Characterizations of fuzzy finite automata

In this paper, we establish some important concepts in fuzzy finite automata (FFAs) with bifuzzy property, and clarify their essential relationships. First we present a number of basic definitions and properties in FFAs. We then define a bifuzzy family of subautomata, bifuzzy source and successor operators, as well as demonstrate the bifuzzy topological characterization of FFAs. Also we clarify the relationships between our results and previous ones, and thus derive a relation figure visualizing their connections. Afterwards, we investigate further the bifuzzy separability, retrievability, and homomorphism; fuzzy continuous mapping and open mapping are also incorporated. In particular, we prove a number of equivalent characterizations for these concepts, and expound the relationships amongst them. This also concludes that both the bifuzzy source and successor operators are fuzzy closure operators. We discover that our investigation generalizes, to some extent, the algebraic fuzzy automata elaborated by Malik, Mordeson, and others. Finally, the main results obtained are summarized; the potential applications are indicated, and a number of related problems for further study are addressed.