119865119886)(1199051 1199051)


wehave 119905

119865119886(119910) ge 1199051 and 1 minus 119905


119865119886(119910) ge 1199051 and thus min119865

119886(119908) 119865

119886(119909) = 1199051 le 119865

119886(119910)


119886(119909) le 119865

119886(119910) Therefore



Now (119865119886)(119905119905)



119909 119910 isin (119865119886)(1199052 1199052)



119886(119911) ge 1199052 As a result


119886(119911) 119911 isin 119909 ∘ 119910 which








119886is a vague


Then 119905119865119886(119909) ge 120572

1 minus 119891119865119886(119909) ge 120573 and 119905

119865119886(119910) ge 120572 1 minus 119891

119865119886(119910) ge 120573 Thus the


min 119905119865119886(119909) 119905119865119886(119910) ge min 120572 120572 = 120572

min 1 minus 119891119865119886(119909) 1minus119891

119865119886(119910) ge min 120573 120573 = 120573

(18)



119865119886(119911) 119911 isin 119909+119910 ge 120573


andthus 119909 + 119910 sube (119865 119860)


119911 + (119865 119860)(120572120573)

sube (119865 119860)(120572120573)



and min119865119886(119909) 119865119886(119911) le 119865

119886(119910) Now let 119909 119911 isin (119865 119860)

(120572120573)


min 119905119865119886(119909) 119905119865119886(119911) ge min 120572 120572 = 120572

min 1 minus 119891119865119886(119909) 1minus119891

119865119886(119910) ge min 120573 120573 = 120573

(19)


119865119886(119910) ge 120573 which implies

that119910 isin (119865 119860)(120572120573)


we have


119909 isin 119911 + 119910 and 119910 isin (119865 119860)(120572120573)


(120572120573)sube 119911 + (119865 119860)


any 119911 isin (119865 119860)(120572120573)

119911 + (119865 119860)(120572120573)

= (119865 119860)(120572120573)





(120572120573) Then min119905

119865119886(119909) 119905119865119886(119910) ge min120572 120572 = 120572


119865119886(119910) ge min120573 120573 = 120573 and this

results in

inf 119905119865119886(119911) 119911 isin 119909 ∘ 119910 ge min 119905

119865119886(119909) 119905119865119886(119910) ge 120572

inf 1 minus 119891119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119910) ge 120573

(20)




(120572120573)


(120572120573)) As such (119865 119860)

(120572120573)is










119886



(119909) = 119905(119865119886)119879

(119910) = 119904 and 119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119910) = 119908which means that (119865

119886)119879(119909) = (119865

119886)119879(119910) = 119904 Therefore

min(119865119886)119879(119909) (119865

119886)119879(119910) = min119904 119904 = 119904 and so inf(119865

119886)119879(119911)



Now let 119909 119911 isin 119879 Then 119905(119865119886)119879

(119909) = 119905(119865119886)119879

(119911) = 119904 and119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119911) = 119908 and so (119865119886)119879(119909) = (119865

119886)119879(119911) =



and min(119865119886)119879(119909) (119865

119886)119879(119911) le (119865

119886)119879(119910) Thus we obtain

min(119865119886)119879(119909) (119865





(119865119886)119879(119909) = 119905

(119865119886)119879(119910) =

119904 and 119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119910) = 119908 and thus (119865119886)119879(119909) =


119886is a


119886)119879(119909) (119865

119886)119879(119910) = min119904 119904 = 119904 and
















119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and


min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911) 119911 isin 119909 ∘ 119910 (resp

119905119865119886(119909) le inf119905

119865119886(119911) 119911 isin 119909 ∘ 119910) and 1 minus 119891

119865119886(119910) le


119865119886(119909) le


119886(119910) le inf119865

119886(119911)

119911 isin 119909 ∘ 119910 (resp 119865119886(119909) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910)



119886is a vague


119886


condition 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 (resp 119865

119886(119909) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910)



119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911)


119865119886(119910) le


119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910










max 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= max 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le max 119905119865120572(119909) 119905119865120572(119910) capmax 119905

119866120573(119909) 119905119866120573

(119910)

le inf 119905119865120572(119911) 119911 isin 119909 ∘ 119910

cap inf 119905119866120573

(119911) 119911 isin 119909 ∘ 119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 ∘ 119910

= inf 119905(120572120573)

119911 isin 119909 ∘ 119910

(21)


(120572120573)(119909) 1 minus119891

(120572120573)(119910) le inf1 minus 119891

(120572120573) 119911 isin 119909 ∘ 119910


(119909) (120572120573)

(119910) le

inf(120572120573)







Now let 119909 119910 isin (119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and

1 minus 119891119865119886(119909) 1 minus 119891


119886(119909) ge 119905 and


119886is a vague




119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(22)



Then 119905119865119886(119910) ge 119905 and


119886(119910) ge 119905 Since 119865

119886is a



119905119865119886(119910) le inf 119905

119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119911) 119911 isin 119909 ∘ 119910

(23)




119886(119910) ge 119905 This proves

that 119911 isin (119865119886)(119905119905)


119886)(119905119905)


weobtain 119877 ∘ (119865

119886)(119905119905)

sube (119865119886)(119905119905)



(119905119905)is a soft left







(lArr) Let (119865 119860)(119905119905)


119886)(119905119905)


119886)(119905119905)





119886)(119905119905)


119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(24)


or 119909 isin (119865119886)(119905119905)


119886)(119905119905)

and 119910 ∘ 119909 sube (119865119886)(119905119905)


119886)(119905119905)


119886)(119905119905)

Then119865119886(119909) ge 119905 and 119865


119886(119911) 119911 isin


119886(119911) 119911 isin 119909 ∘ 119910 ge 119865

119886(119910) ge 119905


119886(119911) 119911 isin 119909 ∘ 119910 and 119865

119886(119910) le


max119865119886(119909) 119865119886(119910) le inf119865








(120572120573)is a



to prove that 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

(resp (119865 119860)(120572120573)

∘

119877 sube (119865 119860)(120572120573)


Then119905119865119886(119910) ge 120572 and 1 minus 119891

119865119886(119910) ge 120573 Since 119865



119865119886(119910) le


119865119886(119910) le inf1 minus 119891

119865119886(119911) 119911 isin


119865119886(119910) ge 120572


119865119886(119910) ge 120573 which




119910 isin (119865 119860)(120572120573)

we obtain 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

Hence(119865 119860)


(120572120573)


119865119886(119909) ge 120573 Similarly


119865119886(119909) ge 120572 and


119865119886(119909) ge 120573 This implies

that 119911 isin (119865 119860)(120572120573)



(120572120573)and119910 isin 119877

we obtain (119865 119860)(120572120573)

∘ 119877 sube (119865 119860)(120572120573)

which means (119865 119860)(120572120573)









119886(119909) 119865119886(119910) le inf119865

119886(119911)


119865119886(119910) = 119904 which means that 119865

119886(119910) = 119904 Since



119865119886(119910) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

119865119886 (119909) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

(25)



119886(119910) = 119904 which means




119865119886= 119904 which means that 119865

119886(119909) = 119904 Similarly


119886(119909) = 119904 which means










119891(119909) +2 119891(119910) and 119891(119909 ∘1 119910) sube 119891(119909) ∘2 119891(119910)






(120593 120595) (119865 119860) = (120593 (119865) 120595 (119860)) (26)

where

120593 (119865119886) (120593 (119909)) = (120593 (119865))

120595(119886)

(119910) (27)



(120593 120595)minus1(119866 119861) = (120593

minus1(119866) 120595

minus1(119861)) (28)

where

120593minus1(119866119887) (120593minus1(119910)) = (120593

minus1(119866))120595minus1(119887)

(119909) (29)






1and 119877








Supp (120593 (119865) 120595 (119860)) sube 120595 (Supp (119865 119860)) (30)



min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119906) 119906 isin1199081 +2 1199082

= inf 120593 (119865119886) (120593 (119911))

(31)





min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119887) 1199062 isin 1199061 +2 119887

= inf 120593 (119865119886) (120593 (119910))

(32)


min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119888) 1199062 isin 119888 +2 1199061

