Fuzzy homomorphisms of algebras (Q1040916)

In this paper one of the approaches to fuzzy homomorphisms is presented. In the works of most authors, the role of fuzzy homomorphisms was assumed by ordinary functions and homomorphisms, which were studied in connection with diverse fuzzy concepts on algebraic structures. The authors of this paper consider fuzzy relations compatible with algebraic operations, which are called fuzzy relational morphisms. Fuzzy relational morphisms trace their origin to relational morphisms of semigroups, which were introduced to solve some problems related to the wreath product decomposition of finite semigroups. In this paper those fuzzy relational morphisms which are uniform fuzzy relations and those which are partially uniform F-functions are considered. Fuzzy relational morphisms which are uniform fuzzy relations are called uniform fuzzy relational morphisms and they are characterized in the paper in terms of the fuzzy congruences studied by the authors. Some necessary and sufficient conditions for the existence of a uniform fuzzy relational morphism from \(A\) to \(B\) which induce a given pair of fuzzy congruences \(E\) on \(A\) and \(F\) on \(B\) are determined. Uniform fuzzy relational morphisms are interesting because they can be considered as fuzzy congruences which relate elements of two possibly different algebras. Moreover, they have important applications. In the paper, the authors give an example which demonstrates an application of uniform fuzzy relational morphisms in a study of equivalence of states of deterministic automata with fuzzy sets of terminal states. Furthermore, fuzzy homomorphisms (those fuzzy relational morphisms which are partially uniform F-functions) are studied and some basic properties of them are presented. A correspondence between fuzzy homomorphisms and fuzzy congruences, analogous to the well-known correspondence between crisp homomorphisms and crisp congruences is given. Some homomorphism and isomorphism theorems concerning fuzzy homomorphisms and fuzzy congruences are presented. Finally, similarities and differences between two concepts of fuzzy homomorphisms are studied.