On categorical semigroups

Abstract

A semigroup S is called categorical if every ideal I of S has the property that abc є I (a, b, c є S) implies ab є I or be bc є I Necessary and sufficient conditions for an orthodox semigroup to be categorical are found and then used to characterize those bands which appear as an isomorphic copy of the band of idempotents of some orthodox categorical semigroup and to simplify the proof of a theorem of Mario Petrich. The structure of commutative categorical semigroups is found modulo the structure of abelian groups and categorical semilattices.