Formal power series and regular operations on fuzzy languages

In this paper we study formal power series over a quantale with coefficients in the algebra of all languages over a given alphabet, and representation of fuzzy languages by these formal power series. This representation generalizes the well-known representation of fuzzy languages by their cut and kernel languages. We show that regular operations on fuzzy languages can be represented by regular operations on power series which are defined by means of operations on ordinary languages. We use power series in study of fuzzy languages which are recognized by fuzzy finite automata and deterministic finite automata, and we study closure properties of the set of polynomials and the set of polynomials with regular coefficients under regular operations on power series.