Eventually regular semigroups

Abstract

A semigroup is called eventually regular if each of its elements hassome power that is regular. Thus the class of all eventually regular semi-groups includes both the class of all regular semigroups and the class ofall group-bound semigroups and so in particular includes the class of allfinite semigroups [J].Many results that hold for all regular semigroups also hold for allfinite semigroups; often this occurrence is not just a coincidence but isnecessarily the case since the results concerned hold for eventuallyregular semigroups. We show that many results may be generalized fromregular semigroups to eventually regular semigroups. In particularLallement's Lemma that for every congruence p on a regular semigroup 5 ,every idempotent p-class contains an idempotent is shown to hold foreventually regular semigroups [I].We define a relation that we denote by \i = u(S) on an arbitrarysemigroup S and show that u is an idempotent-separating congruence onS . For an eventually regular semigroup 5 it is shown that \i is themaximum idempotent-separating congruence on S . Let S be an arbitrarysemigroup. We show that the semigroup S/p is finite if and only if theset of idempotents of S , E(S) is finite [1]. The semigroup S iscalled fundamental if the only idempotent-separating congruence on S isu . It is shown that p is the identity congruence on S/]i(S) [2].(Using the previous result David Easdown has shown that for any semigroupReceived 16 Augus 198t U . Thesis submitted to Monash University February198U. Degree approved August 198U. Supervisor: Professor G.B. Preston.Copyright Clearance Centre, Inc. Serial-fee code: OOOU-9727/85$A2.00 + 0.00.157