关键词: 自同态半环, 正则半群, 幂等元

Abstract: This paper studies the properties of the endomorphism semiring over the direct product of two finite chains. Using two binary operations on the direct product of two finite chains, a sufficient and necessary condition under which the subset of the direct product of two finite chains is the codomain of the endomorphism is given. Next, it proves that the multiplicative reduct of the endomorphism semiring is a regular semigroup. Finally, by decomposing the endomorphism of the direct product of two finite chains, it obtains that the endomorphism semiring can be generated by the idempotents of its multiplicative reduct. Some known results about the endomorphism semigroup over a finite chain are expanded.

Key words: endomorphism semiring, regular semigroup, idempotent