On the notion of fuzzy adjunctions between fuzzy orders

On the notion of fuzzy adjunctions between fuzzy orders I. P. Cabrera, P. Cordero, F. Garc´ıa-Pardo, and M. Ojeda-Aciego 1. Introduction Adjunctions (also called isotone Galois connections) between two mathematical structures provide a means of linking both theories allowing for mutual cooperative advantages. A number of results can be found in the literature concerning sufficient or necessary conditions for a Galois connection between ordered structures to exist. In previous works [8], the authors studied the existence and construction of the right adjoint to a given mapping f, but in a more general framework: the initial setting is to consider a mapping f:A→Bfrom a (fuzzy or pre-) ordered set Ainto an unstructured set B, and then characterize those situations in which Bcan be (fuzzily or pre) ordered and an isotone mapping g:B→Acan be built such that the pair (f, g) is an adjunction. Afuzzy order is understood as a fuzzy relation satisfying reflexivity, antisymmetry and ⊗-transitivity. It is worth to recall that, in a fuzzy setting, reflexivity and antisymmetry are conflicting properties [3] and some authors [5] opted for dropping reflexivity from the ordered structures used. Our choice in [7, 6] was to introduce the notion of fuzzy Galois connection in a straightforward way for a fuzzy order, together with the following very specific version of antisymmetry: Antisymmetry: Condition ρU(a, b) = ρU(b, a) = 1 implies a=b, for all a, b ∈U. The definition given in [7] was the expected extension of that in the crisp case. Namely, Definition 1. Let A= (A, ρA), B= (B, ρB) be fuzzy orders, and two mappings f:A→Band g:B→A. The pair (f, g) forms a fuzzy adjunction between Aand B, denoted (f, g) : A⇌Bif, for all a∈Aand b∈B, the equality ρA(a, g(b)) = ρB(f(a), b) holds. Preprint submitted to Elsevier October 5, 2015 In principle, the term fuzzy adjunction is not fully justified, since mappings fand gare both crisp. In this paper, we explain the way in which the given definition is related to fuzzy mappings, hence, explaining the suitability of this notion to work in a completely fuzzy environment. 2. Preliminary definitions On underlying fuzzy framework is that of the L-fuzzy sets, where L= (L, ∨,∧,⊤,⊥,⊗,→) is a residuated lattice. A L-fuzzy set Xis a mapping from the universe set, say A, to the lattice L, i.e. X:A→L, where X(u) means the degree in which ubelongs to X. A fuzzy binary relation on Ais a fuzzy subset of A×A, that is R:A× A→L, and it is said to be: •Reflexive if R(a, a) = ⊤for all a∈A. • ⊗-Transitive if R(a, b)⊗R(b, c)≤R(a, c) for all a, b, c ∈A. •Symmetric if R(a, b) = R(b, a) for all a, b ∈A. From now on, when no confusion arises, we will omit the prefix “L-”. Definition 2. A fuzzy relation Ron Ais said to be a: •Fuzzy equivalence if Ris a reflexive, ⊗-transitive and symmetric fuzzy relation on A. •Fuzzy equality if Ris a fuzzy equivalence relation satisfying that R(a, b) = ⊤implies a=b, for all a, b ∈A. We will use the infix notation for fuzzy equivalence relation, that is: for ≈:A×A→La fuzzy equivalence relation, we denote a1≈a2to refer to ≈(a1, a2). Our approach to fuzzy ordered structures is based on the following definitions, see [4]: Definition 3. Let ≈Abe a fuzzy equivalence relation on A. A fuzzy binary relation ρA:A×A→Lis said to be • ≈A-reflexive if (a1≈Aa2)≤ρA(a1, a2) for all a1, a2∈A. • ⊗-≈A-antisymmetric if ρA(a1, a2)⊗ρA(a2, a1)≤(a1≈Aa2) for all a1, a2∈A. 2 Afuzzy order with respect to ⊗and ≈, shortly ⊗-≈Afuzzy order, is a fuzzy binary relation that is ≈A-reflexive, ⊗-≈A-antisymmetric, and ⊗- transitive. The triplet A= (A, ≈A, ρA) will be called ⊗-≈A-fuzzy ordered set or simply fuzzy ordered set, when no confusion can arise. 