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 multilat- tice (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 2 A. cepts of ordered and algebraic multilattice were in- • Reproductivity: aA = Aa = A for all a 2 A. That troduced in (Benado, 1954): a multilattice is an alge- is, for all a;b 2 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 up- per 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 2 A. which is more closely related to that of lattice, allow- • Transposition Property: for all a;b;c;d 2 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 fx j a 2 xbg. 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 2 A and X ⊆ A then aX will tion 2, we introduce the definitions of some hyper- denote fax j x 2 Xg and Xa will denote fxa j x 2 Xg. 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 2 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 2 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 = fa;cg 6= a(bc) = fa;b;cg).