Abstract
We study fuzzy finite automata in which all fuzzy sets are defined by membership functions whose codomain forms a lattice-ordered monoid
L
. For these
L
-fuzzy finite automata (
L
-FFA, for short), we provide necessary and sufficient conditions for the extendability of the state-transition function. It is shown that nondeterministic
L
-FFA (N
L
-FFA, for short) are more powerful than deterministic
L
-FFA (D
L
-FFA, for short). Then, we give necessary and sufficient conditions for the simulation of an N
L
-FFA by an equivalent D
L
-FFA. Next, we turn to the closure properties of languages defined by
L
-FFAs: we establish closure under the regular operations and provide conditions for closure under intersection and reversal, in particular we show that the family of fuzzy languages accepted by D
L
-FFAs is not closed under Kleene closure operation, and the family of fuzzy languages accepted by N
L
-FFAs is not closed under complement operation. Furthermore, we introduce the notions of
L
-fuzzy regular expressions and give the Kleene theorem for N
L
-FFAs. The description of D
L
-FFAs by
L
-fuzzy regular expressions is also given. Finally, we investigate the level structures of
L
-FFAs. Our results provide some insight as to what extend properties of
L
-FFAs and their languages depend on the algebraic properties of
L
.