Algebraic Theory of Machines. I. Prime Decomposition Theorem for Finite Semigroups and Machines

Abstract

Introduction. In the following all semigroups are of finite order. One semigroup Si is said to divide another semigroup S2, written Si| S2, if Si is a homomorphic image of a subsemigroup of S2. The semidirect product of S2 by Si, with connecting homomorphism Y, is written S2 Xy Si. See Definition 1.6. A semigroup S is called irreducible if for all finite semigroups S2 and Si and all connecting homomorphisms Y, S\textbackslash(S2Xy Si) implies S| S2 or S| Si. It is shown that S is irreducible if and only if either:(i) S is a nontrivial simple group, in …