= inf 120593 (119865119886) (120593 (119911))

(33)






min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119901) 119901 isin1199081 ∘2 1199082

= inf 120593 (119865119886) (120593 (119911))

(34)





119886) is




minus1(119866) 120595

minus1(119861))






6 Conclusion




Acknowledgments


References





















119909 ∘ (119910 + 119911) cap ((119909 ∘ 119910) + (119909 ∘ 119911)) = 120601

(119909 + 119910) ∘ 119911 cap ((119909 ∘ 119911) + (119910 ∘ 119911)) = 120601

(10)













119865119886(119911) 119911 isin


119865119886(119910) le inf1 minus 119891

119865119886(119911) 119911 isin

119909 ∘ 119910 that is min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910




119865119886(119911)


119865119886(119910) le inf1minus

119891119865119886(119911) 119911 isin 119909 ∘ 119910 that is min119865

119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)




Supp(119865 119860)



119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911)


119865119886(119910) le


119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910



119886is a vague









(119865 119860)+

= (119865119886)+

= 119909 isin119877 119905(119865119886)+ (119909) = 119905

119865119886(119909) + 1

minus 119905119865119886(119890) 1 minus 119891

(119865119886)+ (119909) = 1 minus 119891

119865119886(119909)

+119891119865119886(119890) ie (119865119886)

+

(119909) = 119865119886 (119909) + 1minus119865119886 (119890)

(11)





givenby

(119865 119860)120572= 119909 isin119867 119865

119886 (119909) ge 120572 (12)






119905(120572120573)

(119909) = min 119905119865120572(119909) 119905119866120573

(119909)

1minus119891(120572120573)

(119909) = min 1minus119891119865120572(119909) 1minus119891

119866120573(119909)

(13)

that is

(120572120573) (119909) = 119865

120572 (119909) cap119866120573 (119909) (14)





(120572120573)given by 119905


(120572120573)can


obtain

min 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= min 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le min 119905119865120572(119909) cap 119905

119865120572(119910) 119905

119866120573(119909) cap 119905

119866120573(119910)

le min 119905119865120572(119909) 119905119865120572(119910) capmin 119905

119866120573(119909) 119905119866120573

(119910)

le inf 119905119865120572(119911) 119911 isin 119909 +119910

cap inf 119905119866120573

(119911) 119911 isin 119909 +119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 +119910

= inf 119905(120572120573)

119911 isin 119909 +119910

(15)


min 119905(120572120573)

(119908) 119905(120572120573)

(119909)

= min 119905119865120572(119908) cap 119905

119866120573(119908) 119905

119865120572(119909) cap 119905

119866120573(119909)

le min 119905119865120572(119908) cap 119905

119865120572(119909) 119905119866120573

(119908) cap 119905119866120573

(119909)

le min 119905119865120572(119908) 119905

119865120572(119909) capmin 119905

119866120573(119908) 119905

119866120573(119909)

le 119905119865120572(119910) cap 119905

119866120573(119910) = 119905

(120572120573)(119910)

(16)


(120572120573)(119908) 119905(120572120573)

(119909) le 119905(120572120573)

(119911) where 119911 isin 119877




min 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= min 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le min 119905119865120572(119909) cap 119905

119865120572(119910) 119905

119866120573(119909) cap 119905

119866120573(119910)

le min 119905119865120572(119909) 119905119865120572(119910) capmin 119905

119866120573(119909) 119905119866120573

(119910)


le inf 119905119865120572(119911) 119911 isin 119909 ∘ 119910

cap inf 119905119866120573

(119911) 119911 isin 119909 ∘ 119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 ∘ 119910

= inf 119905(120572120573)

119911 isin 119909 ∘ 119910

(17)





[0 1] (119865 119860)(119905119905)




of 119877 Now let 119909 119910 isin (119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and

1 minus 119891119865119886(119909) 1 minus 119891


119886(119909) ge 119905 and




119886(119909) 119865119886(119910) ge 119905



119886)(119905119905)


we obtain 119911 + (119865

119886)(119905119905)

sube (119865119886)(119905119905)

Now let 119909 119911 isin (119865119886)(119905119905)

Then 119905

119865119886(119909) 119905119865119886(119911) ge 119905 and 1 minus 119891

119865119886(119909) 1 minus 119891

119865119886(119911) ge 119905 which

means that 119865119886(119909) ge 119905 and 119865




119886(119910) Since

119909 119911 isin (119865119886)(119905119905)



119886)(119905119905)


sube 119911+(119865119886)(119905119905)


119886)(119905119905)

= (119865119886)(119905119905)

As a result (119865119886)(119905119905)





119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 Now let 119909 119910 isin

(119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and 1 minus 119891

119865119886(119909) 1 minus 119891

119865119886(119910) ge


119886(119910) ge 119905 Therefore



119911 isin (119865119886)(119905119905)



119886)(119905119905)


119886)(119905119905)



(lArr)Assume that (119865 119860)(119905119905)


119886)(119905119905)


119886(119909) 119865119886(119910) le 119865

119886(119909)

ormin119865119886(119909) 119865119886(119910) le 119865

119886(119910) So if we letmin119865

119886(119909) 119865119886(119910) =

1199050 then 119909 119910 isin (119865119886)(1199050 1199050)



119886(119911) ge 1199050 which


119886(119911) 119911 isin 119909 +


119886(119908) 119865

119886(119909) = 1199051 This implies

that 119908 119909 isin (119865119886)(1199051 1199051)


(119865119886)(1199051 1199051)


wehave 119905

119865119886(119910) ge 1199051 and 1 minus 119905


119865119886(119910) ge 1199051 and thus min119865

119886(119908) 119865

119886(119909) = 1199051 le 119865

119886(119910)


119886(119909) le 119865

119886(119910) Therefore



Now (119865119886)(119905119905)



119909 119910 isin (119865119886)(1199052 1199052)



119886(119911) ge 1199052 As a result


119886(119911) 119911 isin 119909 ∘ 119910 which








119886is a vague


Then 119905119865119886(119909) ge 120572

1 minus 119891119865119886(119909) ge 120573 and 119905

119865119886(119910) ge 120572 1 minus 119891

119865119886(119910) ge 120573 Thus the


min 119905119865119886(119909) 119905119865119886(119910) ge min 120572 120572 = 120572

min 1 minus 119891119865119886(119909) 1minus119891

119865119886(119910) ge min 120573 120573 = 120573

(18)



119865119886(119911) 119911 isin 119909+119910 ge 120573


andthus 119909 + 119910 sube (119865 119860)


119911 + (119865 119860)(120572120573)

sube (119865 119860)(120572120573)



and min119865119886(119909) 119865119886(119911) le 119865

119886(119910) Now let 119909 119911 isin (119865 119860)

(120572120573)


min 119905119865119886(119909) 119905119865119886(119911) ge min 120572 120572 = 120572

min 1 minus 119891119865119886(119909) 1minus119891

119865119886(119910) ge min 120573 120573 = 120573

(19)


119865119886(119910) ge 120573 which implies

that119910 isin (119865 119860)(120572120573)


we have


119909 isin 119911 + 119910 and 119910 isin (119865 119860)(120572120573)


(120572120573)sube 119911 + (119865 119860)


any 119911 isin (119865 119860)(120572120573)

119911 + (119865 119860)(120572120573)

= (119865 119860)(120572120573)





(120572120573) Then min119905

119865119886(119909) 119905119865119886(119910) ge min120572 120572 = 120572


119865119886(119910) ge min120573 120573 = 120573 and this

results in

inf 119905119865119886(119911) 119911 isin 119909 ∘ 119910 ge min 119905

119865119886(119909) 119905119865119886(119910) ge 120572

inf 1 minus 119891119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119910) ge 120573

(20)




(120572120573)


(120572120573)) As such (119865 119860)

(120572120573)is










119886



(119909) = 119905(119865119886)119879

(119910) = 119904 and 119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119910) = 119908which means that (119865

119886)119879(119909) = (119865

119886)119879(119910) = 119904 Therefore

min(119865119886)119879(119909) (119865

119886)119879(119910) = min119904 119904 = 119904 and so inf(119865

119886)119879(119911)



