research article relational demonic fuzzy...

Research ArticleRelational Demonic Fuzzy Refinement

Fairouz Tchier

Mathematics Department King Saud University PO Box 22452 Riyadh 11495 Saudi Arabia

Correspondence should be addressed to Fairouz Tchier ftchierksuedusa

Received 25 February 2014 Accepted 8 July 2014 Published 19 August 2014

Academic Editor Carlos Conca

Copyright copy 2014 Fairouz TchierThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We use relational algebra to define a refinement fuzzy order called demonic fuzzy refinement and also the associated fuzzy operatorswhich are fuzzy demonic join (⊔fuz) fuzzy demonic meet (⊓fuz) and fuzzy demonic composition (◻fuz) Our definitions andproperties are illustrated by some examples using mathematica software (fuzzy logic)

1 Introduction

11 Motivation Fuzzy set theory provides a major newerparadigm in modeling and reasoning with uncertaintyZadeh a professor at University of California at Berkeley wasthe first to propose a theory of fuzzy sets and an associatedlogic namely fuzzy logic in [1] Essentially a fuzzy set is a setwhose members may have degrees of membership between0 and 1 as opposed to classical sets where each elementmust have either 0 or 1 as the membership degree if 0 theelement is completely outside the set if 1 the element iscompletely in the set As classical logic is based on classical settheory fuzzy logic is based on fuzzy set theory Since Zadehrsquosinvention the concept of fuzzy sets has been extensivelyinvestigated in mathematics science and engineering Thenotion of fuzzy relations is also a basic one in processingfuzzy information in relational structures see for examplePedrycz [2] Goguen [3] generalized the concepts of fuzzy setsand relations taking values on partially ordered sets Fuzzyrelations were initiated and applied to medical models ofdiagnosis by Sanchez [4] Fuzzy set theory is now appliedto problems in engineering business medical and relatedhealth sciences and the natural sciences Fuzzy relations playan important role in fuzzy modeling fuzzy diagnosis andfuzzy control They also have applications in fields such aspsychology medicine economics and sociology

There are countless applications for fuzzy logic In factsome claim that fuzzy logic is the encompassing theory overall types of logic The items in this list are more commonapplications that one may encounter in everyday life

(a) Temperature Control (HeatingCooling) [5ndash7] I do notthink the university has figured this one out yet The trickin temperature control is to keep the room at the sametemperature consistently Well that seems pretty easy rightBut how much does a room have to cool off before the heatkicks in again There must be some standard so the heat(or air conditioning) is not in a constant state of turning onand off Therein lies the fuzzy logic The set is determinedby what the temperature is actually set to Membership inthat set weakens as the room temperature varies from the settemperature Once membership weakens to a certain pointtemperature control kicks in to get the room back to thetemperature it should be

(b) Medical Diagnosis [8] How many of what kinds ofsymptoms will yield a diagnosis How often are doctors inerror Surely everyone has seen those lists of symptoms fora horrible disease that say ldquoif you have at least 5 of thesesymptoms you are at riskrdquo It is a hypochondriacrsquos havenThequestion is as follows how do doctors go from that list ofsymptoms to a diagnosis Fuzzy logicThere is no guaranteedsystem to reach a diagnosis If there was we would not hearabout cases of medical misdiagnosis The diagnosis can onlybe some degree within the fuzzy set

Fuzzy set theory appeared in 1965 [1] Since then it hasreceived increasing attention by the scientific communityand applied in almost all the general disciplines [9ndash11]Fuzzy set theory provides a strict mathematical framework(there is nothing fuzzy about fuzzy set theory) in whichvague conceptual phenomena can be precisely and rigorously

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 140707 17 pageshttpdxdoiorg1011552014140707

2 Journal of Applied Mathematics

studied It can also be considered as amodeling language wellsuited for situations in which fuzzy relations criteria andphenomena exist It will mean different things dependingon the application area and the way it is measured Inthe meantime numerous authors have contributed to thistheory In 1984 as many as 4000 publications may alreadyexist The first publications in fuzzy set theory by Zadeh[1] and Goguen [3 12] show the intention of the authorsto generalize the classical notion of a set Zadeh [1] writesldquoThe notion of a fuzzy set provides a convenient point ofdeparture for the construction of a conceptual frameworkwhich parallels in many respects the framework used in thecase of ordinary sets but is more general than the latterand potentially may prove to have a much wider scopeof applicability particularly in the fields of mathematicsand computer science (pattern classification and informationprocessing) Fuzzy logic is a superset of conventional logicthat has been extended to handle the concept of partialtruth-truth values between ldquocompletely truerdquo and ldquocompletelyfalserdquo As its name suggests it is the logic underlying modesof reasoning which are approximate rather than exact Theimportance of fuzzy theory derives from the fact that mostmodes of human reasoning and especially common sensereasoning are approximate in naturerdquo

The calculus of relations has been an important compo-nent of the development of logic and algebra since themiddleof the nineteenth century [13ndash15] The main advantages ofthe relational formalization are uniformity and modularityActually once problems in these fields are formalized in termsof relational calculus these problems can be considered byusing formulae of relations that is we need only calculusof relations in order to solve the problems In the contextof software development one important approach is thatof developing programs from specifications by stepwiserefinement see for example [16ndash20] One point of view isthat a specification is a relation constraining the input-output(resp argument-result) behaviour of programs

The demonic calculus of relations [21 22] views anyrelation 119877 from a set 119860 to another set 119861 as specifying thoseprograms that terminate for all 119886 isin 119860 wherever 119877 associatesany values from 119861 with 119886 and then the program may onlyreturn values 119887 for which (119886 119887) isin 119877 Consequently a relation119877 refines another relation 119878 if 119877 specifies a larger domainof termination and fewer possibilities for return values Thedemonic calculus of relations has the advantage that thedemonic operations are defined on top of the conventionalrelation algebraic operations and can easily and usefully bemixed with the latter allowing the application of numerousalgebraic properties

In Section 2 we present our mathematical tool namelyrelational algebra First we will give certain notions aboutelementary theory on relations and ordered structuresTherea concept of type can be defined that allows an abstracttreatment of the domain (of definedness) of an element andalso of assertions After some auxiliary results (Section 3on relational calculus and Section 4 on fuzzy calculus) wepresent in Section 5 notions andproperties of demonic fuzzyoperators Finally we conclude our work in Section 6

2 Elementary Theory on Relations

Using Tarski [15] approach we distinguish two levels ofabstraction in the study of binary relations the elementarytheory of relations and relational calculus First level definesthe relations as sets of (pairs) and second level defines therelations as elementary objects on which the operations aredefined and studied in an algebraic point of view

In this paper we need both levels of abstraction The firstone will give us our examples and the second one is useful forour proofs and formulas So in this section we will presentboth levels A relation 119877 from a set 119883 to a set 119884 is a subset ofpairs (119909 119910) where 119909 isin 119883 and 119910 isin 119884 Formally

119877 sube 119883 times 119884 = (119909 119910) | 119909 isin 119883 and 119910 isin 119884 (1)

If119883 = 119884 then 119877 is homogeneous on119883 The relations on finitesets can be represented by matrices For example the relation119877 = (119886 119887) (119886 119888) (119887 119889) (119888 119889) (119889 119890) can be represented by

119877 =

119886

119887

119888

119889

119890

119886 119887 119888 119889 119890

(

0 1 1 0 0

0 0 0 1 0

0 0 0 1 0

0 0 0 0 1

0 0 0 0 0

) (2)

The graphs and relations are closely linkedEvery finite relation can be interpreted as a representation

graph and vice versa Graph (1) which corresponds to therelation 119877 is shown in Figure 1

In matrix notation an entry 0 corresponds to the absenceof an edge between two vertices of the graph (the absence ofthe pair in the relation) and an entry 1 means the opposite

As relations are sets they are ordered by inclusion Theleast relation between sets 119883 and 119884 is the empty (also calledzero) noted Oslash

119883119884 and the greatest one called universal

relation and noted 119871119883119884

A particular relation defined forevery set 119883 is the identity relation noted 119868

119883

def= (119909 119909) |

119909 isin 119883 The set of elements of 119883 whose images by 119877 is calleddomain of 119877 noted dom(119877) and the set of images is notedimg(119877) Formally

dom (119877)def= 119909 | (exist119910 (119909 119910) isin 119877)

img (119877)def= 119910 | (exist119909 (119909 119910) isin 119877)

(3)

As relations are particular sets we can apply the usualsets operations which are union (cup) intersection (cap) andcomplementation (

mdash) Relations are ordered by inclusion

More their structure helps us to define other operationswhich are as follows

(a) Inverse of a relation 119877 denoted 119877S

119877S

= (119909 119910) | (119910 119909) isin 119877 (4)

(b) For 119877 sube 119883 times 119885 and 119878 sube 119885 times 119884 we define thecomposition of 119877 and 119878 noted 119877 ∘ 119878 as follows

119877 ∘ 119878 = (119909 119910) | (exist119911 119911 isin 119885 (119909 119911) isin 119877 and (119911 119910) isin 119878)

(5)

Journal of Applied Mathematics 3

a

b

c

d e

Figure 1 The graph associated to relation 119877

We remark that 119877 ∘ 119878 sube 119883 times 119884

The composition operator symbol (∘) will be omitted (ie wewrite (119877119878) for (119877 ∘ 119878))

We can now define the abstract algebraic structureshaving many properties of the relations They are basedon boolean algebras and other operators which are thecomposition (∘) and the inverse (

S) and also a particular

element 119868 (identity relation)

3 Relational Calculus

The origin of relational calculus goes back to last centurywith the work of de Morgan [23 24] and Strohlein [14] alsoin the beginning of present century with the work of Peirce[25 26] Their study has been revived by the work of ChinandTarski [15 27] Formore details on the algebra of relationssee [27ndash33] In what follows we will give a definition of thealgebra of relations and some important models of axiomscharacterizing this algebra Most of our definitions are from[34]

Definition 1 An abstract heterogeneous relational algebra is analgebraic structure (R cup cap

mdashS ∘Oslash 119871 119868) on a nonempty setR of elements called relations such that the next conditionsare satisfied

(a) Every structure (R cup capmdashOslash 119871) is a complete atomicboolean algebra with zero element Oslash universal ele-ment 119871 and order sube

(b) Every relation 119877 has an inverse 119877S(c) (R ∘) is a semigroup with precisely one unit element

119868(d) Schroder rule119875∘119876 sube 119877 hArr 119875S ∘119877 sube 119876 hArr 119877∘119876S sube 119875

is valid(e) Tarski rule is valid 119871 ∘ 119877 ∘ 119871 = 119871 if and if only 119877 = Oslash

The precedence of the relational operators from highestto lowest is the following mdashS bind equally followed by(∘) followed by cap and finally by cup The scope of ⋃

119894and ⋂

119894

goes to the right as far as possible The relation 119877S is calledthe converse of 119877 From Definition 1 the usual rules of thecalculus of relations can be derived (see eg [27 34ndash36])We assume these rules to be known and simply recall a few of

them We will present certain examples of models satisfyingthese axioms

Example 2 (a) The algebra of binary relations on differentsets is an important relation algebra because it is very usefulLet 1198781 119878

119899be sets Then

Rdef= 119877 | 119877 sube 119878

119894times 119878119895 1 le 119894 119895 le 119899 (6)

with relation operators is a relation algebraThe operations cupand cap between relations 119876 and 119877 are defined if and if only 119876

and 119877 have the same type A relation is homogeneous if andif only 119877 119878

119894harr 119878119894for a certain 119894 The composition 119876 ∘ 119877 is

defined if and if only 119876 119878119894harr 119878119895and 119877 119878

119895harr 119878119896for certain

119894 119895 119896(b) The set of all homogeneous binary relations on a set

119883 denoted Rel(119883)def= (P(119883 times 119883) cup capSmdash ∘Oslash 119883 times 119883 119868

119883)

is a relation algebra(c) The algebra of boolean matrices is another important

relation algebraWe recall by the next examples how some of the operators

are applied to boolean matrices To respect the usual conven-tion we will use the boolean values 0 1 instead of the valuesOslash 119871 Consider

1198682times2

= (1 0

0 1) Oslash

1times2= (0 0)

1198712times3

= (1 1 1

1 1 1) (

1 0

0 1

1 0

)

S

= (1 0 1

0 1 0)

(1 0 1

0 1 0)(

1 0

0 1

0 1

) = (1 1

0 1)

(7)

(d) We will give another example The set of matriceswhose entries are relations constitutes a relation algebra [34]with the operators defined as follows

(119877 cup 119878)119894119895 = 119877119894119895cup 119878119894119895 (119877)

119894119895= 119877119894119895

(119877119878)119894119895 = ⋃119896

119877119894119896119878119896119895 (119877 cap 119878)119894119895 = 119877

119894119895cap 119878119894119895

(119877S

)119894119895 = (119877119895119894)S

(8)

The constant relations are defined as follows

119871119894119895

= 119871 Oslash119894119895

= Oslash 119868119894119895

= 119868 if 119894 = 119895

Oslash otherwise(9)

where 119877119894119895denotes entry 119894 119895 of matrix 119877 Of course 119877cup 119878 and

119877cap 119878 exist only if matrices 119877 and 119878 have the same dimensionthe composition 119877119878 exists only if the number of columns of119877 is the same as the number of rows of 119878 The entries of theidentity matrix (which is square) are 0 except those of thediagonal which are 1 The entries of the zero matrix are Oslashand those of the universal matrix are 119871

From Definition 1 the usual rules of the calculus ofrelations can be derived (see eg [27 34ndash38]) We assumethese rules to be known and simply recall a few of them


Theorem 3 Let 119875119876 119877 be relations and let 119883 be an arbitraryindex set Consider the following

(1) ⋃119894isin119883

119877119894= ⋂119894isin119883

119877119894

(2) ⋂119894isin119883

119877119894= ⋃119894isin119883

119877119894

(3) (119876 cap 119877) cup 119877 = 119876 cup 119877(4) 119875 cap 119876 sube 119877 hArr 119875 sube 119876 cup 119877(5) 119876 sube 119877 hArr 119877 sube 119876(6) 119876(⋃

119894isin119883119877119894) = ⋃

119894isin119883119876119877119894

(7) 119875(119876 cup 119877) = 119875119876 cup 119875119877(8) (⋃

119894isin119883119876119894)119877 = ⋃

119894isin119883119876119894119877

(9) (119875 cup 119876)119877 = 119875119877 cup 119876119877(10) 119876(⋂

119894isin119883119877119894) sube ⋂

119894isin119883119876119877119894

(11) (⋂119894isin119883

119876119894)119877 sube ⋂

119894isin119883119876119894119877

(12) 119876 sube 119877 rArr 119875119876 sube 119875119877(13) 119875 sube 119876 rArr 119875119877 sube 119876119877(14) (⋃

119894isin119883119877119894)S

= ⋃119894isin119883

119877S119894

(15) (⋂119894isin119883

119877119894)S

= ⋂119894isin119883

119877S119894

(16) (119876119877)S

= 119877S119876S(17) 119877

S= 119877S

(18) 119875119876 cap 119877 sube (119875 cap 119877119876S)(119876 cap 119875S119877)(19) 119875119876 cap 119877 sube 119875(119876 cap 119875S119877)(20) 119875119876 cap 119877 sube (119875 cap 119877119876S)119876(21) 119871119871 = 119871(22) (⋂

119894isin119883119877119894119871)119871 = ⋂

119894isin119883119877119894119871

(23) (⋃119894isin119883

119877119894119871)119871 = ⋃

119894isin119883119877119894119871

(24) (119875 cap 119876119871)119877 = 119875119877 cap 119876119871(25) (119875 cap 119871119876S)119877 = 119875(119877 cap 119876119871)(26) 119876119871119877 = 119876119871 cap 119871119877(27) 119877119871119871 = 119877119871(28) 119877 = (119868 cap 119877119877S)119877

Sometimes instead of refering to laws 6 7 8 and 9 werefer to the operation (∘) as distributive Then we have (∘)which is monotonic instead of refering to laws 12 and 13

In the following we will give the definitions of certainproperties

Definition 4 A relation 119877 is as follows

(a) deterministic if and if only 119877S119877 sube 119868

(b) total if and if only 119871 = 119877119871 (equivalent to 119868 sube 119877119877S)(c) an application if and if only it is total and determinis-

tic(d) injective if and if only119877S is deterministic (ie119877119877S sube

119868)(e) surjective if and if only 119877S is total (ie 119871119877 = 119871 or also

119868 sube 119877S119877)

(f) a partial identity if and if only 119877 sube 119868 (subidentity)(g) a vector if and if only 119877 = 119877119871 (the vectors are usually

denoted by the letter V)(h) a point if and if only 119877 = Oslash 119877 = 119877119871 and 119877119877

S sube 119868

A function is a deterministic relationThe vectors 119877119871 and119877S119871 are particular vectors characterizing respectively thedomain and codomain of 119877 The set of vectors of a certaintype is a complete boolean algebra [34 36 38]

In an algebra of boolean matrices a vector is a matrix inwhich the rows are constant and a point is a vector with anonzero row

Example 5 Let 119883 = 0 1 2 and 119881 = 0 1 Then

V def= 119881 times 119883 = (0 0) (0 1) (0 2) (1 0) (1 1) (1 2) (10)

is a vector that corresponds to a set of points 119881 The partialidentity which corresponds to V is 119886 = (0 0) (1 1)

Let 119877 def= (0 1) (0 2) (2 1) and let V and 119886 be the vector

and the partial identity given before

(i) Prerestriction of 119877 to V (or to 119886) is as follows

V cap 119877 = 119886119877 = (0 1) (0 2) (11)

(ii) Postrestriction of 119877 to V (or to 119886) is as follows

VS cap 119877 = 119877119886 = (0 1) (2 1) (12)

(iii) Domain of 119877 is

119877119871 = (0 0) (0 1) (0 2) (2 0) (2 1) (2 2) (13)

The vector represents the subset 0 2(iv) The relation 119877S119871 = (1 0) (1 1) (1 2) (2 0) (2 1)

(2 2) is the vector characterizing the subset 1 2which is the codomain of 119877

4 Fuzzy Relational Calculus

Fuzzy set theory has been studied extensively over the past 30years Most of the early interest in fuzzy set theory pertainedto representing uncertainty in human cognitive processes [1]

The concept of a fuzzy relation on a set was defined byZadeh [1 39] and other authors like Rosenfeld [40] Tamuraet al [41] and Yeh and Bang [5] considered it furtherFuzzy relations are of fundamental importance in fuzzy logicand fuzzy set theory including particularly fuzzy preferencemodeling fuzzy mathematics fuzzy inference and manymore

Let 119860 119861 sube 119880 be two sets a fuzzy relation on 119860 times 119861 isdefined by = ((119909 119910) 120583

(119909 119910)) | (119909 119910) isin 119860times119861 where the

map 120583

119860 times 119861 rarr [0 1] is called membership function andthe value 120583

(119909 119910) isin [0 1] is called the degree of membership

of (119909 119910) in [40 41]As fuzzy relations are sets they are ordered by inclusion

The least fuzzy relation between sets 119860 and 119861 is the empty


(also called zero) noted Oslash119860119861 and the greatest one called

universal fuzzy relation and noted 119860119861 A particular fuzzy

relation defined for every set 119860 is the identity fuzzy relationnoted 119868

119860

def= ((119909 119910) 120583

119868(119909 119910)) | 119909 isin 119860 where 120583

119868(119909 119910) = 1 if

119909 = 119910 and 0 otherwiseThe domain of denoted by dom()is defined as follows

dom ()def= sup

119910isin119861((119909 119910) 120583

(119909 119910)) | forall119909 isin 119860 (14)

and the codomain of denoted by codom() is expressedby finding the maximal value of a long 119860

codom ()def= sup

119909isin119860((119909 119910) 120583

(119909 119910)) | forall119910 isin 119861 (15)

