NAFIPS 2005 - 2005 Annual Meeting of the North American Fuzzy Information Processing Society

Characterizing the Result of theDivision of Fuzzy relations

P. Bosc, 0. Pivert, D. RocacherIRISA/ENSSAT

Technopole ANTICIPA BP 8051822305 Lannion Cedex Francebosclpivertlrocacher@enssat.fr

Abstract - The role and properties of the division operatorare well known in the framework of queries addressed to regularrelational databases. However, Boolean queries may turn out tobe too restrictive to answer some user needs and it is desirable toconsider extended queries by introducing preferences insideselection conditions. In this paper, the extension of the divisionoperator is investigated in the context of graded relations, i.e.,whose tuples are weighted. Several interpretations of the divisionare possible and they mainly depend on the roles of the gradesattached to tuples of input relations. Their properties areexamined in the perspective of a characterization of the resultobtained as a quotient, similarly to that obtained for integers.

I. INTRODUCTION

The database domain is an important field of research anddevelopment and many works aim at enriching databasemanagement systems (DBMSs) capabilities. The researchreported in this paper is intended for allowing the expressionof flexible queries, i.e., where preferences intervene inselection conditions instead of Boolean ones. This view isillustrated by the query: "find the affordable restaurantslocated close to the seashore". In such a situation,discrimination among restaurants has to take into account boththe price of the menu(s) and the location of the restaurants(and optionally levels of importance attached to each of thesecriteria).

Several works devoted to the expression and theinterpretation of fuzzy queries in the relational framework [4]have been undertaken (in particular [1, 8, 10]). Selection,projection, Cartesian product, join as well as set-orientedoperations have been studied in order to take into accountlevels of preference. On the contrary, the division operationhas not been so much investigated [2, 3, 5, 7, 9, 11, 12] anddifferent extensions have been proposed with variousmotivations and contexts, in particular depending on thenature of the relations involved and the meaning of degreesassociated with tuples.

In the remainder of this paper, the division of fuzzyrelations is investigated. The principal objective is to discussthe properties of the result delivered by a division operation.Indeed, this result depends on the approach adopted for theextension of the division as it is mentioned in the worksreported in [2, 3, 7]. One would like to determine if the resultobtained is a quotient in the sense of the properties which holdwhen the division oftwo integers is performed (which turn out

to hold with the division of regular relations). The key pointbehind this is of a semantic nature, because a negative answerwould mean that the term division is inappropriate.

The rest of the paper is organized as follows. In section 2,the definition of the division of regular relations is recalled aswell as the two characteristic properties of a quotient. Theprinciple for adapting the division to fuzzy relations, whichrelies on the notion of a degree of inclusion (instead of a usualBoolean inclusion) is described in section 3. The next sectionis devoted to the study of the division of fuzzy relations in alogical framework, i.e., the degree of inclusion is based onfuzzy implications, while section 5 concerns a cardinality-based approach for defining the degree of inclusion. Finally,section 6 concludes the paper in two respects: the major resultsobtained are recalled and some perspectives for future worksare outlined.

II. SOME REMINDERS ABOuT THE DIVISION

The relational division, i.e., the division of relation rwhose schema is R(A, X) by relation s whose schema is S(B)where A and B are compatible sets of attributes (i.e., definedon the same domains ofvalues) is defined as:

div(r, s, A, B) = {x (x E dom (X)) A (s c Kr(x))} (1)

where Kr(x) = {a <a, x> E r}. In other words, an element xbelongs to the result of the division of r by s if and only if it isassociated in r with at least all the values a appearing in s. Thejustification of the term "division" assigned to this operationrelies on the fact that a property similar to that of the quotientof integers holds. Indeed, the resulting relation t obtained withexpression (1) has the double characteristic of a quotient:

prod(s, t) c r

lttl, (tlIv t) => (prod(s, tlI) a r)(2a)

(2b)

with prod(s, t) being the Cartesian product of the two relationss and t.

Proof. Case 1. Neither the result of the division, nor thedivisor relation is empty. Let x be an element of t and a be anelement of s. Let us suppose that <x, a> does not belong to r,then x would not be associated with all the values of s and itwould not be in the result of the division of r by s, hence

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inclusion (2a) holds. Now, let us consider relation tl = t u {y}(y z t). The Cartesian product of tl and s contains a tuple <y,b> which does not belong to r, otherwise y would beassociated with any value a of s and it would have been in t. Itfollows that property (2b) holds.

