Automata theory based on unsharp quantum logic

By studying two unsharp quantum structures, namely extended lattice ordered effect algebras and lattice ordered QMV algebras, we obtain some characteristic theorems of MV algebras. We go on to discuss automata theory based on these two unsharp quantum structures. In particular, we prove that an extended lattice ordered effect algebra (or a lattice ordered QMV algebra) is an MV algebra if and only if a certain kind of distributive law holds for the sum operation. We introduce the notions of (quantum) finite automata based on these two unsharp quantum structures, and discuss closure properties of languages and the subset construction of automata. We show that the universal validity of some important properties (such as sum, concatenation and subset constructions) depend heavily on the above distributive law. These generalise results about automata theory based on sharp quantum logic.