Grammar theory based on lattice-ordered monoid

Motivated by Li and Pedryczʹs work on fuzzy finite automata and fuzzy regular expressions with membership values in lattice-ordered monoids and inspired by the close relationship between the automata theory and the theory of formal grammars, we have established a fundamental framework of L -valued grammar. It is shown that the set of L -valued regular languages coincides with the set of L -languages recognized by nondeterministic L -fuzzy finite automata and every L -language recognized by a deterministic L -fuzzy finite automaton is an L -valued regular language.