Case 2. The result of the division is empty but the divisoris not empty. Property (2a) holds since the Cartesian productof t and s is empty and then included in any relation. Noelement x is associated with all the elements of s and if y isadded to t, property (2b) does not hold since the Cartesianproduct of {y} with s involves elements which are not in r.

Case 3. The divisor s is empty. The solution returned by(1) is the (possibly infinite) set of the values in the domain ofX. Properties (2a) and (2b) are both satisfied since theCartesian product of t and s is empty and t cannot beaugmented..

Remark. When the divisor is empty, the theoreticalsolution of the division is the entire domain of X. In practice,such a solution cannot be computed since the domains of theattributes are not represented (and are thus unknown) indatabase systems. To overcome this problem, a solution is toadapt the definition of the division by constraining thepossible elements of the result to belong to the dividendrelation. So, the practical computation of the result can beperformed even if the divisor is empty and the definition of thedivision becomes:

div(r, s, A, B) = {x (x E proj(r, X)) A (s c Kr(x)} (3)

where proj(r, X) stands for the projection of relation r overattribute X defined as:

proj(r, X) = {x 3t, (t E r) A (t.X = x)} (4).

The characterization of a quotient is changed into:

Vx, (x E t) => (prod(s, {x})) c r) (5a)

Vtl, (tl = t u {x}) A (x E proj(r, X)) =(prod(s, {x}) z r)) (5b).

Expressions (5a) and (5b) express the fact that the relation (t)resulting from the division is a quotient, i.e., the largestrelation whose Cartesian product with the divisor returns aresult smaller than or equal to the dividend (according toregular set inclusion).

Example 1. Let us take a database involving the tworelations order (o) and product (p) with respective schemasO(np, store, qty) and P(np, price). Tuples <n, s, q> of o and<n, pr> ofp state that the product whose number is n has beenordered to store s in quantity q and that its price is pr. Thequery aiming at retrieving the stores which have been orderedall the products priced under $127 in a quantity greater than35, can be expressed thanks to a division as:

div(o-g35, p-u127, {np}, {np})

where relation o-g35 corresponds to pairs (n, s) such thatproduct n has been ordered to store s in a quantity over 35 and

relation p-u127 gathers products whose price is under $127.From the following extensions ofrelations o and p:

0 np store qty

15 32 5012 32 6834 32 4926 32 7826 7 12078 7 3012 7 96

p up price15 1024 20012 8726 5978 34534 258

the relations o-g35 and p-u127 obtained are:

o-g35 np151234262612

store

3232323277

p-u127 Dp7 |

151226-

whose division using formula (3) leads to a result made of thesingle element {32}. It can easily be checked that this resultsatisfies expressions (2a) and (2b), or alternatively (5a) and(5b).*

III. APPROACHES TO THE DIVISION OF Fuzzy RELATIONS

A. Fuzzy Queries and Fuzzy RelationsThe context considered now is that of flexible queries

where conditions call on preferences instead of Booleancriteria. The answer to such a query is made of a set ofelements rank-ordered according to their accordance with thepreferences. From now on, predicates of flexible queries areassumed to be modeled by fuzzy sets [1] and fuzzy relationsare used instead of regular ones.

Formally, a fuzzy relation is defined as a fuzzy subset ofthe Cartesian product of domains of values. Hence, a fuzzyrelation r whose schema is R(A, B, C) is made of a set ofweighted triples denoted by g, (t)/t, where t = <a, b, c> and gk(t) stands for the membership degree of t in relation r, i.e., itscompatibility with the fuzzy concept associated with thisrelation. It is worth noticing that a regular relation is just aspecial case of a fuzzy relation where the degree attached toevery tuple equals 1.

