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 Sq 𝜖 Q, s (q) = t (q) s = t. A state machine or a semiautomation is
an ordered triple M = (Q, Σ, F ), where Q and are finite sets and F : Q X Σ 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.