The domain and codomain are regarded as the height of rowand columns of the fuzzy relation [42]

41 Operations on Fuzzy Relations As fuzzy relations are par-ticular fuzzy sets we can apply the usual fuzzy sets operationswhich are union (cup) intersection (cap) and complementation(mdash

) given in [43] More their structure helps us to defineother operations which are as follows

Definition 6 (see [1 5 39]) Let 119860 119861 and 119862 be sets and let and 119878 be fuzzy relations defined respectively on 119860 times 119861 and119861 times 119862 One has

= [(119909 119910) 120583(119909 119910)] | (119909 119910) isin 119860 times 119861

119878 = ((119910 119911) 120583119878(119910 119911)) | (119910 119911) isin 119861 times 119862

(16)

(a) Inverse of a fuzzy relation denoted S

S

= [(119909 119910) 120583(119909 119910)] | (119909 119910) isin 119860 times 119861 (17)

such that

= [(119910 119909) 120583(119910 119909)] | (119910 119909) isin 119861 times 119860 (18)

(b) The max-min composition ∘ 119878 is the fuzzy relation

∘ 119878 = [(119909 119911) or119910isin119861 120583 (119909 119910) and 120583119878(119910 119911)] | 119909 isin 119860

119910 isin 119861 119911 isin 119862

(19)

(i) From now on the composition operator symbol(∘) will be omitted (ie we write (119878) for (∘119878))

(c) The semiscalars multiplication 119896 of a fuzzy relation by a scalar 119896(isin [0 1]) is a fuzzy relation such that

119896 = [(119909 119910) 119896120583(119909 119910)] | (119909 119910) isin 119860 times 119861 (20)

Example 7 (i) Let and 119878 be two fuzzy relations given asfollows and 119896 = 03

(ii) Let 119860 = 1199091 1199092 1199093 119861 = 119910

1 1199102 1199103 1199104 = ldquo119909

considerably larger than 119910rdquo and 119878 = ldquo119910 very close to 119909rdquo then

=1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

08 1 01 07

0 08 0 0

09 1 07 08

)

119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

04 0 09 06

09 04 05 07

03 0 08 05

)

(21)

We have

(i) cup 119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

08 1 09 07

09 08 05 07

09 1 08 08

)

(ii) S

=

1199101

1199102

1199103

1199104

1199091

1199092

1199093

(

08 0 09

1 08 1

01 0 07

07 0 08

)

(iii) cap 119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

04 0 01 06

0 04 0 0

03 0 07 05

)

(iv) 119896119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

012 0 027 018

027 012 015 021

09 0 024 015

)

(22)

These operations are illustrated respectively by Figures 23 4 5 and 6

Remark 8 Different versions of ldquocompositionrdquo have beensuggested which differ in their results and also with respectto their mathematical properties The max-min compositionhas become the best known and the most frequently usedHowever often the so-called max-product or max-averagecompositions lead to results that aremore appealing see [40]

(a) The max-prod composition (∘sdot119878) is defined as fol-

lows∘sdot119878 = [(119909 119911) or119910 120583

1

(119909 119910) sdot 1205832

(119910 119911)] | 119909 isin 119860

119910 isin 119861 119911 isin 119862

(23)

(b) The max-av composition ( ∘av 119878) is defined as fol-lows

∘av119878 = [(119909 119911) 1

2or119910120583(119909 119910) + 120583

119878(119910 119911)] | 119909 isin 119860

119910 isin 119861 119911 isin 119862

(24)


4

6

100

2

4

6

8

10

0

025

05

075

1

02

8

Figure 2 Fuzzy relation in Example 9

4

6

100

2

4

6

8

10

0

025

05

075

1

0

2

8

Figure 3 Fuzzy relations 119878 in Example 9

4

6

100

2

4

6

8

10

0

025

05

075

1

0

2

8

Figure 4 The max-min composition of fuzzy relations and 119878

0

025

05

075

1

0

2

4

6

8

10

4

6

10

0

2

8

Figure 5 The max-product composition of fuzzy relations and 119878


0

025

05

075

1

4

6

10

0

2

8

0

2

4

6

8

10

Figure 6 The max-av composition of fuzzy relations and 119878

Example 9 Let and 119878 be defined by the following matrices[44]

=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

)

119878 =

1199101

1199102

1199103

1199104

1199105

1199111

1199112

1199113

1199114

(

09 0 03 04

02 1 08 0

08 0 07 1

04 02 03 0

0 1 0 08

)

(25)

We will first compute the max-min composition 119878 Wewill show in details the determination for 119909 = 119909

1 119911 = 119911

1 Let

119909 = 1199091 119911 = 119911

1 and 119910 = 119910

119894 1 le 119894 le 5

(i) and120583(1199091 1199101) 120583119878(1199101 1199111) = and01 09 = 01

(ii) and120583(1199091 1199102) 120583119878(1199102 1199111) = and02 02 = 02

(iii) and120583(1199091 1199103) 120583119878(1199103 1199111) = and0 08 = 0

(iv) and120583(1199091 1199104) 120583119878(1199104 1199111) = and1 04 = 04

(v) and120583(1199091 1199105) 120583119878(1199105 1199111) = and07 0 = 0

119878 (1199091 1199111) = ((119909

1 1199111) 120583119878

(1199091 1199111))

= ((1199091 1199111) or 01 02 0 04 0)

= ((1199091 1199111) 04)

(26)

In analogy to the above computation we now deter-mine the grades of membership for all pairs (119909

119894 119911119894)

1 le 119894 le 3 1 le 119895 le 4 and finally we have

119878 =1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

04 07 03 07

03 1 05 08

08 03 07 1

) (27)

For the max-prod we obtain119909 = 119909

1 119911 = 119911

1 and 119910 = 119910

119894 1 le 119894 le 5

(vi) 120583(1199091 1199101) sdot 120583119878(1199101 1199111) = 01 sdot 09 = 009

(vii) 120583(1199091 1199102) sdot 120583119878(1199102 1199111) = 02 sdot 02 = 004

(viii) 120583(1199091 1199103) sdot 120583119878(1199103 1199111) = 0 sdot 08 = 0

(ix) 120583(1199091 1199104) sdot 120583119878(1199104 1199111) = 1 sdot 04 = 04

(x) 120583(1199091 1199105) sdot 120583119878(1199105 1199111) = 07 sdot 0 = 0

then

119878 (1199091 1199111) = ((119909

1 1199111) 120583119878

(1199091 1199111))

= ((1199091 1199111) 009 or 004 or 0 or 04 or 0)

= ((1199091 1199111) 04)

(28)

After performing the remaining computations weobtain

∘sdot119878 =

1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

04 07 03 056

027 1 04 08

08 03 07 1

) (29)

The max-av composition finally yieldsi 120583 (119909

1 119910119894) + 120583 (119910

119894 1199111)

1 12 043 084 145 07

then

1

2sdot or119910120583(1199091 119910119894) + 120583119878(119910119894 1199111) =

1

2sdot (14) = 07

∘av119878 =1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

07 085 065 075

06 1 065 09

09 065 085 1

)

(30)

These operations are respectively illustrated by Figures 7 89 10 11 12 13 14 15 16 17 18 19 20 21 and 22

Remark 10 The vectors and S

are particular vectorscharacterizing respectively the domain and codomain of which are defined in (14) and (15)


0

025

05

075

1

Grade

3

V

1

2

1

2

3

4

5

W

Figure 7 The fuzzy relation

0

025

05

075

1

Grade

1

2

3

4

5

W

3

V

1

2

Figure 8 The complement of fuzzy relation

0

025

05

075

1

Grade

3

V

1

2

1

2

3

4

5

W

Figure 9 The fuzzy relation cup =

0

025

05

075

1

Grade

3

V

1

2

1

2

3

4

5

W

Figure 10 The fuzzy relation cap = 0


0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8

Figure 11 The demonic fuzzy relation 119876

0

025

05

075

1

4

6

10

02

80

2

4

6

8

10

Figure 12 A demonic fuzzy relation

0

025

05

075

1

4

6

10

02

80

2

4

6

8

10

Figure 13 Demonic union of fuzzy relations 119876 and

0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8

Figure 14 Angelic union of fuzzy relations 119876 and


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10

Figure 15 Demonic intersection of fuzzy relations 119876 and

0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8

Figure 16 Angelic intersection of fuzzy relations 119876 and

50

1

2

4

5

0

1

2

3

3

4

0

025

05

075

1


0

025

05

075

1

0

1

2

4

5

3

5

0

1

2

34

Figure 18 The demonic fuzzy relation


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

0

1

2

4

5

3

5

0

1

2

3

4


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3



42 Properties of Fuzzy Relations Just as for relationsthe properties of commutativity associativity distributivityinvolution and idempotency all hold for fuzzy relationsMoreover DeMorganrsquos principles hold for fuzzy relations justas they do for relations and the empty relation 0 and theuniversal relation are analogous to the empty set and thewhole set in set-theoretic form respectively Fuzzy relationsare not constrained as is the case for fuzzy sets in generalby the excluded middle axioms Since a fuzzy relation isalso a fuzzy set there is overlap between a relation and itscomplement [45] hence

cup = cap = 0 (31)

Example 11 Let

=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

) (32)

Then

(i) =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 0 03

07 05 1 08 0

02 1 0 06 07

)

(ii) cap =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 0 03

03 05 0 02 0

02 0 0 04 03

)= 0

(iii) cup =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 1 07

07 05 1 08 1

08 1 1 06 07

)=

(33)

In the following we will give some properties of fuzzyrelations

Definition 12 Let 1and

2be fuzzy relations on 119860 times 119861

one have 1= ((119909 119910) 120583

1

(119909 119910)) 2= ((119909 119910) 120583

2

(119909 119910))Then one has the following

(a) Equality

1= 2if and only if 120583

1

(119909 119910) = 1205832

(119909 119910) (34)

(b) Inclusion

(i) If 1205831

(119909 119910) le 1205832

(119909 119910) the relation 1is

included in 2or 2is larger

1 denoted by

1sube 2

(ii) If 1sube 2and in addition if for at least one pair

(119909 119910)

1205831

(119909 119910) lt 1205832

(119909 119910) (35)

Then one has the proper inclusion 1sube 2

The following properties in Theorem 13 and Proposition 14have been proved to hold for fuzzy relations (see [11 46])

Theorem 13 Let 119878 and be fuzzy relations Then

(a) associativity of composition (119878) = (119878)

(b) distributivity over union (119878 cup ) = (119878) cup ()

(c) weak distributivity over intersection (119878cap) sube (119878)cap

()

(d) commutativity cap 119878 = 119878 cap cup 119878 = 119878 cup

(e) associativity cap (119878 cap ) = ( cap 119878) cap cup (119878 cup ) =

( cup 119878) cup

(f) distributivity cap(119878cup) = (cap119878)cup(cap) cup(119878cap) =

( cup 119878) cap ( cup )

(g) idempotency cup = cap =

(h) identity cap 120601 = 120601 cup 120601 = cap = cup =

(i) involution =

(j) De Morgans law cup 119878 = cap 119878 cap 119878 = cup 119878

Proposition 14 A fuzzy relation sube 119860 times 119860 is as follows

(a) reflexive if and if only 119868 sube that is (forall119909 120583(119909 119909) =

1)

(b) transitive if and if only sube that is (forall119909 119910 119911

120583(119909 119911) ge and120583

(119909 119910) 120583

(119910 119911))

(c) symmetric if and if only sube S that is (forall119909 119910

120583(119909 119910) = 120583

(119910 119909)

(d) antisymmetric if and if only S sube cup 119868 that is

forall119909 119910 120583(119909 119910) = 120583

(119910 119909)

119900119903 120583(119909 119910) = 120583

(119910 119909) = 0

(36)

(e) equivalence if and if only verifies properties (a) (b)and (c)

(f) order if and if only verifies properties (a) (b) and (d)

(g) preorder if and if only verifies properties (a) and (b)


Example 15 Let 119878 119876 and be fuzzy relations

(i) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

1 0 02 03

0 1 01 1

02 07 1 04

0 1 04 1

)

(ii) 119878 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

02 1 04 04

0 06 03 0

0 1 03 0

01 1 1 01

)

(iii) =

1199091

1199092

1199093

1199094

1199095

1199096

1199101

1199102

1199103

1199104

1199105

1199106

(

(

1 02 1 06 02 06

02 1 02 02 08 02

1 02 1 06 02 06

06 02 06 1 02 08

02 08 02 02 1 02

06 02 06 08 02 1

)

)

(iv) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

04 0 07 0

0 1 09 06

08 04 07 04

0 01 0 0

)

(v) 119876 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

0 01 0 01

01 1 02 03

0 02 08 08

01 03 08 1

)

(37)

is a reflexive fuzzy relation 119878 is a transitive fuzzy relation119876 is a symmetric fuzzy relation is an antisymmetric fuzzyrelation and is an equivalence fuzzy relation

5 Demonic Fuzzy Order andFuzzy Demonic Operators

First we will give the rationale behind the definition ofrefinement called the refinement ordering If we consider arelation 119877 as a specification of the input-output behavior of aprogram 119901 then 119901 refines 119877 (or that 119901 is correct with respectto 119877) if

(i) for any input 119894 in the domain of 119877 1198941015840 is a possible

output of 119901 only if (119894 1198941015840) isin 119877(ii) 119901 always terminates for any input belonging to the

domain of119877 [47] For an input that does not belong tothe domain of specification 119877 program 119901 may returnany result or return no result that is the specifier doesnot care what happens following the submission ofsuch an input

In the following we will define the refinement fuzzyordering (demonic fuzzy inclusion) The associated fuzzyoperators are fuzzy demonic join (⊔) fuzzy demonic meet

(⊓) and fuzzy demonic composition (◻) We will give thedefinitions and needed properties of these operators Wewill illustrate them with simple examples using mathematica(fuzzy logic)

Definition 16 One says that a fuzzy relation 119876 fuzzy refines afuzzy relation denoted by 119876 ⊑ if and if only

or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(38)

where 120583and 120583

are respectively the membership functions

of and 119876

In other words 119876 fuzzy refines if and only if theprerestriction of 119876 to the domain of is included in thismeans that 119876 must not produce results not allowed by forthose states that are in the domain of

It is easy to show that this definition is equivalent todefinition (given [43]) In other words

119876 sube and 119876 cap sube (39)

if and only if

or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(40)

Example 17 Let

119876 = (01 0

1 02) = (

01 01

04 04) (41)

We have

or119910isin119861

120583

= (01

04) sube (

01

1) = or

119910isin119861120583

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 0

04 02)

sube (01 01

04 04)

= 120583

(42)

Then

119876 ⊑ (43)

Let

= (

03 02 05

04 05 09

01 02 07

) 119878 = (

03 02 04

07 08 08

03 05 06

) (44)


We have

or119910isin119861

120583119878= (

04

08

06

) sube (

05

09

07

) = or119910isin119861

120583

and 120583(119909 119910) or

119910isin119861120583119878(119909 119910) = (

03 02 04

04 05 08

01 02 06

)

sube (

03 02 04

07 08 08

03 05 06

)

= 120583119878

(45)

Then

⊑ 119878 (46)

Let

119876 = (01 02 04

05 07 09) = (

02 02 03

04 05 08)

119876 ⊑

(47)

because

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 02 03

05 07 08)

sube (02 02 03

04 05 08)

= 120583

(48)

Let

= (05 02 07

07 0 03) 119878 = (

01 03 04

09 1 05)

⊑ 119878

(49)

because

or119910isin119861

120583119878= (

04

1) sube (

07

07) = or

119910isin119861sube 120583 (50)

Theorem 18 The fuzzy relation ⊑ is a partial order

51 Fuzzy Demonic Operators and Illustration with Mathe-matica In this subsection we will present fuzzy demonicoperators and some of their properties

Definition 19 Let 119876 and be fuzzy relations

(a) Their supremum is 119876 ⊔ and their membership is

120583(⊔)

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)] and [or

119910isin119861(120583(119909 119910))]

and [or119910isin119861

(120583(119909 119910))]

(51)

and satisfies

or119910isin119861

120583( ⊔ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(52)

The operator (⊔) is called fuzzy demonic union Thisdefinition is equivalent to the definition ([43]) Inother words

119876 ⊔ 119877 = (119876 cup 119877) cap 119876119871 cap 119877119871 (53)

if and only if

120583( ⊔ )

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)]

and [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(54)

(b) Their infimum if it exists is119876 ⊓ and their member-ship is

120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(55)

and it satisfies

or119910isin119861

120583( ⊓ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] or [or

119910isin119861(120583(119909 119910))]

(56)

The operator (⊓) is called fuzzy demonic intersectionThis definition is equivalent to the definition given in([18 19 36 43 48ndash50]) In other words

119876 ⊓ 119877 = 119876 cap 119877 cup 119876 cap 119877119871 cup 119877 cap 119876119871 (57)

if and only if

120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(58)

For 119876 ⊓ to exist one has to verify

120583(119909 119910) sube or

119910isin119861[(120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))

and (120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))]

(59)


This condition is equivalent to

(or119910isin119861

120583(119909 119910)) and (or

119910isin119861120583(119909 119910))

sube or119910isin119861

(120583(119909 119910) and 120583

(119909 119910))

(60)

which can be interpreted as follows the existencecondition simply means that on the intersection oftheir domains 119876 and have to agree for at least onevalue

Example 20 We know that119876 ⊔ = 119876cup and119876 ⊓ = 119876cap

(a) Let 119876 = (01 0 02

03 08 1

0 1 07) and = (

0 1 0

03 05 04

09 07 02)

(i) 119876 ⊔ = (01 02 02

03 05 05

09 09 07) but 119876 cup = (

01 1 02

03 08 1

09 1 07)

(ii) 119876 ⊓ = (0 08 0

03 05 05

0 07 02) but 119876 cap = (

0 0 0

03 05 04

0 07 02)

(b) Let 119876 = ( 03 0102 05

) and = ( 01 01 07

)

(i) 119876 ⊔ = ( 01 0105 05

) but 119876 cup = ( 03 011 07

)

(ii) 119876 ⊓ = ( 03 0105 05

) but 119876 cap = ( 01 002 05

)

These demonic operations are illustrated respectively byFigures 23 and 24

Now we need to define the relative fuzzy implicationIn what follows we will give our definition of the relative

fuzzy implication and some examples

Definition 21 The binary operator (⊳) is called relative fuzzyimplication and its a membership function is defined asfollows

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (61)

This definition is equivalent to definition ([18 19 36 48ndash50])which is

119876 ⊳ 119877def= 119876119877 (62)

if and only if

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (63)

Example 22 (a) Let 119876 = ( 0 0103 05

) and = ( 09 0207 1

)

(i) 119876 ⊳ = ( 0 0103 05

)

(b) Let 119876 = (01 02 04

05 05 1

0 0 03) and = (

07 08 09

1 0 01

06 02 03)

(i) 119876 ⊳ = (06 06 06

06 02 03

07 07 07)

In what follows we will give the definition of the fuzzydemonic composition

0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3

Figure 23 Demonic composition in Example 24 (a)

0

025

05

075

1

4

6

10

02

80

2

4

6

8

10

Figure 24 Demonic composition in Example 24 (b)


Definition 23 The fuzzy demonic composition of relations 119876

and is (119876 ◻ ) and its membership function is given by

120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)

1 minus or119910isin119861

and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)

Example 24 Consider the following

(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)

