On Congruences and Homomorphisms on Some Non-Deterministic Algebras

ON CONGRUENCES AND HOMOMORPHISMS ON SOME NON-DETERMINISTIC ALGEBRAS

I. P. Cabrera, P. Cordero, G. Gutierrez,´ J. Mart´ınez and M. Ojeda-Aciego Dept. Applied Mathematics, University of Malaga,´ Spain

Keywords: L-fuzzy sets, Non-deterministic algebra, Congruence, Homomorphism.

Abstract: Starting with the underlying motivation of developing a general theory of L-fuzzy sets where L is a multilattice (a particular case of non-deterministic algebra), we study the relationship between the crisp notions of congruence, homomorphism and substructure on some non-deterministic algebras which have been used in the literature, i.e. hypergroups, and join spaces. Moreover, we provide suitable extensions of these notions to the fuzzy case.

1 INTRODUCTION and its relationships with other logics of uncertainty. More focused on the theoretical aspects of Computer This paper follows the trend of developing fuzzy Science, some authors (Belohlˇ avek,´ 2002),(Petkovic,´ versions of crisp concepts in mathematics. Specif- 2006) have pointed out the relation between congru- ically, we focus on congruence relations, substruc- ences, fuzzy automata and determinism. tures and homomorphisms in the framework of sev- More on the practical side, applications of the con- eral hyperstructures, which are non-deterministic al- cept of congruence can be seen in the World Wide gebraic structures in the sense that the operations are Web. Concerning web applications, some authors not single-valued but set-valued. have argued on the convenience of using Answer Set The study of congruence relations and homomor- Programming (ASP) in the Semantic Web. Congru- phisms between given hyperstructures plays an im- ence relations have been used in the study of modu- portant role in the general theory of algebraic fuzzy larization of ASP as a way to structure and ease the systems. The underlying idea here is to apply meth- program development process. Specifically, compo- ods of the algebraic theory of ordinary congruences sition of modules has been formalized in(Oikarinen and homomorphisms between classical structures in and Janhunen, 2008; Janhunen et al., 2007) in terms studying suitable extensions for specific hyperstruc- of equivalence relations which are proper congruence tures, such as hypergroups, and join spaces. relations. One can find a number of extensions of classical The previous paragraphs have shown the useful- algebraic structures to a fuzzy framework in the liter- ness of the theory of (crisp) congruences regarding ature, all of them based more or less in similar ideas. practical applications. At this point, it is important to However, the fuzzy extension of the notion of func- recall that the problem of providing suitable fuzzifi- tion has been studied from several standpoints, and cations of crisp concepts is an important topic which this fact complicates the choice of the most suitable has attracted the attention of a number of researchers. definition of fuzzy homomorphism: the most conve- Since the inception of fuzzy sets and fuzzy logic, nient definition seems to depend on particular details there have been approaches to consider underlying of the underlying algebraic structure under consider- sets of truth-values more general than the unit inter- ation. val; for instance, consider the L-fuzzy sets introduced The study of congruences is important both from by Goguen in (Goguen, 1967), where L is a complete a theoretical standpoint and for its applications in the lattice. field of logic-based approaches to uncertainty. Re- There are more general structures than a com- garding applications, the notion of congruence is in- plete lattice which could host a suitable extension timately related to the foundations of fuzzy reasoning of the notion of fuzzy set, for instance, the multi-

59 Cabrera I., Cordero P., Gutiérrez G., Martínez J. and Ojeda-Aciego M. (2009). ON CONGRUENCES AND HOMOMORPHISMS ON SOME NON-DETERMINISTIC ALGEBRAS. In Proceedings of the International Joint Conference on Computational Intelligence, pages 59-67 DOI: 10.5220/0002313300590067 Copyright c SciTePress IJCCI 2009 - International Joint Conference on Computational Intelligence

