Automata theory based on quantum logic II

We establish the pumping lemma in automata theory based on quantum logic un der certain conditions on implication, and discuss the recognizability by t he product and union of orthomodular lattice-valued (quantum) automata. In particular, we show that the equivalence between the recognizabilty by the product of automata and the conjunction of the recognizabilities by the fac tor automata is equivalent to the distributivity of meet over union in the truth-value set.