3. Fuzzy functions and fuzzy adjunction A number of different approaches to the notion of fuzzy function can be found in the literature. The main problem with the definition resides in that the fuzziness of the function would imply that the function itself should be a fuzzy set (in some sense). This difficulty can be overcome with the use of suitable fuzzy equivalences in the domain and the codomain of the function. Thus, one arrives to the following definition [1, 9]: Definition 4. Let ≈Aand ≈Bbe fuzzy equivalence relations on the sets Aand B, respectively. A partial fuzzy function from Ato Bis a mapping µ:A×B→Lsatisfying the following conditions: (Ext1) µ(a1, b)⊗(a1≈Aa2)≤µ(a2, b) for all a1, a2∈Aand b∈B. (Ext2) µ(a, b1)⊗(b1≈Bb2)≤µ(a, b2) for all a∈Aand b1, b2∈B. (Part) µ(a, b1)⊗µ(a, b2)≤(b1≈Bb2) for all a∈Aand b1, b2∈B Moreover, µis said to be a perfect fuzzy function whenever the following condition holds: (Tot) For all a∈Athere exists b∈Bsuch that µ(a, b) = ⊤. An alternative approach was introduced in [2], which used the notion of compatibility. Definition 5. Let ≈Aand ≈Bbe fuzzy equivalence relations on the sets A and B, respectively. A mapping µ:A×B→Lis said to be compatible wrt ≈Aand ≈Bif the following condition holds: (Comp) (a1≈Aa2)⊗(b1≈Bb2)⊗µ(a1, b1)≤µ(a2, b2) for all a1, a2∈A and b1, b2∈B. 3 It is not difficult to show that compatibility is an equivalent version of the two types of extensionality properties in Definition 4. Specifically, we have the following Lemma 6. Let ≈Aand ≈Bbe fuzzy equivalence relations on the sets Aand B, respectively and µ:A×B→La fuzzy relation. Then µsatisfies (Ext1) and (Ext2) if and only if µsatisfies (Comp). Proof. Suppose that µverifies (Ext1) and (Ext2). Then, (a1≈Aa2)⊗(b1≈Bb2)⊗µ(a1, b1) = (b1≈Bb2)⊗(µ(a1, b1)⊗(a1≈Aa2)) ≤(b1≈Bb2)⊗µ(a2, b1)≤µ(a2, b2) Conversely, assume now that µverifies (Comp). Then, µ(a1, b)⊗(a1≈Aa2) = µ(a1, b)⊗(a1≈Aa2)⊗(b≈Bb)≤µ(a2, b) and µ(a, b1)⊗(b1≈Bb2) = µ(a, b1)⊗(b1≈Bb2)⊗(a≈Aa)≤µ(a, b2) The interesting part of the previous approaches is that, given a fuzzy function, there always exists a crisp function which, somehow, represents it. Formally: Definition 7. Let µ:A×B→Lbe a fuzzy function. A crisp description of µis a partial mapping f:A→Bsuch that dom(f) = {a∈A|µ(a, b) = ⊤for some b∈B}and µ(a, f(a)) = ⊤for all a∈dom(f). It is worth to take into account the following facts concerning a fuzzy function, say µ:A×B→L, and its crisp description: •There is always a partial mapping f:A→Bsuch that fis a crisp description of µ. •Obviously, µis a perfect fuzzy function, i.e. it satisfies (Tot), if and only if any crisp description of µis a total map. •In general, a fuzzy function need not have a unique crisp description. However, if the fuzzy relation ≈Bon B is a fuzzy equality, then there exists just one crisp description for µ. Indeed, given f1and f2two crisp descriptions for µ, observe that ⊤=µ(a, f1(a)) = µ(a, f2(a)), then by (Part), ⊤=µ(a, f1(a)) ⊗µ(a, f2(a)) ≤(f1(a)≈Bf2(a)) which implies that (f1(a)≈Bf2(a)) = ⊤, thus f1(a) = f2(a). 