Properties And Structures Of Some Semirings

Guo Y Q,Shum K P and Sen M K defined left Clifford semirings and studied their structures,Zhang J J,Li Y H and Zhao X Z defined rectangular Clifford semirings and described their structures,Han J,Li G defined quasi Clifford semirings and studied their properties and structures.In order to improve the research on Clifford hierarchy of semirings,based on the existing research results and with the help of Clifford hierarchy of semigroups,in this paper,we define LR-regular Clifford semirings and weak left Clifford semirings,their properties and structures are also studied,the obtained results extend the corresponding results of left(right)Clifford semirings.This dissertation is divided into three chapters,the main content of which is in follows:Chapter 1: Firstly,the definition of LR-regular Clifford semirings is given,these semirings are proper generalizations of left(right)Clifford semirings.Secondly,by using the properties of left Clifford semirings and rectangular Clifford semirings,a necessary and sufficient condition for a semiring to be LR-regular Clifford semiring is given.Finally,by means of the structure of LR-regular Clifford semigroups,the spined product structure of LR-regular Clifford semirings is obtained.Chapter 2: The definition of the weak left Clifford semirings is given,properties and structures of the weak left Clifford semirings are studied.Chapter 3: Furthermore,we give another structure of weak left Clifford semirings,that is,Δ-product structure.