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

摘要： 研究两条有限链直积上自同态半环的性质。利用有限链直积上的两种二元运算，给出了两条有限链直积的子集构成自同态像集的充要条件，证明了自同态半环的乘法半群是正则半群。通过对有限链直积上的自同态进行分解，得到了自同态半环可由其乘法半群的幂等元集生成；推广了有限链上自同态半群的一些结果。

关键词: 自同态半环, 正则半群, 幂等元