Reversibility in monadic algebras and automata

If we regard reversibility (in the intuitive sense) as the ability to return to a previous state, monadic algebras possess varying types and degrees of reversibility. In this paper, we define several properties of monadic algebras, each describing a type of reversibility, and study their relative strength and whether they are preserved under generalized homomorphisms. We show that, in the case of automata (finite monadic algebras) five of these properties are equivalent. We define the reverse, Rev(A), of an algebra A and study its relationship to A.