Figures 23 and 24 represent fuzzy demonic composition oftwo relations

6 Conclusion

In this paper we have presented the notion of relationalfuzzy calculus specially a fuzzy refinement order (⊑) also thedefinitions of the operators associated to this order which are(⊳) fuzzy demonic operators (⊓ ⊔) and fuzzy composition(◻) and give some of their properties These operators havebeen illustrated by mathematica (fuzzy logic)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research project was supported by a grant from theResearch Center of the Center for Female Scientific andMedical Colleges Deanship of Scientific Research King SaudUniversity

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[2] W Pedrycz ldquoProcessing in relational structures fuzzy relationalequationsrdquo Fuzzy Sets and Systems vol 40 no 1 pp 77ndash1061991

[3] J A Goguen ldquo119871-fuzzy setsrdquo Journal of Mathematical Analysisand Applications vol 18 pp 145ndash174 1967

[4] E Sanchez ldquoResolution of composite fuzzy relation equationsrdquoInformation and Computation vol 30 no 1 pp 38ndash48 1976

[5] R T Yeh and S Y Bang ldquoFuzzy graphs fuzzy relations andtheir applications to cluster analysisrdquo in Fuzzy Sets and theirApplications to Cognitive and Decision Processes L A Zadeh KS Fu K Tanaka and M Shimura Eds pp 125ndash149 AcademicPress New York NY USA 1975

[6] H J Zimmermann Fuzzy Set Theory And its ApplicationsKluwer Academic Boston Mass USA 2nd edition 1990

[7] H Zimmermann ldquoTestability and meaning of mathematicalmodels in social sciencesrdquo Mathematical Modelling vol 1 no1 pp 123ndash139 1980

[8] M A Vila and M Delgado ldquoOn medical diagnosis usingpossibility measuresrdquo Fuzzy Sets and Systems vol 10 no 3 pp211ndash222 1983

[9] H Furusawa Algebraic Formalisations of Fuzzy Relations andTheir Representation Theorems Department of InformaticsKyushu University Fukuoka Japan 1998

[10] E H Mamdani Advances in the Linguistic Syntesis of FuzzyControllers in Fuzzy Reasoning and its Applications AcademicPress London UK 1981

[11] A Kaufmann Theorie des Sous-Ensembles Flous (Fuzzy SetTheory) vol 1 Masson Paris France 2nd edition 1977

[12] J A Goguen ldquoThe logic of inexact conceptsrdquo Synthese vol 19no 3-4 pp 325ndash373 1969

[13] G Boole The Mathematical Analysis of Logic Being an EssayToward a Calculus of Deductive Reasoning Macmillan Cam-bridge Mass USA 1847

[14] E Strohlein Vorlesungen uber die Algebra der Logik (exacteLogik) Part 1 vol 3 of Algebra und Logik der Relative TeubnerLeipzig Germany 1966 2nd edition published byChelsea 1966

[15] A Tarski ldquoOn the calculus of relationsrdquoThe Journal of SymbolicLogic vol 6 no 3 pp 73ndash89 1941

[16] R J R Back ldquoOn correct refinement of programsrdquo Journal ofComputer and System Sciences vol 23 no 1 pp 49ndash68 1981

[17] J M Morris ldquoA theoretical basis for stepwise refinement andthe programming calculusrdquo Science of Computer Programmingvol 9 no 3 pp 287ndash306 1987

[18] F Tchier ldquoDemonic semantics by monotypesrdquo in Proceedingsof the International Arab Conference on Information Technology(Acit rsquo02) University of Qatar Doha Qatar December 2002

[19] F Tchier ldquoWhile loop demonic relational semantics mono-typeresidual stylerdquo in Proceedings of the International Confer-ence on Software Engineering Research and Practice (SERPrsquo03)pp 23ndash26 Las Vegas Nev USA June 2003

[20] N Wirth ldquoProgram development by stepwise refinementrdquoCommunications of the ACM vol 14 no 4 pp 221ndash227 1971

[21] N Boudriga F Elloumi and A Mili ldquoOn the lattice ofspecifications applications to a specification methodologyrdquoFormal Aspects of Computing vol 4 no 6 pp 544ndash571 1992

[22] J Desharnais A Mili and T T Nguyen ldquoRefinement anddemonic semanticsrdquo inRelational methods in Computer ScienceC Brink W Khal and G Schmidt Eds Advances in Comput-ing Springer New York NY USA 1997

[23] A de Morgan ldquoOn the syllogism III and on logic in generalrdquoTransactions of the Cambridge Philosophical Society vol 10 pp173ndash230 1864

[24] A de Morgan ldquoOn the symbols of logic the theory of thesyllogism and in particular of the Copula and the applicationof the theory of probabilities to some questions in the theory of


evidencerdquo Transactions of the Cambridge Philosophical Societyvol 9 pp 79ndash127 1856

[25] C S Peirce ldquoOn the algebra of logic a contribution to thephilosophy of notationrdquo American Journal of Mathematics vol7 no 2 pp 180ndash196 1885

[26] C S Peirce ldquoOn the algebra of logicrdquo American Journal ofMathematics vol 3 no 1 pp 15ndash57 1880

[27] L H Chin and A Tarski Distributive and Modular Laws in theArithmetic of Relation Algebras vol 1 University of CaliforniaPublications 1951

[28] B Jonsson ldquoThe theory of binary relationsrdquo in Algebairc Logicpp 245ndash292 Colloquia Mathematica Societatis AmsterdamThe Netherlands 1991

[29] B Jonsson ldquoVarieties of relation algebrasrdquo Algebra Universalisvol 15 no 3 pp 273ndash298 1982

[30] B Jonnson andA Tarski ldquoBoolean algebraswith operators partIIrdquo The American Journal of Mathematics vol 74 pp 127ndash1621952

[31] B Jonsson and A Tarski ldquoRepresentation problems for relationalgebrardquo Bulletin of the American Mathematical Society vol 801948

[32] R D Maddux ldquoIntroductory course on relations algebrasfinite-dimensional cylindric algebras and their interconnec-tionsrdquo Colloquium Mathematical Society vol 54 pp 361ndash3921991

[33] R D Maddux ldquoThe origin of relation algebras in the devel-opment and axiomatization of the calculus of relationsrdquo StudiaLogica vol 50 no 3-4 pp 421ndash455 1991

[34] G Schmidt and T Strohlein Relations and Graphs EATCSMonographs in Computer Science Springer Berlin Germany1993

[35] R Berghammer and H Zierer ldquoRelational algebraic semanticsof deterministic and nondeterministic programsrdquo TheoreticalComputer Science vol 43 no 2-3 pp 123ndash147 1986

[36] F Tchier ldquoDemonic relational semantics of compound dia-gramsrdquo in Relational Methods in computer Science The QuebecSeminar J Desharnais M Frappier andWMacCaull Eds pp117ndash140 Methods Publishers 2002

[37] G Schmidt and T Strohlein ldquoRelation algebras concept ofpoints and representabilityrdquo Discrete Mathematics vol 54 no1 pp 83ndash92 1985

[38] G Schmidt W G Kahl and C Brink Relational Methods inComputer Science Springer Berlin Germany 1997

[39] L A Zadeh ldquoSimilarity relations and fuzzy orderingsrdquo vol 3pp 177ndash200 1971

[40] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[41] S Tamura S Higuchi and K Tanaka ldquoPattern classificationbased on fuzzy relationsrdquo IEEE Transactions on Systems Manand Cybernetics vol 1 pp 61ndash66 1971

[42] W Pedrycz and F Gomide An Introduction to Fuzzy SetsAnalysis and Design 1998

[43] F Tchier Semantiques relationnelles demoniaques et verificationde boucles non deterministes [Theses of doctorat] Departementde Mathematiques et de Statistique Universite Laval QuebecCanada 1996

[44] A Kaufmann Introduction to the Theory of Fuzzy Subsets vol1 Academic Press New York NY USA 1975

[45] T J Ross Fuzzy Logic with Engineering Applications Universityof New Mexico 2004

[46] Y Kawahara and H Furusawa ldquoAn algebraic formatisation offuzzy relationsrdquo in Proceedings of the 2nd International Seminaron Relational Methods in Computer Science Rio Brazil July1995

[47] A Mili ldquoA relational approach to the design of deterministicprogramsrdquo Acta Informatica vol 20 no 4 pp 315ndash328 1983

[48] J Desharnais BMoller and F Tchier ldquoKleene under a demonicstarrdquo in Proceedings of the 8th International Conference onAlgebraic Methodology And Software Technology (AMAST rsquo00)vol 1816 of Lecture Notes in Computer Science pp 355ndash370Springer Iowa City Iowa USA May 2000

[49] J Desharnais N Belkhiter S Ben Mohamed Sghaier et alldquoEmbedding a demonic semilattice in a relation algebrardquoTheoretical Computer Science vol 149 no 2 pp 333ndash360 1995

[50] J Desharnais and F Tchier ldquoDemonic relational semanticsof sequential programsrdquo Rapport de Recherche DIUL-RR-9406 Departement drsquoInformatique Universite Laval QuebecCanada 1995

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Stochastic AnalysisInternational Journal of


studied It can also be considered as amodeling language wellsuited for situations in which fuzzy relations criteria andphenomena exist It will mean different things dependingon the application area and the way it is measured Inthe meantime numerous authors have contributed to thistheory In 1984 as many as 4000 publications may alreadyexist The first publications in fuzzy set theory by Zadeh[1] and Goguen [3 12] show the intention of the authorsto generalize the classical notion of a set Zadeh [1] writesldquoThe notion of a fuzzy set provides a convenient point ofdeparture for the construction of a conceptual frameworkwhich parallels in many respects the framework used in thecase of ordinary sets but is more general than the latterand potentially may prove to have a much wider scopeof applicability particularly in the fields of mathematicsand computer science (pattern classification and informationprocessing) Fuzzy logic is a superset of conventional logicthat has been extended to handle the concept of partialtruth-truth values between ldquocompletely truerdquo and ldquocompletelyfalserdquo As its name suggests it is the logic underlying modesof reasoning which are approximate rather than exact Theimportance of fuzzy theory derives from the fact that mostmodes of human reasoning and especially common sensereasoning are approximate in naturerdquo

The calculus of relations has been an important compo-nent of the development of logic and algebra since themiddleof the nineteenth century [13ndash15] The main advantages ofthe relational formalization are uniformity and modularityActually once problems in these fields are formalized in termsof relational calculus these problems can be considered byusing formulae of relations that is we need only calculusof relations in order to solve the problems In the contextof software development one important approach is thatof developing programs from specifications by stepwiserefinement see for example [16ndash20] One point of view isthat a specification is a relation constraining the input-output(resp argument-result) behaviour of programs

The demonic calculus of relations [21 22] views anyrelation 119877 from a set 119860 to another set 119861 as specifying thoseprograms that terminate for all 119886 isin 119860 wherever 119877 associatesany values from 119861 with 119886 and then the program may onlyreturn values 119887 for which (119886 119887) isin 119877 Consequently a relation119877 refines another relation 119878 if 119877 specifies a larger domainof termination and fewer possibilities for return values Thedemonic calculus of relations has the advantage that thedemonic operations are defined on top of the conventionalrelation algebraic operations and can easily and usefully bemixed with the latter allowing the application of numerousalgebraic properties

In Section 2 we present our mathematical tool namelyrelational algebra First we will give certain notions aboutelementary theory on relations and ordered structuresTherea concept of type can be defined that allows an abstracttreatment of the domain (of definedness) of an element andalso of assertions After some auxiliary results (Section 3on relational calculus and Section 4 on fuzzy calculus) wepresent in Section 5 notions andproperties of demonic fuzzyoperators Finally we conclude our work in Section 6

2 Elementary Theory on Relations

Using Tarski [15] approach we distinguish two levels ofabstraction in the study of binary relations the elementarytheory of relations and relational calculus First level definesthe relations as sets of (pairs) and second level defines therelations as elementary objects on which the operations aredefined and studied in an algebraic point of view

In this paper we need both levels of abstraction The firstone will give us our examples and the second one is useful forour proofs and formulas So in this section we will presentboth levels A relation 119877 from a set 119883 to a set 119884 is a subset ofpairs (119909 119910) where 119909 isin 119883 and 119910 isin 119884 Formally

119877 sube 119883 times 119884 = (119909 119910) | 119909 isin 119883 and 119910 isin 119884 (1)

If119883 = 119884 then 119877 is homogeneous on119883 The relations on finitesets can be represented by matrices For example the relation119877 = (119886 119887) (119886 119888) (119887 119889) (119888 119889) (119889 119890) can be represented by

119877 =

119886

119887

119888

119889

119890

119886 119887 119888 119889 119890

(

0 1 1 0 0

0 0 0 1 0

0 0 0 1 0

0 0 0 0 1

0 0 0 0 0

) (2)

The graphs and relations are closely linkedEvery finite relation can be interpreted as a representation

graph and vice versa Graph (1) which corresponds to therelation 119877 is shown in Figure 1

In matrix notation an entry 0 corresponds to the absenceof an edge between two vertices of the graph (the absence ofthe pair in the relation) and an entry 1 means the opposite

As relations are sets they are ordered by inclusion Theleast relation between sets 119883 and 119884 is the empty (also calledzero) noted Oslash

119883119884 and the greatest one called universal

relation and noted 119871119883119884

A particular relation defined forevery set 119883 is the identity relation noted 119868

119883

def= (119909 119909) |

119909 isin 119883 The set of elements of 119883 whose images by 119877 is calleddomain of 119877 noted dom(119877) and the set of images is notedimg(119877) Formally

dom (119877)def= 119909 | (exist119910 (119909 119910) isin 119877)

img (119877)def= 119910 | (exist119909 (119909 119910) isin 119877)

(3)

As relations are particular sets we can apply the usualsets operations which are union (cup) intersection (cap) andcomplementation (

mdash) Relations are ordered by inclusion

More their structure helps us to define other operationswhich are as follows

(a) Inverse of a relation 119877 denoted 119877S

119877S

= (119909 119910) | (119910 119909) isin 119877 (4)

(b) For 119877 sube 119883 times 119885 and 119878 sube 119885 times 119884 we define thecomposition of 119877 and 119878 noted 119877 ∘ 119878 as follows

119877 ∘ 119878 = (119909 119910) | (exist119911 119911 isin 119885 (119909 119911) isin 119877 and (119911 119910) isin 119878)

(5)


a

b

c

d e
















119894and ⋂

119894




119899be sets Then

Rdef= 119877 | 119877 sube 119878

119894times 119878119895 1 le 119894 119895 le 119899 (6)








119883)




1198682times2

= (1 0

0 1) Oslash

1times2= (0 0)

1198712times3

= (1 1 1

1 1 1) (

1 0

0 1

1 0

)

S

= (1 0 1

0 1 0)

(1 0 1

0 1 0)(

1 0

0 1

0 1

) = (1 1

0 1)

(7)


(119877 cup 119878)119894119895 = 119877119894119895cup 119878119894119895 (119877)

119894119895= 119877119894119895

(119877119878)119894119895 = ⋃119896

119877119894119896119878119896119895 (119877 cap 119878)119894119895 = 119877

119894119895cap 119878119894119895

(119877S

)119894119895 = (119877119895119894)S

(8)


119871119894119895

= 119871 Oslash119894119895

= Oslash 119868119894119895

= 119868 if 119894 = 119895

Oslash otherwise(9)






(1) ⋃119894isin119883

119877119894= ⋂119894isin119883

119877119894

(2) ⋂119894isin119883

119877119894= ⋃119894isin119883

119877119894


119894isin119883119877119894) = ⋃

119894isin119883119876119877119894

(7) 119875(119876 cup 119877) = 119875119876 cup 119875119877(8) (⋃

119894isin119883119876119894)119877 = ⋃

119894isin119883119876119894119877

(9) (119875 cup 119876)119877 = 119875119877 cup 119876119877(10) 119876(⋂

119894isin119883119877119894) sube ⋂

119894isin119883119876119877119894

(11) (⋂119894isin119883

119876119894)119877 sube ⋂

119894isin119883119876119894119877


119894isin119883119877119894)S

= ⋃119894isin119883

119877S119894

(15) (⋂119894isin119883

119877119894)S

= ⋂119894isin119883

119877S119894

(16) (119876119877)S

= 119877S119876S(17) 119877

S= 119877S


119894isin119883119877119894119871)119871 = ⋂

119894isin119883119877119894119871

(23) (⋃119894isin119883

119877119894119871)119871 = ⋃

119894isin119883119877119894119871









119868 sube 119877S119877)



S sube 119868




V def= 119881 times 119883 = (0 0) (0 1) (0 2) (1 0) (1 1) (1 2) (10)





V cap 119877 = 119886119877 = (0 1) (0 2) (11)


VS cap 119877 = 119877119886 = (0 1) (2 1) (12)


119877119871 = (0 0) (0 1) (0 2) (2 0) (2 1) (2 2) (13)








map 120583









119860

def= ((119909 119910) 120583

119868(119909 119910)) | 119909 isin 119860 where 120583

119868(119909 119910) = 1 if


dom ()def= sup

119910isin119861((119909 119910) 120583

(119909 119910)) | forall119909 isin 119860 (14)


codom ()def= sup

119909isin119860((119909 119910) 120583

(119909 119910)) | forall119910 isin 119861 (15)





= [(119909 119910) 120583(119909 119910)] | (119909 119910) isin 119860 times 119861

119878 = ((119910 119911) 120583119878(119910 119911)) | (119910 119911) isin 119861 times 119862

(16)


S

= [(119909 119910) 120583(119909 119910)] | (119909 119910) isin 119860 times 119861 (17)

such that

= [(119910 119909) 120583(119910 119909)] | (119910 119909) isin 119861 times 119860 (18)


∘ 119878 = [(119909 119911) or119910isin119861 120583 (119909 119910) and 120583119878(119910 119911)] | 119909 isin 119860

119910 isin 119861 119911 isin 119862

(19)



119896 = [(119909 119910) 119896120583(119909 119910)] | (119909 119910) isin 119860 times 119861 (20)


(ii) Let 119860 = 1199091 1199092 1199093 119861 = 119910

1 1199102 1199103 1199104 = ldquo119909


=1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

08 1 01 07

0 08 0 0

09 1 07 08

)

119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

04 0 09 06

09 04 05 07

03 0 08 05

)

(21)

We have

(i) cup 119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

08 1 09 07

09 08 05 07

09 1 08 08

)

(ii) S

=

1199101

1199102

1199103

1199104

1199091

1199092

1199093

(

08 0 09

1 08 1

01 0 07

07 0 08

)

(iii) cap 119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

04 0 01 06

0 04 0 0

03 0 07 05

)

(iv) 119896119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

012 0 027 018

027 012 015 021

09 0 024 015

)

(22)




lows∘sdot119878 = [(119909 119911) or119910 120583

1

(119909 119910) sdot 1205832

(119910 119911)] | 119909 isin 119860

119910 isin 119861 119911 isin 119862

(23)


∘av119878 = [(119909 119911) 1

2or119910120583(119909 119910) + 120583

119878(119910 119911)] | 119909 isin 119860

119910 isin 119861 119911 isin 119862

(24)


4

6

100

2

4

6

8

10

0

025

05

075

1

02

8


4

6

100

2

4

6

8

10

0

025

05

075

1

0

2

8


4

6

100

2

4

6

8

10

0

025

05

075

1

0

2

8


0

025

05

075

1

0

2

4

6

8

10

4

6

10

0

2

8



0

025

05

075

1

4

6

10

0

2

8

0

2

4

6

8

10



=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

)

119878 =

1199101

1199102

1199103

1199104

1199105

1199111

1199112

1199113

1199114

(

09 0 03 04

02 1 08 0

08 0 07 1

04 02 03 0

0 1 0 08

)

(25)


