A generalization of Myhill-Nerode theorem for fuzzy languages

The well-known Myhill-Nerode Theorem provides a necessary and sufficient condition for a language to be regular. In the context of fuzzy languages and automata theory, Myhill-Nerode type theorems have been proved for fuzzy languages with finite range. This paper introduces a new right equivalence relation on the free monoid of an alphabet based on the notion of factorization of fuzzy languages. The index of this relation for a fuzzy language with infinite range can be finite. This fact allows us to generalize the Myhill-Nerode Theorem for any kind of fuzzy languages. In this paper is proved that the following two conditions are mutually equivalent for a given fuzzy language X: (i) there exists a factorization such that the right equivalence relation of X (defined via the factorization) has a finite index; (ii) the fuzzy language X is recognized by a fuzzy deterministic finite automaton.