4 The previous considerations lead us to discuss the potential one-to-one correspondence of fuzzy functions and their crisp descriptions together with possibly extra conditions. Definition 8. Let ≈Aand ≈Bbe fuzzy equivalence relations on the sets Aand B, respectively. A mapping f:A→Bis said to be compatible with ≈Aand ≈Bif (a1≈Aa2)≤(f(a1)≈Bf(a2)) for all a1, a2∈A. The following two technical lemmas are the key to the proof of the correspondence between fuzzy functions and crisp descriptions. Lemma 9. Let ≈Aand ≈Bbe fuzzy equivalence relations on the sets Aand B, respectively and let f:A→Bbe a mapping which is compatible with ≈A and ≈B. Then, there exists a perfect fuzzy function µ:A×B→Ldefined by µ(a, b) = (f(a)≈Bb)for all a∈Aand b∈Bsuch that fis a crisp description of µ. Proof. Trivially, fis a crisp description of µbecause ≈Bis reflexive: µ(a, f(a)) = (f(a)≈Af(a)) = ⊤for all a∈A. Furthermore, this equality states that property (Tot) holds. 5 Let us see now that µis a fuzzy function: (Ext1) Since fis a map which is compatible with ≈Aand ≈B, for all a1, a2∈ Aand b∈B, µ(a1, b)⊗(a1≈Aa2) = (f(a1)≈Bb)⊗(a1≈Aa2)≤ (f(a1)≈Bb)⊗(f(a1)≈Bf(a2)). Applying symmetry and ⊗-transitivity of ≈B, we can rewrite (f(a1)≈Bb)⊗(f(a1)≈Bf(a2)) = (f(a2)≈Bf(a1))⊗(f(a1)≈Bb)≤ (f(a2)≈Bb) = µ(a2, b). (Ext2) By definition of µand the ⊗-transitive property of ≈B µ(a, b1)⊗(b1≈Bb2) = (f(a)≈Bb1)⊗(b1≈Bb2)≤(f(a)≈Bb2) = µ(a, b2) (Part) By definition of µand the symmetric and ⊗-transitive properties of ≈B, µ(a, b1)⊗µ(a, b2) = (f(a)≈Bb1)⊗(f(a)≈Bb2) = (b1≈Bf(a)) ⊗(f(a)≈Bb2)≤(b1≈Bb2). Lemma 10. Let ≈Aand ≈Bbe fuzzy equivalence relations on the sets A and B, respectively and let µ:A×B→Lbe a perfect fuzzy function. Then, every crisp description f:A→Bof µis compatible with both ≈Aand ≈B. Proof. Given a1, a2∈A, since fis a crisp description of µ, we have that µ(ai, f(ai)) = ⊤for i∈ {1,2}. Then, (a1≈Aa2) = µ(a1, f(a1)) ⊗µ(a2, f(a2)) ⊗(a1≈Aa2). Now, by the condition (Ext1), we have that µ(a1, f(a1)) ⊗(a1≈Aa2)≤ µ(a2, f(a1)). Thus, we obtain that µ(a1, f(a1)) ⊗µ(a2, f(a2)) ⊗(a1≈Aa2)≤µ(a2, f(a1)) ⊗µ(a2, f(a2)). And, by the condition (Part), µ(a2, f(a1)) ⊗µ(a2, f(a2)) ≤(f(a1)≈Bf(a2)). Therefore, (a1≈Aa2)≤(f(a1)≈Bf(a2)), for all a1, a2∈A. 6 We are now in situation to state and prove the promised result about equivalence between fuzzy mappings and their crisp descriptions. Theorem 11. Let ≈Abe a fuzzy equivalence on Aand let ≈Bbe a fuzzy equality on B. There exists a bijection between the perfect fuzzy functions defined from Ato Band the crisp mappings from Ato Bwhich are compatible with ≈Aand ≈B. Proof. Given µ:A×B→La perfect fuzzy function, the crisp description f:A→Bof µis compatible with ≈Aand ≈B, by Lemma 10. Conversely, given a mapping f:A→Bwhich is compatible with ≈Aand ≈B, by Lemma 9, the fuzzy relation µ∈LA×Bdefined by µ(a, b) = (f(a)≈B b), for all a∈Aand b∈B, is a fuzzy function, whose crisp description is precisely f, since we have that µ(a, f(a)) = (f(a)≈Bf(a)) = ⊤. Moreover, the crisp description f:A→Bof a perfect fuzzy function µ satisfies that µ(a, b) = (f(a)≈Bb). In effect, µ(a, b) = µ(a, b)⊗µ(a, f(a)) ≤ (b≈Bf(a)); on the other hand, (b≈Bf(a)) = (b≈Bf(a)) ⊗µ(a, f(a)) ≤ µ(a, b). It is worth to remark that, under the hypotheses of the theorem, for every perfect fuzzy function µ:A×B→Lthere exists a unique mapping f:A→Bsuch that µ(a, f(a)) = ⊤and µ(a, b) = (f(a)≈Bb), for all a∈A and b∈B. As a result, we can safely work with crisp mappings which are compatible wrt the fuzzy equivalences. Now, a reasonable approach to the fuzzified notion of adjunction would be the following: Definition 12. Let A= (A, ≈A, ρA) and B= (B, ≈B, ρB) be two fuzzy ordered sets. Let f:A→Band g:B→Abe two mappings which are compatible with ≈Aand ≈B. The pair (f, g) is said to be a fuzzy adjunction between Aand Bif the following conditions hold (G1) (a1≈Aa2)⊗ρA(a2, g(b)) ≤ρB(f(a1), b) (G2) (b1≈Bb2)⊗ρB(f(a), b1)≤ρA(a,g(b2)) for all a, a1, a2∈Aand b, b1, b2∈B. It turns out that the previous definition is equivalent to the na¨ı ve and straightforward definition given in [7]. 