B. Principlefor Extending the DivisionBy analogy with a query calling on a division such as that

of example 1, one may envisage the query aiming at thedetermination of the extent to which any store has beenordered all the fairly cheap products in a high quantity, whichis expressed thanks to a division of fuzzy relations, namely:

div(hq-o, fcp-p, {np}, {np})

where the degree attached to any tuple of hq-o (resp. fcp-p)expresses the compatibility of the quantity (resp. price) with

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high (resp. fairly cheap).The extension of the division to fuzzy relations is based

on the adaptation of formula (3) where:

* the regular inclusion is replaced by a fuzzy one (i.e., adegree of inclusion),

* the expression of the restriction of the calculus to thevalues present in the dividend accounts for the factthat the divisor is a fuzzy relation,

which yields:

Vx E proj(supp(r, X)),9div(r, s, A, B) (X) = deg(s c Kr(x)) (6)

where supp(r) denotes the support of the fuzzy relation r, i.e.,the regular relation {t g, (t) > 0}, proj(supp(r, X)) representsthe domain of X restricted to those values appearing in thedividend (r), Kr(x) is defined as:

Kr(x)= {p/a W<x, a> E= r},

and deg(E c F) denotes the degree of inclusion of E in F.Several types of degrees of inclusion exist depending on theapproach adopted. The logical one is based on:

EcF=Vxe U,(xe E)=>(xe F) (7)

where U is the underlying referential. This leads to:

deg(E c F) = inf, kE (X) If-F4 (X) (8)

where Rf is a fuzzy implication. Another one is founded oncardinalities of (finite) fuzzy sets, namely:

E c F 4 card(E r) F) = card(E) (9)

which leads to a degree of inclusion expressing a ratio ofcardinalities:

deg(E c F) = card((E n F) / card(E) (10).

C. Characterizing the Result ofthe Extended DivisionAssessing the fact that the result t of the extended division

is a quotient entails an adaptation of the doublecharacterization conveyed by formulas (5a) and (5b) in orderto take into account that fuzzy relations come into play, whichyields:

Vx, (x E proj(supp(r), X) A gt (x) = d) =>(prod(s, {d/<x>})) c r (1 la)

where cnj denotes a conjunction extending the regularconjunction, as well as the inclusion which is based on that offuzzy sets, i.e.:

E c F X Vx E U, UE(x) < RF(x))}.

In the next two sections, the properties of the result of anextended division is studied in terms of satisfaction ofproperties (1 la) and (1 lb).

IV. A LOGICALVEW OF THE DIVISION OF FUZZY RELATIONS

In this section, the extension of the division is studiedwhen fuzzy implications come into play. More precisely, thecase of R-implications and S-implications is dealt with. First,some connections between these fuzzy implications and twotypes of conjunction operators (triangular norms and a familyof non-commutative conjunctions) are pointed out. This servesas a basis for showing that the division built with theconsidered fuzzy implications returns a quotient, which is thenillustrated through examples.

A. Fuzzy Implications and ConjunctionsThe connection between R-implications (denoted by =:'R-i)

and conjunctions is due to the very definition of an R-implication:

(12)P =*R-i q= sup [0,1] {u T(p, u) < q}

where the chosen continuous norm T(a, b) can be called thegenerator of the R-implication. An alternative formulation ofany R-implication is:

P= >Riq= 1 ifp<q,f(p, q) otherwise,

where f(p, q) expresses a partial satisfaction (a value less than1) when the threshold p is not reached by the conclusion partq. The minimal element of R-implications, called G6delimplication, is obtained by choosing T(a, b) = min(a, b), i.e.,the maximal norm, in formula (12). It is defined as:

P =>G q = 1 ifp < q,q otherwise.

Other representatives of R-implications are Goguen andLukasiewicz implications obtained with T(a, b) = ab and T(a,b) = max(a + b - 1, 0) respectively. Combining expression (6)and the choice of an R-implication in (8), the followingdefinition of the division of fuzzy relations is obtained:

Vx, (x E proj(supp(r), X) A Sk (x) = d) =>(Vdl > d,prod(s, {dl/<x>}) z r) ( lIb).

It is also necessary to specify the Cartesian product of fuzzyrelations as well as the inclusion used in the previous twoexpressions. The Cartesian product is extended to two fuzzyrelations r and s as follows:

" ,s (uv) = cnj(Qt (u), t (v)),

Vx E proj(supp(r, X)), div(r, s, A, B) (x) = d =infs (ps (a) =:>R-i ,Tf(a, x)) (13).