1 119911 = 119911

1 Let

119909 = 1199091 119911 = 119911

1 and 119910 = 119910

119894 1 le 119894 le 5

(i) and120583(1199091 1199101) 120583119878(1199101 1199111) = and01 09 = 01

(ii) and120583(1199091 1199102) 120583119878(1199102 1199111) = and02 02 = 02

(iii) and120583(1199091 1199103) 120583119878(1199103 1199111) = and0 08 = 0

(iv) and120583(1199091 1199104) 120583119878(1199104 1199111) = and1 04 = 04

(v) and120583(1199091 1199105) 120583119878(1199105 1199111) = and07 0 = 0

119878 (1199091 1199111) = ((119909

1 1199111) 120583119878

(1199091 1199111))

= ((1199091 1199111) or 01 02 0 04 0)

= ((1199091 1199111) 04)

(26)


119894 119911119894)


119878 =1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

04 07 03 07

03 1 05 08

08 03 07 1

) (27)


1 119911 = 119911

1 and 119910 = 119910

119894 1 le 119894 le 5

(vi) 120583(1199091 1199101) sdot 120583119878(1199101 1199111) = 01 sdot 09 = 009

(vii) 120583(1199091 1199102) sdot 120583119878(1199102 1199111) = 02 sdot 02 = 004

(viii) 120583(1199091 1199103) sdot 120583119878(1199103 1199111) = 0 sdot 08 = 0

(ix) 120583(1199091 1199104) sdot 120583119878(1199104 1199111) = 1 sdot 04 = 04

(x) 120583(1199091 1199105) sdot 120583119878(1199105 1199111) = 07 sdot 0 = 0

then

119878 (1199091 1199111) = ((119909

1 1199111) 120583119878

(1199091 1199111))

= ((1199091 1199111) 009 or 004 or 0 or 04 or 0)

= ((1199091 1199111) 04)

(28)


∘sdot119878 =

1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

04 07 03 056

027 1 04 08

08 03 07 1

) (29)


1 119910119894) + 120583 (119910

119894 1199111)

1 12 043 084 145 07

then

1

2sdot or119910120583(1199091 119910119894) + 120583119878(119910119894 1199111) =

1

2sdot (14) = 07

∘av119878 =1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

07 085 065 075

06 1 065 09

09 065 085 1

)

(30)





0

025

05

075

1

Grade

3

V

1

2

1

2

3

4

5

W


0

025

05

075

1

Grade

1

2

3

4

5

W

3

V

1

2


0

025

05

075

1

Grade

3

V

1

2

1

2

3

4

5

W


0

025

05

075

1

Grade

3

V

1

2

1

2

3

4

5

W



0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10


0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8



0

025

05

075

1

4

6

10

02

80

2

4

6

8

10


0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8


50

1

2

4

5

0

1

2

3

3

4

0

025

05

075

1


0

025

05

075

1

0

1

2

4

5

3

5

0

1

2

34



0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

0

1

2

4

5

3

5

0

1

2

3

4


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3




cup = cap = 0 (31)

Example 11 Let

=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

) (32)

Then

(i) =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 0 03

07 05 1 08 0

02 1 0 06 07

)

(ii) cap =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 0 03

03 05 0 02 0

02 0 0 04 03

)= 0

(iii) cup =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 1 07

07 05 1 08 1

08 1 1 06 07

)=

(33)




one have 1= ((119909 119910) 120583

1

(119909 119910)) 2= ((119909 119910) 120583

2


(a) Equality


1

(119909 119910) = 1205832

(119909 119910) (34)

(b) Inclusion

(i) If 1205831

(119909 119910) le 1205832



1 denoted by

1sube 2


(119909 119910)

1205831

(119909 119910) lt 1205832

(119909 119910) (35)







()



( cup 119878) cup


( cup 119878) cap ( cup )



(i) involution =




1)


120583(119909 119911) ge and120583

(119909 119910) 120583

(119910 119911))


120583(119909 119910) = 120583

(119910 119909)


forall119909 119910 120583(119909 119910) = 120583

(119910 119909)

119900119903 120583(119909 119910) = 120583

(119910 119909) = 0

(36)






(i) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

1 0 02 03

0 1 01 1

02 07 1 04

0 1 04 1

)

(ii) 119878 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

02 1 04 04

0 06 03 0

0 1 03 0

01 1 1 01

)

(iii) =

1199091

1199092

1199093

1199094

1199095

1199096

1199101

1199102

1199103

1199104

1199105

1199106

(

(

1 02 1 06 02 06

02 1 02 02 08 02

1 02 1 06 02 06

06 02 06 1 02 08

02 08 02 02 1 02

06 02 06 08 02 1

)

)

(iv) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

04 0 07 0

0 1 09 06

08 04 07 04

0 01 0 0

)

(v) 119876 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

0 01 0 01

01 1 02 03

0 02 08 08

01 03 08 1

)

(37)










or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(38)

where 120583and 120583


of and 119876



119876 sube and 119876 cap sube (39)

if and only if

or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(40)

Example 17 Let

119876 = (01 0

1 02) = (

01 01

04 04) (41)

We have

or119910isin119861

120583

= (01

04) sube (

01

1) = or

119910isin119861120583

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 0

04 02)

sube (01 01

04 04)

= 120583

(42)

Then

119876 ⊑ (43)

Let

= (

03 02 05

04 05 09

01 02 07

) 119878 = (

03 02 04

07 08 08

03 05 06

) (44)


We have

or119910isin119861

120583119878= (

04

08

06

) sube (

05

09

07

) = or119910isin119861

120583

and 120583(119909 119910) or

119910isin119861120583119878(119909 119910) = (

03 02 04

04 05 08

01 02 06

)

sube (

03 02 04

07 08 08

03 05 06

)

= 120583119878

(45)

Then

⊑ 119878 (46)

Let

119876 = (01 02 04

05 07 09) = (

02 02 03

04 05 08)

119876 ⊑

(47)

because

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 02 03

05 07 08)

sube (02 02 03

04 05 08)

= 120583

(48)

Let

= (05 02 07

07 0 03) 119878 = (

01 03 04

09 1 05)

⊑ 119878

(49)

because

or119910isin119861

120583119878= (

04

1) sube (

07

07) = or

119910isin119861sube 120583 (50)





120583(⊔)

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)] and [or

119910isin119861(120583(119909 119910))]

and [or119910isin119861

(120583(119909 119910))]

(51)

and satisfies

or119910isin119861

120583( ⊔ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(52)


119876 ⊔ 119877 = (119876 cup 119877) cap 119876119871 cap 119877119871 (53)

if and only if

120583( ⊔ )

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)]

and [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(54)


120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(55)

and it satisfies

or119910isin119861

120583( ⊓ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] or [or

119910isin119861(120583(119909 119910))]

(56)



if and only if

120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(58)


120583(119909 119910) sube or

119910isin119861[(120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))

and (120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))]

(59)



(or119910isin119861

120583(119909 119910)) and (or

119910isin119861120583(119909 119910))


(120583(119909 119910) and 120583

(119909 119910))

(60)



(a) Let 119876 = (01 0 02

03 08 1

0 1 07) and = (

0 1 0

03 05 04

09 07 02)

(i) 119876 ⊔ = (01 02 02

03 05 05

09 09 07) but 119876 cup = (

01 1 02

03 08 1

09 1 07)

(ii) 119876 ⊓ = (0 08 0

03 05 05

0 07 02) but 119876 cap = (

0 0 0

03 05 04

0 07 02)

(b) Let 119876 = ( 03 0102 05

) and = ( 01 01 07

)

(i) 119876 ⊔ = ( 01 0105 05

) but 119876 cup = ( 03 011 07

)

(ii) 119876 ⊓ = ( 03 0105 05

) but 119876 cap = ( 01 002 05

)





120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (61)


119876 ⊳ 119877def= 119876119877 (62)

if and only if

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (63)

Example 22 (a) Let 119876 = ( 0 0103 05

) and = ( 09 0207 1

)

(i) 119876 ⊳ = ( 0 0103 05

)

(b) Let 119876 = (01 02 04

05 05 1

0 0 03) and = (

07 08 09

1 0 01

06 02 03)

(i) 119876 ⊳ = (06 06 06

06 02 03

07 07 07)


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10





120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)


(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)


6 Conclusion




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










a

b

c

d e
















119894and ⋂

119894




119899be sets Then

Rdef= 119877 | 119877 sube 119878

119894times 119878119895 1 le 119894 119895 le 119899 (6)








119883)




1198682times2

= (1 0

0 1) Oslash

1times2= (0 0)

1198712times3

= (1 1 1

1 1 1) (

1 0

0 1

1 0

)

S

= (1 0 1

0 1 0)

(1 0 1

0 1 0)(

1 0

0 1

0 1

) = (1 1

0 1)

(7)


(119877 cup 119878)119894119895 = 119877119894119895cup 119878119894119895 (119877)

119894119895= 119877119894119895

(119877119878)119894119895 = ⋃119896

119877119894119896119878119896119895 (119877 cap 119878)119894119895 = 119877

119894119895cap 119878119894119895

(119877S

)119894119895 = (119877119895119894)S

(8)


119871119894119895

= 119871 Oslash119894119895

= Oslash 119868119894119895

= 119868 if 119894 = 119895

Oslash otherwise(9)






(1) ⋃119894isin119883

119877119894= ⋂119894isin119883

119877119894

(2) ⋂119894isin119883

119877119894= ⋃119894isin119883

119877119894


119894isin119883119877119894) = ⋃

119894isin119883119876119877119894

(7) 119875(119876 cup 119877) = 119875119876 cup 119875119877(8) (⋃

119894isin119883119876119894)119877 = ⋃

119894isin119883119876119894119877

(9) (119875 cup 119876)119877 = 119875119877 cup 119876119877(10) 119876(⋂

119894isin119883119877119894) sube ⋂

119894isin119883119876119877119894

(11) (⋂119894isin119883

119876119894)119877 sube ⋂

119894isin119883119876119894119877


119894isin119883119877119894)S

= ⋃119894isin119883

119877S119894

(15) (⋂119894isin119883

119877119894)S

= ⋂119894isin119883

119877S119894

(16) (119876119877)S

= 119877S119876S(17) 119877

S= 119877S


119894isin119883119877119894119871)119871 = ⋂

119894isin119883119877119894119871

(23) (⋃119894isin119883

119877119894119871)119871 = ⋃

119894isin119883119877119894119871









119868 sube 119877S119877)



S sube 119868




V def= 119881 times 119883 = (0 0) (0 1) (0 2) (1 0) (1 1) (1 2) (10)





V cap 119877 = 119886119877 = (0 1) (0 2) (11)


VS cap 119877 = 119877119886 = (0 1) (2 1) (12)


119877119871 = (0 0) (0 1) (0 2) (2 0) (2 1) (2 2) (13)








map 120583









119860

def= ((119909 119910) 120583

119868(119909 119910)) | 119909 isin 119860 where 120583

119868(119909 119910) = 1 if


dom ()def= sup

119910isin119861((119909 119910) 120583

(119909 119910)) | forall119909 isin 119860 (14)


codom ()def= sup

119909isin119860((119909 119910) 120583

(119909 119910)) | forall119910 isin 119861 (15)





= [(119909 119910) 120583(119909 119910)] | (119909 119910) isin 119860 times 119861

119878 = ((119910 119911) 120583119878(119910 119911)) | (119910 119911) isin 119861 times 119862

(16)


S

= [(119909 119910) 120583(119909 119910)] | (119909 119910) isin 119860 times 119861 (17)

such that

= [(119910 119909) 120583(119910 119909)] | (119910 119909) isin 119861 times 119860 (18)


∘ 119878 = [(119909 119911) or119910isin119861 120583 (119909 119910) and 120583119878(119910 119911)] | 119909 isin 119860

119910 isin 119861 119911 isin 119862

(19)



119896 = [(119909 119910) 119896120583(119909 119910)] | (119909 119910) isin 119860 times 119861 (20)


(ii) Let 119860 = 1199091 1199092 1199093 119861 = 119910

1 1199102 1199103 1199104 = ldquo119909


=1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

08 1 01 07

0 08 0 0

09 1 07 08

)

119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

04 0 09 06

09 04 05 07

03 0 08 05

)

(21)

We have

(i) cup 119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

08 1 09 07

09 08 05 07

09 1 08 08

)

(ii) S

=

1199101

1199102

1199103

1199104

1199091

1199092

1199093

(

08 0 09

1 08 1

01 0 07

07 0 08

)

(iii) cap 119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

04 0 01 06

0 04 0 0

03 0 07 05

)

(iv) 119896119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

012 0 027 018

027 012 015 021

09 0 024 015

)

(22)




lows∘sdot119878 = [(119909 119911) or119910 120583

1

(119909 119910) sdot 1205832

(119910 119911)] | 119909 isin 119860

119910 isin 119861 119911 isin 119862

(23)


∘av119878 = [(119909 119911) 1

2or119910120583(119909 119910) + 120583

119878(119910 119911)] | 119909 isin 119860

119910 isin 119861 119911 isin 119862

(24)


4

6

100

2

4

6

8

10

0

025

05

075

1

02

8


4

6

100

2

4

6

8

10

0

025

05

075

1

0

2

8


4

6

100

2

4

6

8

10

0

025

05

075

1

0

2

8


0

025

05

075

1

0

2

4

6

8

10

4

6

10

0

2

8



0

025

05

075

1

4

6

10

0

2

8

0

2

4

6

8

10



=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

)

119878 =

1199101

1199102

1199103

1199104

1199105

1199111

1199112

1199113

1199114

(

09 0 03 04

02 1 08 0

08 0 07 1

04 02 03 0

0 1 0 08

)

(25)


1 119911 = 119911

1 Let

119909 = 1199091 119911 = 119911

1 and 119910 = 119910

119894 1 le 119894 le 5

(i) and120583(1199091 1199101) 120583119878(1199101 1199111) = and01 09 = 01

(ii) and120583(1199091 1199102) 120583119878(1199102 1199111) = and02 02 = 02

(iii) and120583(1199091 1199103) 120583119878(1199103 1199111) = and0 08 = 0

(iv) and120583(1199091 1199104) 120583119878(1199104 1199111) = and1 04 = 04

(v) and120583(1199091 1199105) 120583119878(1199105 1199111) = and07 0 = 0

119878 (1199091 1199111) = ((119909

1 1199111) 120583119878

(1199091 1199111))

= ((1199091 1199111) or 01 02 0 04 0)

= ((1199091 1199111) 04)

(26)


119894 119911119894)


119878 =1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

04 07 03 07

03 1 05 08

08 03 07 1

) (27)


1 119911 = 119911

1 and 119910 = 119910

119894 1 le 119894 le 5

(vi) 120583(1199091 1199101) sdot 120583119878(1199101 1199111) = 01 sdot 09 = 009

(vii) 120583(1199091 1199102) sdot 120583119878(1199102 1199111) = 02 sdot 02 = 004

(viii) 120583(1199091 1199103) sdot 120583119878(1199103 1199111) = 0 sdot 08 = 0

(ix) 120583(1199091 1199104) sdot 120583119878(1199104 1199111) = 1 sdot 04 = 04

(x) 120583(1199091 1199105) sdot 120583119878(1199105 1199111) = 07 sdot 0 = 0

then

119878 (1199091 1199111) = ((119909

1 1199111) 120583119878

(1199091 1199111))

= ((1199091 1199111) 009 or 004 or 0 or 04 or 0)

= ((1199091 1199111) 04)

(28)


∘sdot119878 =

1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

04 07 03 056

027 1 04 08

08 03 07 1

) (29)


1 119910119894) + 120583 (119910

119894 1199111)

1 12 043 084 145 07

then

1

2sdot or119910120583(1199091 119910119894) + 120583119878(119910119894 1199111) =

1

2sdot (14) = 07

∘av119878 =1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

07 085 065 075

06 1 065 09

09 065 085 1

)

(30)





0

025

05

075

1

Grade

3

V

1

2

1

2

3

4

5

W


0

025

05

075

1

Grade

1

2

3

4

5

W

3

V

1

2


0

025

05

075

1

Grade

3

V

1

2

1

2

3

4

5

W


0

025

05

075

1

Grade

3

V

1

2

1

2

3

4

5

W



0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10


0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8



0

025

05

075

1

4

6

10

02

80

2

4

6

8

10


0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8


50

1

2

4

5

0

1

2

3

3

4

0

025

05

075

1


0

025

05

075

1

0

1

2

4

5

3

5

0

1

2

34



0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

0

1

2

4

5

3

5

0

1

2

3

4


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3




cup = cap = 0 (31)

Example 11 Let

=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

) (32)

Then

(i) =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 0 03

07 05 1 08 0

02 1 0 06 07

)

(ii) cap =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 0 03

03 05 0 02 0

02 0 0 04 03

)= 0

(iii) cup =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 1 07

07 05 1 08 1

08 1 1 06 07

)=

(33)




one have 1= ((119909 119910) 120583

1

(119909 119910)) 2= ((119909 119910) 120583

2


(a) Equality


1

(119909 119910) = 1205832

(119909 119910) (34)

(b) Inclusion

(i) If 1205831

(119909 119910) le 1205832



1 denoted by

1sube 2


(119909 119910)

1205831

(119909 119910) lt 1205832

(119909 119910) (35)







()



( cup 119878) cup


( cup 119878) cap ( cup )



(i) involution =




1)


120583(119909 119911) ge and120583

(119909 119910) 120583

(119910 119911))


120583(119909 119910) = 120583

(119910 119909)


forall119909 119910 120583(119909 119910) = 120583

(119910 119909)

119900119903 120583(119909 119910) = 120583

(119910 119909) = 0

(36)






(i) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

1 0 02 03

0 1 01 1

02 07 1 04

0 1 04 1

)

(ii) 119878 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

02 1 04 04

0 06 03 0

0 1 03 0

01 1 1 01

)

(iii) =

1199091

1199092

1199093

1199094

1199095

1199096

1199101

1199102

1199103

1199104

1199105

1199106

(

(

1 02 1 06 02 06

02 1 02 02 08 02

1 02 1 06 02 06

06 02 06 1 02 08

02 08 02 02 1 02

06 02 06 08 02 1

)

)

(iv) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

04 0 07 0

0 1 09 06

08 04 07 04

0 01 0 0

)

(v) 119876 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

0 01 0 01

01 1 02 03

0 02 08 08

01 03 08 1

)

(37)










or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(38)

where 120583and 120583


of and 119876



119876 sube and 119876 cap sube (39)

if and only if

or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(40)

Example 17 Let

119876 = (01 0

1 02) = (

01 01

04 04) (41)

We have

or119910isin119861

120583

= (01

04) sube (

01

1) = or

119910isin119861120583

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 0

04 02)

sube (01 01

04 04)

= 120583

(42)

Then

119876 ⊑ (43)

Let

= (

03 02 05

04 05 09

01 02 07

) 119878 = (

03 02 04

07 08 08

03 05 06

) (44)


We have

or119910isin119861

120583119878= (

04

08

06

) sube (

05

09

07

) = or119910isin119861

120583

and 120583(119909 119910) or

119910isin119861120583119878(119909 119910) = (

03 02 04

04 05 08

01 02 06

)

sube (

03 02 04

07 08 08

03 05 06

)

= 120583119878

(45)

Then

⊑ 119878 (46)

Let

119876 = (01 02 04

05 07 09) = (

02 02 03

04 05 08)

119876 ⊑

(47)

because

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 02 03

05 07 08)

sube (02 02 03

04 05 08)

= 120583

(48)

Let

= (05 02 07

07 0 03) 119878 = (

01 03 04

09 1 05)

⊑ 119878

(49)

because

or119910isin119861

120583119878= (

04

1) sube (

07

07) = or

119910isin119861sube 120583 (50)





120583(⊔)

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)] and [or

119910isin119861(120583(119909 119910))]

and [or119910isin119861

(120583(119909 119910))]

(51)

and satisfies

or119910isin119861

120583( ⊔ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(52)