7 Theorem 13. Let A= (A, ≈A, ρA)and B= (B, ≈B, ρB)be two fuzzy ordered sets. Let f:A→Band g:B→Abe two mappings which are compatible with ≈Aand ≈B, respectively. Then, the pair (f, g)is a fuzzy adjunction between Aand Bif and only if ρA(a, g(b)) = ρB(f(a), b)for all a∈Aand b∈B. Proof. Assume that for all a∈Aand b∈Bthe equality ρA(a, g(b)) = ρB(f(a), b) holds. Let a1, a2∈Aand b∈B. Since fis a map which is compatible with ≈Aand ≈B, then (a1≈Aa2)⊗ρA(a2, g(b)) ≤(f(a1)≈Bf(a2)) ⊗ρA(a2, g(b)). By the hypothesis, we obtain that (f(a1)≈Bf(a2)) ⊗ρA(a2, g(b)) ≤(f(a1)≈Bf(a2)) ⊗ρB(f(a2), b). As ρBis ≈B-reflexive and transitive, we have that (f(a1)≈Bf(a2))⊗ρB(f(a2), b)≤ρB(f(a1), f(a2))⊗ρB(f(a2), b)≤ρB(f(a1), b). Therefore, (a1≈Aa2)⊗ρA(a2, g(b)) ≤ρB(f(a1), b) for all a1, a2∈Aand b∈B. Analogously, the condition (G2) holds. Conversely, assume now that conditions (G1) and (G2) hold. Applying condition (G1), for a∈Aand b∈B, we have that (a≈Aa)⊗ρA(a, g(b)) ≤ ρB(f(a), b). Being ≈Areflexive, it is deduced that ρA(a, g(b)) ≤ρB(f(a), b) for all a∈Aand b∈B. Analogously, ρB(f(a), b)≤ρA(a, g(b)) for all a∈A and b∈B. Therefore, ρA(a, g(b)) = ρB(f(a), b) for all a∈Aand b∈B. 4. Conclusions and future work Theorem 13 above states that the straightforward approach to the fuzzy notion of adjunction (or isotone Galois connection) makes perfect sense and, moreover, opens up two different ways to the generalization, depending on whether one would consider underlying fuzzy equalities/equivalences within the fuzzy order or not. A thorough study of both possibilities will be developed as future work. 8 Acknowledgements This work has been partially supported by the Spanish Science Ministry projects TIN12-39353-C04-01 and TIN11-28084. [1] M. Demirci. Fuzzy functions and their applications. J. Math. Anal. Appl, 252:495– 517, 2000. [2] R. Belohlavek Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic Publishers, Norwell, MA, USA. 2002 [3] U. Bodenhofer. A similarity-based generalization of fuzzy orderings preserving the classical axioms. International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems 8(5):593–610, 2000. [4] U. Bodenhofer, B. De Baets and J. Fodor. A compendium of fuzzy weak orders: Representations and constructions. Fuzzy Sets and Systems 158(8):811–829, 2007. [5] J. Fodor and M. Roubens. Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht, 1994 [6] F. Garc´ıa-Pardo, I.P. Cabrera, P. Cordero, and M. Ojeda-Aciego. On adjunctions between fuzzy preordered sets: necessary conditions. Lecture Notes in Artificial Intelligence 8536:211–221, 2014. [7] F. Garc´ıa-Pardo, I.P. Cabrera, P. Cordero, and M. Ojeda-Aciego. On the construction of fuzzy Galois connections. Proc. of XVII Spanish Conference on Fuzzy Logic and Technology, pages 99-102, 2014. [8] F. Garc´ıa-Pardo, I.P. Cabrera, P. Cordero, M. Ojeda-Aciego, and F.J. Rodr´ıguez. On the definition of suitable orderings to generate adjunctions over an unstructured codomain. Information Sciences 286: 173–187, 2014. [9] S. Gottwald. Fuzzy Sets and Fuzzy Logic. Foundations of Application—from a Mathematical Point of View. Vieweg, Braunschweig, Wiesbaden 1993. 9