With such a definition and that given for an R-implication(formula (12)), the result delivered is a quotient. Indeed, let usdenote by b one of the values of s for which d is obtained, i.e.:

d = (j4 (b) =>R-i ji (b, X)).

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O s=sUp [0, 1] {u ncc2(p, u) < q}

Va E supp(s), T(p. (a), d) < g, (a, x),

Vdl > d, T(gs (b), dl)> gr(b, x),

where T is the norm used to generate the considered R-implication. Since this is true for any x of the dividend, thisensures that the result of the division delivered by formula(13) satisfies properties (1 la) and (1 Ib).

As to S-implications, they are also strongly tied to normsthrough their definition since any S-implication writes:

P =>s-i q = -T(p, 1-q) (14).The most common representatives of this family are Kleene-Dienes (p RK-D q = max(1 - p, q)) and Reichenbach (p =>Rb q= 1 - p + pq) implications, obtained with T(a, b) = min(a, b)and T(a, b) = ab respectively. Let us also recall thatLukasiewicz implication is both an R and an S-implication.

It is clear that formula (14) differs significantly fromexpression (12) which serves for defining R-implications.Consequently, one cannot expect that formulas (1 la) and(1 lb) hold with a Cartesian product based on the triangularnorm used in expression (14). Nevertheless, it has been shownin [6] that any S-implication generated by a continuous norm(in the sense of formula (14)) can be rewritten as:

p>s-iq=sup[o,1] {uIncc(p,u)<q} (15)

where ncc(a, b) is a non-commutative conjunction defined as:

ncc(a, b) = 1 - (a =>R-i (1 - b))

where the underlying R-implication is the one generated bythe norm associated with the S-implication. More precisely,any such non-commutative conjunction has the followingproperties:

* it coincides with the usual conjunction when a and bare the usual truth values (represented by 0 and 1),

* it is non associative,* 1 is its right-hand side neutral element,* it is monotonically increasing with respect to both

arguments.

Example 2. Kleene-Dienes implication can be written:

P =>K-D q= 1 - min(p, 1 - q)=SUp[O,l] {uInccl(p,u)<q}

where ncc I (a, b) = 1 - (a =>oG(l - b))=O if (a + b) < 1, b otherwise.

Similarly, for Reichenbach implication, one has:

p=Rb q= 1 -p(l -q)

with ncc2(a, b) = 0 if (a + b) < 1,(a +b - 1) / a otherwise.

For Lukasiewicz implication, the non-commutativeconjunction operator turns out to be a norm, since thisparticular implication is also an R-implication. Last, let usmention that no such operator exists for the maximal S-implication generated by the minimal norm Tm(a, b) = a if b =1, b if a = 1, 0 otherwise), which is not continuous.*

It is then obvious that if one uses the appropriate non-commutative conjunction operator (in the sense of expression(15)) for the Cartesian product, formulas (1 la) and (1 lb) hold,which means that the result delivered by the division based onS-implications is a quotient. It is worth noticing that the orderof the arguments of the Cartesian product matters (the divisoris the first operand and the result of the division the secondone).

Remark 1. It should be mentioned that the semantics ofinclusion conveyed by S-implications (in general) is notcompatible with the intuitive view according to which deg(Ec F) equals 1 when E is included in F in Zadeh's sense. Forexample, with Kleene-Dienes implication, 1 is obtained iff thesupport ofE is included in the core of F.

Remark 2. It turns out that:

infs (gs (a) rs-i 4 (a, x)) =infs 1 - T(,U (a), 1 - i, (a, x))=1 - sups T(k (a), 1 - PlKr (x) (a)) =1 - h(s - Kr(x)),

where h(E) denotes the height of the fuzzy set E, i.e., thehighest degree of its elements and the difference between twofuzzy sets E and F is given by:

gE-F (X) = T(gE (X), 1 - gF (x)) (16).

The last expression (1 - h(s - Kr(x))) is nothing but what isobtained from formula (6) if the degree of inclusion chosen is:

deg(E c F) = 1 - h(E - F),

which stems from: E c F X ((E - F) = 0). This proves thatthe logical approach based on S-implications captures also thesemantics of a degree of inclusion built from the standarddifference between fuzzy sets according to formula (16).