Now let 119909 119911 isin 119879 Then 119905(119865119886)119879

(119909) = 119905(119865119886)119879

(119911) = 119904 and119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119911) = 119908 and so (119865119886)119879(119909) = (119865

119886)119879(119911) =



and min(119865119886)119879(119909) (119865

119886)119879(119911) le (119865

119886)119879(119910) Thus we obtain

min(119865119886)119879(119909) (119865





(119865119886)119879(119909) = 119905

(119865119886)119879(119910) =

119904 and 119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119910) = 119908 and thus (119865119886)119879(119909) =


119886is a


119886)119879(119909) (119865

119886)119879(119910) = min119904 119904 = 119904 and
















119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and


min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911) 119911 isin 119909 ∘ 119910 (resp

119905119865119886(119909) le inf119905

119865119886(119911) 119911 isin 119909 ∘ 119910) and 1 minus 119891

119865119886(119910) le


119865119886(119909) le


119886(119910) le inf119865

119886(119911)

119911 isin 119909 ∘ 119910 (resp 119865119886(119909) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910)



119886is a vague


119886


condition 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 (resp 119865

119886(119909) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910)



119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911)


119865119886(119910) le


119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910










max 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= max 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le max 119905119865120572(119909) 119905119865120572(119910) capmax 119905

119866120573(119909) 119905119866120573

(119910)

le inf 119905119865120572(119911) 119911 isin 119909 ∘ 119910

cap inf 119905119866120573

(119911) 119911 isin 119909 ∘ 119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 ∘ 119910

= inf 119905(120572120573)

119911 isin 119909 ∘ 119910

(21)


(120572120573)(119909) 1 minus119891

(120572120573)(119910) le inf1 minus 119891

(120572120573) 119911 isin 119909 ∘ 119910


(119909) (120572120573)

(119910) le

inf(120572120573)







Now let 119909 119910 isin (119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and

1 minus 119891119865119886(119909) 1 minus 119891


119886(119909) ge 119905 and


119886is a vague




119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(22)



Then 119905119865119886(119910) ge 119905 and


119886(119910) ge 119905 Since 119865

119886is a



119905119865119886(119910) le inf 119905

119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119911) 119911 isin 119909 ∘ 119910

(23)




119886(119910) ge 119905 This proves

that 119911 isin (119865119886)(119905119905)


119886)(119905119905)


weobtain 119877 ∘ (119865

119886)(119905119905)

sube (119865119886)(119905119905)



(119905119905)is a soft left







(lArr) Let (119865 119860)(119905119905)


119886)(119905119905)


119886)(119905119905)





119886)(119905119905)


119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(24)


or 119909 isin (119865119886)(119905119905)


119886)(119905119905)

and 119910 ∘ 119909 sube (119865119886)(119905119905)


119886)(119905119905)


119886)(119905119905)

Then119865119886(119909) ge 119905 and 119865


119886(119911) 119911 isin


119886(119911) 119911 isin 119909 ∘ 119910 ge 119865

119886(119910) ge 119905


119886(119911) 119911 isin 119909 ∘ 119910 and 119865

119886(119910) le


max119865119886(119909) 119865119886(119910) le inf119865








(120572120573)is a



to prove that 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

(resp (119865 119860)(120572120573)

∘

119877 sube (119865 119860)(120572120573)


Then119905119865119886(119910) ge 120572 and 1 minus 119891

119865119886(119910) ge 120573 Since 119865



119865119886(119910) le


119865119886(119910) le inf1 minus 119891

119865119886(119911) 119911 isin


119865119886(119910) ge 120572


119865119886(119910) ge 120573 which




119910 isin (119865 119860)(120572120573)

we obtain 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

Hence(119865 119860)


(120572120573)


119865119886(119909) ge 120573 Similarly


119865119886(119909) ge 120572 and


119865119886(119909) ge 120573 This implies

that 119911 isin (119865 119860)(120572120573)



(120572120573)and119910 isin 119877

we obtain (119865 119860)(120572120573)

∘ 119877 sube (119865 119860)(120572120573)

which means (119865 119860)(120572120573)









119886(119909) 119865119886(119910) le inf119865

119886(119911)


119865119886(119910) = 119904 which means that 119865

119886(119910) = 119904 Since



119865119886(119910) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

119865119886 (119909) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

(25)



119886(119910) = 119904 which means




119865119886= 119904 which means that 119865

119886(119909) = 119904 Similarly


119886(119909) = 119904 which means










119891(119909) +2 119891(119910) and 119891(119909 ∘1 119910) sube 119891(119909) ∘2 119891(119910)






(120593 120595) (119865 119860) = (120593 (119865) 120595 (119860)) (26)

where

120593 (119865119886) (120593 (119909)) = (120593 (119865))

120595(119886)

(119910) (27)



(120593 120595)minus1(119866 119861) = (120593

minus1(119866) 120595

minus1(119861)) (28)

where

120593minus1(119866119887) (120593minus1(119910)) = (120593

minus1(119866))120595minus1(119887)

(119909) (29)






1and 119877








Supp (120593 (119865) 120595 (119860)) sube 120595 (Supp (119865 119860)) (30)



min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119906) 119906 isin1199081 +2 1199082

= inf 120593 (119865119886) (120593 (119911))

(31)





min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119887) 1199062 isin 1199061 +2 119887

= inf 120593 (119865119886) (120593 (119910))

(32)


min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119888) 1199062 isin 119888 +2 1199061

= inf 120593 (119865119886) (120593 (119911))

(33)






min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119901) 119901 isin1199081 ∘2 1199082

= inf 120593 (119865119886) (120593 (119911))

(34)





119886) is




minus1(119866) 120595

minus1(119861))






6 Conclusion




Acknowledgments


References





















(119865 119860)+

= (119865119886)+

= 119909 isin119877 119905(119865119886)+ (119909) = 119905

119865119886(119909) + 1

minus 119905119865119886(119890) 1 minus 119891

(119865119886)+ (119909) = 1 minus 119891

119865119886(119909)

+119891119865119886(119890) ie (119865119886)

+

(119909) = 119865119886 (119909) + 1minus119865119886 (119890)

(11)





givenby

(119865 119860)120572= 119909 isin119867 119865

119886 (119909) ge 120572 (12)






119905(120572120573)

(119909) = min 119905119865120572(119909) 119905119866120573

(119909)

1minus119891(120572120573)

(119909) = min 1minus119891119865120572(119909) 1minus119891

119866120573(119909)

(13)

that is

(120572120573) (119909) = 119865

120572 (119909) cap119866120573 (119909) (14)





(120572120573)given by 119905


(120572120573)can


obtain

min 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= min 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le min 119905119865120572(119909) cap 119905

119865120572(119910) 119905

119866120573(119909) cap 119905

119866120573(119910)

le min 119905119865120572(119909) 119905119865120572(119910) capmin 119905

119866120573(119909) 119905119866120573

(119910)

le inf 119905119865120572(119911) 119911 isin 119909 +119910

cap inf 119905119866120573

(119911) 119911 isin 119909 +119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 +119910

= inf 119905(120572120573)

119911 isin 119909 +119910

(15)


min 119905(120572120573)

(119908) 119905(120572120573)

(119909)

= min 119905119865120572(119908) cap 119905

119866120573(119908) 119905

119865120572(119909) cap 119905

119866120573(119909)

le min 119905119865120572(119908) cap 119905

119865120572(119909) 119905119866120573

(119908) cap 119905119866120573

(119909)

le min 119905119865120572(119908) 119905

119865120572(119909) capmin 119905

119866120573(119908) 119905

119866120573(119909)

le 119905119865120572(119910) cap 119905

119866120573(119910) = 119905

(120572120573)(119910)

(16)


(120572120573)(119908) 119905(120572120573)

(119909) le 119905(120572120573)

(119911) where 119911 isin 119877




min 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= min 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le min 119905119865120572(119909) cap 119905