lattices and other general hyperstructures. The con- • Associativity: (ab)c = a(bc) for all a,b,c ∈ A. cepts of ordered and algebraic multilattice were in- • Reproductivity: aA = Aa = A for all a ∈ A. That troduced in (Benado, 1954): a multilattice is an alge- is, for all a,b ∈ A, the equations ax = b y xa = b braic structure in which the restrictions imposed on a have solutions. (complete) lattice, namely, the “existence of least upper bounds and greatest lower bounds” are relaxed to (A,·) is said to be a join space if it satisfies associa- the“existence of minimal upper bounds and maximal tivity, reproductivity and, moreover, the two following lower bounds”. An alternative algebraic definition of properties hold: multilattice was proposed in (Medina et al., 2005), • Commutativity: ab = ba for all a,b,c ∈ A. which is more closely related to that of lattice, allow- • Transposition Property: for all a,b,c,d ∈ A, ing for natural definitions of related structures such a/b ∩ c/d 6= ∅ implies ad ∩ bc 6= ∅ where a/b = that multisemilattices and, in addition, is better suited {x | a ∈ xb}. for applications. For instance, a general approach to fuzzy logic programming based on a multilattice as It is remarkable that in the context of multilattices underlying set of truth-values was presented in (Med- it is admissible to consider the empty set in the range ina et al., 2007). of the hyperoperation, hence interest arises in devel- A number of papers have been published on the oping a suitable extension of the concept of hyper- lattice of fuzzy congruences on different classical al- groupoid, the so-called nd-groupoid. Despite of the gebraic structures, and even in some hyperstructures, small change regarding the empty set, it is noticeable for instance (Cordero et al., 2008) studies congru- that the resulting theory differs substantially. ences on a multilattice. In this paper, we will focus Definition 3. An nd-groupoid is a pair (A,·) con- on congruences and homomorphisms in the more gen- sisting of a non-empty set A together with an nd- eral setting of hyperstructures (Corsini and Leoreanu, operation · : A × A → 2A. 2003). The structure of the paper is as follows: in Sec- Remark 4. As usual, if a ∈ A and X ⊆ A then aX will tion 2, we introduce the definitions of some hyper- denote {ax | x ∈ X} and Xa will denote {xa | x ∈ X}. structures, together with some notational convention In particular, a∅ = ∅a = ∅. to be used in the rest of the paper; then, in Sec- Note, as well, that in the rest of the paper we will tion 3, we concentrate on the definition of homo- frequently write singletons without braces. morphism between nd-groupoids and how it preserves the different subhyperstructures. Later, in Section 4, the relation between the introduced notion of nd- homomorphism and (crisp) congruence on a hyper- 3 ON THE DEFINITION OF structure is investigated. It is in Section 5 where the ND-GROUPOID notion of fuzzy homomorphism is defined and the HOMOMORPHISM canonical decomposition theorem is presented. Fi- nally, we present some conclusions and prospects for This section introduces the extension of the existing future work. results about homomorphisms to the more general framework of nd-grupoids. Firstly, we begin by dis- cussing the different versions of the concept of ho- 2 PRELIMINARY DEFINITIONS momorphism on hypergroupoids (also called multi- groupoids) appearing in the literature. They are usu- Firstly, let us introduce the preliminary concepts used ally associated to particular classes of hypergroupoids in this paper: such as those of hypergroups and join spaces. Definition 1. A hypergroupoid is a pair (A,·) con- Some authors that deal with these and other hy- sisting of a non-empty set A together with a hyperop- perstructures use the following definitions of homo- A eration · : A × A → 2 r ∅. morphism (Corsini, 2003). Traditionally, authors working with hyperstruc- Definition 5. Let (A,·) and (B,·) be hypergroupoids. tures have considered the natural restriction of the A map h: A → B is said to be: images of the operation to be non-empty sets, for instance, the structures of hypergroup or join • Benado-homomorphism if h(ab) ⊆ h(a)h(b), for space (Corsini and Leoreanu, 2003). all a,b ∈ A. Definition 2. A hypergroupoid (A,·) is said to be a • Algebraic-homomorphism if h(ab) = h(a)h(b), hypergroup if the following properties hold: for all a,b ∈ A.

