Properties Of Some Particular Non-regular Semigroups

This master dissertation consists of four chapters. We investigate the properties, constructions and congruences of eventually regular semigroups especially, GV-semigroups. And some new results on characterization of semilattices of nil-extensions of left groups are studied.The chapter 1 is a brief introduction. We show the research background, research way and research content of the paper.In chapter 2, the rectangular group congruence on an eventually regular semigroup S is first described by means of the rectangular group congruence pairs. Furthermore, we can obtain a lattice isomorphism between the rectangular group congruences and the rectangular group congruence pairs. Finally, the minimum group congruence on an eventually regular semigroup S is investigated in the way of weak inverse. Then the representation of the minimum group congruence on S and an equivalent condition of the minimum group congruence are shown.The aim of chapter 3 is to study the basic properties of GV-semigroups . We first define a family function and use it to characterize the structure of GV-semigroups. Furthermore, the equivalent conditions of the isomorphism of GV-semigroups are offered. Then, we study the relations between the GV-semigroup and its various ideals, particular subsemigroups. And we obtain the completely regularity of a GV-semigroup is a hereditary property concerning all the kind of ideals. Finally, we give some particular properties of GV-semigroups, which are a generation of corresponding properties of completely regular semigroups.Chapter 4 is devoted to investigate some new results on a characterization of semilattices of nil-extensions of left groups by means of two new definitions of left group. By the approach, we can get many new equivalent conditions which contribute to know the structure of semilattices of nil-extensions of left groups clearly.