119876 ⊔ 119877 = (119876 cup 119877) cap 119876119871 cap 119877119871 (53)

if and only if

120583( ⊔ )

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)]

and [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(54)


120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(55)

and it satisfies

or119910isin119861

120583( ⊓ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] or [or

119910isin119861(120583(119909 119910))]

(56)



if and only if

120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(58)


120583(119909 119910) sube or

119910isin119861[(120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))

and (120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))]

(59)



(or119910isin119861

120583(119909 119910)) and (or

119910isin119861120583(119909 119910))


(120583(119909 119910) and 120583

(119909 119910))

(60)



(a) Let 119876 = (01 0 02

03 08 1

0 1 07) and = (

0 1 0

03 05 04

09 07 02)

(i) 119876 ⊔ = (01 02 02

03 05 05

09 09 07) but 119876 cup = (

01 1 02

03 08 1

09 1 07)

(ii) 119876 ⊓ = (0 08 0

03 05 05

0 07 02) but 119876 cap = (

0 0 0

03 05 04

0 07 02)

(b) Let 119876 = ( 03 0102 05

) and = ( 01 01 07

)

(i) 119876 ⊔ = ( 01 0105 05

) but 119876 cup = ( 03 011 07

)

(ii) 119876 ⊓ = ( 03 0105 05

) but 119876 cap = ( 01 002 05

)





120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (61)


119876 ⊳ 119877def= 119876119877 (62)

if and only if

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (63)

Example 22 (a) Let 119876 = ( 0 0103 05

) and = ( 09 0207 1

)

(i) 119876 ⊳ = ( 0 0103 05

)

(b) Let 119876 = (01 02 04

05 05 1

0 0 03) and = (

07 08 09

1 0 01

06 02 03)

(i) 119876 ⊳ = (06 06 06

06 02 03

07 07 07)


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10





120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)


(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)


6 Conclusion




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(1) ⋃119894isin119883

119877119894= ⋂119894isin119883

119877119894

(2) ⋂119894isin119883

119877119894= ⋃119894isin119883

119877119894


119894isin119883119877119894) = ⋃

119894isin119883119876119877119894

(7) 119875(119876 cup 119877) = 119875119876 cup 119875119877(8) (⋃

119894isin119883119876119894)119877 = ⋃

119894isin119883119876119894119877

(9) (119875 cup 119876)119877 = 119875119877 cup 119876119877(10) 119876(⋂

119894isin119883119877119894) sube ⋂

119894isin119883119876119877119894

(11) (⋂119894isin119883

119876119894)119877 sube ⋂

119894isin119883119876119894119877


119894isin119883119877119894)S

= ⋃119894isin119883

119877S119894

(15) (⋂119894isin119883

119877119894)S

= ⋂119894isin119883

119877S119894

(16) (119876119877)S

= 119877S119876S(17) 119877

S= 119877S


119894isin119883119877119894119871)119871 = ⋂

119894isin119883119877119894119871

(23) (⋃119894isin119883

119877119894119871)119871 = ⋃

119894isin119883119877119894119871









119868 sube 119877S119877)



S sube 119868




V def= 119881 times 119883 = (0 0) (0 1) (0 2) (1 0) (1 1) (1 2) (10)





V cap 119877 = 119886119877 = (0 1) (0 2) (11)


VS cap 119877 = 119877119886 = (0 1) (2 1) (12)


119877119871 = (0 0) (0 1) (0 2) (2 0) (2 1) (2 2) (13)








map 120583









119860

def= ((119909 119910) 120583

119868(119909 119910)) | 119909 isin 119860 where 120583

119868(119909 119910) = 1 if


dom ()def= sup

119910isin119861((119909 119910) 120583

(119909 119910)) | forall119909 isin 119860 (14)


codom ()def= sup

119909isin119860((119909 119910) 120583

(119909 119910)) | forall119910 isin 119861 (15)





= [(119909 119910) 120583(119909 119910)] | (119909 119910) isin 119860 times 119861

119878 = ((119910 119911) 120583119878(119910 119911)) | (119910 119911) isin 119861 times 119862

(16)


S

= [(119909 119910) 120583(119909 119910)] | (119909 119910) isin 119860 times 119861 (17)

such that

= [(119910 119909) 120583(119910 119909)] | (119910 119909) isin 119861 times 119860 (18)


∘ 119878 = [(119909 119911) or119910isin119861 120583 (119909 119910) and 120583119878(119910 119911)] | 119909 isin 119860

119910 isin 119861 119911 isin 119862

(19)



119896 = [(119909 119910) 119896120583(119909 119910)] | (119909 119910) isin 119860 times 119861 (20)


(ii) Let 119860 = 1199091 1199092 1199093 119861 = 119910

1 1199102 1199103 1199104 = ldquo119909


=1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

08 1 01 07

0 08 0 0

09 1 07 08

)

119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

04 0 09 06

09 04 05 07

03 0 08 05

)

(21)

We have

(i) cup 119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

08 1 09 07

09 08 05 07

09 1 08 08

)

(ii) S

=

1199101

1199102

1199103

1199104

1199091

1199092

1199093

(

08 0 09

1 08 1

01 0 07

07 0 08

)

(iii) cap 119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

04 0 01 06

0 04 0 0

03 0 07 05

)

(iv) 119896119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

012 0 027 018

027 012 015 021

09 0 024 015

)

(22)




lows∘sdot119878 = [(119909 119911) or119910 120583

1

(119909 119910) sdot 1205832

(119910 119911)] | 119909 isin 119860

119910 isin 119861 119911 isin 119862

(23)


∘av119878 = [(119909 119911) 1

2or119910120583(119909 119910) + 120583

119878(119910 119911)] | 119909 isin 119860

119910 isin 119861 119911 isin 119862

(24)


4

6

100

2

4

6

8

10

0

025

05

075

1

02

8


4

6

100

2

4

6

8

10

0

025

05

075

1

0

2

8


4

6

100

2

4

6

8

10

0

025

05

075

1

0

2

8


0

025

05

075

1

0

2

4

6

8

10

4

6

10

0

2

8



0

025

05

075

1

4

6

10

0

2

8

0

2

4

6

8

10



=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

)

119878 =

1199101

1199102

1199103

1199104

1199105

1199111

1199112

1199113

1199114

(

09 0 03 04

02 1 08 0

08 0 07 1

04 02 03 0

0 1 0 08

)

(25)


1 119911 = 119911

1 Let

119909 = 1199091 119911 = 119911

1 and 119910 = 119910

119894 1 le 119894 le 5

(i) and120583(1199091 1199101) 120583119878(1199101 1199111) = and01 09 = 01

(ii) and120583(1199091 1199102) 120583119878(1199102 1199111) = and02 02 = 02

(iii) and120583(1199091 1199103) 120583119878(1199103 1199111) = and0 08 = 0

(iv) and120583(1199091 1199104) 120583119878(1199104 1199111) = and1 04 = 04

(v) and120583(1199091 1199105) 120583119878(1199105 1199111) = and07 0 = 0

119878 (1199091 1199111) = ((119909

1 1199111) 120583119878

(1199091 1199111))

= ((1199091 1199111) or 01 02 0 04 0)

= ((1199091 1199111) 04)

(26)


119894 119911119894)


119878 =1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

04 07 03 07

03 1 05 08

08 03 07 1

) (27)


1 119911 = 119911

1 and 119910 = 119910

119894 1 le 119894 le 5

(vi) 120583(1199091 1199101) sdot 120583119878(1199101 1199111) = 01 sdot 09 = 009

(vii) 120583(1199091 1199102) sdot 120583119878(1199102 1199111) = 02 sdot 02 = 004

(viii) 120583(1199091 1199103) sdot 120583119878(1199103 1199111) = 0 sdot 08 = 0

(ix) 120583(1199091 1199104) sdot 120583119878(1199104 1199111) = 1 sdot 04 = 04

(x) 120583(1199091 1199105) sdot 120583119878(1199105 1199111) = 07 sdot 0 = 0

then

119878 (1199091 1199111) = ((119909

1 1199111) 120583119878

(1199091 1199111))

= ((1199091 1199111) 009 or 004 or 0 or 04 or 0)

= ((1199091 1199111) 04)

(28)


∘sdot119878 =

1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

04 07 03 056

027 1 04 08

08 03 07 1

) (29)


1 119910119894) + 120583 (119910

119894 1199111)

1 12 043 084 145 07

then

1

2sdot or119910120583(1199091 119910119894) + 120583119878(119910119894 1199111) =

1

2sdot (14) = 07

∘av119878 =1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

07 085 065 075

06 1 065 09

09 065 085 1

)

(30)





0

025

05

075

1

Grade

3

V

1

2

1

2

3

4

5

W


0

025

05

075

1

Grade

1

2

3

4

5

W

3

V

1

2


0

025

05

075

1

Grade

3

V

1

2

1

2

3

4

5

W


0

025

05

075

1

Grade

3

V

1

2

1

2

3

4

5

W



0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10


0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8



0

025

05

075

1

4

6

10

02

80

2

4

6

8

10


0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8


50

1

2

4

5

0

1

2

3

3

4

0

025

05

075

1


0

025

05

075

1

0

1

2

4

5

3

5

0

1

2

34



0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

0

1

2

4

5

3

5

0

1

2

3

4


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3




cup = cap = 0 (31)

Example 11 Let

=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

) (32)

Then

(i) =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 0 03

07 05 1 08 0

02 1 0 06 07

)

(ii) cap =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 0 03

03 05 0 02 0

02 0 0 04 03

)= 0

(iii) cup =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 1 07

07 05 1 08 1

08 1 1 06 07

)=

(33)




one have 1= ((119909 119910) 120583

1

(119909 119910)) 2= ((119909 119910) 120583

2


(a) Equality


1

(119909 119910) = 1205832

(119909 119910) (34)

(b) Inclusion

(i) If 1205831

(119909 119910) le 1205832



1 denoted by

1sube 2


(119909 119910)

1205831

(119909 119910) lt 1205832

(119909 119910) (35)







()



( cup 119878) cup


( cup 119878) cap ( cup )



(i) involution =




1)


120583(119909 119911) ge and120583

(119909 119910) 120583

(119910 119911))


120583(119909 119910) = 120583

(119910 119909)


forall119909 119910 120583(119909 119910) = 120583

(119910 119909)

119900119903 120583(119909 119910) = 120583

(119910 119909) = 0

(36)






(i) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

1 0 02 03

0 1 01 1

02 07 1 04

0 1 04 1

)

(ii) 119878 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

02 1 04 04

0 06 03 0

0 1 03 0

01 1 1 01

)

(iii) =

1199091

1199092

1199093

1199094

1199095

1199096

1199101

1199102

1199103

1199104

1199105

1199106

(

(

1 02 1 06 02 06

02 1 02 02 08 02

1 02 1 06 02 06

06 02 06 1 02 08

02 08 02 02 1 02

06 02 06 08 02 1

)

)

(iv) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

04 0 07 0

0 1 09 06

08 04 07 04

0 01 0 0

)

(v) 119876 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

0 01 0 01

01 1 02 03

0 02 08 08

01 03 08 1

)

(37)










or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(38)

where 120583and 120583


of and 119876



119876 sube and 119876 cap sube (39)

if and only if

or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(40)

Example 17 Let

119876 = (01 0

1 02) = (

01 01

04 04) (41)

We have

or119910isin119861

120583

= (01

04) sube (

01

1) = or

119910isin119861120583

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 0

04 02)

sube (01 01

04 04)

= 120583

(42)

Then

119876 ⊑ (43)

Let

= (

03 02 05

04 05 09

01 02 07

) 119878 = (

03 02 04

07 08 08

03 05 06

) (44)


We have

or119910isin119861

120583119878= (

04

08

06

) sube (

05

09

07

) = or119910isin119861

120583

and 120583(119909 119910) or

119910isin119861120583119878(119909 119910) = (

03 02 04

04 05 08

01 02 06

)

sube (

03 02 04

07 08 08

03 05 06

)

= 120583119878

(45)

Then

⊑ 119878 (46)

Let

119876 = (01 02 04

05 07 09) = (

02 02 03

04 05 08)

119876 ⊑

(47)

because

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 02 03

05 07 08)

sube (02 02 03

04 05 08)

= 120583

(48)

Let

= (05 02 07

07 0 03) 119878 = (

01 03 04

09 1 05)

⊑ 119878

(49)

because

or119910isin119861

120583119878= (

04

1) sube (

07

07) = or

119910isin119861sube 120583 (50)





120583(⊔)

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)] and [or

119910isin119861(120583(119909 119910))]

and [or119910isin119861

(120583(119909 119910))]

(51)

and satisfies

or119910isin119861

120583( ⊔ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(52)


119876 ⊔ 119877 = (119876 cup 119877) cap 119876119871 cap 119877119871 (53)

if and only if

120583( ⊔ )

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)]

and [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(54)


120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(55)

and it satisfies

or119910isin119861

120583( ⊓ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] or [or

119910isin119861(120583(119909 119910))]

(56)



if and only if

120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(58)


120583(119909 119910) sube or

119910isin119861[(120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))

and (120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))]

(59)



(or119910isin119861

120583(119909 119910)) and (or

119910isin119861120583(119909 119910))


(120583(119909 119910) and 120583

(119909 119910))

(60)



(a) Let 119876 = (01 0 02

03 08 1

0 1 07) and = (

0 1 0

03 05 04

09 07 02)

(i) 119876 ⊔ = (01 02 02

03 05 05

09 09 07) but 119876 cup = (

01 1 02

03 08 1

09 1 07)

(ii) 119876 ⊓ = (0 08 0

03 05 05

0 07 02) but 119876 cap = (

0 0 0

03 05 04

0 07 02)

(b) Let 119876 = ( 03 0102 05

) and = ( 01 01 07

)

(i) 119876 ⊔ = ( 01 0105 05

) but 119876 cup = ( 03 011 07

)

(ii) 119876 ⊓ = ( 03 0105 05

) but 119876 cap = ( 01 002 05

)





120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (61)


119876 ⊳ 119877def= 119876119877 (62)

if and only if

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (63)

Example 22 (a) Let 119876 = ( 0 0103 05

) and = ( 09 0207 1

)

(i) 119876 ⊳ = ( 0 0103 05

)

(b) Let 119876 = (01 02 04

05 05 1

0 0 03) and = (

07 08 09

1 0 01

06 02 03)

(i) 119876 ⊳ = (06 06 06

06 02 03

07 07 07)


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10





120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)


(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)


6 Conclusion




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119860

def= ((119909 119910) 120583

119868(119909 119910)) | 119909 isin 119860 where 120583

119868(119909 119910) = 1 if


dom ()def= sup

119910isin119861((119909 119910) 120583

(119909 119910)) | forall119909 isin 119860 (14)


codom ()def= sup

119909isin119860((119909 119910) 120583

(119909 119910)) | forall119910 isin 119861 (15)





= [(119909 119910) 120583(119909 119910)] | (119909 119910) isin 119860 times 119861

119878 = ((119910 119911) 120583119878(119910 119911)) | (119910 119911) isin 119861 times 119862

(16)


S

= [(119909 119910) 120583(119909 119910)] | (119909 119910) isin 119860 times 119861 (17)

such that

= [(119910 119909) 120583(119910 119909)] | (119910 119909) isin 119861 times 119860 (18)


∘ 119878 = [(119909 119911) or119910isin119861 120583 (119909 119910) and 120583119878(119910 119911)] | 119909 isin 119860

119910 isin 119861 119911 isin 119862

(19)



119896 = [(119909 119910) 119896120583(119909 119910)] | (119909 119910) isin 119860 times 119861 (20)


(ii) Let 119860 = 1199091 1199092 1199093 119861 = 119910

1 1199102 1199103 1199104 = ldquo119909


=1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

08 1 01 07

0 08 0 0

09 1 07 08

)

119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

04 0 09 06

09 04 05 07

03 0 08 05

)

(21)

We have

(i) cup 119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

08 1 09 07

09 08 05 07

09 1 08 08

)

(ii) S

=

1199101

1199102

1199103

1199104

1199091

1199092

1199093

(

08 0 09

1 08 1

01 0 07

07 0 08

)

(iii) cap 119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

04 0 01 06

0 04 0 0

03 0 07 05

)

(iv) 119896119878 =1199091

1199092

1199093

1199101

1199102

1199103

1199104

(

012 0 027 018

027 012 015 021

09 0 024 015

)

(22)




lows∘sdot119878 = [(119909 119911) or119910 120583

1

(119909 119910) sdot 1205832

(119910 119911)] | 119909 isin 119860

119910 isin 119861 119911 isin 119862

(23)


∘av119878 = [(119909 119911) 1

2or119910120583(119909 119910) + 120583

119878(119910 119911)] | 119909 isin 119860

119910 isin 119861 119911 isin 119862

(24)


4

6

100

2

4

6

8

10

0

025

05

075

1

02

8


4

6

100

2

4

6

8

10

0

025

05

075

1

0

2

8


4

6

100

2

4

6

8

10

0

025

05

075

1

0

2

8


0

025

05

075

1

0

2

4

6

8

10

4

6

10

0

2

8



0

025

05

075

1

4

6

10

0

2

8

0

2

4

6

8

10



=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

)

119878 =

1199101

1199102

1199103

1199104

1199105

1199111

1199112

1199113

1199114

(

09 0 03 04

02 1 08 0

08 0 07 1

04 02 03 0

0 1 0 08

)

(25)


1 119911 = 119911

1 Let

119909 = 1199091 119911 = 119911

1 and 119910 = 119910

119894 1 le 119894 le 5

(i) and120583(1199091 1199101) 120583119878(1199101 1199111) = and01 09 = 01

(ii) and120583(1199091 1199102) 120583119878(1199102 1199111) = and02 02 = 02

(iii) and120583(1199091 1199103) 120583119878(1199103 1199111) = and0 08 = 0

(iv) and120583(1199091 1199104) 120583119878(1199104 1199111) = and1 04 = 04

(v) and120583(1199091 1199105) 120583119878(1199105 1199111) = and07 0 = 0

119878 (1199091 1199111) = ((119909

1 1199111) 120583119878

(1199091 1199111))

= ((1199091 1199111) or 01 02 0 04 0)

= ((1199091 1199111) 04)

(26)


119894 119911119894)


119878 =1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

04 07 03 07

03 1 05 08

08 03 07 1

) (27)


1 119911 = 119911

1 and 119910 = 119910

119894 1 le 119894 le 5

(vi) 120583(1199091 1199101) sdot 120583119878(1199101 1199111) = 01 sdot 09 = 009

(vii) 120583(1199091 1199102) sdot 120583119878(1199102 1199111) = 02 sdot 02 = 004

(viii) 120583(1199091 1199103) sdot 120583119878(1199103 1199111) = 0 sdot 08 = 0

(ix) 120583(1199091 1199104) sdot 120583119878(1199104 1199111) = 1 sdot 04 = 04

(x) 120583(1199091 1199105) sdot 120583119878(1199105 1199111) = 07 sdot 0 = 0

then

119878 (1199091 1199111) = ((119909

1 1199111) 120583119878

(1199091 1199111))

= ((1199091 1199111) 009 or 004 or 0 or 04 or 0)

= ((1199091 1199111) 04)

(28)


∘sdot119878 =

1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

04 07 03 056

027 1 04 08

08 03 07 1

) (29)


1 119910119894) + 120583 (119910

119894 1199111)

1 12 043 084 145 07

then

1

2sdot or119910120583(1199091 119910119894) + 120583119878(119910119894 1199111) =

1

2sdot (14) = 07

∘av119878 =1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

07 085 065 075

06 1 065 09

09 065 085 1

)

(30)