119865120572(119910) 119905

119866120573(119909) cap 119905

119866120573(119910)

le min 119905119865120572(119909) 119905119865120572(119910) capmin 119905

119866120573(119909) 119905119866120573

(119910)


le inf 119905119865120572(119911) 119911 isin 119909 ∘ 119910

cap inf 119905119866120573

(119911) 119911 isin 119909 ∘ 119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 ∘ 119910

= inf 119905(120572120573)

119911 isin 119909 ∘ 119910

(17)





[0 1] (119865 119860)(119905119905)




of 119877 Now let 119909 119910 isin (119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and

1 minus 119891119865119886(119909) 1 minus 119891


119886(119909) ge 119905 and




119886(119909) 119865119886(119910) ge 119905



119886)(119905119905)


we obtain 119911 + (119865

119886)(119905119905)

sube (119865119886)(119905119905)

Now let 119909 119911 isin (119865119886)(119905119905)

Then 119905

119865119886(119909) 119905119865119886(119911) ge 119905 and 1 minus 119891

119865119886(119909) 1 minus 119891

119865119886(119911) ge 119905 which

means that 119865119886(119909) ge 119905 and 119865




119886(119910) Since

119909 119911 isin (119865119886)(119905119905)



119886)(119905119905)


sube 119911+(119865119886)(119905119905)


119886)(119905119905)

= (119865119886)(119905119905)

As a result (119865119886)(119905119905)





119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 Now let 119909 119910 isin

(119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and 1 minus 119891

119865119886(119909) 1 minus 119891

119865119886(119910) ge


119886(119910) ge 119905 Therefore



119911 isin (119865119886)(119905119905)



119886)(119905119905)


119886)(119905119905)



(lArr)Assume that (119865 119860)(119905119905)


119886)(119905119905)


119886(119909) 119865119886(119910) le 119865

119886(119909)

ormin119865119886(119909) 119865119886(119910) le 119865

119886(119910) So if we letmin119865

119886(119909) 119865119886(119910) =

1199050 then 119909 119910 isin (119865119886)(1199050 1199050)



119886(119911) ge 1199050 which


119886(119911) 119911 isin 119909 +


119886(119908) 119865

119886(119909) = 1199051 This implies

that 119908 119909 isin (119865119886)(1199051 1199051)


(119865119886)(1199051 1199051)


wehave 119905

119865119886(119910) ge 1199051 and 1 minus 119905


119865119886(119910) ge 1199051 and thus min119865

119886(119908) 119865

119886(119909) = 1199051 le 119865

119886(119910)


119886(119909) le 119865

119886(119910) Therefore



Now (119865119886)(119905119905)



119909 119910 isin (119865119886)(1199052 1199052)



119886(119911) ge 1199052 As a result


119886(119911) 119911 isin 119909 ∘ 119910 which








119886is a vague


Then 119905119865119886(119909) ge 120572

1 minus 119891119865119886(119909) ge 120573 and 119905

119865119886(119910) ge 120572 1 minus 119891

119865119886(119910) ge 120573 Thus the


min 119905119865119886(119909) 119905119865119886(119910) ge min 120572 120572 = 120572

min 1 minus 119891119865119886(119909) 1minus119891

119865119886(119910) ge min 120573 120573 = 120573

(18)



119865119886(119911) 119911 isin 119909+119910 ge 120573


andthus 119909 + 119910 sube (119865 119860)


119911 + (119865 119860)(120572120573)

sube (119865 119860)(120572120573)



and min119865119886(119909) 119865119886(119911) le 119865

119886(119910) Now let 119909 119911 isin (119865 119860)

(120572120573)


min 119905119865119886(119909) 119905119865119886(119911) ge min 120572 120572 = 120572

min 1 minus 119891119865119886(119909) 1minus119891

119865119886(119910) ge min 120573 120573 = 120573

(19)


119865119886(119910) ge 120573 which implies

that119910 isin (119865 119860)(120572120573)


we have


119909 isin 119911 + 119910 and 119910 isin (119865 119860)(120572120573)


(120572120573)sube 119911 + (119865 119860)


any 119911 isin (119865 119860)(120572120573)

119911 + (119865 119860)(120572120573)

= (119865 119860)(120572120573)





(120572120573) Then min119905

119865119886(119909) 119905119865119886(119910) ge min120572 120572 = 120572


119865119886(119910) ge min120573 120573 = 120573 and this

results in

inf 119905119865119886(119911) 119911 isin 119909 ∘ 119910 ge min 119905

119865119886(119909) 119905119865119886(119910) ge 120572

inf 1 minus 119891119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119910) ge 120573

(20)




(120572120573)


(120572120573)) As such (119865 119860)

(120572120573)is










119886



(119909) = 119905(119865119886)119879

(119910) = 119904 and 119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119910) = 119908which means that (119865

119886)119879(119909) = (119865

119886)119879(119910) = 119904 Therefore

min(119865119886)119879(119909) (119865

119886)119879(119910) = min119904 119904 = 119904 and so inf(119865

119886)119879(119911)



Now let 119909 119911 isin 119879 Then 119905(119865119886)119879

(119909) = 119905(119865119886)119879

(119911) = 119904 and119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119911) = 119908 and so (119865119886)119879(119909) = (119865

119886)119879(119911) =



and min(119865119886)119879(119909) (119865

119886)119879(119911) le (119865

119886)119879(119910) Thus we obtain

min(119865119886)119879(119909) (119865





(119865119886)119879(119909) = 119905

(119865119886)119879(119910) =

119904 and 119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119910) = 119908 and thus (119865119886)119879(119909) =


119886is a


119886)119879(119909) (119865

119886)119879(119910) = min119904 119904 = 119904 and
















119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and


min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911) 119911 isin 119909 ∘ 119910 (resp

119905119865119886(119909) le inf119905

119865119886(119911) 119911 isin 119909 ∘ 119910) and 1 minus 119891

119865119886(119910) le


119865119886(119909) le


119886(119910) le inf119865

119886(119911)

119911 isin 119909 ∘ 119910 (resp 119865119886(119909) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910)



119886is a vague


119886


condition 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 (resp 119865

119886(119909) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910)



119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911)


119865119886(119910) le


119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910










max 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= max 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le max 119905119865120572(119909) 119905119865120572(119910) capmax 119905

119866120573(119909) 119905119866120573

(119910)

le inf 119905119865120572(119911) 119911 isin 119909 ∘ 119910

cap inf 119905119866120573

(119911) 119911 isin 119909 ∘ 119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 ∘ 119910

= inf 119905(120572120573)

119911 isin 119909 ∘ 119910

(21)


(120572120573)(119909) 1 minus119891

(120572120573)(119910) le inf1 minus 119891

(120572120573) 119911 isin 119909 ∘ 119910


(119909) (120572120573)

(119910) le

inf(120572120573)







Now let 119909 119910 isin (119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and

1 minus 119891119865119886(119909) 1 minus 119891


119886(119909) ge 119905 and


119886is a vague




119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(22)



Then 119905119865119886(119910) ge 119905 and


119886(119910) ge 119905 Since 119865

119886is a



119905119865119886(119910) le inf 119905

119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119911) 119911 isin 119909 ∘ 119910

(23)




119886(119910) ge 119905 This proves

that 119911 isin (119865119886)(119905119905)


119886)(119905119905)


weobtain 119877 ∘ (119865

119886)(119905119905)

sube (119865119886)(119905119905)



(119905119905)is a soft left







(lArr) Let (119865 119860)(119905119905)


119886)(119905119905)


119886)(119905119905)





119886)(119905119905)


119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(24)


or 119909 isin (119865119886)(119905119905)


119886)(119905119905)

and 119910 ∘ 119909 sube (119865119886)(119905119905)


119886)(119905119905)


119886)(119905119905)

Then119865119886(119909) ge 119905 and 119865


119886(119911) 119911 isin


119886(119911) 119911 isin 119909 ∘ 119910 ge 119865

119886(119910) ge 119905


119886(119911) 119911 isin 119909 ∘ 119910 and 119865

119886(119910) le


max119865119886(119909) 119865119886(119910) le inf119865








(120572120573)is a



to prove that 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

(resp (119865 119860)(120572120573)