Finally, it appears that the division of fuzzy relationsbased on any R-implication (resp. any S-implication generatedby a continuous norm) delivers a quotient provided that theCartesian product used for the characterization makes use ofthe norm which serves for generating the R-implicationthrough formula (12) (resp. the non-commutative conjunctionassociated with this S-implication via formula (15)).

B. IllustrationsLet us consider the two following fuzzy relations r and s

whose respective schemas are R(A, X) and S(B). The result t

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One has:

of the division of r and s is successively computed withdifferent R and S-implications. With Godel implication, ityields:

p4(x) = inf(I =>G( 0.7, 0.5 =*G 0.4, 0.3 =:>G 1) = 0.4

p (y) = inf(l =:>G, 1, 0.5 =>G3 0.6, 0.3 rG6 0.2) = 0.2.

r A X A |

al x 0.7

a2 x 0.4

a3 x 1

al y 1a2 y 0.6

a3 y 0.2

s B

a2 0.5

a3 0.3

When performing the Cartesian product of s and t with thenorm "minimum", one gets the relation:

{0.4/<al, x>, 0.4/<a2, x>, 0.3/<a3, x>,0.2/<al, y>, 0.2/<a2, y>, 0.2/<a3, y>}

which is strictly included in r (formula (1 la) holds). It is easyto check that formula ( lib) holds as well, because of thepresence of the tuples <a2, x> and <a3, y> whose grades equalthose of r.

If Goguen implication is used, the result of the division is:

p4(x) = inf(1 ='Gg 0.7, 0.5 ='Gg 0.4, 0.3 =>Gg 1) 0.7

.tt (y) = inf(l RGg 1, 0.5 =>Gg 0.6, 0.3 =>Gg 0.2) = 0.67.

If the Cartesian product of s and t is performed with the norm"product", one gets the relation:

{0.7/<al, x>, 0.35/<a2, x>, 0.21/<a3, x>,0.67/<al, y>, 0.33/<a2, y>, 0.2/<a3, y>}

which is strictly included in r. Here again, it is easy to checkthat the result of the division is maximal. Then, formulas (1 la)and (1 Ib) hold.

Similarly, with Kleene-Dienes implication, the result ofthe division of r by s is:

4(x) = inf(1 RK-D O.7, 0.5 rK-D O.4, 0.3 =>K-D 1) = 0.5,p4 (y) = inf(1 =>K-D 1, 0.5 'K-D 0.6, 0.3 :K-D 0.2) = 0.6.

When performing the Cartesian product of s and t with theappropriate non-commutative conjunction (nccl), one gets:

{0.5/<al, x>, 0.6/<al, y>, 0.6/<a2, y>}

which is strictly included in r. Relation t is maximal because if0.5 is increased to 0.5+, the value nccl(0.5, 0.5+) leads toassign the degree 0.5+ to <a2, x> which is over 0.4 andformula (1 Ib) holds as well.

If Reichenbach implication is used, the result of thedivision of r by s is:

p4(x) = inf(1 R'Rb 0.7, 0.5 RRb 0.4, 0.3 =>Rb 1) = 0.7,

S4 (y) = inf(I =:Rb 1, 0.5 RRb 0.6, 0.3 =>Rb 0.2) = 0.76.

The Cartesian product of s and t with ncc2 returns the relation:

{0.7/<al, x>, 0.4/<a2, x>,0.76/<al, y>, 0.52/<a2, y>, 0.2/<a3, y>}

which is strictly included in r. In addition, it is easy to checkthat relation t is maximal.

Last, let us use Lukasiewicz implication. The result of thedivision is:

A4 (x) = inf(l =Lu 0.7, 0.5 =>LU 0.4, 0.3 =>Lu 1) = 0.7,

pt (y) = inf(l =LU 1, 0.5 'Lu 0.6, 0.3 ='Lu 0.2) = 0.9.

When performing the Cartesian product of s and t with theassociated norm T(a, b) = max(a + b - 1, 0), one gets:

{0.7/<al, x>, 0.2/<a2, x>,0.9/<al, y>, 0.4/<a2, y>, 0.2/<a3, y>}

which is strictly included in r. In addition, t is maximal andonce again, formulas (1 la) and (1 Ib) hold.