60 ON CONGRUENCES AND HOMOMORPHISMS ON SOME NON-DETERMINISTIC ALGEBRAS

(B,·) Recall that this definition extends without modifi- (A,·) cation to the framework of nd-groupoids. · a b c d · a b c Regarding the terminology, we depart here a bit a a b b,c A a a b c from the usual one. The first one was the original def- b b b a,c A b b b c inition by Benado (Benado, 1954), which has been c c,d c,d a,c A c c c c used in several recent papers (Davvaz, 2000; Gen- d A A A A tile, 2006; Corsini, 2003). However, it is noticeable that, finally, the authors concentrate mostly on the (i(A),·) equality-based definition. This choice is partly due to · a b c the excessive generality of Benado’s definition, which a a b b,c limits the possibility of obtaining interesting theoreti- b b b a,c cal results. The terminology used in those papers is to c c c a,c call homomorphism to Benado’s ones and call good The initial hypergrupoid (A,·) is commutative, idem- (or strong) homomorphism to algebraic ones. potent and associative. However (i(A),·) is neither We have adopted the term algebraic instead of commutative nor idempotent nor associative (for in- good or strong because this type of homomorphism stance, note that (ab)c = {a,c} 6= a(bc) = {a,b,c}). immediate allows the lifting of classical homomorphisms to the so-called powerset extension. Ob- viously, the advantage of using algebraic homo- When searching for a suitable definition which morphisms is that one can transfer properties from allows to transfer properties to the codomain, it the powerset to the nd-groupoid very easily, so is noticeable the following interesting property of that the presentation of multivalued (namely, non- multisemilattices homomorphism in the case of the deterministic) concepts is greatly simplified. operations are actually hyperoperations (not nd- Theorem 6. Let (A,·) and (B,·) be nd-groupoids and operations). h: A → B be a Benado-homomorphism. Let (2A,·), Theorem 8. Let (A,·) and (B,·) be multisemilattices (2B,·) and H : 2A → 2B be the usual power extension and h: A → B be a Benado-homomorphism. If ab 6= of h, i.e. ∅, for all a,b ∈ A, then h(ab) = h(a)h(b) ∩ h(A), for all a,b ∈ A. X ·Y = [ xy and H(X) = {h(x) | x ∈ X} x∈X Based on the previous result, we propose a new y∈Y definition of homomorphism on nd-grupoids, which Then, h is an algebraic-homomorphism if and only if is stronger than Benado’s one and weaker than the al- H is a homomorphism. gebraic one. Furthermore, the underlying idea follows the categorical meaning of morphism. Proof. Straightforward. Definition 9. Let (A,·) and (B,·) be nd-groupoids. A It should be noticed that the definition of algebraic map h: A → B is said to be nd-homomorphism if, for homomorphism, when applied to the lifted powerset all a,b ∈ A, h(ab) = h(a)h(b) ∩ h(A). version, collapses to the classical algebraic results on the powerset. Hence, the condition limits too much Note that, h is nd-homomorphism if, for all the non-deterministic interpretation of the concept of a,b,c ∈ A, the following conditions hold: morphism in that does not provide any added value to 1. c ∈ ab implies h(c) ∈ h(a)h(b), that is, h is a the theory. Benado-homomorphism. Moreover, the term homomorphism should induce 2. h(c) ∈ h(a)h(b) implies that there exists c0 ∈ ab the properties of the initial hypergroupoid on the im- such that h(c0) = h(c). age set. It can be easily checked that this is the case for algebraic-homomorphisms but, in general, it is not The following examples show that the notion of true for Benado-homomorphisms as the following ex- nd-homomorphism is different from the two other amples show. definitions given previously. Example 7. Let (A,·) and (B,·) be the hyper- Example 10. Consider the hypergroupoid (A,·) be- groupoids A = {a,b,c} and B = {a,b,c,d} with the ing A = [0,1] and the hyperoperation · : A × A → 2A hyperoperations defined as in the tables. The inclusion map i from A to B is a Benado-homomorphism. given by a · b = [0,max{a,b}]. Notice that this hy- Let us consider the restriction of the operation in B to pergroupoid is a join space (it is commutative, as- the codomain i(A) = A. sociative, reproductive and verifies the transposition axiom). Let us consider a homomorphism h: A → A

61 IJCCI 2009 - International Joint Conference on Computational Intelligence

defined by Definition 12. Let (A,·) be an nd-groupoid, X be a ( non-empty subset of A and ∗: X × X → 2X an nd- 0 if x = 0 h(x) = operation defined on X. 1 if x 6= 0 • (X,∗) is a Benado-subgroupoid of (A,·) if a∗b ⊆ we have that a · b, for all a,b ∈ X. ( {0} if a = b = 0 • (X,∗) is a Birkhoff-subgroupoid of (A,·) if a ∗ h(a · b) = b = a · b, for all a,b ∈ X. {0,1} otherwise • (X,∗) is an nd-subgroupoid of (A,·) if a ∗ b = a · and b ∩ X, for all a,b ∈ X. ( {0} if a = b = 0 Therefore, (X,∗) is a (Benado, Birkhoff, nd)- h(a) · h(b) = [0,1] otherwise subgroupoid if the inclusion map i: X → A is a (Benado, algebraic, nd)-homomorphism. Like- and as a result, one obtains the equality h(a · b) = wise, the image space of a (Benado, algebraic, h(a)·h(b)∩h(A) but, in general, the proper inclusion nd)-homomorphism is a (Benado, Birkhoff, nd)- h(a · b) h(a) · h(b) holds. subgroupoid of the codomain. Therefore, h is an nd-homomorphism but it is not Let (A,·) and (B,·) be nd-groupoids and an algebraic-homomorphism. Lemma 13. h: A → B be an nd-homomorphism. Let us consider Example 11. Let us consider the set H = {a,b,c} h(A) and the operation ∗ defined as a0 ∗ b0 = a0b0 ∩ with the hyperoperation defined by the following ta- h(A). ble: · a b c 1. If (A,·) is commutative then (h(A),∗) is commu- a c c c tative. b c c c 2. If (A,·) is associative then (h(A),∗) is associative. c c c H 3. If (A,·) is idempotent then (h(A),∗) is idempotent. (H,·) is a commutative hypergrupoid. Let h : H → 4. If (A,·) is reproductive then (h(A),∗) is reproduc- H be the homomorphism tive. ( c if x = a 5. If (A,·) satisfies the transposition property then h(x) = x otherwise (h(A),∗) satisfies the transposition property. Im(h) = {b,c} and we have that: Proof. It is a matter of systematic calculations. As an example we provide the proof of the case of associa- h(a · a) = h(c) = c h(a) · h(a) ∩ Im(h) = Im(h) tivity: h(a · b) = h(c) = c = h(a) · h(b) ∩ Im(h) h(a · c) = h(c) = c h(a) · h(c) ∩ Im(h) = Im(h) h(a) ∗ h(b) ∗ h(c) = h(a)h(b) ∩ h(A) ∗ h(c) = h(b · b) = h(c) = c = h(b) · h(b) ∩ Im(h) = h(ab) ∗ h(c) = h(ab)h(c) ∩ h(A) = h(b · c) = h(c) = c = h(b) · h(c) ∩ Im(h) h(c · c) = h(H) = {b,c} = h(c) · h(c) ∩ Im(h) = h(ab)c = ha(bc) = Therefore, h is a Benado-homomorphism but it is = h(a)h(bc) ∩ h(A) = h(a) ∗ h(bc) = not an nd-homomorphism. = h(a) ∗ h(b)h(c) ∩ h(A) We next discuss the different possible generaliza- =h(a) ∗ h(b) ∗ h(c) tions of the substructures of an nd-groupoid that con- nect with the corresponding definitions of homomorphism. The more general definition of subalgebra in Proposition 14. Let (A,·) and (B,·) be nd-groupoids a non-deterministic setting was introduced by Be- and h: A → B be an nd-homomorphism. Let us con- nado(Benado, 1954) for multilattices, and later used sider h(A) and the operation ∗ defined as a0 ∗ b0 = by several authors (Davvaz, 2000; Gentile, 2006; a0b0 ∩ h(A). Corsini, 2003). A second definition follows the line of 1. If (A,·) is a hypergroup then (h(A),∗) is a hyper- the algebraic homomorphism and corresponds to the group. embedding of the classical (deterministic) notion of subgroupoid into the framework of nd-subgroupoids. 2. If (A,·) is a join space then (h(A),∗) is a join We introduce a third alternative definition below. space.

