Some Studies On Several Classes Of Idempotent Semirings

The class of idempotent semirings is a very important class of semirings. Many experts and scholars thorough, painstakingly and systematically investigate it. In this paper we mainly study some important classes of idempotent semirings. First, we give certain characterizations of the members of them from all kinds of angles; second, we also using the structure of sturdy frame of type (2,2) algebras obtain some equivalent depicts of this semirings which are members of them.This dissertation can be divided into four parts. The first chapter, we simply introduce the background and the present state of the theory of semirings. for the sake of convenience,we introduce the basic notations and results of idempotent semirings. The second and third chapter, we separate study the idempotent semirings which satisfies the identity x + yx + x≈y + x,xy + y≈xy and x + xy + x≈xy,they also are subvarieties of the variety idempotent semiring, respectively. Firstly, by using Green - relations on the addition reduct of idempotent semirings and partial relation on idempotent semirings, we give many characterizations of the members of them ; Secondly, we study the structure of the members of them in virtue of sturdy frame of type (2,2) algebras. In chapter four,we study the idempotent semiring with involution, they also are subvarieties of the variety of idempotent semirings. We discuss the properties and the structure theorems for the semirings in it.