0

025

05

075

1

Grade

3

V

1

2

1

2

3

4

5

W


0

025

05

075

1

Grade

1

2

3

4

5

W

3

V

1

2


0

025

05

075

1

Grade

3

V

1

2

1

2

3

4

5

W


0

025

05

075

1

Grade

3

V

1

2

1

2

3

4

5

W



0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10


0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8



0

025

05

075

1

4

6

10

02

80

2

4

6

8

10


0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8


50

1

2

4

5

0

1

2

3

3

4

0

025

05

075

1


0

025

05

075

1

0

1

2

4

5

3

5

0

1

2

34



0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

0

1

2

4

5

3

5

0

1

2

3

4


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3




cup = cap = 0 (31)

Example 11 Let

=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

) (32)

Then

(i) =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 0 03

07 05 1 08 0

02 1 0 06 07

)

(ii) cap =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 0 03

03 05 0 02 0

02 0 0 04 03

)= 0

(iii) cup =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 1 07

07 05 1 08 1

08 1 1 06 07

)=

(33)




one have 1= ((119909 119910) 120583

1

(119909 119910)) 2= ((119909 119910) 120583

2


(a) Equality


1

(119909 119910) = 1205832

(119909 119910) (34)

(b) Inclusion

(i) If 1205831

(119909 119910) le 1205832



1 denoted by

1sube 2


(119909 119910)

1205831

(119909 119910) lt 1205832

(119909 119910) (35)







()



( cup 119878) cup


( cup 119878) cap ( cup )



(i) involution =




1)


120583(119909 119911) ge and120583

(119909 119910) 120583

(119910 119911))


120583(119909 119910) = 120583

(119910 119909)


forall119909 119910 120583(119909 119910) = 120583

(119910 119909)

119900119903 120583(119909 119910) = 120583

(119910 119909) = 0

(36)






(i) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

1 0 02 03

0 1 01 1

02 07 1 04

0 1 04 1

)

(ii) 119878 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

02 1 04 04

0 06 03 0

0 1 03 0

01 1 1 01

)

(iii) =

1199091

1199092

1199093

1199094

1199095

1199096

1199101

1199102

1199103

1199104

1199105

1199106

(

(

1 02 1 06 02 06

02 1 02 02 08 02

1 02 1 06 02 06

06 02 06 1 02 08

02 08 02 02 1 02

06 02 06 08 02 1

)

)

(iv) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

04 0 07 0

0 1 09 06

08 04 07 04

0 01 0 0

)

(v) 119876 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

0 01 0 01

01 1 02 03

0 02 08 08

01 03 08 1

)

(37)










or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(38)

where 120583and 120583


of and 119876



119876 sube and 119876 cap sube (39)

if and only if

or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(40)

Example 17 Let

119876 = (01 0

1 02) = (

01 01

04 04) (41)

We have

or119910isin119861

120583

= (01

04) sube (

01

1) = or

119910isin119861120583

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 0

04 02)

sube (01 01

04 04)

= 120583

(42)

Then

119876 ⊑ (43)

Let

= (

03 02 05

04 05 09

01 02 07

) 119878 = (

03 02 04

07 08 08

03 05 06

) (44)


We have

or119910isin119861

120583119878= (

04

08

06

) sube (

05

09

07

) = or119910isin119861

120583

and 120583(119909 119910) or

119910isin119861120583119878(119909 119910) = (

03 02 04

04 05 08

01 02 06

)

sube (

03 02 04

07 08 08

03 05 06

)

= 120583119878

(45)

Then

⊑ 119878 (46)

Let

119876 = (01 02 04

05 07 09) = (

02 02 03

04 05 08)

119876 ⊑

(47)

because

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 02 03

05 07 08)

sube (02 02 03

04 05 08)

= 120583

(48)

Let

= (05 02 07

07 0 03) 119878 = (

01 03 04

09 1 05)

⊑ 119878

(49)

because

or119910isin119861

120583119878= (

04

1) sube (

07

07) = or

119910isin119861sube 120583 (50)





120583(⊔)

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)] and [or

119910isin119861(120583(119909 119910))]

and [or119910isin119861

(120583(119909 119910))]

(51)

and satisfies

or119910isin119861

120583( ⊔ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(52)


119876 ⊔ 119877 = (119876 cup 119877) cap 119876119871 cap 119877119871 (53)

if and only if

120583( ⊔ )

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)]

and [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(54)


120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(55)

and it satisfies

or119910isin119861

120583( ⊓ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] or [or

119910isin119861(120583(119909 119910))]

(56)



if and only if

120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(58)


120583(119909 119910) sube or

119910isin119861[(120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))

and (120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))]

(59)



(or119910isin119861

120583(119909 119910)) and (or

119910isin119861120583(119909 119910))


(120583(119909 119910) and 120583

(119909 119910))

(60)



(a) Let 119876 = (01 0 02

03 08 1

0 1 07) and = (

0 1 0

03 05 04

09 07 02)

(i) 119876 ⊔ = (01 02 02

03 05 05

09 09 07) but 119876 cup = (

01 1 02

03 08 1

09 1 07)

(ii) 119876 ⊓ = (0 08 0

03 05 05

0 07 02) but 119876 cap = (

0 0 0

03 05 04

0 07 02)

(b) Let 119876 = ( 03 0102 05

) and = ( 01 01 07

)

(i) 119876 ⊔ = ( 01 0105 05

) but 119876 cup = ( 03 011 07

)

(ii) 119876 ⊓ = ( 03 0105 05

) but 119876 cap = ( 01 002 05

)





120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (61)


119876 ⊳ 119877def= 119876119877 (62)

if and only if

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (63)

Example 22 (a) Let 119876 = ( 0 0103 05

) and = ( 09 0207 1

)

(i) 119876 ⊳ = ( 0 0103 05

)

(b) Let 119876 = (01 02 04

05 05 1

0 0 03) and = (

07 08 09

1 0 01

06 02 03)

(i) 119876 ⊳ = (06 06 06

06 02 03

07 07 07)


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10





120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)


(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)


6 Conclusion




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4

6

100

2

4

6

8

10

0

025

05

075

1

02

8


4

6

100

2

4

6

8

10

0

025

05

075

1

0

2

8


4

6

100

2

4

6

8

10

0

025

05

075

1

0

2

8


0

025

05

075

1

0

2

4

6

8

10

4

6

10

0

2

8



0

025

05

075

1

4

6

10

0

2

8

0

2

4

6

8

10



=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

)

119878 =

1199101

1199102

1199103

1199104

1199105

1199111

1199112

1199113

1199114

(

09 0 03 04

02 1 08 0

08 0 07 1

04 02 03 0

0 1 0 08

)

(25)


1 119911 = 119911

1 Let

119909 = 1199091 119911 = 119911

1 and 119910 = 119910

119894 1 le 119894 le 5

(i) and120583(1199091 1199101) 120583119878(1199101 1199111) = and01 09 = 01

(ii) and120583(1199091 1199102) 120583119878(1199102 1199111) = and02 02 = 02

(iii) and120583(1199091 1199103) 120583119878(1199103 1199111) = and0 08 = 0

(iv) and120583(1199091 1199104) 120583119878(1199104 1199111) = and1 04 = 04

(v) and120583(1199091 1199105) 120583119878(1199105 1199111) = and07 0 = 0

119878 (1199091 1199111) = ((119909

1 1199111) 120583119878

(1199091 1199111))

= ((1199091 1199111) or 01 02 0 04 0)

= ((1199091 1199111) 04)

(26)


119894 119911119894)


119878 =1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

04 07 03 07

03 1 05 08

08 03 07 1

) (27)


1 119911 = 119911

1 and 119910 = 119910

119894 1 le 119894 le 5

(vi) 120583(1199091 1199101) sdot 120583119878(1199101 1199111) = 01 sdot 09 = 009

(vii) 120583(1199091 1199102) sdot 120583119878(1199102 1199111) = 02 sdot 02 = 004

(viii) 120583(1199091 1199103) sdot 120583119878(1199103 1199111) = 0 sdot 08 = 0

(ix) 120583(1199091 1199104) sdot 120583119878(1199104 1199111) = 1 sdot 04 = 04

(x) 120583(1199091 1199105) sdot 120583119878(1199105 1199111) = 07 sdot 0 = 0

then

119878 (1199091 1199111) = ((119909

1 1199111) 120583119878

(1199091 1199111))

= ((1199091 1199111) 009 or 004 or 0 or 04 or 0)

= ((1199091 1199111) 04)

(28)


∘sdot119878 =

1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

04 07 03 056

027 1 04 08

08 03 07 1

) (29)


1 119910119894) + 120583 (119910

119894 1199111)

1 12 043 084 145 07

then

1

2sdot or119910120583(1199091 119910119894) + 120583119878(119910119894 1199111) =

1

2sdot (14) = 07

∘av119878 =1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

07 085 065 075

06 1 065 09

09 065 085 1

)

(30)





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cup = cap = 0 (31)

Example 11 Let

=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

) (32)

Then

(i) =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 0 03

07 05 1 08 0

02 1 0 06 07

)

(ii) cap =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 0 03

03 05 0 02 0

02 0 0 04 03

)= 0

(iii) cup =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 1 07

07 05 1 08 1

08 1 1 06 07

)=

(33)




one have 1= ((119909 119910) 120583

1

(119909 119910)) 2= ((119909 119910) 120583

2


(a) Equality


1

(119909 119910) = 1205832

(119909 119910) (34)

(b) Inclusion

(i) If 1205831

(119909 119910) le 1205832



1 denoted by

1sube 2


(119909 119910)

1205831

(119909 119910) lt 1205832

(119909 119910) (35)







()



( cup 119878) cup


( cup 119878) cap ( cup )



(i) involution =




1)


120583(119909 119911) ge and120583

(119909 119910) 120583

(119910 119911))


120583(119909 119910) = 120583

(119910 119909)


forall119909 119910 120583(119909 119910) = 120583

(119910 119909)

119900119903 120583(119909 119910) = 120583

(119910 119909) = 0

(36)






(i) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

1 0 02 03

0 1 01 1

02 07 1 04

0 1 04 1

)

(ii) 119878 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

02 1 04 04

0 06 03 0

0 1 03 0

01 1 1 01

)

(iii) =

1199091

1199092

1199093

1199094

1199095

1199096

1199101

1199102

1199103

1199104

1199105

1199106

(

(

1 02 1 06 02 06

02 1 02 02 08 02

1 02 1 06 02 06

06 02 06 1 02 08

02 08 02 02 1 02

06 02 06 08 02 1

)

)

(iv) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

04 0 07 0

0 1 09 06

08 04 07 04

0 01 0 0

)

(v) 119876 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

0 01 0 01

01 1 02 03

0 02 08 08

01 03 08 1

)

(37)










or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(38)

where 120583and 120583


of and 119876



119876 sube and 119876 cap sube (39)

if and only if

or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(40)

Example 17 Let

119876 = (01 0

1 02) = (

01 01

04 04) (41)

We have

or119910isin119861

120583

= (01

04) sube (

01

1) = or

119910isin119861120583

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 0

04 02)

sube (01 01

04 04)

= 120583

(42)

Then

119876 ⊑ (43)

Let

= (

03 02 05

04 05 09

01 02 07

) 119878 = (

03 02 04

07 08 08

03 05 06

) (44)


We have

or119910isin119861

120583119878= (

04

08

06

) sube (

05

09

07

) = or119910isin119861

120583

and 120583(119909 119910) or

119910isin119861120583119878(119909 119910) = (

03 02 04

04 05 08

01 02 06

)

sube (

03 02 04

07 08 08

03 05 06

)

= 120583119878

(45)

Then

⊑ 119878 (46)

Let

119876 = (01 02 04

05 07 09) = (

02 02 03

04 05 08)

119876 ⊑

(47)

because

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 02 03

05 07 08)

sube (02 02 03

04 05 08)

= 120583

(48)

Let

= (05 02 07

07 0 03) 119878 = (

01 03 04

09 1 05)

⊑ 119878

(49)

because

or119910isin119861

120583119878= (

04

1) sube (

07

07) = or

119910isin119861sube 120583 (50)





120583(⊔)

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)] and [or

119910isin119861(120583(119909 119910))]

and [or119910isin119861

(120583(119909 119910))]

(51)

and satisfies

or119910isin119861

120583( ⊔ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(52)


119876 ⊔ 119877 = (119876 cup 119877) cap 119876119871 cap 119877119871 (53)

if and only if

120583( ⊔ )

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)]

and [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(54)


120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(55)

and it satisfies

or119910isin119861

120583( ⊓ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] or [or

119910isin119861(120583(119909 119910))]

(56)



if and only if

120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(58)


120583(119909 119910) sube or

119910isin119861[(120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))

and (120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))]

(59)



(or119910isin119861

120583(119909 119910)) and (or

119910isin119861120583(119909 119910))


(120583(119909 119910) and 120583

(119909 119910))

(60)



(a) Let 119876 = (01 0 02

03 08 1

0 1 07) and = (

0 1 0

03 05 04

09 07 02)

(i) 119876 ⊔ = (01 02 02

03 05 05

09 09 07) but 119876 cup = (

01 1 02

03 08 1

09 1 07)

(ii) 119876 ⊓ = (0 08 0

03 05 05

0 07 02) but 119876 cap = (

0 0 0

03 05 04

0 07 02)

(b) Let 119876 = ( 03 0102 05

) and = ( 01 01 07

)

(i) 119876 ⊔ = ( 01 0105 05

) but 119876 cup = ( 03 011 07

)

(ii) 119876 ⊓ = ( 03 0105 05

) but 119876 cap = ( 01 002 05

)





120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (61)


119876 ⊳ 119877def= 119876119877 (62)

if and only if

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (63)

Example 22 (a) Let 119876 = ( 0 0103 05

) and = ( 09 0207 1

)

(i) 119876 ⊳ = ( 0 0103 05

)

(b) Let 119876 = (01 02 04

05 05 1

0 0 03) and = (

07 08 09

1 0 01

06 02 03)

(i) 119876 ⊳ = (06 06 06

06 02 03

07 07 07)


0

025

05

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120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)


(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)


6 Conclusion




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

025

05

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1

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0

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0

2

4

6

8

10



=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

)

119878 =

1199101

1199102

1199103

1199104

1199105

1199111

1199112

1199113

1199114

(

09 0 03 04

02 1 08 0

08 0 07 1

04 02 03 0

0 1 0 08

)

(25)


1 119911 = 119911

1 Let

119909 = 1199091 119911 = 119911

1 and 119910 = 119910

119894 1 le 119894 le 5

(i) and120583(1199091 1199101) 120583119878(1199101 1199111) = and01 09 = 01

(ii) and120583(1199091 1199102) 120583119878(1199102 1199111) = and02 02 = 02

(iii) and120583(1199091 1199103) 120583119878(1199103 1199111) = and0 08 = 0

(iv) and120583(1199091 1199104) 120583119878(1199104 1199111) = and1 04 = 04

(v) and120583(1199091 1199105) 120583119878(1199105 1199111) = and07 0 = 0

119878 (1199091 1199111) = ((119909

1 1199111) 120583119878

(1199091 1199111))

= ((1199091 1199111) or 01 02 0 04 0)

= ((1199091 1199111) 04)

(26)


119894 119911119894)


119878 =1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

04 07 03 07

03 1 05 08

08 03 07 1

) (27)


1 119911 = 119911

1 and 119910 = 119910

119894 1 le 119894 le 5

(vi) 120583(1199091 1199101) sdot 120583119878(1199101 1199111) = 01 sdot 09 = 009

(vii) 120583(1199091 1199102) sdot 120583119878(1199102 1199111) = 02 sdot 02 = 004

(viii) 120583(1199091 1199103) sdot 120583119878(1199103 1199111) = 0 sdot 08 = 0

(ix) 120583(1199091 1199104) sdot 120583119878(1199104 1199111) = 1 sdot 04 = 04

(x) 120583(1199091 1199105) sdot 120583119878(1199105 1199111) = 07 sdot 0 = 0

then

119878 (1199091 1199111) = ((119909

1 1199111) 120583119878

(1199091 1199111))

= ((1199091 1199111) 009 or 004 or 0 or 04 or 0)

= ((1199091 1199111) 04)

(28)


∘sdot119878 =

1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

04 07 03 056

027 1 04 08

08 03 07 1

) (29)


1 119910119894) + 120583 (119910

119894 1199111)

1 12 043 084 145 07

then

1

2sdot or119910120583(1199091 119910119894) + 120583119878(119910119894 1199111) =

1

2sdot (14) = 07

∘av119878 =1199091

1199092

1199093

1199111

1199112

1199113

1199114

(

07 085 065 075

06 1 065 09

09 065 085 1

)

(30)





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cup = cap = 0 (31)

Example 11 Let

=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

) (32)

Then

(i) =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 0 03

07 05 1 08 0

02 1 0 06 07

)

(ii) cap =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 0 03

03 05 0 02 0

02 0 0 04 03

)= 0

(iii) cup =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 1 07

07 05 1 08 1

08 1 1 06 07

)=

(33)




one have 1= ((119909 119910) 120583

1

(119909 119910)) 2= ((119909 119910) 120583

2


(a) Equality


1

(119909 119910) = 1205832

(119909 119910) (34)

(b) Inclusion

(i) If 1205831

(119909 119910) le 1205832



1 denoted by

1sube 2


(119909 119910)

1205831

(119909 119910) lt 1205832

(119909 119910) (35)







()



( cup 119878) cup


( cup 119878) cap ( cup )



(i) involution =




1)


120583(119909 119911) ge and120583

(119909 119910) 120583

(119910 119911))


120583(119909 119910) = 120583

(119910 119909)


forall119909 119910 120583(119909 119910) = 120583

(119910 119909)

119900119903 120583(119909 119910) = 120583

(119910 119909) = 0

(36)






(i) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

1 0 02 03

0 1 01 1

02 07 1 04

0 1 04 1

)

(ii) 119878 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

02 1 04 04

0 06 03 0

0 1 03 0

01 1 1 01

)

(iii) =

1199091

1199092

1199093

1199094

1199095

1199096

1199101

1199102

1199103

1199104

1199105

1199106

(

(

1 02 1 06 02 06

02 1 02 02 08 02

1 02 1 06 02 06

06 02 06 1 02 08

02 08 02 02 1 02

06 02 06 08 02 1

)

)

(iv) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

04 0 07 0

0 1 09 06

08 04 07 04

0 01 0 0

)

(v) 119876 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

0 01 0 01

01 1 02 03

0 02 08 08

01 03 08 1

)

(37)










or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(38)

where 120583and 120583


of and 119876



119876 sube and 119876 cap sube (39)

if and only if

or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(40)

Example 17 Let

119876 = (01 0

1 02) = (

01 01

04 04) (41)

We have

or119910isin119861

120583

= (01

04) sube (

01

1) = or

119910isin119861120583

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 0

04 02)

sube (01 01

04 04)

= 120583

(42)

Then

119876 ⊑ (43)

Let

= (

03 02 05

04 05 09

01 02 07

) 119878 = (

03 02 04

07 08 08

03 05 06

) (44)


We have

or119910isin119861

120583119878= (

04

08

06

) sube (

05

09

07

) = or119910isin119861

120583

and 120583(119909 119910) or

119910isin119861120583119878(119909 119910) = (

03 02 04

04 05 08

01 02 06

)

sube (

03 02 04

07 08 08

03 05 06

)

= 120583119878

(45)

Then

⊑ 119878 (46)

Let

119876 = (01 02 04

05 07 09) = (

02 02 03

04 05 08)

119876 ⊑

(47)

because

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 02 03

05 07 08)

sube (02 02 03

04 05 08)

= 120583

(48)

Let

= (05 02 07

07 0 03) 119878 = (

01 03 04

09 1 05)

⊑ 119878

(49)

because

or119910isin119861

120583119878= (

04

1) sube (

07

07) = or

119910isin119861sube 120583 (50)





120583(⊔)