62 ON CONGRUENCES AND HOMOMORPHISMS ON SOME NON-DETERMINISTIC ALGEBRAS

Proof. It is a consequence of the previous lemma. exists y ∈ Y such that x ≡ y and for all y ∈ Y there exists x ∈ X such that x ≡ y. In general, given a Benado-homomorphism between nd-groupoids, the properties of the initial nd-groupoid The following result is a direct consequence of the cannot be induced on the codomain, even with the definition. map being bijective. Furthermore, the inverse map Proposition 18. Let (A,·) and (B,·) be nd-groupoids of a bijective Benado-homomorphism needs not be a and h: A → B be an nd-homomorphism. The kernel Benado-homomorphism. relation ≡h, defined as Example 15. Let A = {a,b,c} and the two operations defined as x · y = {x,y} and x ? y = A. a ≡h b if and only if h(a) = h(b), Obviously, the identity mapping, I, is a Benado- is a congruence relation. homomorphism from (A,·) to (A,?) which is not an nd-homomorphism. However, I−1 is not a Benado- It is obvious that an algebraic-homomorphism homomorphism: I−1(a ? b) = I−1(A) = A 6⊆ I−1(a) · provides a congruence relation as well. However, be- −1 ing an nd-homomorphism is not a necessary condition I (b) = a · b = {a,b}. to provide a congruence relation. Theorem 16. Let h be a bijective mapping between nd-groupoids. The following conditions are equiva- Example 19. Let (Z,·) with a · b = {a,b} and (N,?) lent: with a ? b = {0,a,b}. Let us define h: Z → N as h(a) = |a|. Then, h is a surjective Benado- (i) h and h−1 are Benado-homomorphisms. homomorphism but is neither an algebraic or nd- (ii) h is an algebraic-homomorphism. homomorphism. Straightforwardly, the kernel rela- (iii) h is an nd-homomorphism. tion a ≡h b iff |a| = |b| is a congruence relation. Proof. Obviously items (ii) and (iii) are equivalent Theorem 20. Let h: (A,·) → (B,·) be a Benado- because the map is bijective. homomorphism between nd-groupoids. The kernel re- (i) ⇒ (ii) Since h−1 is a Benado-homomorphism, lation is a congruence relation if and only if h(ab) = − − one certainly has that h−1h(a) · h(b) ⊆ h−1h(a) · h h 1(h(a)) · h 1(h(b)) , for all a,b ∈ A h−1h(b) = ab. Thus, for an element x ∈ h(a) · h(b), Proof. Assume that the kernel relation is a congru- we have h−1(x) ⊆ a · b. As a consequence, x = ence. Since for all a ∈ A we have a ∈ h−1(h(a)), hh−1(x) ∈ h(a · b). and h is a Benado-homomorphism, then h(ab) ⊆ (ii) ⇒ (i) x ∈ h−1(a0b0) h(x) ∈ a0b0 = Let . Then hh−1(h(a)) · h−1(h(b)). hh−1(a0) · hh−1(b0) h and, since is algebraic ho- h(x) ∈ −1 0 −1 0 −1 0 For the other inclusion, consider momorphism, h h (a ) · h h (b ) = h h (a ) · h(h−1(h(a)) · h−1(h(b))), there exist y ∈ h−1(h(a)) h−1(b0). Thus, x ∈ h−1 hh−1(a0) · h−1(b0) = and z ∈ h−1(h(b)) such that x ∈ yz. Since h(y) = h(a) and h(z) = h(b), under the assumption that the kernel h−1(a0) · h−1(b0). relation is a congruence, yz ≡ ab. So, for x ∈ yz, there As a consequence of the previous result, we will exists x0 ∈ ab such that h(x) = h(x0) ∈ h(ab). call nd-isomorphism to any bijective algebraic homo- Conversely, assume the equality and let us prove morphism. that the kernel is a congruence. It is sufficient to note the following chain of equalities for all a,b,c,d ∈ A such that h(a) = h(b) and h(c) = h(d):

