Fuzzy Closure Operators

Closure operators (and closure systems) play a significant role in both pure and applied mathematics. In the framework of fuzzy set theory, several particular examples of closure operators and systems have been considered (e.g. so-called fuzzy subalgebras, fuzzy congruences, fuzzy topology etc.). Recently, fuzzy closure operators and fuzzy closure systems themselves (i.e. operators which map fuzzy sets to fuzzy sets and the corresponding systems of closed fuzzy sets) have been studied by Gerla et al., see e.g. [3, 4, 6, 7]. As a matter of fact, a fuzzy set A is usually defined as a mapping from a universe set X into the real interval [0, 1] in the above mentioned works. Therefore, the structure of truth values of the “logic behind” is fixed to [0, 1] equipped usually with minimum as the operation corresponding to logical conjunction. As it appeared recently in the investigations of fuzzy logic in narrow sense [9, 10] (i.e. fuzzy logic as a many-valued logical calculus), there are several logical calculi formalizing the intuitive idea of “fuzzy reasoning” which are complete with respect to the semantics over special structures of truth values. Among these structures, the most general one is that of a residuated lattice (it is worth noticing that residuated lattices (introduced originally in [12] as an abstraction in the study of ideal systems of rings) have been proposed as a suitable structure of truth values by Goguen in [8]). From this point of view, the need for a general notion of a “fuzzy closure” concept becomes apparent. The aim of this paper is to outline a general theory of fuzzy closure operators and fuzzy closure systems. In the next section we introduce the necessary concepts. In Section 3, fuzzy closure operators and systems are