Semigroup. Multiplication Inverse Semigroup Semiring

Algebric theory of semirings is a very important branch of algebra. There are significant values of principle and applications in the study of semirings.In this dissertation, we consider semirings whose additive reducts are semilattices and multiplicative reducts are inverse semigroups, dliberate the nature and structure of semirings and the subclasses of given class of semirings. In chapter one, we introduce related knowledge of semirings and notes uesed in the following. In chapter two, we research inverse semirings, obtain the conditions what the semirings satisfy and related propositions. We give some necessary and sufficient conditions that semirings become distributive lattices. In chapter three, we obtain completely regular semirings are Clifford semirings and 0-group semirings are only subdirectly irreducible Clifford semirings. By using of theory of Mcalister cone, we construct partial orders on semirings and get some interesting results; In chapter four, we investigate semirings whose additive reducts are semilattices and multiplicative reducts are E-unitary Cliford semigroups, give structure theory and decompositon of subdirect product about these semirings.