A CONDITION ON THE NATURAL ORDER FOR REGULAR SEMIGROUPS
Mario Petrich · Communications in Algebra · 2002
Let S be a regular semigroup with set of idempotents E(S). We say that S is pure if, for all E(S) and , in the natural order implies that . We study the influence of purity on various constructions which produce semigroups belonging to some familiar classes of regular semigroups as well as examine the relationship of purity and -unitariness. This condition appears most natural on locally inverse semigroups where it is equivalent to the greatest idempotent pure congruence being a completely simple congruence. We also construct a relatively free completely regular semigroup generated by where and apply this result to obtain a criterion for purity for completely regular semigroups.