∘

119877 sube (119865 119860)(120572120573)


Then119905119865119886(119910) ge 120572 and 1 minus 119891

119865119886(119910) ge 120573 Since 119865



119865119886(119910) le


119865119886(119910) le inf1 minus 119891

119865119886(119911) 119911 isin


119865119886(119910) ge 120572


119865119886(119910) ge 120573 which




119910 isin (119865 119860)(120572120573)

we obtain 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

Hence(119865 119860)


(120572120573)


119865119886(119909) ge 120573 Similarly


119865119886(119909) ge 120572 and


119865119886(119909) ge 120573 This implies

that 119911 isin (119865 119860)(120572120573)



(120572120573)and119910 isin 119877

we obtain (119865 119860)(120572120573)

∘ 119877 sube (119865 119860)(120572120573)

which means (119865 119860)(120572120573)









119886(119909) 119865119886(119910) le inf119865

119886(119911)


119865119886(119910) = 119904 which means that 119865

119886(119910) = 119904 Since



119865119886(119910) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

119865119886 (119909) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

(25)



119886(119910) = 119904 which means




119865119886= 119904 which means that 119865

119886(119909) = 119904 Similarly


119886(119909) = 119904 which means










119891(119909) +2 119891(119910) and 119891(119909 ∘1 119910) sube 119891(119909) ∘2 119891(119910)






(120593 120595) (119865 119860) = (120593 (119865) 120595 (119860)) (26)

where

120593 (119865119886) (120593 (119909)) = (120593 (119865))

120595(119886)

(119910) (27)



(120593 120595)minus1(119866 119861) = (120593

minus1(119866) 120595

minus1(119861)) (28)

where

120593minus1(119866119887) (120593minus1(119910)) = (120593

minus1(119866))120595minus1(119887)

(119909) (29)






1and 119877








Supp (120593 (119865) 120595 (119860)) sube 120595 (Supp (119865 119860)) (30)



min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119906) 119906 isin1199081 +2 1199082

= inf 120593 (119865119886) (120593 (119911))

(31)





min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119887) 1199062 isin 1199061 +2 119887

= inf 120593 (119865119886) (120593 (119910))

(32)


min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119888) 1199062 isin 119888 +2 1199061

= inf 120593 (119865119886) (120593 (119911))

(33)






min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119901) 119901 isin1199081 ∘2 1199082

= inf 120593 (119865119886) (120593 (119911))

(34)





119886) is




minus1(119866) 120595

minus1(119861))






6 Conclusion




Acknowledgments


References

















le inf 119905119865120572(119911) 119911 isin 119909 ∘ 119910

cap inf 119905119866120573

(119911) 119911 isin 119909 ∘ 119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 ∘ 119910

= inf 119905(120572120573)

119911 isin 119909 ∘ 119910

(17)





[0 1] (119865 119860)(119905119905)




of 119877 Now let 119909 119910 isin (119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and

1 minus 119891119865119886(119909) 1 minus 119891


119886(119909) ge 119905 and




119886(119909) 119865119886(119910) ge 119905



119886)(119905119905)


we obtain 119911 + (119865

119886)(119905119905)

sube (119865119886)(119905119905)

Now let 119909 119911 isin (119865119886)(119905119905)

Then 119905

119865119886(119909) 119905119865119886(119911) ge 119905 and 1 minus 119891

119865119886(119909) 1 minus 119891

119865119886(119911) ge 119905 which

means that 119865119886(119909) ge 119905 and 119865




119886(119910) Since

119909 119911 isin (119865119886)(119905119905)



119886)(119905119905)


sube 119911+(119865119886)(119905119905)


119886)(119905119905)

= (119865119886)(119905119905)

As a result (119865119886)(119905119905)





119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 Now let 119909 119910 isin

(119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and 1 minus 119891

119865119886(119909) 1 minus 119891

119865119886(119910) ge


119886(119910) ge 119905 Therefore



119911 isin (119865119886)(119905119905)



119886)(119905119905)


119886)(119905119905)



(lArr)Assume that (119865 119860)(119905119905)


119886)(119905119905)


119886(119909) 119865119886(119910) le 119865

119886(119909)

ormin119865119886(119909) 119865119886(119910) le 119865

119886(119910) So if we letmin119865

119886(119909) 119865119886(119910) =

1199050 then 119909 119910 isin (119865119886)(1199050 1199050)



119886(119911) ge 1199050 which


119886(119911) 119911 isin 119909 +


119886(119908) 119865

119886(119909) = 1199051 This implies

that 119908 119909 isin (119865119886)(1199051 1199051)


(119865119886)(1199051 1199051)


wehave 119905

119865119886(119910) ge 1199051 and 1 minus 119905


119865119886(119910) ge 1199051 and thus min119865

119886(119908) 119865

119886(119909) = 1199051 le 119865

119886(119910)


119886(119909) le 119865

119886(119910) Therefore



Now (119865119886)(119905119905)



119909 119910 isin (119865119886)(1199052 1199052)



119886(119911) ge 1199052 As a result


119886(119911) 119911 isin 119909 ∘ 119910 which








119886is a vague


Then 119905119865119886(119909) ge 120572

1 minus 119891119865119886(119909) ge 120573 and 119905

119865119886(119910) ge 120572 1 minus 119891

119865119886(119910) ge 120573 Thus the


min 119905119865119886(119909) 119905119865119886(119910) ge min 120572 120572 = 120572

min 1 minus 119891119865119886(119909) 1minus119891

119865119886(119910) ge min 120573 120573 = 120573

(18)



119865119886(119911) 119911 isin 119909+119910 ge 120573


andthus 119909 + 119910 sube (119865 119860)


119911 + (119865 119860)(120572120573)

sube (119865 119860)(120572120573)



and min119865119886(119909) 119865119886(119911) le 119865

119886(119910) Now let 119909 119911 isin (119865 119860)

(120572120573)


min 119905119865119886(119909) 119905119865119886(119911) ge min 120572 120572 = 120572

min 1 minus 119891119865119886(119909) 1minus119891

119865119886(119910) ge min 120573 120573 = 120573

(19)


119865119886(119910) ge 120573 which implies

that119910 isin (119865 119860)(120572120573)


we have


119909 isin 119911 + 119910 and 119910 isin (119865 119860)(120572120573)


(120572120573)sube 119911 + (119865 119860)


any 119911 isin (119865 119860)(120572120573)

119911 + (119865 119860)(120572120573)

= (119865 119860)(120572120573)





(120572120573) Then min119905

119865119886(119909) 119905119865119886(119910) ge min120572 120572 = 120572


119865119886(119910) ge min120573 120573 = 120573 and this

results in

inf 119905119865119886(119911) 119911 isin 119909 ∘ 119910 ge min 119905

119865119886(119909) 119905119865119886(119910) ge 120572

inf 1 minus 119891119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119910) ge 120573

(20)




(120572120573)


(120572120573)) As such (119865 119860)

(120572120573)is










119886



(119909) = 119905(119865119886)119879

(119910) = 119904 and 119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119910) = 119908which means that (119865

119886)119879(119909) = (119865

119886)119879(119910) = 119904 Therefore

min(119865119886)119879(119909) (119865

119886)119879(119910) = min119904 119904 = 119904 and so inf(119865

119886)119879(119911)



Now let 119909 119911 isin 119879 Then 119905(119865119886)119879

(119909) = 119905(119865119886)119879

(119911) = 119904 and119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119911) = 119908 and so (119865119886)119879(119909) = (119865

119886)119879(119911) =



and min(119865119886)119879(119909) (119865

119886)119879(119911) le (119865

119886)119879(119910) Thus we obtain

min(119865119886)119879(119909) (119865





(119865119886)119879(119909) = 119905

(119865119886)119879(119910) =

119904 and 119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119910) = 119908 and thus (119865119886)119879(119909) =


119886is a


119886)119879(119909) (119865

119886)119879(119910) = min119904 119904 = 119904 and
















119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and


min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911) 119911 isin 119909 ∘ 119910 (resp

119905119865119886(119909) le inf119905

119865119886(119911) 119911 isin 119909 ∘ 119910) and 1 minus 119891

119865119886(119910) le


119865119886(119909) le


119886(119910) le inf119865

119886(119911)

