Algebraic properties of L-fuzzy finite automata

Fuzzy automata theory on lattice-ordered monoids was introduced by Li and Pedrycz. Dropping the distributive laws, fuzzy finite automata (L-FFAs for short) based on a more generalized structure L, named a po-monoid, are presented and investigated from the view of algebra in this paper. The notions of (strong) successor and source operators, fuzzy successor and source operators which are shown to be closure operators on certain conditions are introduced and discussed in detail. Using the weak primary submachines, a unique decomposition theorem of a fuzzy finite automaton based on a lattice-ordered monoid is obtained. Taking L as a quantale, fuzzy subsystems are proved to be the same as fuzzy submachines of an L-FFA. In particular, intrinsic connections between algebraic properties of L and properties of some operators of an L-FFA are discovered. It is shown that the join-preserving property of fuzzy successor and source operators can be fully characterized by the right and left distributive laws respectively, and the idempotence of successor operator can be characterized equivalently by the nonexistence of zero divisors when L is a lattice-ordered monoid.