Eventually regular semigroups

A semigroup is said to be eventually regular if each of its elements has some power that is regular. Regular and group-bound semigroups are each eventually regular. Idempotent-surjective semigroups are semigroups such that all idempotent congruence classes contain idempotents; eventually regular semigroups are idempotent-surjective. Many results for regular semigroups also hold for eventually regular semigroups or even for idempotent-surjective semigroups and so in particular are also valid for group-bound semigroups. Lallement's lemma is generalized to eventually regular semigroups and the maximum idempotent-separating congruence on such a semigroup is found. Other congruences are considered and the results obtained are applied to yield results on biordered sets.