Algebraic aspects of families of fuzzy languages

摘要

We study operations on fuzzy languages such as union, concatenation, Kleene ★, intersection with regular fuzzy languages, and several kinds of (iterated) fuzzy substitution. Then we consider families of fuzzy languages, closed under a fixed collection of these operations, which results in the concept of fall abstract family of fuzzy languages or lull AFFL. This algebraic structure is the fuzzy counterpart of the notion of fall abstract family of languages that has been encountered frequently in investigating families of crisp (i.e., non-fuzzy) languages. Some simpler and more complicated algebraic structures (such as fall substitution-closed AFFL, fall super-AFFL, fall hyper-AFFL) will be considered as well. In the second part of the paper we focus our attention to fall AFFLs closed under iterated parallel fuzzy substitution, where the iterating process is prescribed by given crisp control languages. Proceeding inductively over the family of these control languages, yields an infinite sequence of lull AFFL-structures with increasingly stronger closure properties.