119911 isin 119909 ∘ 119910 (resp 119865119886(119909) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910)



119886is a vague


119886


condition 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 (resp 119865

119886(119909) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910)



119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911)


119865119886(119910) le


119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910










max 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= max 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le max 119905119865120572(119909) 119905119865120572(119910) capmax 119905

119866120573(119909) 119905119866120573

(119910)

le inf 119905119865120572(119911) 119911 isin 119909 ∘ 119910

cap inf 119905119866120573

(119911) 119911 isin 119909 ∘ 119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 ∘ 119910

= inf 119905(120572120573)

119911 isin 119909 ∘ 119910

(21)


(120572120573)(119909) 1 minus119891

(120572120573)(119910) le inf1 minus 119891

(120572120573) 119911 isin 119909 ∘ 119910


(119909) (120572120573)

(119910) le

inf(120572120573)







Now let 119909 119910 isin (119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and

1 minus 119891119865119886(119909) 1 minus 119891


119886(119909) ge 119905 and


119886is a vague




119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(22)



Then 119905119865119886(119910) ge 119905 and


119886(119910) ge 119905 Since 119865

119886is a



119905119865119886(119910) le inf 119905

119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119911) 119911 isin 119909 ∘ 119910

(23)




119886(119910) ge 119905 This proves

that 119911 isin (119865119886)(119905119905)


119886)(119905119905)


weobtain 119877 ∘ (119865

119886)(119905119905)

sube (119865119886)(119905119905)



(119905119905)is a soft left







(lArr) Let (119865 119860)(119905119905)


119886)(119905119905)


119886)(119905119905)





119886)(119905119905)


119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(24)


or 119909 isin (119865119886)(119905119905)


119886)(119905119905)

and 119910 ∘ 119909 sube (119865119886)(119905119905)


119886)(119905119905)


119886)(119905119905)

Then119865119886(119909) ge 119905 and 119865


119886(119911) 119911 isin


119886(119911) 119911 isin 119909 ∘ 119910 ge 119865

119886(119910) ge 119905


119886(119911) 119911 isin 119909 ∘ 119910 and 119865

119886(119910) le


max119865119886(119909) 119865119886(119910) le inf119865








(120572120573)is a



to prove that 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

(resp (119865 119860)(120572120573)

∘

119877 sube (119865 119860)(120572120573)


Then119905119865119886(119910) ge 120572 and 1 minus 119891

119865119886(119910) ge 120573 Since 119865



119865119886(119910) le


119865119886(119910) le inf1 minus 119891

119865119886(119911) 119911 isin


119865119886(119910) ge 120572


119865119886(119910) ge 120573 which




119910 isin (119865 119860)(120572120573)

we obtain 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

Hence(119865 119860)


(120572120573)


119865119886(119909) ge 120573 Similarly


119865119886(119909) ge 120572 and


119865119886(119909) ge 120573 This implies

that 119911 isin (119865 119860)(120572120573)



(120572120573)and119910 isin 119877

we obtain (119865 119860)(120572120573)

∘ 119877 sube (119865 119860)(120572120573)

which means (119865 119860)(120572120573)









119886(119909) 119865119886(119910) le inf119865

119886(119911)


119865119886(119910) = 119904 which means that 119865

119886(119910) = 119904 Since



119865119886(119910) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

119865119886 (119909) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

(25)



119886(119910) = 119904 which means




119865119886= 119904 which means that 119865

119886(119909) = 119904 Similarly


119886(119909) = 119904 which means










119891(119909) +2 119891(119910) and 119891(119909 ∘1 119910) sube 119891(119909) ∘2 119891(119910)






(120593 120595) (119865 119860) = (120593 (119865) 120595 (119860)) (26)

where

120593 (119865119886) (120593 (119909)) = (120593 (119865))

120595(119886)

(119910) (27)



(120593 120595)minus1(119866 119861) = (120593

minus1(119866) 120595

minus1(119861)) (28)

where

120593minus1(119866119887) (120593minus1(119910)) = (120593

minus1(119866))120595minus1(119887)

(119909) (29)






1and 119877








Supp (120593 (119865) 120595 (119860)) sube 120595 (Supp (119865 119860)) (30)



min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119906) 119906 isin1199081 +2 1199082

= inf 120593 (119865119886) (120593 (119911))

(31)





min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119887) 1199062 isin 1199061 +2 119887

= inf 120593 (119865119886) (120593 (119910))

(32)


min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119888) 1199062 isin 119888 +2 1199061

= inf 120593 (119865119886) (120593 (119911))

(33)






min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119901) 119901 isin1199081 ∘2 1199082

= inf 120593 (119865119886) (120593 (119911))

(34)





119886) is




minus1(119866) 120595

minus1(119861))






6 Conclusion




Acknowledgments


References

















119909 isin 119911 + 119910 and 119910 isin (119865 119860)(120572120573)


(120572120573)sube 119911 + (119865 119860)


any 119911 isin (119865 119860)(120572120573)

119911 + (119865 119860)(120572120573)

= (119865 119860)(120572120573)





(120572120573) Then min119905

119865119886(119909) 119905119865119886(119910) ge min120572 120572 = 120572


119865119886(119910) ge min120573 120573 = 120573 and this

results in

inf 119905119865119886(119911) 119911 isin 119909 ∘ 119910 ge min 119905

119865119886(119909) 119905119865119886(119910) ge 120572

inf 1 minus 119891119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119910) ge 120573

(20)




(120572120573)


(120572120573)) As such (119865 119860)

(120572120573)is










119886



(119909) = 119905(119865119886)119879

(119910) = 119904 and 119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119910) = 119908which means that (119865

119886)119879(119909) = (119865

119886)119879(119910) = 119904 Therefore

min(119865119886)119879(119909) (119865

119886)119879(119910) = min119904 119904 = 119904 and so inf(119865

119886)119879(119911)



Now let 119909 119911 isin 119879 Then 119905(119865119886)119879

(119909) = 119905(119865119886)119879

(119911) = 119904 and119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119911) = 119908 and so (119865119886)119879(119909) = (119865

119886)119879(119911) =



and min(119865119886)119879(119909) (119865

119886)119879(119911) le (119865

119886)119879(119910) Thus we obtain

min(119865119886)119879(119909) (119865





(119865119886)119879(119909) = 119905

(119865119886)119879(119910) =

119904 and 119891(119865119886)119879

(119909) = 119891(119865119886)119879

(119910) = 119908 and thus (119865119886)119879(119909) =


119886is a


119886)119879(119909) (119865

119886)119879(119910) = min119904 119904 = 119904 and
















119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and


min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911) 119911 isin 119909 ∘ 119910 (resp

119905119865119886(119909) le inf119905

119865119886(119911) 119911 isin 119909 ∘ 119910) and 1 minus 119891

119865119886(119910) le


119865119886(119909) le


119886(119910) le inf119865

119886(119911)

119911 isin 119909 ∘ 119910 (resp 119865119886(119909) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910)



119886is a vague


119886


condition 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 (resp 119865

119886(119909) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910)



119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911)


119865119886(119910) le


119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910










max 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= max 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le max 119905119865120572(119909) 119905119865120572(119910) capmax 119905

119866120573(119909) 119905119866120573

(119910)

le inf 119905119865120572(119911) 119911 isin 119909 ∘ 119910

cap inf 119905119866120573

(119911) 119911 isin 119909 ∘ 119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 ∘ 119910

= inf 119905(120572120573)

119911 isin 119909 ∘ 119910

(21)


(120572120573)(119909) 1 minus119891

(120572120573)(119910) le inf1 minus 119891

(120572120573) 119911 isin 119909 ∘ 119910


(119909) (120572120573)

(119910) le

inf(120572120573)







Now let 119909 119910 isin (119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and

1 minus 119891119865119886(119909) 1 minus 119891


119886(119909) ge 119905 and


119886is a vague




119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(22)



Then 119905119865119886(119910) ge 119905 and


119886(119910) ge 119905 Since 119865

119886is a



119905119865119886(119910) le inf 119905

119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119911) 119911 isin 119909 ∘ 119910

