Some Studies For Completely π-Regular Semigroups

The study on completely regular semigroups and completely n -regular semigroups is an important topic in the theory of semigroups. Completely regular semigroups were first studied by A. H. Clifford, and its structure on the semilattice decomposition was given in [26]. Thus, completely regular semigroups can be expressed as semilattice of completely simple semigroups. A perfect structure of completely regular semigroup was given by Petrich in [10], by using Clifford's representation of a completely regular semigroup. Congruences on completely regular semigroups were also considered laterly. In particular, M .Petrich described the properties of congruences and congruence lattices on the semigroups S which are ideal extension of some semigroups. Moreover, he pointed out that any congruence on such semigroups can be uniquely determined by an admissible triple on S.In recent years, the class of completely n -regular semigroups which are nil-extensions of a completely regular semigroup has attracted the attention of a number of authors. For example, Bogdanovic[2], B.J.YU[3] Yuqi Guo and Xueming Ren[7] have investigated the structure on nil-extensions of completely regular semigroups of various kinds. Then Yuqi Guo and Xueming Ren described the properties of congruence on completely π -regular semigroups in [4] by using the concept of admissible congruence pair.On this base, the isomorphism problem on strictly π -regular semigroups which are nil-extensions of completely regular semigroups is studied in this thesis firstly, and the necessary and sufficient condition under which strictly π -regular semigroups are isomorphic is obtained. Then group congruences, the least group congruence, regular congruence, and idempotent-seperating congruence on such semigroups are obtained.There are three chapters in this thesis.In the first chapter, we give some notations and preparations.In the second chapter, we consider the isomorphism problem between two strictly n -regularsemigroups and give the necessary and sufficient condition under which strictly π-regular semigroups are isomorphic by using the structure of strictly n -regular semigroups.Lastly, we study some congruences on such semigroups, namely, group congruences, the least group congruence, regular congruences, idempotent-seperating congruences, and give their descriptions respectively.