Transformation Semigroups and State Machines

Abstract

A transformation semigroup is a pair (Q; S) consisting of a finite set Q, a finite semigroup S and a semigroup action λ : Q X S Q, (q, s  s (q) which means : i) q ε Q, s, t ε S : st (q) = s (t (q)) , and (ii) s, t ε Sq ε Q, s (q) = t (q) s = t. A state machine or a semiautomation is an ordered triple M = (Q, ∑, F ), where Q and are finite sets and F : Q X ∑ Q is a partial function. This paper provides the construction of state machines associate a direct product, the cascade product, and wreath product of transformations semigroups.