Studies On The Congruence Y~* On Completely Regular Semigroups

In this thesis, we investigate such a problem which was proposed by Petrich and Reilly in [2] in 1999: What can be said about (?)~*? This dissertation consists of four chapters in all.Some definitions and properties for Semigroups and completely regular semigroups are proposed in Chapter 1. And then, we fix some notation which is used later.The aim of Chapter 2 is to give the congruence internal of (?)~* on completely regular semigroups. Using properties of normal subgroups, we describe the least group congruence on completely simple semigroups. We prove (?)v by the construction of completely simple semigroups. We prove y is the most idempotent pure congruence on completely simple semigroups. We give the sufficiency and necessity of (?)~*=(?).Chapter 3 is devoted to character (?)~* on completely simple semigroups. We find that (?)~*=(?) in this case.We investigated (?)~* on normal cryptogroups in the final chapter. Using the construction of normal cryptogroups, we construct a binary relation on normal cryptogroups. And then we prove that the transitive closure of the relation is just y~*.