On regular semigroups whose idempotents form a subsemigroup

Abstract

For brevity the semigroups in the title are called orthodox semigroups. The finest inverse semigroup congruence on an orthodox semigroup is shown to have a simple form and conversely, regular semigroups whose finest inverse congruence has this simple form are shown to be orthodox. Next ideal extensions of orthodox semigroups by orthodox semigroups are shown to be also orthodox, whence a finite semigroup is orthodox if and only if each principal factor is orthodox and completely O-simple or simple. Finally it is determined which completely O-simple semigroups are orthodox.