Minimization of lattice automata

Theories of minimization of fuzzy automata have been developed by several authors, and most of which applied methods of algebraic theory, more specifically, the equivalence relation, the quotient space and equivalence class, congruence and homomorphism, in studying fuzzy automata. In this paper, we also apply the algebraic theory in the study of lattice automata, and obtain some results similar to the ones of fuzzy automata. In this paper, concepts of refining equivalence and refining congruence are defined, the quotient lattice automaton with respect to refining congruence is formulated, the concept of lattice automaton is reviewed, the equivalence of a lattice automaton and its quotient automaton is proved, the minimal property of quotient automaton is shown, and the minimization algorithm of lattice automata is proposed. The main idea of this paper is that by putting forward the concepts of refining equivalence and refining congruence, we can derive the quotient lattice automaton with respect to refining congruence, and via showing that the quotient lattice automaton is not only equivalent to the lattice automaton but also a minimal automaton, we obtain the minimization of a lattice automaton, and thus get the minimization algorithm of lattice automata.