Abstract
Introduction. In the following all semigroups are of finite order. One semigroup S, is said to divide another semigroup S2, written SlIS2, if S, is a homomorphic image of a subsemigroup of S2. The semidirect product of S2 by Sl, with connecting homomorphism Y, is written S2 X y Sl. See Definition 1.6. A semigroup S is called irreducible if for all finite semigroups S2 and Si and all connecting homomorphisms Y, S I (S2 X Y SJ) implies S I S2 or S I S1. It is shown that S is irreducible if and only if either:
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