4 CONGRUENCES AND h(ac) =h(h−1(h(a)) · h−1(h(c))) HOMOMORPHISMS IN =h(h−1(h(b)) · h−1(h(d))) = h(bd) ND-GROUPOIDS In this section, we concentrate on the relation between Finally, it is worth to note that every homomorphism homomorphisms and congruence relations. To begin h defining a congruence relation can be canonically with, we recall below the notion of congruence rela- decomposed as i ◦ h¯ ◦ p, where tion on an nd-groupoid that we are interested in. Definition 17. Let (A,·) be an nd-groupoid. A con- • p is the projection homomorphism p(x) = [x] and gruence on A is an equivalence relation ≡ which for [a] · [b] = {[x] | x ∈ ab}. all a,b,c ∈ A satisfies that if a ≡ b, then ac ≡ bc and • h is the isomorphism defined as h([x]) = h(x) and −1 −1 ca ≡ cb, where X ≡ Y if and only if for all x ∈ X there a ·h b = h(h (a) · h (b)).

63 IJCCI 2009 - International Joint Conference on Computational Intelligence

• i is the inclusion homomorphism correspond- ext2 ϕ(a,b) ∧ σ(b,b0) ≤ ϕ(a,b0) ing to the type of the homomorphism of h. part ϕ(a,b) ∧ ϕ(a,b0) ≤ σ(b,b0) That is, if h is a Benado-homomorphism (re- If, in addition, the following condition holds: spectively, algebraic or nd-homomorphism) then f1 For all a ∈ A there is b ∈ B such that ϕ(a,b) = 1 (h(A),·h ) is a Benado-subgroupoid (respectively, Birkhoff or nd-subgroupoid) and i is a Benado- then we say that ϕ is a perfect fuzzy function. monomorphism (respectively, algebraic or nd- It is not difficult to show that the element b in con- monomorphism). dition (f1) above is unique. As a result, every perfect fuzzy function defines a crisp mapping from A to B called crisp description of ϕ. 5 FUZZY HOMOMORPHISMS Definition 23. Let (A,·) and (B,·) be nd-groupoids ON ND-GROUPOIDS endowed with fuzzy equalities ρ and σ, respectively. A perfect fuzzy function ϕ ∈ [0,1]A×B is said to A fuzzy relation is a mapping ϕ from A×B into [0,1], be a fuzzy homomorphism if for all a,a0 ∈ A and that is to say, any fuzzy subset of A×B. The powerset b,b0 ∈ B, the following inequality holds: A extension of a fuzzy relation is defined as, ϕb : 2 × compat ϕ(a,b) ∧ ϕ(a0,b0) ≤ ϕ(aa0,bb0) 2B → [0,1] with b Moreover, ϕ is said to be complete if the two fol- ^ _ ^ _ lowing conditions hold: ϕb(X,Y) = ϕ(x,y) ∧ ϕ(x,y) W x∈X y∈Y y∈Y x∈X 1. if y∈Y ϕ(a,y) = 1, then there exists y ∈ Y such that (a,y) = . The composition of fuzzy relations ϕ and ψ is defined ϕ 1 W as follows: 2. if x∈X ϕ(x,b) = 1, then there exists x ∈ X such _ that ϕ(x,b) = 1. (ψ ◦ ϕ)(a,c) = ϕ(a,b) ∧ ψ(b,c) Remark: Hereafter, unless stated otherwise, we will b∈B always consider that we are working with a complete A fuzzy relation ρ on A × A is said to be fuzzy homomorphism ϕ between nd-groupoids A and 1. reflexive if ρ(x,x) = 1, for every x ∈ A B and fuzzy equalities ρ and σ, respectively. 2. symmetric if ρ(x,y) = ρ(y,x), for all x,y ∈ A Proposition 24. Given ϕ between A and B, the crisp description h of ϕ is an algebraic homomorphism. 3. transitive if for all x,a,y ∈ A we have Proof. Let a1,a2 ∈ A. As ϕ(a1,h(a1)) ∧ ρ(x,a) ∧ ρ(a,y) ≤ ρ(x,y) ϕ(a2,h(a2)) ≤ ϕb(a1a2,h(a1)h(a2)), we have that ^ _ ^ _ A reflexive, symmetric and transitive fuzzy rela- ϕ(a,b)∧ ϕ(a,b) = 1 tion on A is called a fuzzy equivalence. A fuzzy a∈a1a2 b∈h(a1)h(a2) b∈h(a1)h(a2) a∈a1a2 equivalence ρ on A is called a fuzzy equality if for any x,y ∈ A, ρ(x,y) = 1 implies x = y. and so, for all a ∈ a1a2 we have that W ϕ(a,b) = 1, and by completeness, Definition 21 ((Cabrera, 2009)). A fuzzy equiva- b∈h(a1)h(a2) there exist b ∈ h(a )h(a ) such that ϕ(a,b) = 1 and, lence relation ρ on an nd-groupoid (A,·) is said to 1 2 therefore, b = h(a). So h(a a ) ⊆ h(a )h(a ). be a fuzzy congruence relation if ρ(a a ,b b ) ≥ 1 2 1 2 b 1 2 1 2 Conversely for all b ∈ h(a )h(a ), we have that ρ(a ,b ) ∧ ρ(a ,b ) for all a ,a ,b ,b ∈ A. 1 2 1 1 2 2 1 2 1 2 W (a,b) = a∈a1a2 ϕ 1 and by completeness, there ex- The fuzzification of the concept of function that ist a ∈ a1a2 such that h(a) = b, and so h(a1)h(a2) ⊆ we adopt has been introduced in (Klawonn, 2000), h(a1a2). and also studied in (Demirci, 2000; Demirci, 2003; Demirci, 2001), and more recently in (Ciri´ c´ et al., The notion of fuzzy homomorphism between nd- 2009). We will introduce the extension of the notion groupoids behaves properly with respect to the com- of perfect fuzzy function. position of fuzzy relations, in that the composition of fuzzy homomorphisms is a fuzzy homomorphism. Definition 22 ((Demirci, 2000)). Let ρ and σ be fuzzy Furthermore, the composition is associative and there equalities defined on the sets A and B, respectively. A exists an identity for this composition. As a result, the partial fuzzy function ϕ from A to B is a mapping class of nd-groupoids together with the fuzzy homo- ϕ: A × B → [0,1] satisfying the following conditions 0 0 morphisms between them forms a category. for all a,a ∈ A and b,b ∈ B: Let us concentrate now on the relationship be- ext1 ϕ(a,b) ∧ ρ(a,a0) ≤ ϕ(a0,b) tween fuzzy homomorphism and congruences.

