标题： 半群同余与直平方的子半群 显示英文标题

标题： Semigroup Congruences and Subsemigroups of the Direct Square

摘要： 我们研究了半群$S$，它具有如下性质：包含对角线$\{ (s,s)\colon s\in S\}$的每一个子半群$S\times S$必然都是$S$上的一个同余关系。我们将这样的$S$称为 DSC 半群。 众所周知，所有有限群都是 DSC 半群，并且容易看出每个 DSC 半群必然是简单的。 基于此，我们证明对于广泛的半群类——包括周期性半群、稳定半群、逆半群以及若干已知的简单半群类型——唯一的 DSC 成员是群。 然而，事实证明存在非群的 DSC 半群，我们通过利用 Byleen 引入的一种构造方法得到了这些例子。这类例子可以进一步是正则的或双单的。 显示英文摘要

摘要： We investigate semigroups $S$ which have the property that every subsemigroup of $S\times S$ which contains the diagonal $\{ (s,s)\colon s\in S\}$ is necessarily a congruence on $S$. We call such $S$ a DSC semigroup. It is well known that all finite groups are DSC, and easy to see that every DSC semigroup must be simple. Building on this, we show that for broad classes of semigroups -- including periodic, stable, inverse and several well-known types of simple semigroups -- the only DSC members are groups. However, it turns out that there exist non-group DSC semigroups, which we obtain utilising a construction introduced by Byleen for the purpose of constructing interesting congruence-free semigroups. Such examples can additionally be regular or bisimple.