The Structure And Properties Of Several Class Power And Etc. Semiring

In this paper, several special classes of the idempotent semiring are studied. The main results are as follows:1. A subclass BR o M of the idempotent semiring variety is studied. Some characterizations of the members of this class BR o M are given by using the following conditions:the partial orders on the idempotent semiring, identities of variety and Green's relations on the additive reduct and multiplicative reduct of the idempotent semiring. At the same time, it is indicated that BR o M is a variety. In addition, the structure of the members of this variety are achieved from the angles of the subdirect product decomposition and Mal'cev product decomposition.2. Two subclasses GBR o D and GBR o M of the idempotent semiring variety are studied. It is proved that they are the proper subvarieties of the idempotent semiring variety by way of studying the structure of them, and some structure are achieved about them. Then some properties of the members of these two subvarieties are explained. The necessary and sufficient conditions for the members of GBR o D respectively in R o D, R o D and BR o D are given, and the necessary and sufficient conditions for the members of GBR o M re-spectively in R o M, R o M and BR o M are given.3. Two subclasses R o (D V M) and R o (D∨M) of the idempotent semir-ing variety are studied. Some properties of the members of these two idempotent semiring classes are given. It is proved that D is the least D V M-congruence on the member of the class R o (D V M) and∨is the least D V M-congruence on the member of the class R o (D V M).