The universal fuzzy automaton

In this paper, we introduce the notion of universal fuzzy automaton with membership values in a complete residuated lattice, whose states are the factorizations of this fuzzy language and transition function is defined using the inclusion degree of related fuzzy languages. Next, we define the homomorphism of fuzzy automata, prove that every automaton accepting a fuzzy language can canonically map into the universal fuzzy automaton of this language, which is called the universal property. For a fuzzy language, the connections between the universal fuzzy automaton and fuzzy minimal automata of the given fuzzy language are exploited. Finally, we give a method to construct the universal fuzzy automaton by a deterministic fuzzy automaton accepting the given fuzzy language, which is effective in the case that this deterministic fuzzy automaton is finite.