Analogues of the bicyclic semigroup in simple semigroups without idempotents

Abstract

SynopsisBy an important theorem of Andersen, any semigroup, containing idempotents, which is simple but not completely simple contains a copy of the bicyclic semigroup B = 〈a, b | ab = 1〉. In this paper the semigroups A = 〈a, b | a2b = a〉 and C = 〈a, b | a2b = a, ab2 = b〉 are shown to play a similar role in various classes of simple semigroups without idempotents, particularly in those for which Green's relation is nontrivial. For example it is shown that every right simple semigroup without idempotents is a union of copies of A; every finitely generated simple semigroup without idempotents contains either A or C. In a generalisation of a different sort it is shown that the bicyclic semigroup divides every simple semigroup without idempotents.Similar results are obtained for 0-simple semigroups without nonzero idempotents.