V. A CARDINALrrY-BASED APPROACH TO THE DIVISION OFFuzzY RELATIONS

If formula (10) is used as a basis for extending thedivision, the definition hereafter is obtained:

Wdiv(r, s, A, B) (x) = card(s rTh Kr (x)) / card(s)

=Ys T4([ (a), pI. (a, x)) /sXs (a) (17).

The question is once again to assess whether or not thisview of the division of fuzzy relations delivers a quotient. Wewill see that this is not the case, whatever the type ofconjunction used for the Cartesian product, i.e., a norm or anon-commutative conjunction.

Let us consider the extensions of the dividend and divisorrelations r and s given hereafter:

r A X g

al x 0.4

a2 x 1

a3 x 1

s B |A7|al I 1

a2 1

Using the minimal norm Tm in expression (17), the degreeassigned to x by the division is:

(Tm(0.4, 1) + Tm(l, 1) + Tm(l, 1)) / (1 + 1 + 1) = 0.8.

If the Cartesian product of s and t = 0.8/x is performed withthe same norm, one gets:

{0.8/<al, x>, 0.8/<a2, x>, 0.8/a3, x>}

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which is not included in the dividend r.Similarly, let us take the following extensions of the

dividend and divisor relations r and s:

r A X _

al x 0.6

a2 x 0.1

a3 x 0.1

s B Aal I

a2 0.3

a3 0.3

Using the norm "minimum", which is the largest norm, inexpression (16), the degree assigned to x by the division is:

(min(O.6, 1) + min(0.1, 0.3) + min(0.1, 0.3))!(1 + 0.3 + 0.3) = 0.5.

If the Cartesian product of s and t = 0.5/x is performed withthe largest non-commutative conjunction (the one associatedwith Kleene-Dienes implication), one gets: {0.5/<al, x>},which is included in the dividend r, but 0.5/x is not maximal,since the product of s and t' = 0.6/x would give: {0.6/<al,x>},which is also included in r.

Finally, it appears that using triangular norms or non-commutative conjunctions to perform the Cartesian product:i) the smallest Cartesian product of the divisor and the smallestresult of a division may lead to a relation which is not includedin the dividend, and ii) the largest result of a division may notbe maximal. These two facts allow to conclude that this typeof division does not comply with properties (1 la) and (1 lb)which characterize a quotient.

VI. CONCLUSION

The topic of this paper is the extension of the division tofuzzy relations. The key point dealt with concerns theproperties of the result delivered by different approaches to theextended division. More precisely, we are interested inassessing whether the result is a quotient or not, i.e., thelargest fuzzy relation which, once composed with the divisor,does not exceed the dividend. Such a property is acharacteristic of the result of the division of integers andjustifies the appropriateness of the term division.

Starting with the definition of the division of regularrelations which calls on an inclusion, three main lines ofextension are envisaged depending on the replacement of theinclusion by a degree of inclusion based on: i) an R-implication, ii) an S-implication or iii) a ratio of cardinalities.It turns out that the first two approaches constitute a soundextension, because both implications (denoted by Rfhereafter) can be expressed under a residuated form of thetype:

p fq=sup[o,l] {uIcnj(p,u)<q}

R-implication is used for the extended division, the Cartesianproduct serving for the characterization must be performedwith the triangular norm generating the R-implication. For S-implications, things are somewhat similar, except that theCartesian product has to be done with a specific conjunctionoperator, called a non-commutative conjunction. In addition,this works only for fuzzy implications generated by acontinuous norm (or co-norm). The approach founded on theuse of a degree of inclusion expressing a ratio of cardinalitiesdoes not deliver a quotient, whatever the norm used tocompute the ratio on the one hand and the norm or the non-commutative conjunction used for the Cartesian product on theother hand.

This work opens a number of perspectives. In particular,the division considered so far can be called a non-fuzzy onesince only the operand relations are fuzzy. An orthogonalapproach for extending the division would be to soften theuniversal quantifier so as to define a truly fuzzy division basedon the fuzzy linguistic quantifier "almost all". The questionwould then be to determine under which assumptions theresult returned by such an approximate division is a quotient.

The same type of question would arise if the operands ofthe division operation are no longer relations, but multi-relations, or even fuzzy multi-relations.

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where cnj(a, b) is a conjunction operator generalizing the usualone. Such an expression guarantees that the result of thedivision is both maximal and such that its product with thedivisor is included in the dividend. More precisely, when an

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