We apply lattice ordered monoids in the study of lattice finite automata, and obtain some results similar to the ones of fuzzy finite automata. In this paper, the concepts of lattice minimal automata and lattice reduced automata are put forward, and the relationship of them associated with an L-language is discussed. Moreover, the L-fuzzy equivalence relation on states is defined, and an algorithm to minimize the lattice finite automata is presented based on clustering states.