The Structure And Properties Of Some Non-regular Semigroups And Strong Semilattices

As a natural generalization of the theories of groups and rings,algebraic semigroups theo-ry has become a systematic algebra subject after nearly one hundred years of development and research.Completely regular semigroups can be expressed as semilattice of completely single semigroups,are one of the main research objects of algebraic semigroups theory.The structure of strong semilattice is a better semigroup structure than the semilattice structure.For example,as two special completely regular semigroups,Clifford semigroups and normal cryptographic semi-groups all have the structure of strong semilattice.The class of π-regular semigroups is a generalization of regular semigroups in the range of non-regular semigroups.And π-regular semigroups are concerned by many semigroup scholars,the ideal nil-extension of regular semigroup is a kind of π-regular semigroup.In this thesis,We focus on the properties,structures and strong semilattice of ideal nil-extension of regular semi-groups.We use the θ-product to describe the structure of the ideal nil-extension of normal band,then give necessary and sufficient conditions of a class of ideal nil-extensions of normal band with strong semilattice structures.Finally we study some congruences on π-regular semigroups with strong semilattice structures.This thesis consists of four chapters.We give the main research backgrounds,basic concepts and preliminary knowledges in the first chapter.In the second chapter,we discuss the properties and structures of ideal nil-extension of nor-mal bands.In the first section,we give the structure of θ-product and the characteristics of ideal-nil extension of normal band.In the second section,the structures of ideal nil-extensions of normal band are discussed.In the third chapter,we discuss the strong semilattices structures of ideal nil-extension of rectangle band.In the first section,we use Y-structure two-part homomorphisms to describe the strong semilattice structures of ideal nil-extension of normal band,then we give the necessary and sufficient conditions of a class of special ideal nil-extensions of normal band with strong semilattice structures.In the second section,we will show that an inflation of normal band has the strong semilattice structure.In the fourth chapter,we consider some congruences on some non-regular semigroups with strong semilattice structures.