The Nature And Structure Of Several Kinds Of Eventually Regular Semigroups

The present dissertation is devoted exclusively to the nature and structure of several kinds of eventually regular semigroups. It contains three chapters.Chapter 1 is a brief introduction, we mainly introduce the development of eventually regular semigroups of domestic and abroad, the problems which exist at the moment , the significance of the subject and the role of the project research.In Chapter 2, we study eventually regular PI semigroups. It can be divided into three sections. Section 1, we introduce some basic preliminaries, regular semigroups whose idempotents satisfy permutation identities have been studied. A semigroup is strongly eventually inverse if it is eventually regular and idempotent elements commute. In section 2, we study eventually regular semigroup whose idempotents satisfy permutation identities, we know a regular semigroup is PI -semigroup if and only of it is isomorphic to the direct product of normal band and Clifford semigroups which can be exchanged. We generalize the structure of quasi-direct product to eventually regular semigroups and characterize the new definition of quasi-direct product on eventually regular semigroups, so we obtain the structure and nature of an eventually regular semigroup whose idempotents satisfy permutation identities. An eventually regular semigroup is an eventually regular semigroup whose idempotents satisfy permutation identities if and only if it is isomorphic to the quasidirect product of a left eventually normal band , strongly eventually inverse and right eventually normal band. In section 3, we mainly study the nature and structure of eventually regular PI semigroups. The concept of a complete eventually inverse semigroup, a generalized complete eventually inverse semigroup and a generalized weak complete eventually inverse semigroup are introduced in order to study the nature and structure of eventually regular PI semigroups. We give a semigroup S is an eventually regular PI semigroup if and only if S is a generalized complete eventually inverse semigroup and study some nature of eventually regular semigroup S whose satisfy permutation identities.In Chapter 3, we study orthodox congruences on a GV-semigroup. It contains three sections. In section 1, we state briefly most of the needed notion, terminology and preliminary. In section 2, we characterize the structure of an orthodox congruence pair on on GV-semigroups. Orthodox congruences on a regular semigroup have been studied, an orthodox congruence pair on regular semigroups is uniquely determined by its kernel and hyper-trace. We give an new definiton of an orthodox congruence pair. If a certain congruence pair (ξ,K) on GV-semigroups is an orthodox congruence pair, then it will satisfy some conditions, whereξis a certain congruence on subsemigroup E ( S), K is a certain normal subsemigroup. But the difference is that definite conditions can be exchanged, We separately give the new definiton of normal congruenceξon subsemigroup E ( S) and normal subsemigroup K by means of weak inverse in this section, we characterize an orthodox congruence pair on on a GV-semigroup by using the same method. In section 3, we study the nature of orthodox congruences on GV-semigroups. The results of orthodox congruences on regular semigroups generalize to GV-semigroup. We characterize a binary relationρ(ξ,K) by means of weak inverse: It is also shown that an orthodox congruence pair on GV-semigroups is uniquely determined by its kernel and hyper-trace, we give some nature of an orthodox congruence pair and the theorem relations between an orthodox congruence pair and an orthodox congruence.