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A semigroup is said to be eventually regular if each of i t s elements has some power that is regular. Regular and group-bound semigroups are each eventually regular.

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A semigroup is called eventually regular if each of i t s elements has some power that is regular. Thus the class of a l l eventually regular semigroups includes both the class of a l l regular semigroups and the class of a l l group-bound semigroups and so in particular includes the class of all finite semigroups [ J ] .Many results that hold for a l l regular semigroups also hold for a l l finite semigroups; often this occurrence is not just a coincidence but is necessarily the case since the results concerned hold for eventually regular semigroups. We show that many results may be generalized from regular semigroups to eventually regular semigroups. In particular Lallement's Lemma that for every congruence p on a regular semigroup 5 , every idempotent p-class contains an idempotent is shown to hold for eventually regular semigroups [ I ] .We define a relation that we denote by \i = u(S) on an arbitrary semigroup S and show that u is an idempotent-separating congruence on S . For an eventually regular semigroup 5 i t is shown that \i is the maximum idempotent-separating congruence on S . Let S be an arbitrary (Using the previous result David Easdown has shown that for any semigroup