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)] and [or

119910isin119861(120583(119909 119910))]

and [or119910isin119861

(120583(119909 119910))]

(51)

and satisfies

or119910isin119861

120583( ⊔ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(52)


119876 ⊔ 119877 = (119876 cup 119877) cap 119876119871 cap 119877119871 (53)

if and only if

120583( ⊔ )

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)]

and [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(54)


120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(55)

and it satisfies

or119910isin119861

120583( ⊓ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] or [or

119910isin119861(120583(119909 119910))]

(56)



if and only if

120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(58)


120583(119909 119910) sube or

119910isin119861[(120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))

and (120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))]

(59)



(or119910isin119861

120583(119909 119910)) and (or

119910isin119861120583(119909 119910))


(120583(119909 119910) and 120583

(119909 119910))

(60)



(a) Let 119876 = (01 0 02

03 08 1

0 1 07) and = (

0 1 0

03 05 04

09 07 02)

(i) 119876 ⊔ = (01 02 02

03 05 05

09 09 07) but 119876 cup = (

01 1 02

03 08 1

09 1 07)

(ii) 119876 ⊓ = (0 08 0

03 05 05

0 07 02) but 119876 cap = (

0 0 0

03 05 04

0 07 02)

(b) Let 119876 = ( 03 0102 05

) and = ( 01 01 07

)

(i) 119876 ⊔ = ( 01 0105 05

) but 119876 cup = ( 03 011 07

)

(ii) 119876 ⊓ = ( 03 0105 05

) but 119876 cap = ( 01 002 05

)





120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (61)


119876 ⊳ 119877def= 119876119877 (62)

if and only if

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (63)

Example 22 (a) Let 119876 = ( 0 0103 05

) and = ( 09 0207 1

)

(i) 119876 ⊳ = ( 0 0103 05

)

(b) Let 119876 = (01 02 04

05 05 1

0 0 03) and = (

07 08 09

1 0 01

06 02 03)

(i) 119876 ⊳ = (06 06 06

06 02 03

07 07 07)


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10





120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)


(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)


6 Conclusion




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

025

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075

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025

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1

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1

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025

05

075

1


0

025

05

075

1

0

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1

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34



0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

0

1

2

4

5

3

5

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1

2

3

4


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3




cup = cap = 0 (31)

Example 11 Let

=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

) (32)

Then

(i) =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 0 03

07 05 1 08 0

02 1 0 06 07

)

(ii) cap =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 0 03

03 05 0 02 0

02 0 0 04 03

)= 0

(iii) cup =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 1 07

07 05 1 08 1

08 1 1 06 07

)=

(33)




one have 1= ((119909 119910) 120583

1

(119909 119910)) 2= ((119909 119910) 120583

2


(a) Equality


1

(119909 119910) = 1205832

(119909 119910) (34)

(b) Inclusion

(i) If 1205831

(119909 119910) le 1205832



1 denoted by

1sube 2


(119909 119910)

1205831

(119909 119910) lt 1205832

(119909 119910) (35)







()



( cup 119878) cup


( cup 119878) cap ( cup )



(i) involution =




1)


120583(119909 119911) ge and120583

(119909 119910) 120583

(119910 119911))


120583(119909 119910) = 120583

(119910 119909)


forall119909 119910 120583(119909 119910) = 120583

(119910 119909)

119900119903 120583(119909 119910) = 120583

(119910 119909) = 0

(36)






(i) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

1 0 02 03

0 1 01 1

02 07 1 04

0 1 04 1

)

(ii) 119878 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

02 1 04 04

0 06 03 0

0 1 03 0

01 1 1 01

)

(iii) =

1199091

1199092

1199093

1199094

1199095

1199096

1199101

1199102

1199103

1199104

1199105

1199106

(

(

1 02 1 06 02 06

02 1 02 02 08 02

1 02 1 06 02 06

06 02 06 1 02 08

02 08 02 02 1 02

06 02 06 08 02 1

)

)

(iv) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

04 0 07 0

0 1 09 06

08 04 07 04

0 01 0 0

)

(v) 119876 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

0 01 0 01

01 1 02 03

0 02 08 08

01 03 08 1

)

(37)










or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(38)

where 120583and 120583


of and 119876



119876 sube and 119876 cap sube (39)

if and only if

or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(40)

Example 17 Let

119876 = (01 0

1 02) = (

01 01

04 04) (41)

We have

or119910isin119861

120583

= (01

04) sube (

01

1) = or

119910isin119861120583

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 0

04 02)

sube (01 01

04 04)

= 120583

(42)

Then

119876 ⊑ (43)

Let

= (

03 02 05

04 05 09

01 02 07

) 119878 = (

03 02 04

07 08 08

03 05 06

) (44)


We have

or119910isin119861

120583119878= (

04

08

06

) sube (

05

09

07

) = or119910isin119861

120583

and 120583(119909 119910) or

119910isin119861120583119878(119909 119910) = (

03 02 04

04 05 08

01 02 06

)

sube (

03 02 04

07 08 08

03 05 06

)

= 120583119878

(45)

Then

⊑ 119878 (46)

Let

119876 = (01 02 04

05 07 09) = (

02 02 03

04 05 08)

119876 ⊑

(47)

because

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 02 03

05 07 08)

sube (02 02 03

04 05 08)

= 120583

(48)

Let

= (05 02 07

07 0 03) 119878 = (

01 03 04

09 1 05)

⊑ 119878

(49)

because

or119910isin119861

120583119878= (

04

1) sube (

07

07) = or

119910isin119861sube 120583 (50)





120583(⊔)

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)] and [or

119910isin119861(120583(119909 119910))]

and [or119910isin119861

(120583(119909 119910))]

(51)

and satisfies

or119910isin119861

120583( ⊔ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(52)


119876 ⊔ 119877 = (119876 cup 119877) cap 119876119871 cap 119877119871 (53)

if and only if

120583( ⊔ )

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)]

and [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(54)


120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(55)

and it satisfies

or119910isin119861

120583( ⊓ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] or [or

119910isin119861(120583(119909 119910))]

(56)



if and only if

120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(58)


120583(119909 119910) sube or

119910isin119861[(120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))

and (120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))]

(59)



(or119910isin119861

120583(119909 119910)) and (or

119910isin119861120583(119909 119910))


(120583(119909 119910) and 120583

(119909 119910))

(60)



(a) Let 119876 = (01 0 02

03 08 1

0 1 07) and = (

0 1 0

03 05 04

09 07 02)

(i) 119876 ⊔ = (01 02 02

03 05 05

09 09 07) but 119876 cup = (

01 1 02

03 08 1

09 1 07)

(ii) 119876 ⊓ = (0 08 0

03 05 05

0 07 02) but 119876 cap = (

0 0 0

03 05 04

0 07 02)

(b) Let 119876 = ( 03 0102 05

) and = ( 01 01 07

)

(i) 119876 ⊔ = ( 01 0105 05

) but 119876 cup = ( 03 011 07

)

(ii) 119876 ⊓ = ( 03 0105 05

) but 119876 cap = ( 01 002 05

)





120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (61)


119876 ⊳ 119877def= 119876119877 (62)

if and only if

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (63)

Example 22 (a) Let 119876 = ( 0 0103 05

) and = ( 09 0207 1

)

(i) 119876 ⊳ = ( 0 0103 05

)

(b) Let 119876 = (01 02 04

05 05 1

0 0 03) and = (

07 08 09

1 0 01

06 02 03)

(i) 119876 ⊳ = (06 06 06

06 02 03

07 07 07)


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10





120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)


(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)


6 Conclusion




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

025

05

075

1

0

2

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05

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05

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05

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05

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025

05

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1

2

4

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0

1

2

3

3

4

0

025

05

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1


0

025

05

075

1

0

1

2

4

5

3

5

0

1

2

34



0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

0

1

2

4

5

3

5

0

1

2

3

4


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3




cup = cap = 0 (31)

Example 11 Let

=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

) (32)

Then

(i) =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 0 03

07 05 1 08 0

02 1 0 06 07

)

(ii) cap =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 0 03

03 05 0 02 0

02 0 0 04 03

)= 0

(iii) cup =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 1 07

07 05 1 08 1

08 1 1 06 07

)=

(33)




one have 1= ((119909 119910) 120583

1

(119909 119910)) 2= ((119909 119910) 120583

2


(a) Equality


1

(119909 119910) = 1205832

(119909 119910) (34)

(b) Inclusion

(i) If 1205831

(119909 119910) le 1205832



1 denoted by

1sube 2


(119909 119910)

1205831

(119909 119910) lt 1205832

(119909 119910) (35)







()



( cup 119878) cup


( cup 119878) cap ( cup )



(i) involution =




1)


120583(119909 119911) ge and120583

(119909 119910) 120583

(119910 119911))


120583(119909 119910) = 120583

(119910 119909)


forall119909 119910 120583(119909 119910) = 120583

(119910 119909)

119900119903 120583(119909 119910) = 120583

(119910 119909) = 0

(36)






(i) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

1 0 02 03

0 1 01 1

02 07 1 04

0 1 04 1

)

(ii) 119878 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

02 1 04 04

0 06 03 0

0 1 03 0

01 1 1 01

)

(iii) =

1199091

1199092

1199093

1199094

1199095

1199096

1199101

1199102

1199103

1199104

1199105

1199106

(

(

1 02 1 06 02 06

02 1 02 02 08 02

1 02 1 06 02 06

06 02 06 1 02 08

02 08 02 02 1 02

06 02 06 08 02 1

)

)

(iv) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

04 0 07 0

0 1 09 06

08 04 07 04

0 01 0 0

)

(v) 119876 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

0 01 0 01

01 1 02 03

0 02 08 08

01 03 08 1

)

(37)










or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(38)

where 120583and 120583


of and 119876



119876 sube and 119876 cap sube (39)

if and only if

or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(40)

Example 17 Let

119876 = (01 0

1 02) = (

01 01

04 04) (41)

We have

or119910isin119861

120583

= (01

04) sube (

01

1) = or

119910isin119861120583

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 0

04 02)

sube (01 01

04 04)

= 120583

(42)

Then

119876 ⊑ (43)

Let

= (

03 02 05

04 05 09

01 02 07

) 119878 = (

03 02 04

07 08 08

03 05 06

) (44)


We have

or119910isin119861

120583119878= (

04

08

06

) sube (

05

09

07

) = or119910isin119861

120583

and 120583(119909 119910) or

119910isin119861120583119878(119909 119910) = (

03 02 04

04 05 08

01 02 06

)

sube (

03 02 04

07 08 08

03 05 06

)

= 120583119878

(45)

Then

⊑ 119878 (46)

Let

119876 = (01 02 04

05 07 09) = (

02 02 03

04 05 08)

119876 ⊑

(47)

because

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 02 03

05 07 08)

sube (02 02 03

04 05 08)

= 120583

(48)

Let

= (05 02 07

07 0 03) 119878 = (

01 03 04

09 1 05)

⊑ 119878

(49)

because

or119910isin119861

120583119878= (

04

1) sube (

07

07) = or

119910isin119861sube 120583 (50)





120583(⊔)

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)] and [or

119910isin119861(120583(119909 119910))]

and [or119910isin119861

(120583(119909 119910))]

(51)

and satisfies

or119910isin119861

120583( ⊔ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(52)


119876 ⊔ 119877 = (119876 cup 119877) cap 119876119871 cap 119877119871 (53)

if and only if

120583( ⊔ )

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)]

and [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(54)


120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(55)

and it satisfies

or119910isin119861

120583( ⊓ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] or [or

119910isin119861(120583(119909 119910))]

(56)



if and only if

120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(58)


120583(119909 119910) sube or

119910isin119861[(120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))

and (120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))]

(59)



(or119910isin119861

120583(119909 119910)) and (or

119910isin119861120583(119909 119910))


(120583(119909 119910) and 120583

(119909 119910))

(60)



(a) Let 119876 = (01 0 02

03 08 1

0 1 07) and = (

0 1 0

03 05 04

09 07 02)

(i) 119876 ⊔ = (01 02 02

03 05 05

09 09 07) but 119876 cup = (

01 1 02

03 08 1

09 1 07)

(ii) 119876 ⊓ = (0 08 0

03 05 05

0 07 02) but 119876 cap = (

0 0 0

03 05 04

0 07 02)

(b) Let 119876 = ( 03 0102 05

) and = ( 01 01 07

)

(i) 119876 ⊔ = ( 01 0105 05

) but 119876 cup = ( 03 011 07

)

(ii) 119876 ⊓ = ( 03 0105 05

) but 119876 cap = ( 01 002 05

)





120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (61)


119876 ⊳ 119877def= 119876119877 (62)

if and only if

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (63)

Example 22 (a) Let 119876 = ( 0 0103 05

) and = ( 09 0207 1

)

(i) 119876 ⊳ = ( 0 0103 05

)

(b) Let 119876 = (01 02 04

05 05 1

0 0 03) and = (

07 08 09

1 0 01

06 02 03)

(i) 119876 ⊳ = (06 06 06

06 02 03

07 07 07)


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10





120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)


(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)


6 Conclusion




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

025

05

075

1

4

6

10

02

80

2

4

6

8

10


0

025

05

075

1

0

2

4

6

8

10

4

6

10

02

8


50

1

2

4

5

0

1

2

3

3

4

0

025

05

075

1


0

025

05

075

1

0

1

2

4

5

3

5

0

1

2

34



0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

0

1

2

4

5

3

5

0

1

2

3

4


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3




cup = cap = 0 (31)

Example 11 Let

=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

) (32)

Then

(i) =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 0 03

07 05 1 08 0

02 1 0 06 07

)

(ii) cap =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 0 03

03 05 0 02 0

02 0 0 04 03

)= 0

(iii) cup =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 1 07

07 05 1 08 1

08 1 1 06 07

)=

(33)




one have 1= ((119909 119910) 120583

1

(119909 119910)) 2= ((119909 119910) 120583

2


(a) Equality


1

(119909 119910) = 1205832

(119909 119910) (34)

(b) Inclusion

(i) If 1205831

(119909 119910) le 1205832



1 denoted by

1sube 2


(119909 119910)

1205831

(119909 119910) lt 1205832

(119909 119910) (35)







()



( cup 119878) cup


( cup 119878) cap ( cup )



(i) involution =




1)


120583(119909 119911) ge and120583

(119909 119910) 120583

(119910 119911))


120583(119909 119910) = 120583

(119910 119909)


forall119909 119910 120583(119909 119910) = 120583

(119910 119909)

119900119903 120583(119909 119910) = 120583

(119910 119909) = 0

(36)






(i) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

1 0 02 03

0 1 01 1

02 07 1 04

0 1 04 1

)

(ii) 119878 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

02 1 04 04

0 06 03 0

0 1 03 0

01 1 1 01

)

(iii) =

1199091

1199092

1199093

1199094

1199095

1199096

1199101

1199102

1199103

1199104

1199105

1199106

(

(

1 02 1 06 02 06

02 1 02 02 08 02

1 02 1 06 02 06

06 02 06 1 02 08

02 08 02 02 1 02

06 02 06 08 02 1

)

)

(iv) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

04 0 07 0

0 1 09 06

08 04 07 04

0 01 0 0

)

(v) 119876 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

0 01 0 01

01 1 02 03

0 02 08 08

01 03 08 1

)

(37)










or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(38)

where 120583and 120583


of and 119876



119876 sube and 119876 cap sube (39)

if and only if

or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(40)

Example 17 Let

119876 = (01 0

1 02) = (

01 01

04 04) (41)

We have

or119910isin119861

120583

= (01

04) sube (

01

1) = or

119910isin119861120583

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 0

04 02)

sube (01 01

04 04)

= 120583

(42)

Then

119876 ⊑ (43)

Let

= (

03 02 05

04 05 09

01 02 07

) 119878 = (

03 02 04

07 08 08

03 05 06

) (44)


We have

or119910isin119861

120583119878= (

04

08

06

) sube (

05

09

07

) = or119910isin119861

120583

and 120583(119909 119910) or

119910isin119861120583119878(119909 119910) = (

03 02 04

04 05 08

01 02 06

)

sube (

03 02 04

07 08 08

03 05 06

)

= 120583119878

(45)

Then

⊑ 119878 (46)

Let

119876 = (01 02 04

05 07 09) = (

02 02 03

04 05 08)

119876 ⊑

(47)

because

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 02 03

05 07 08)

sube (02 02 03

04 05 08)

= 120583

(48)

Let

= (05 02 07

07 0 03) 119878 = (

01 03 04

09 1 05)

⊑ 119878

(49)

because

or119910isin119861

120583119878= (

04

1) sube (

07

07) = or

119910isin119861sube 120583 (50)





120583(⊔)

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)] and [or

119910isin119861(120583(119909 119910))]

and [or119910isin119861

(120583(119909 119910))]

(51)

and satisfies

or119910isin119861

120583( ⊔ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(52)


119876 ⊔ 119877 = (119876 cup 119877) cap 119876119871 cap 119877119871 (53)

if and only if

120583( ⊔ )

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)]

and [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(54)


120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(55)

and it satisfies

or119910isin119861

120583( ⊓ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] or [or

119910isin119861(120583(119909 119910))]

(56)



if and only if

120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(58)


120583(119909 119910) sube or

119910isin119861[(120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))

and (120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))]

(59)



(or119910isin119861

120583(119909 119910)) and (or

119910isin119861120583(119909 119910))


(120583(119909 119910) and 120583

(119909 119910))

(60)



(a) Let 119876 = (01 0 02

03 08 1

0 1 07) and = (

0 1 0

03 05 04

09 07 02)

(i) 119876 ⊔ = (01 02 02

03 05 05

09 09 07) but 119876 cup = (

01 1 02

03 08 1

09 1 07)

(ii) 119876 ⊓ = (0 08 0

03 05 05

0 07 02) but 119876 cap = (

0 0 0

03 05 04

0 07 02)

(b) Let 119876 = ( 03 0102 05

) and = ( 01 01 07

)

(i) 119876 ⊔ = ( 01 0105 05

) but 119876 cup = ( 03 011 07

)

(ii) 119876 ⊓ = ( 03 0105 05

) but 119876 cap = ( 01 002 05

)





120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (61)


119876 ⊳ 119877def= 119876119877 (62)

if and only if

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (63)

Example 22 (a) Let 119876 = ( 0 0103 05

) and = ( 09 0207 1

)

(i) 119876 ⊳ = ( 0 0103 05

)

(b) Let 119876 = (01 02 04

05 05 1

0 0 03) and = (

07 08 09

1 0 01

06 02 03)

(i) 119876 ⊳ = (06 06 06

06 02 03

07 07 07)


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10





120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)


(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)


6 Conclusion




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

0

1

2

4

5

3

5

0

1

2

3

4


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3




cup = cap = 0 (31)

Example 11 Let

=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

) (32)

Then

(i) =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 0 03

07 05 1 08 0

02 1 0 06 07

)

(ii) cap =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 0 03

03 05 0 02 0

02 0 0 04 03

)= 0

(iii) cup =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 1 07

07 05 1 08 1

08 1 1 06 07

)=

(33)




one have 1= ((119909 119910) 120583

1

(119909 119910)) 2= ((119909 119910) 120583

2


(a) Equality


1

(119909 119910) = 1205832

(119909 119910) (34)

(b) Inclusion

(i) If 1205831

(119909 119910) le 1205832



1 denoted by

1sube 2


(119909 119910)

1205831

(119909 119910) lt 1205832

(119909 119910) (35)







()



( cup 119878) cup


( cup 119878) cap ( cup )



(i) involution =




1)


120583(119909 119911) ge and120583

(119909 119910) 120583

(119910 119911))


120583(119909 119910) = 120583

(119910 119909)


forall119909 119910 120583(119909 119910) = 120583

(119910 119909)

