(W) Several Studies On The Nil-extension Of LRC-semigroup Ideals

As a generalization of the theories of groups and rings,the theory of algebraic semigroups has developed as a systematic algebraic branch.Regular semigroups are the main research objects of algebraic semigroup theory because of their regularity.Completely regular semigroups and inverse semigroups are two large subclasses of regular semigroups.Orthogroups play an impor-tant role in the completely regular semigroups.As two important subclasses,LRC-semigroups and LRN-semigroups have become the main research objects.The class of-regular semigroups is a generalization of regular semigroups in the range of non-regular semigroups.And-regular semigroups are concerned by more semigroup scholars in recent years.The ideal nil-extension of?W?LRC-semigroups,as a generalization of?W?LRC-semigroups in the range of-regular semi-groups,is an important kind of-regular semigroups.In this thesis,we describe the properties of the ideal nil-extension of?W?LRC-semigroups with the Green Star relations and the congruence relations.Also we characterize the structures of the ideal nil-extension of?W?LRC-semigroups by using the spined product representation.This thesis consists of four chapters.We give the main research backgrounds,basic concepts and preliminary in the first chapter.In the second chapter,we discuss the properties and structures of the ideal nil-extension of regular orthogroups.In the first section,we define ?*,?₁*,and ?₂*,and we prove that they are congruence relations.In the second section,we give the spined product representation of the ideal nil-extension of regular orthogroup.In the third chapter,we discuss the properties and structures of the ideal nil-extension of?W?LRC-semigeoups.Firstly,we obtain the properties of the ideal nil-extension of?W?LRC-semigroups with help of the Green Star relations.Then we obtain some properties and structural representations of the ideal nil-extension of?W?LRC-semigroups by the spined product of the ideal nil-extension of regular orthogroups.In the last chapter,we discuss the properties and structures of the ideal nil-extension of LRN-semigroups.