Monads and a common framework for fuzzy type automata

Automata defined by monads in categories are introduced as special examples of monoids actions on free -algebras, where is a monad in a category. Morphisms between monads are introduced as special functors between Kleisli categories. Any morphism generates a functor between the corresponding categories of monadic automata. The relationship between morphisms of monads and functors of corresponding monadic automata categories gives a common framework in the theory of automata defined by monads. The proposed framework unifies many of well-known automata types and transformation processes of one type automata to other type. The notion of a monadic automaton with input and output morphisms, and a language accepted by this monadic automaton are introduced. An acceptance of a language is preserved by morphisms between monadic automata with input and output morphisms and it is also preserved by morphisms between monads.