119900119903 120583(119909 119910) = 120583

(119910 119909) = 0

(36)






(i) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

1 0 02 03

0 1 01 1

02 07 1 04

0 1 04 1

)

(ii) 119878 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

02 1 04 04

0 06 03 0

0 1 03 0

01 1 1 01

)

(iii) =

1199091

1199092

1199093

1199094

1199095

1199096

1199101

1199102

1199103

1199104

1199105

1199106

(

(

1 02 1 06 02 06

02 1 02 02 08 02

1 02 1 06 02 06

06 02 06 1 02 08

02 08 02 02 1 02

06 02 06 08 02 1

)

)

(iv) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

04 0 07 0

0 1 09 06

08 04 07 04

0 01 0 0

)

(v) 119876 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

0 01 0 01

01 1 02 03

0 02 08 08

01 03 08 1

)

(37)










or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(38)

where 120583and 120583


of and 119876



119876 sube and 119876 cap sube (39)

if and only if

or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(40)

Example 17 Let

119876 = (01 0

1 02) = (

01 01

04 04) (41)

We have

or119910isin119861

120583

= (01

04) sube (

01

1) = or

119910isin119861120583

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 0

04 02)

sube (01 01

04 04)

= 120583

(42)

Then

119876 ⊑ (43)

Let

= (

03 02 05

04 05 09

01 02 07

) 119878 = (

03 02 04

07 08 08

03 05 06

) (44)


We have

or119910isin119861

120583119878= (

04

08

06

) sube (

05

09

07

) = or119910isin119861

120583

and 120583(119909 119910) or

119910isin119861120583119878(119909 119910) = (

03 02 04

04 05 08

01 02 06

)

sube (

03 02 04

07 08 08

03 05 06

)

= 120583119878

(45)

Then

⊑ 119878 (46)

Let

119876 = (01 02 04

05 07 09) = (

02 02 03

04 05 08)

119876 ⊑

(47)

because

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 02 03

05 07 08)

sube (02 02 03

04 05 08)

= 120583

(48)

Let

= (05 02 07

07 0 03) 119878 = (

01 03 04

09 1 05)

⊑ 119878

(49)

because

or119910isin119861

120583119878= (

04

1) sube (

07

07) = or

119910isin119861sube 120583 (50)





120583(⊔)

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)] and [or

119910isin119861(120583(119909 119910))]

and [or119910isin119861

(120583(119909 119910))]

(51)

and satisfies

or119910isin119861

120583( ⊔ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(52)


119876 ⊔ 119877 = (119876 cup 119877) cap 119876119871 cap 119877119871 (53)

if and only if

120583( ⊔ )

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)]

and [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(54)


120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(55)

and it satisfies

or119910isin119861

120583( ⊓ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] or [or

119910isin119861(120583(119909 119910))]

(56)



if and only if

120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(58)


120583(119909 119910) sube or

119910isin119861[(120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))

and (120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))]

(59)



(or119910isin119861

120583(119909 119910)) and (or

119910isin119861120583(119909 119910))


(120583(119909 119910) and 120583

(119909 119910))

(60)



(a) Let 119876 = (01 0 02

03 08 1

0 1 07) and = (

0 1 0

03 05 04

09 07 02)

(i) 119876 ⊔ = (01 02 02

03 05 05

09 09 07) but 119876 cup = (

01 1 02

03 08 1

09 1 07)

(ii) 119876 ⊓ = (0 08 0

03 05 05

0 07 02) but 119876 cap = (

0 0 0

03 05 04

0 07 02)

(b) Let 119876 = ( 03 0102 05

) and = ( 01 01 07

)

(i) 119876 ⊔ = ( 01 0105 05

) but 119876 cup = ( 03 011 07

)

(ii) 119876 ⊓ = ( 03 0105 05

) but 119876 cap = ( 01 002 05

)





120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (61)


119876 ⊳ 119877def= 119876119877 (62)

if and only if

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (63)

Example 22 (a) Let 119876 = ( 0 0103 05

) and = ( 09 0207 1

)

(i) 119876 ⊳ = ( 0 0103 05

)

(b) Let 119876 = (01 02 04

05 05 1

0 0 03) and = (

07 08 09

1 0 01

06 02 03)

(i) 119876 ⊳ = (06 06 06

06 02 03

07 07 07)


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10





120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)


(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)


6 Conclusion




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











cup = cap = 0 (31)

Example 11 Let

=1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 1 07

03 05 0 02 1

08 0 1 04 03

) (32)

Then

(i) =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 0 03

07 05 1 08 0

02 1 0 06 07

)

(ii) cap =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

01 02 0 0 03

03 05 0 02 0

02 0 0 04 03

)= 0

(iii) cup =1199091

1199092

1199093

1199101

1199102

1199103

1199104

1199105

(

09 08 1 1 07

07 05 1 08 1

08 1 1 06 07

)=

(33)




one have 1= ((119909 119910) 120583

1

(119909 119910)) 2= ((119909 119910) 120583

2


(a) Equality


1

(119909 119910) = 1205832

(119909 119910) (34)

(b) Inclusion

(i) If 1205831

(119909 119910) le 1205832



1 denoted by

1sube 2


(119909 119910)

1205831

(119909 119910) lt 1205832

(119909 119910) (35)







()



( cup 119878) cup


( cup 119878) cap ( cup )



(i) involution =




1)


120583(119909 119911) ge and120583

(119909 119910) 120583

(119910 119911))


120583(119909 119910) = 120583

(119910 119909)


forall119909 119910 120583(119909 119910) = 120583

(119910 119909)

119900119903 120583(119909 119910) = 120583

(119910 119909) = 0

(36)






(i) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

1 0 02 03

0 1 01 1

02 07 1 04

0 1 04 1

)

(ii) 119878 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

02 1 04 04

0 06 03 0

0 1 03 0

01 1 1 01

)

(iii) =

1199091

1199092

1199093

1199094

1199095

1199096

1199101

1199102

1199103

1199104

1199105

1199106

(

(

1 02 1 06 02 06

02 1 02 02 08 02

1 02 1 06 02 06

06 02 06 1 02 08

02 08 02 02 1 02

06 02 06 08 02 1

)

)

(iv) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

04 0 07 0

0 1 09 06

08 04 07 04

0 01 0 0

)

(v) 119876 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

0 01 0 01

01 1 02 03

0 02 08 08

01 03 08 1

)

(37)










or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(38)

where 120583and 120583


of and 119876



119876 sube and 119876 cap sube (39)

if and only if

or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(40)

Example 17 Let

119876 = (01 0

1 02) = (

01 01

04 04) (41)

We have

or119910isin119861

120583

= (01

04) sube (

01

1) = or

119910isin119861120583

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 0

04 02)

sube (01 01

04 04)

= 120583

(42)

Then

119876 ⊑ (43)

Let

= (

03 02 05

04 05 09

01 02 07

) 119878 = (

03 02 04

07 08 08

03 05 06

) (44)


We have

or119910isin119861

120583119878= (

04

08

06

) sube (

05

09

07

) = or119910isin119861

120583

and 120583(119909 119910) or

119910isin119861120583119878(119909 119910) = (

03 02 04

04 05 08

01 02 06

)

sube (

03 02 04

07 08 08

03 05 06

)

= 120583119878

(45)

Then

⊑ 119878 (46)

Let

119876 = (01 02 04

05 07 09) = (

02 02 03

04 05 08)

119876 ⊑

(47)

because

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 02 03

05 07 08)

sube (02 02 03

04 05 08)

= 120583

(48)

Let

= (05 02 07

07 0 03) 119878 = (

01 03 04

09 1 05)

⊑ 119878

(49)

because

or119910isin119861

120583119878= (

04

1) sube (

07

07) = or

119910isin119861sube 120583 (50)





120583(⊔)

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)] and [or

119910isin119861(120583(119909 119910))]

and [or119910isin119861

(120583(119909 119910))]

(51)

and satisfies

or119910isin119861

120583( ⊔ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(52)


119876 ⊔ 119877 = (119876 cup 119877) cap 119876119871 cap 119877119871 (53)

if and only if

120583( ⊔ )

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)]

and [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(54)


120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(55)

and it satisfies

or119910isin119861

120583( ⊓ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] or [or

119910isin119861(120583(119909 119910))]

(56)



if and only if

120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(58)


120583(119909 119910) sube or

119910isin119861[(120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))

and (120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))]

(59)



(or119910isin119861

120583(119909 119910)) and (or

119910isin119861120583(119909 119910))


(120583(119909 119910) and 120583

(119909 119910))

(60)



(a) Let 119876 = (01 0 02

03 08 1

0 1 07) and = (

0 1 0

03 05 04

09 07 02)

(i) 119876 ⊔ = (01 02 02

03 05 05

09 09 07) but 119876 cup = (

01 1 02

03 08 1

09 1 07)

(ii) 119876 ⊓ = (0 08 0

03 05 05

0 07 02) but 119876 cap = (

0 0 0

03 05 04

0 07 02)

(b) Let 119876 = ( 03 0102 05

) and = ( 01 01 07

)

(i) 119876 ⊔ = ( 01 0105 05

) but 119876 cup = ( 03 011 07

)

(ii) 119876 ⊓ = ( 03 0105 05

) but 119876 cap = ( 01 002 05

)





120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (61)


119876 ⊳ 119877def= 119876119877 (62)

if and only if

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (63)

Example 22 (a) Let 119876 = ( 0 0103 05

) and = ( 09 0207 1

)

(i) 119876 ⊳ = ( 0 0103 05

)

(b) Let 119876 = (01 02 04

05 05 1

0 0 03) and = (

07 08 09

1 0 01

06 02 03)

(i) 119876 ⊳ = (06 06 06

06 02 03

07 07 07)


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10





120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)


(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)


6 Conclusion




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(i) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

1 0 02 03

0 1 01 1

02 07 1 04

0 1 04 1

)

(ii) 119878 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

02 1 04 04

0 06 03 0

0 1 03 0

01 1 1 01

)

(iii) =

1199091

1199092

1199093

1199094

1199095

1199096

1199101

1199102

1199103

1199104

1199105

1199106

(

(

1 02 1 06 02 06

02 1 02 02 08 02

1 02 1 06 02 06

06 02 06 1 02 08

02 08 02 02 1 02

06 02 06 08 02 1

)

)

(iv) =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

04 0 07 0

0 1 09 06

08 04 07 04

0 01 0 0

)

(v) 119876 =

1199091

1199092

1199093

1199094

1199101

1199102

1199103

1199104

(

0 01 0 01

01 1 02 03

0 02 08 08

01 03 08 1

)

(37)










or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(38)

where 120583and 120583


of and 119876



119876 sube and 119876 cap sube (39)

if and only if

or119910isin119861

120583(119909 119910) sube or

119910isin119861120583(119909 119910)

and 120583(119909 119910) or

119910isin119861120583(119909 119910) sube 120583

(119909 119910)

(40)

Example 17 Let

119876 = (01 0

1 02) = (

01 01

04 04) (41)

We have

or119910isin119861

120583

= (01

04) sube (

01

1) = or

119910isin119861120583

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 0

04 02)

sube (01 01

04 04)

= 120583

(42)

Then

119876 ⊑ (43)

Let

= (

03 02 05

04 05 09

01 02 07

) 119878 = (

03 02 04

07 08 08

03 05 06

) (44)


We have

or119910isin119861

120583119878= (

04

08

06

) sube (

05

09

07

) = or119910isin119861

120583

and 120583(119909 119910) or

119910isin119861120583119878(119909 119910) = (

03 02 04

04 05 08

01 02 06

)

sube (

03 02 04

07 08 08

03 05 06

)

= 120583119878

(45)

Then

⊑ 119878 (46)

Let

119876 = (01 02 04

05 07 09) = (

02 02 03

04 05 08)

119876 ⊑

(47)

because

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 02 03

05 07 08)

sube (02 02 03

04 05 08)

= 120583

(48)

Let

= (05 02 07

07 0 03) 119878 = (

01 03 04

09 1 05)

⊑ 119878

(49)

because

or119910isin119861

120583119878= (

04

1) sube (

07

07) = or

119910isin119861sube 120583 (50)





120583(⊔)

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)] and [or

119910isin119861(120583(119909 119910))]

and [or119910isin119861

(120583(119909 119910))]

(51)

and satisfies

or119910isin119861

120583( ⊔ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(52)


119876 ⊔ 119877 = (119876 cup 119877) cap 119876119871 cap 119877119871 (53)

if and only if

120583( ⊔ )

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)]

and [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(54)


120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(55)

and it satisfies

or119910isin119861

120583( ⊓ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] or [or

119910isin119861(120583(119909 119910))]

(56)



if and only if

120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(58)


120583(119909 119910) sube or

119910isin119861[(120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))

and (120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))]

(59)



(or119910isin119861

120583(119909 119910)) and (or

119910isin119861120583(119909 119910))


(120583(119909 119910) and 120583

(119909 119910))

(60)



(a) Let 119876 = (01 0 02

03 08 1

0 1 07) and = (

0 1 0

03 05 04

09 07 02)

(i) 119876 ⊔ = (01 02 02

03 05 05

09 09 07) but 119876 cup = (

01 1 02

03 08 1

09 1 07)

(ii) 119876 ⊓ = (0 08 0

03 05 05

0 07 02) but 119876 cap = (

0 0 0

03 05 04

0 07 02)

(b) Let 119876 = ( 03 0102 05

) and = ( 01 01 07

)

(i) 119876 ⊔ = ( 01 0105 05

) but 119876 cup = ( 03 011 07

)

(ii) 119876 ⊓ = ( 03 0105 05

) but 119876 cap = ( 01 002 05

)





120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (61)


119876 ⊳ 119877def= 119876119877 (62)

if and only if

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (63)

Example 22 (a) Let 119876 = ( 0 0103 05

) and = ( 09 0207 1

)

(i) 119876 ⊳ = ( 0 0103 05

)

(b) Let 119876 = (01 02 04

05 05 1

0 0 03) and = (

07 08 09

1 0 01

06 02 03)

(i) 119876 ⊳ = (06 06 06

06 02 03

07 07 07)


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10





120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)


(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)


6 Conclusion




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










We have

or119910isin119861

120583119878= (

04

08

06

) sube (

05

09

07

) = or119910isin119861

120583

and 120583(119909 119910) or

119910isin119861120583119878(119909 119910) = (

03 02 04

04 05 08

01 02 06

)

sube (

03 02 04

07 08 08

03 05 06

)

= 120583119878

(45)

Then

⊑ 119878 (46)

Let

119876 = (01 02 04

05 07 09) = (

02 02 03

04 05 08)

119876 ⊑

(47)

because

and 120583(119909 119910) or

119910isin119861120583(119909 119910) = (

01 02 03

05 07 08)

sube (02 02 03

04 05 08)

= 120583

(48)

Let

= (05 02 07

07 0 03) 119878 = (

01 03 04

09 1 05)

⊑ 119878

(49)

because

or119910isin119861

120583119878= (

04

1) sube (

07

07) = or

119910isin119861sube 120583 (50)





120583(⊔)

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)] and [or

119910isin119861(120583(119909 119910))]

and [or119910isin119861

(120583(119909 119910))]

(51)

and satisfies

or119910isin119861

120583( ⊔ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(52)


119876 ⊔ 119877 = (119876 cup 119877) cap 119876119871 cap 119877119871 (53)

if and only if

120583( ⊔ )

(119909 119910) = [120583(119909 119910) or 120583

(119909 119910)]

and [or119910isin119861

(120583(119909 119910))] and [or

119910isin119861(120583(119909 119910))]

(54)


120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(55)

and it satisfies

or119910isin119861

120583( ⊓ )

(119909 119910)

= [or119910isin119861

(120583(119909 119910))] or [or

119910isin119861(120583(119909 119910))]

(56)



if and only if

120583( ⊓ )

(119909 119910) = [120583(119909 119910) and 120583

(119909 119910)]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

or [120583(119909 119910) and (1 minus or

119910isin119861(120583(119909 119910)))]

(58)


120583(119909 119910) sube or

119910isin119861[(120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))

and (120583(119909 119910) or (1 minus or

119910isin119861120583(119909 119910)))]

(59)



(or119910isin119861

120583(119909 119910)) and (or

119910isin119861120583(119909 119910))


(120583(119909 119910) and 120583

(119909 119910))

(60)



(a) Let 119876 = (01 0 02

03 08 1

0 1 07) and = (

0 1 0

03 05 04

09 07 02)

(i) 119876 ⊔ = (01 02 02

03 05 05

09 09 07) but 119876 cup = (

01 1 02

03 08 1

09 1 07)

(ii) 119876 ⊓ = (0 08 0

03 05 05

0 07 02) but 119876 cap = (

0 0 0

03 05 04

0 07 02)

(b) Let 119876 = ( 03 0102 05

) and = ( 01 01 07

)

(i) 119876 ⊔ = ( 01 0105 05

) but 119876 cup = ( 03 011 07

)

(ii) 119876 ⊓ = ( 03 0105 05

) but 119876 cap = ( 01 002 05

)





120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (61)


119876 ⊳ 119877def= 119876119877 (62)

if and only if

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (63)

Example 22 (a) Let 119876 = ( 0 0103 05

) and = ( 09 0207 1

)

(i) 119876 ⊳ = ( 0 0103 05

)

(b) Let 119876 = (01 02 04

05 05 1

0 0 03) and = (

07 08 09

1 0 01

06 02 03)

(i) 119876 ⊳ = (06 06 06

06 02 03

07 07 07)


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10





120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)


(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)


6 Conclusion




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(or119910isin119861

120583(119909 119910)) and (or

119910isin119861120583(119909 119910))


(120583(119909 119910) and 120583

(119909 119910))

(60)



(a) Let 119876 = (01 0 02

03 08 1

0 1 07) and = (

0 1 0

03 05 04

09 07 02)

(i) 119876 ⊔ = (01 02 02

03 05 05

09 09 07) but 119876 cup = (

01 1 02

03 08 1

09 1 07)

(ii) 119876 ⊓ = (0 08 0

03 05 05

0 07 02) but 119876 cap = (

0 0 0

03 05 04

0 07 02)

(b) Let 119876 = ( 03 0102 05

) and = ( 01 01 07

)

(i) 119876 ⊔ = ( 01 0105 05

) but 119876 cup = ( 03 011 07

)

(ii) 119876 ⊓ = ( 03 0105 05

) but 119876 cap = ( 01 002 05

)





120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (61)


119876 ⊳ 119877def= 119876119877 (62)

if and only if

120583 ⊳

(119909 119911) = 1 minus or119910isin119861

and 120583(119909 119910) 1 minus 120583

(119910 119911) (63)

Example 22 (a) Let 119876 = ( 0 0103 05

) and = ( 09 0207 1

)

(i) 119876 ⊳ = ( 0 0103 05

)

(b) Let 119876 = (01 02 04

05 05 1

0 0 03) and = (

07 08 09

1 0 01

06 02 03)

(i) 119876 ⊳ = (06 06 06

06 02 03

07 07 07)


0

025

05

075

1

5

0

1

2

3

40

1

2

4

5

3


0

025

05

075

1

4

6

10

02

80

2

4

6

8

10





120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)


(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)


6 Conclusion




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120583 ◻

(119909 119910)

= and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(64)

In other words

119876◻119877 = 119876119877 cap 119876 ⊳ 119877119871 (65)

if and only if

and [or119910isin119861

and 120583(119909 119910) 120583

(119910 119911)


and 120583(119909 119910) 1 minus or

119910isin119861(120583(119909 119910))]

(66)


(a) ( 03 0102 05

) ◻ ( 01 01 07

) = ( 01 0105 05

)

(b) (01 0 02

03 08 1

0 1 07) ◻ (

0 1 0

03 05 04

09 07 02) = (

02 02 02

05 05 04

05 05 04)


6 Conclusion




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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