Some Idempotent Semiring Clusters Of Characterizations

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 obtain the subdirect product decomposition of some semirings which are members of them.In chapter 1, we simply introduce the background and the present state of the theory of semirings; for the sake of convenience, we introduce some fundamental knowledge about idempotent semirings. In chapter 2, 3 we study the subvarieties P and R|· ○ M(R|+ ○ M) of the variety of idempotent semirings, respectively. Firstly, by using Green-relations on the addition reduct and the multiplication 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 chapter 4 we study the subvariety GBSl of the variety of idempotent semirings. We discuss the properties and the structure theorems for the semirings in it, moreover, we give the number of members which the lattice L(GBSl) of subvarieties of GBSl contains.