关于幂等元半环理论中的一个问题

On a Problem Related to Idempotent Semiring

幂等元半环 S的加法半群上的 Green关系 D+ 是半环 S的同余 ,然而 S的乘法半群上的Green D-关系 D未必是 S上的同余 .Pastijin证明了 :D是 S上的同余的幂等元半环 S的全体构成了幂等元半环簇的一个子簇 ,进一步还给出了这个子簇的含有 3个变量的簇等式组 ,当时不知道这个子簇是否有两个变量的簇等式组 .从而 ,提出了一个公开问题 :D是否为由两个自由生成元生成的幂等元半环上的同余 ,本文给出了这个问题的一个肯定的回答 .

A Greens D relation D+ of the additive redect of an idempotent semiring S is a congruence of the semiring S, but Greens D relation  of the multiplicative redect of S need not be. In fact, it was proved by Pastijn that the idempotent semirings for which  is a congruence form a proper subvariety of the variety of all idempotent semirings. Also the defining equations were given with three free variables of this variety. Pastijn left the question open whether  is a congruence on the free idempotent semiring generated by two free generators. In this paper, the open question will be solved.