(23)




119886(119910) ge 119905 This proves

that 119911 isin (119865119886)(119905119905)


119886)(119905119905)


weobtain 119877 ∘ (119865

119886)(119905119905)

sube (119865119886)(119905119905)



(119905119905)is a soft left







(lArr) Let (119865 119860)(119905119905)


119886)(119905119905)


119886)(119905119905)





119886)(119905119905)


119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(24)


or 119909 isin (119865119886)(119905119905)


119886)(119905119905)

and 119910 ∘ 119909 sube (119865119886)(119905119905)


119886)(119905119905)


119886)(119905119905)

Then119865119886(119909) ge 119905 and 119865


119886(119911) 119911 isin


119886(119911) 119911 isin 119909 ∘ 119910 ge 119865

119886(119910) ge 119905


119886(119911) 119911 isin 119909 ∘ 119910 and 119865

119886(119910) le


max119865119886(119909) 119865119886(119910) le inf119865








(120572120573)is a



to prove that 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

(resp (119865 119860)(120572120573)

∘

119877 sube (119865 119860)(120572120573)


Then119905119865119886(119910) ge 120572 and 1 minus 119891

119865119886(119910) ge 120573 Since 119865



119865119886(119910) le


119865119886(119910) le inf1 minus 119891

119865119886(119911) 119911 isin


119865119886(119910) ge 120572


119865119886(119910) ge 120573 which




119910 isin (119865 119860)(120572120573)

we obtain 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

Hence(119865 119860)


(120572120573)


119865119886(119909) ge 120573 Similarly


119865119886(119909) ge 120572 and


119865119886(119909) ge 120573 This implies

that 119911 isin (119865 119860)(120572120573)



(120572120573)and119910 isin 119877

we obtain (119865 119860)(120572120573)

∘ 119877 sube (119865 119860)(120572120573)

which means (119865 119860)(120572120573)









119886(119909) 119865119886(119910) le inf119865

119886(119911)


119865119886(119910) = 119904 which means that 119865

119886(119910) = 119904 Since



119865119886(119910) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

119865119886 (119909) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

(25)



119886(119910) = 119904 which means




119865119886= 119904 which means that 119865

119886(119909) = 119904 Similarly


119886(119909) = 119904 which means










119891(119909) +2 119891(119910) and 119891(119909 ∘1 119910) sube 119891(119909) ∘2 119891(119910)






(120593 120595) (119865 119860) = (120593 (119865) 120595 (119860)) (26)

where

120593 (119865119886) (120593 (119909)) = (120593 (119865))

120595(119886)

(119910) (27)



(120593 120595)minus1(119866 119861) = (120593

minus1(119866) 120595

minus1(119861)) (28)

where

120593minus1(119866119887) (120593minus1(119910)) = (120593

minus1(119866))120595minus1(119887)

(119909) (29)






1and 119877








Supp (120593 (119865) 120595 (119860)) sube 120595 (Supp (119865 119860)) (30)



min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119906) 119906 isin1199081 +2 1199082

= inf 120593 (119865119886) (120593 (119911))

(31)





min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119887) 1199062 isin 1199061 +2 119887

= inf 120593 (119865119886) (120593 (119910))

(32)


min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119888) 1199062 isin 119888 +2 1199061

= inf 120593 (119865119886) (120593 (119911))

(33)






min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119901) 119901 isin1199081 ∘2 1199082

= inf 120593 (119865119886) (120593 (119911))

(34)





119886) is




minus1(119866) 120595

minus1(119861))






6 Conclusion




Acknowledgments


References

















min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911) 119911 isin 119909 ∘ 119910 (resp

119905119865119886(119909) le inf119905

119865119886(119911) 119911 isin 119909 ∘ 119910) and 1 minus 119891

119865119886(119910) le


119865119886(119909) le


119886(119910) le inf119865

119886(119911)

119911 isin 119909 ∘ 119910 (resp 119865119886(119909) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910)



119886is a vague


119886


condition 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 ∘ 119910 (resp 119865

119886(119909) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910)



119865119886(119911)


119865119886(119910) le


min119865119886(119909) 119865119886(119910) le inf119865

119886(119911) 119911 isin 119909 + 119910


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119910) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119910) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119910)


119865119886(119908) 119905119865119886(119909) le 119905

119865119886(119911) and

min1 minus 119891119865119886(119908) 1 minus 119891

119865119886(119909) le 1 minus 119891

119865119886(119911) that is

min119865119886(119908) 119865

119886(119909) le 119865

119886(119911)


119865119886(119911)


119865119886(119910) le


119886(119909) 119865119886(119910) le

inf119865119886(119911) 119911 isin 119909 ∘ 119910










max 119905(120572120573)

(119909) 119905(120572120573)

(119910)

= max 119905119865120572(119909) cap 119905

119866120573(119909) 119905119865120572(119910) cap 119905

119866120573(119910)

le max 119905119865120572(119909) 119905119865120572(119910) capmax 119905

119866120573(119909) 119905119866120573

(119910)

le inf 119905119865120572(119911) 119911 isin 119909 ∘ 119910

cap inf 119905119866120573

(119911) 119911 isin 119909 ∘ 119910

le inf 119905119865120572(119911) cap 119905

119866120573(119911) 119911 isin 119909 ∘ 119910

= inf 119905(120572120573)

119911 isin 119909 ∘ 119910

(21)


(120572120573)(119909) 1 minus119891

(120572120573)(119910) le inf1 minus 119891

(120572120573) 119911 isin 119909 ∘ 119910


(119909) (120572120573)

(119910) le

inf(120572120573)







Now let 119909 119910 isin (119865119886)(119905119905)

Then 119905119865119886(119909) 119905119865119886(119910) ge 119905 and

1 minus 119891119865119886(119909) 1 minus 119891


119886(119909) ge 119905 and


119886is a vague




119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(22)



Then 119905119865119886(119910) ge 119905 and


119886(119910) ge 119905 Since 119865

119886is a



119905119865119886(119910) le inf 119905

119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119911) 119911 isin 119909 ∘ 119910

(23)




119886(119910) ge 119905 This proves

that 119911 isin (119865119886)(119905119905)


119886)(119905119905)


weobtain 119877 ∘ (119865

119886)(119905119905)

sube (119865119886)(119905119905)



(119905119905)is a soft left







(lArr) Let (119865 119860)(119905119905)


119886)(119905119905)


119886)(119905119905)





119886)(119905119905)


119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(24)


or 119909 isin (119865119886)(119905119905)


119886)(119905119905)

and 119910 ∘ 119909 sube (119865119886)(119905119905)


119886)(119905119905)


119886)(119905119905)

Then119865119886(119909) ge 119905 and 119865


119886(119911) 119911 isin


119886(119911) 119911 isin 119909 ∘ 119910 ge 119865

119886(119910) ge 119905


119886(119911) 119911 isin 119909 ∘ 119910 and 119865

119886(119910) le


max119865119886(119909) 119865119886(119910) le inf119865








(120572120573)is a



to prove that 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

(resp (119865 119860)(120572120573)

∘

119877 sube (119865 119860)(120572120573)


Then119905119865119886(119910) ge 120572 and 1 minus 119891

119865119886(119910) ge 120573 Since 119865



119865119886(119910) le


119865119886(119910) le inf1 minus 119891

119865119886(119911) 119911 isin


119865119886(119910) ge 120572


119865119886(119910) ge 120573 which




119910 isin (119865 119860)(120572120573)

we obtain 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

Hence(119865 119860)


(120572120573)


119865119886(119909) ge 120573 Similarly


119865119886(119909) ge 120572 and


119865119886(119909) ge 120573 This implies

that 119911 isin (119865 119860)(120572120573)



(120572120573)and119910 isin 119877

we obtain (119865 119860)(120572120573)

∘ 119877 sube (119865 119860)(120572120573)

which means (119865 119860)(120572120573)









119886(119909) 119865119886(119910) le inf119865

119886(119911)


