Completely Regular Semiring Which Multiplicative Reducts Are Normal Orthogroup

In this dissertation, we devote to study an important class of semirings, in which the multiplicative reducts of semirings are normal orthogroups. We diliberate the fundamental properties and structure of semirings in this class. This thesis consists of four chapters. The first chapter includes the basic notations and definitons. We introduce some concepts and theories related to semirings, and recall the notations and results of completely regular semigroups which will be used in the following. In chapter two, we consider a semiring class which the multiplicative reducts of semirings are rectangular group. We characterize the properties and obtain the decomposition of subdirect products of members in this class. In chapter three, we investigate the semirings whose multiplicative reducts are normal orthogroups. We extend a few of congruences in completely regular semigroups to semirings. From different ways, we get the decompositions of subdirect products and Mal'cev products of these semirings by making using of the extended congruences. In the finnal chapter, we give a new interpretation about a problem in [2], characterize completely regular semigroups with property P* and spread some results to semirings.