Regular expressions with truth values in lattice-monoid and their languages

In this study, we introduce the concept of lattice-valued regular expressions. Such expressions become not only the necessary tool for the analysis and synthesis of fuzzy automata, but give rise to a recursive generation of the family of fuzzy languages accepted by fuzzy automata from certain simple fuzzy languages. The equivalence between lattice-valued regular expressions and lattice-valued finite automata is demonstrated. We also show that the family of fuzzy languages accepted by deterministic lattice-valued finite automata (abbreviated as DLA) is not closed under Kleene closure operation. Similarly, we show that in general the family of fuzzy languages accepted by (nondeterministic) lattice-valued finite automata (abbreviated as LA) is not closed under complement operation. These two findings form an essential difference between lattice-valued finite automata and classical finite automata. We also elaborate on regular expressions produced by DLA.