64 ON CONGRUENCES AND HOMOMORPHISMS ON SOME NON-DETERMINISTIC ALGEBRAS

Definition 25. The fuzzy kernel relation induced by The quotient set is defined as A/ρ = {ρ(a) | a ∈ A} 0 0 ϕ in A is defined as ρϕ(a,a ) = ϕ(a,h(a )). and a fuzzy equality ρ can be defined in A/ρ as 0 0 We adopt here the term kernel as an extension of ρ(ρ(a),ρ(a )) = ρ(a,a ). the crisp case because of the inequality The fuzzy projection π from A to A/ρ is defined as 0 0 0 0 π(a,ρ(a )) = ρ(a,a ). ϕ(a,b) ∧ ϕ(a ,b) ≤ ρϕ(a,a ) Proposition 29. Let (A,·) be an nd-groupoid, ρ a which can be checked by direct computation. fuzzy equality in A and ρA be a fuzzy congruence rela- Theorem 26. Consider ϕ between A and B. The fuzzy tion in A that includes ρ. The fuzzy projection π from A to A/ is a surjective fuzzy homomorphism where kernel relation ρϕ is a congruence relation which in- ρA cludes the fuzzy equality ρ in A. the nd-operation in A/ is given by ρA

Proof. Let us see that ρϕ is a congruence relation. ρA (a1) · ρA (a2) = {ρA (d) | d ∈ a1a2} ρϕ(a1a3,a2a4) = b and the fuzzy equality is ρA. ^ _ ^ _ = ρ (a,a0) ∧ ρ (a,a0) ϕ ϕ Proof. We will only prove properties (ext1) and a∈a1a3 a0∈a a a0∈a a a∈a1a3 2 4 2 4 (compat), as the rest of the required properties are ^ _ 0 ^ _ 0 = ϕ(a,h(a )) ∧ ϕ(a,h(a )) more or less straightforward computations. 0 0 a∈a1a3 a ∈a2a4 a ∈a2a4 a∈a1a3 ext1 We will use that ρ ≤ ρA and ρA is symmet- = ϕ(a1a3,h(a2a4)) b ric and transitive: π(a1,ρ (a2)) ∧ ρ(a1,a3) = = (a a ,h(a )h(a )) by Prop. 24 A ϕb 1 3 2 4 ρA(a1,a2) ∧ ρ(a1,a3) ≤ ρA(a1,a2) ∧ ρA(a1,a3) ≤

≥ ϕ(a1,h(a2)) ∧ ϕ(a3,h(a4)) ρA(a3,a2) = π(a3,ρA (a2)).

