Semigroup Separability And Lr-semi-regular Semigroup Structure

Semigroups rised in the 1960s, and in the 1970s, semigroup theory developed rapidly. On this basis, a series of achievements have been made, which covers the sames, structures, clusters and so on. At the same time, through the use of the Green's relationship, some special semigroups were studied, and on this basis, a new Green's relationship and some new semigroups were studied. Along with the continuous deepening of the study, semigroup theory was being enriched accordingly. We know that the concept of splittable is being widely used in topological spaces and other aspects. But in the beginning of this century, D. J. Marron and T. B. M. McMaster introduced the concept of splittable to semigroups and gave the corresponding definition. At the same time, the properties of splittable over a class of semigroups and splittability over some semigroups were studied. On this basis, in 2006, A. J. Hanna and T. B. M. McMaster studied the splittability over 0-simple semigroups and monogenic semigroups. Similarly, in 2005, Professor Yuqi Guo also studied a new type of genetic semigroup, that is, about the nature of the dual. He introduced two new semigroups-LR-regular orthogroups and LR-normal or-thogroups. Meanwhile, he studied the characters and structures of them. In this paper, I further study the splittability over semigroups and give a new concept named strong splittability. At the same time, I study the structure of LR-semiregular sermigroups.This thesis can be divided into three chapters.Chapter 1, we study the strong splittability over some semigroups. Section 1 is a brief introduction and we give the reason of introducing the concept of strong splittability over semigroups. In Section 2, the concept as well as the properties of strong splittability of a semigroup over a class of semigroups are introduced. In Section 3, we study the strong splittability over some semigroups and consider the strong splittability over D-scmigroups.In Chapter 2, we study the splittability over some semigroups and consider the pointwise onto-splittability over 0-simple semigroups. It contains four sections. In Section 1, some background about the splittability over semigroups are introduced. In Section 2, some basic conceptions and preliminaries about the splittability over some semigroups are introduced. In Section 3, the splittability over some semigroups are studied. In Section 4, the pointwise onto-splittability over 0-simple semigroups are characterized.Chapter 3 is devoted to the structure of LR-semiregular sermigroups. It can be divided into 3 sections. In Section 1, we introduce some classes of LR semigroups. In Section 2, the concept as well as the properties of LR-semiregular sermigroups are introduced. In Section 3, the structure of LR-semiregular sermigroups is given.