Regular Congruences on Eventually Regular Semigroups

Let P be an arbitrary full subset of the set E(S) of the idempotents in an eventually regular semigroup S. We shall prove that each regular congruence on eventually regular semigroup S is completely determined by its P-partial kernel normal system and give the P-partial kernel normal system in an eventually regular semigroup an abstract characterization. A semigroup S is called an eventually regular semigroup if for every a ∈ S there exists a positive integer n such that a n is regular. The class of eventually regular semigroups which contains both the class of all regular semigroups and the class of all finite semigroups was introduced by Edwards [1]. The strategy was to generalize known results for regular semigroups and for finite semigroups to eventually regular semigroups. Edwards [1] showed that several aspects of regular semigroups and of finite semigroups have joint natural extensions to eventually regular semigroups. The concept kernel normal system was introduced by Wanger [2] and Preston [3] to describe the congruences on inverse semigroups and then was generalized to the class of orthodox semigroups by Meakin [4]. The kernel normal systems in inverse and orthodox semigroups have been abstractly characterized, we know each congruence on a regular semigroup is completely determined by its kernel normal system. Yong He [5] proved that each congruence on regular semigroup S is completely determined by the congruence classes containing elements in any full subset of E(S). Yanfeng Luo[6] have shown that each regular congruence on eventually regular semigroup S is uniquely determined by its kernel normal system, Yong He [7] have proved that each regular congruence on eventually regular semigroup S is completely determined by its P-partial kernel normal system, but he have not given the P-partial kernel normal system and regular congruence decided by P-partial kernel normal