= ρϕ(a1,a2) ∧ ρϕ(a3,a4) compat π(a1,ρA (a2)) ∧ π(a3,ρA (a4)) = ρA (a1,a2) ∧ ρA (a3,a4) ≤ ρcA (a1a3,a2a4) = Now, let us show that ρ ≤ ρϕ: bπ(a1a3,ρA (a2)ρA (a4)) ρ(a,a0) = ρ(a,a0) ∧ ϕ(a0,h(a0)) 0 0 ≤ ϕ(a,h(a )) = ρϕ(a,a ) by (ext1) Remark: In order to prove that the canonical inclusion from the image of a homomorphism is an injective fuzzy homomorphism, we recall the follow- In the rest of this section we will show the canon- ing result from (Demirci, 2000): given ϕ between ical decomposition theorem for a complete fuzzy ho- A and B, there exists a unique crisp function f such momorphism and a fuzzy congruence relation. For that ϕ(a,b) = σ( f (a),b). This f actually coincides suitable extensions on the notions of injectivity and with the crisp description h of ϕ, which satisfies surjectivity we will rely on the definitions given ϕ(a,h(a)) = 1. in (Demirci, 2000). Lemma 30. Given ϕ between A and B, then the in- Definition 27. A perfect fuzzy function ϕ ∈ [0,1]A×B clusion ι from Im ϕ to B defined as ι(b,b0) = σ(b,b0) is said to be: is an injective fuzzy homomorphism. • surjective if, for all b ∈ B there exists a ∈ A such Proof. We only prove property (compat): that ϕ(a,b) = 1. Let b1,b2 ∈ Im ϕ and a1,a2 ∈ A such that h(a1) = • injective if, for all a,a0 ∈ A and b ∈ B we have b1 and h(a2) = b2. Then ι(b1,b3) ∧ ι(b2,b4) = ϕ(a,b) ∧ ϕ(a0,b) ≤ ρ(a,a0). σ(b1,b3) ∧ σ(b2,b4) = σ(h(a1),b3) ∧ σ(h(a2),b4) = • bijective if it is injective and surjective. ϕ(a1,b3) ∧ ϕ(a2,b4) ≤ ϕb(a1a2,b3b4). Now, by The image set is Im ϕ = {b ∈ B | there exists a ∈ Proposition 24, ϕb(a1a2,b3b4) = bι(h(a1a2),b3b4) = A with ϕ(a,b) = 1}. bι(h(a1)h(a2),b3b4) =bι(b1b2,b3b4). In order to define the different homomorphisms Theorem 31. Any complete fuzzy homomorphism ϕ involved in the decomposition theorem, we have to from A to B can be canonically decomposed as ϕ = introduce the quotient set associated to a fuzzy equiv- ι◦ϕ◦π where π is the fuzzy projection from A to A/ , alence relation. ρϕ ι is the inclusion from Im ϕ to B, and ϕ is the isomor- Definition 28. Let (A,·) be an nd-groupoid and ρ be a phism from A/ρ to Im ϕ defined as ϕ(ρ (a),b) = fuzzy equivalence relation in A. An equivalence class ϕ ϕ ϕ(a,b), and the nd-operation and the fuzzy equality of an element a ∈ A is defined as in Im ϕ being the corresponding restrictions of those ρ(a) ∈ [0,1]A with ρ(a)(a0) = ρ(a,a0) in B.

