Some Studies On Endomorphism Semiring Of Semilattice

The endomorphism of semilattice and its algebra are important topics in the study of algebra.In this paper,the endomorphism of the direct product of two finite chains and its endomorphism semiring is studied.The main results are as follows:First,the properties of endomorphism of the direct product of two finite chains are studied.Using two binary operations on the direct product of two finite chains,a sufficient and necessary conditions is given that a subset of the direct product of two finite chains is a codomain of an endomorphism.Using the relationship between the idempotent endomorphism and fixed points,the characterization of a special subsemiring is given.Second,the structure of endomorphism semiring over the direct product of two finite chains is discussed.The regularity of endomorphism semigroup of a finite chain is extended to the multiplicative reduct of the endomorphism semiring over the direct product of two finite chains.Also,by studying the congruence of the endomorphism semiring,we get that the endomorphism semiring is congruent simple semiring.Third,the set of generators of endomorphism semiring over the direct product of two finite chains is studied.By decomposing the endomorphism of the direct product of two finite chains,we obtain that an endomorphism semiring can be generated from its idempotent set,and the cardinality of the set of generators is given.Finally,some new results have been obtained about centralizer of the set of generators.