119865119886(119910) = 119904 which means that 119865

119886(119910) = 119904 Since



119865119886(119910) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

119865119886 (119909) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

(25)



119886(119910) = 119904 which means




119865119886= 119904 which means that 119865

119886(119909) = 119904 Similarly


119886(119909) = 119904 which means










119891(119909) +2 119891(119910) and 119891(119909 ∘1 119910) sube 119891(119909) ∘2 119891(119910)






(120593 120595) (119865 119860) = (120593 (119865) 120595 (119860)) (26)

where

120593 (119865119886) (120593 (119909)) = (120593 (119865))

120595(119886)

(119910) (27)



(120593 120595)minus1(119866 119861) = (120593

minus1(119866) 120595

minus1(119861)) (28)

where

120593minus1(119866119887) (120593minus1(119910)) = (120593

minus1(119866))120595minus1(119887)

(119909) (29)






1and 119877








Supp (120593 (119865) 120595 (119860)) sube 120595 (Supp (119865 119860)) (30)



min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119906) 119906 isin1199081 +2 1199082

= inf 120593 (119865119886) (120593 (119911))

(31)





min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119887) 1199062 isin 1199061 +2 119887

= inf 120593 (119865119886) (120593 (119910))

(32)


min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119888) 1199062 isin 119888 +2 1199061

= inf 120593 (119865119886) (120593 (119911))

(33)






min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119901) 119901 isin1199081 ∘2 1199082

= inf 120593 (119865119886) (120593 (119911))

(34)





119886) is




minus1(119866) 120595

minus1(119861))






6 Conclusion




Acknowledgments


References


















Then 119905119865119886(119910) ge 119905 and


119886(119910) ge 119905 Since 119865

119886is a



119905119865119886(119910) le inf 119905

119865119886(119911) 119911 isin 119909 ∘ 119910


119865119886(119911) 119911 isin 119909 ∘ 119910

(23)




119886(119910) ge 119905 This proves

that 119911 isin (119865119886)(119905119905)


119886)(119905119905)


weobtain 119877 ∘ (119865

119886)(119905119905)

sube (119865119886)(119905119905)



(119905119905)is a soft left







(lArr) Let (119865 119860)(119905119905)


119886)(119905119905)


119886)(119905119905)





119886)(119905119905)


119877 ∘ (119865119886)(119905119905)

sube (119865119886)(119905119905)

(119865119886)(119905119905)

∘ 119877 sube (119865119886)(119905119905)

(24)


or 119909 isin (119865119886)(119905119905)


119886)(119905119905)

and 119910 ∘ 119909 sube (119865119886)(119905119905)


119886)(119905119905)


119886)(119905119905)

Then119865119886(119909) ge 119905 and 119865


119886(119911) 119911 isin


119886(119911) 119911 isin 119909 ∘ 119910 ge 119865

119886(119910) ge 119905


119886(119911) 119911 isin 119909 ∘ 119910 and 119865

119886(119910) le


max119865119886(119909) 119865119886(119910) le inf119865








(120572120573)is a



to prove that 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

(resp (119865 119860)(120572120573)

∘

119877 sube (119865 119860)(120572120573)


Then119905119865119886(119910) ge 120572 and 1 minus 119891

119865119886(119910) ge 120573 Since 119865



119865119886(119910) le


119865119886(119910) le inf1 minus 119891

119865119886(119911) 119911 isin


119865119886(119910) ge 120572


119865119886(119910) ge 120573 which




119910 isin (119865 119860)(120572120573)

we obtain 119877 ∘ (119865 119860)(120572120573)

sube (119865 119860)(120572120573)

Hence(119865 119860)


(120572120573)


119865119886(119909) ge 120573 Similarly


119865119886(119909) ge 120572 and


119865119886(119909) ge 120573 This implies

that 119911 isin (119865 119860)(120572120573)



(120572120573)and119910 isin 119877

we obtain (119865 119860)(120572120573)

∘ 119877 sube (119865 119860)(120572120573)

which means (119865 119860)(120572120573)









119886(119909) 119865119886(119910) le inf119865

119886(119911)


119865119886(119910) = 119904 which means that 119865

119886(119910) = 119904 Since



119865119886(119910) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

119865119886 (119909) le inf 119865

119886 (119911) 119911 isin 119909 ∘ 119910

(25)



119886(119910) = 119904 which means




119865119886= 119904 which means that 119865

119886(119909) = 119904 Similarly


119886(119909) = 119904 which means










119891(119909) +2 119891(119910) and 119891(119909 ∘1 119910) sube 119891(119909) ∘2 119891(119910)






(120593 120595) (119865 119860) = (120593 (119865) 120595 (119860)) (26)

where

120593 (119865119886) (120593 (119909)) = (120593 (119865))

120595(119886)

(119910) (27)



(120593 120595)minus1(119866 119861) = (120593

minus1(119866) 120595

minus1(119861)) (28)

where

120593minus1(119866119887) (120593minus1(119910)) = (120593

minus1(119866))120595minus1(119887)

(119909) (29)






1and 119877








Supp (120593 (119865) 120595 (119860)) sube 120595 (Supp (119865 119860)) (30)



min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119906) 119906 isin1199081 +2 1199082

= inf 120593 (119865119886) (120593 (119911))

(31)





min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119887) 1199062 isin 1199061 +2 119887

= inf 120593 (119865119886) (120593 (119910))

(32)


min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119888) 1199062 isin 119888 +2 1199061

= inf 120593 (119865119886) (120593 (119911))

(33)






min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119901) 119901 isin1199081 ∘2 1199082

= inf 120593 (119865119886) (120593 (119911))

(34)





119886) is




minus1(119866) 120595

minus1(119861))






6 Conclusion




Acknowledgments


References



















119865119886= 119904 which means that 119865

119886(119909) = 119904 Similarly


119886(119909) = 119904 which means










119891(119909) +2 119891(119910) and 119891(119909 ∘1 119910) sube 119891(119909) ∘2 119891(119910)






(120593 120595) (119865 119860) = (120593 (119865) 120595 (119860)) (26)

where

120593 (119865119886) (120593 (119909)) = (120593 (119865))

120595(119886)

(119910) (27)



(120593 120595)minus1(119866 119861) = (120593

minus1(119866) 120595

minus1(119861)) (28)

where

120593minus1(119866119887) (120593minus1(119910)) = (120593

minus1(119866))120595minus1(119887)

(119909) (29)






1and 119877








Supp (120593 (119865) 120595 (119860)) sube 120595 (Supp (119865 119860)) (30)



min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119906) 119906 isin1199081 +2 1199082

= inf 120593 (119865119886) (120593 (119911))

(31)





min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119887) 1199062 isin 1199061 +2 119887

= inf 120593 (119865119886) (120593 (119910))

(32)


min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119888) 1199062 isin 119888 +2 1199061

= inf 120593 (119865119886) (120593 (119911))

(33)






min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119901) 119901 isin1199081 ∘2 1199082

= inf 120593 (119865119886) (120593 (119911))

(34)





119886) is




minus1(119866) 120595

minus1(119861))






6 Conclusion




Acknowledgments


References



















min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119887) 1199062 isin 1199061 +2 119887

= inf 120593 (119865119886) (120593 (119910))

(32)


min 120593 (119865119886) (120593 (119908)) 120593 (119865119886) (120593 (119909))

= min(120593 (119865))120595(119886)

(1199061) (120593 (119865))120595(119886)

(1199062)

le inf (120593 (119865))120595(119886)

(119888) 1199062 isin 119888 +2 1199061

= inf 120593 (119865119886) (120593 (119911))

(33)






min 120593 (119865119886) (120593 (119909)) 120593 (119865119886) (120593 (119910))

= min(120593 (119865))120595(119886)

(1199081) (120593 (119865))120595(119886)

(1199082)

le inf (120593 (119865))120595(119886)

(119901) 119901 isin1199081 ∘2 1199082

= inf 120593 (119865119886) (120593 (119911))

(34)





119886) is




minus1(119866) 120595

minus1(119861))






6 Conclusion




Acknowledgments


References
















algebraic hyperstructures of vague soft sets associated ...€¦ · researcharticle algebraic...

Documents