65 IJCCI 2009 - International Joint Conference on Computational Intelligence

Proof. Firstly, let us prove (ext1), (inj) and (surj), the them, together with the possible interactions with sev- rest of properties are straightforward: eral applications of these structures already published in the literature. Then, we will analyze the behaviour ext1 ( (a),b) ∧ ( (a), (a0)) = ϕ ρϕ ρϕ ρϕ ρϕ of the extension of a fuzzy homomorphism between 0 0 ϕ(a,b) ∧ ρϕ(a,a ) = ϕ(a,b) ∧ ϕ(a,h(a )) = nd-groupoids to the powerset structures. At the same 0 0 0 ϕ(a,b) ∧ ϕ(a,h(a )) ∧ ϕ(a ,h(a )) ≤ time, we will investigate weaker definitions which ex- 0 0 0 0 0 σ(b,h(a ))∧ϕ(a ,h(a )) ≤ ϕ(a ,b) = ϕ(ρϕ (a ),b) tend that of nd-homomorphism, in order to to obtain 0 0 inj ϕ(ρϕ (a),b) ∧ ϕ(ρϕ (a ),b) = ϕ(a,b) ∧ ϕ(a ,b) ≤ a more abstract and flexible approach. The fuzzy no- 0 0 tions introduced in this paper will be studied in a more ρϕ (a,a ) = ρϕ(ρϕ (a),ρϕ (a )). surj For all b ∈ Im ϕ there exists a ∈ A such that general framework by substituting the unit interval ϕ(a,b) = 1 and then ϕ(ρ (a),b) = 1 with a lattice-based structure. ϕ Finally, it is worth to note that the notion of func- Finally, let us check that ϕ = ι ◦ ϕ ◦ π: tion compatible with certain structure is crucial in the study of fields such as Functional Dependencies (ι◦ϕ ◦ π)(a,b) = in Databases, Attribute Implication in Formal Con- _ 0 0 0 0 = π(a,ρϕ (a )) ∧ ϕ(ρϕ (a ),b ) ∧ ι(b ,b) cept Analysis, Association Rules in Datamining, etc. 0 ρϕ (a )∈A/ρϕ Thus, we will analyze the possible applications de- b0∈Im ϕ rived from our work on these research lines. _ 0 0 0 0 = ρϕ (a,a ) ∧ ϕ(a ,b ) ∧ σ(b ,b) a0∈A b0∈Imϕ (ext2) ACKNOWLEDGEMENTS _ 0 0 ≤ ρϕ (a,a ) ∧ ϕ(a ,b) a0∈A Partially supported by projects TIN2006-15455-C03- (de f ρϕ ) _ = ϕ(a,h(a0)) ∧ ϕ(a0,b) 01 and TIN2007-65819 (Science Ministry of Spain), a0∈A and P06-FQM-02049 (Junta de Andaluc´ıa). ( f 1) _ = ϕ(a,h(a0)) ∧ ϕ(a0,h(a0)) ∧ ϕ(a0,b) a0∈A REFERENCES (part) _ ≤ ϕ(a,h(a0)) ∧ σ(h(a0),b) a0∈A R. Belohlˇ avek.´ Determinism and fuzzy automata. Informa- (ext2) tion Sciences, 143:205–209, 2002. ≤ ϕ(a,b) M. Benado. Les ensembles partiellement ordonnes´ et le ˇ Conversely, ϕ(a,b) = σ(h(a),b) = π(a,ρϕ(a)) ∧ theor´ eme` de raffinement de Schreier. I. Cehoslovack. Mat. Z.ˇ , 4(79):105–129, 1954. ϕ(ρϕ(a),h(a)) ∧ σ(h(a),b) ≤ (ι ◦ ϕ ◦ π)(a,b). I. Cabrera, P. Cordero, G. Gutierrez,´ J. Mart´ınez, and M. Ojeda-Aciego. Fuzzy congruence relations on nd- groupoids. Intl J on Computer Mathematics, 2009. To 6 CONCLUSIONS AND FUTURE appear. WORK M. Ciri´ c,´ J. Ignjatovic,´ and S. Bogdanovic.´ Uniform fuzzy relations and fuzzy functions. Fuzzy Sets and Systems, 160(8):1054–1081, 2009. We have studied the relationship between the crisp notions of congruence, homomorphism and substruc- P. Cordero, G. Gutierrez,´ J. Mart´ınez, M. Ojeda-Aciego, and I. de las Penas.˜ Congruence relations on multilat- ture on some non-deterministic algebras which have tices. In Intl FLINS Conference on Computational In- been used in the literature, i.e. hypergroups, and join telligence in Decision and Control, FLINS’08, pages spaces, as a step towards developing a general theory 139–144, 2008. of L-fuzzy sets where L is a multilattice (a particu- P. Corsini. A new connection between hypergroups and lar case of non-deterministic algebra). Moreover, we fuzzy sets. Southeast Asian Bull. Math., 27(2):221– have provided suitable extensions of these notions to 229, 2003. the fuzzy case. P. Corsini and V. Leoreanu. Applications of hyperstructure As future work, we will study homomorphisms theory. Kluwer, 2003. and congruences in some algebraic nd-structures, B. Davvaz. Some results on congruences on semihyper- such as hyper-rings and hyper-near-rings, with the groups. Bulletin of the Malaysian Mathematical Sci- aim of investigating the adequate notion of ideal for ences Society. Second Series, 23(1):53–58, 2000.

66 ON CONGRUENCES AND HOMOMORPHISMS ON SOME NON-DETERMINISTIC ALGEBRAS

M. Demirci. Fuzzy functions and their applications. J. Math. Anal. Appl, 252:495–517, 2000. M. Demirci. Gradation of being fuzzy function. Fuzzy Sets and Systems, 119(3):383–392, 2001. M. Demirci. Constructions of fuzzy functions based on fuzzy equalities. Soft Computing, 7(3):199–207, 2003. G. Gentile. Multiendomorphisms of hypergroupoids. In- ternational Mathematical Forum, 1(9-12):563–576, 2006. J. Goguen. L-fuzzy sets. J. Math. Anal. Appl., 18:145–174, 1967. T. Janhunen, E. Oikarinen, H. Tompits, and S. Woltran. Modularity aspects of disjunctive stable models. In C. Baral, G. Brewka, and J. S. Schlipf, editors, LP- NMR, volume 4483 of Lecture Notes in Computer Sci- ence, pages 175–187. Springer, 2007. F. Klawonn. Fuzzy points, fuzzy relations and fuzzy function. In V. Novak´ and I. Perﬁlieva, editors, Discover- ing World with Fuzzy Logic, pages 431–453. Physica- Verlag, 2000. J. Mart´ınez, G. Gutierrez,´ I. P. de Guzman,´ and P. Cordero. Generalizations of lattices via non-deterministic oper- ators. Discrete Math., 295(1-3):107–141, 2005. J. Medina, M. Ojeda-Aciego, and J. Ruiz-Calvino.˜ Fuzzy logic programming via multilattices. Fuzzy Sets and Systems, 158(6):674–688, 2007. E. Oikarinen and T. Janhunen. A translation-based approach to the veriﬁcation of modular equivalence. Journal of Logic and Computation, 2008. Advance Access published. T. Petkovic.´ Congruences and homomorphisms of fuzzy automata. Fuzzy Sets and Systems, 157:444–458, 2006.