The conjugacy relation and normal subgroups play an important role in group theory. These notions are linked by the fact that the normal subgroups of a group are precisely the subgroups that are closed under the group conjugacy. There have been several attempts to extend the notion of conjugacy to semigroups. For each semigroup conjugacy, we can define the normal subsemigroups of a semigroup with respect to that conjugacy. In this paper, we study the normal subsemigroups of finite transformation semigroups. We consider four notions of conjugacy for semigroups, which have already appeared in the literature, and the semigroups \(P_n\) of partial transformations, \(T_n\) of full transformations, and \({\mathcal {I}}_n\) of partial injective transformations on a finite set with n elements. We describe the normal subsemigroups of \(P_n\), \(T_n\), and \({\mathcal {I}}_n\), with respect to each of the four conjugacy relations.

共轭关系和正规子群在群论中起着重要作用。这些概念是通过以下事实联系起来的：群的正规子群正是在群共轭下闭合的子群。人们曾多次尝试将共轭的概念扩展到半群。对于每个半群共轭，我们可以定义关于该共轭的半群的正规子半群。在本文中，我们研究有限变换半群的正规子半群。我们考虑文献中已经出现的半群共轭的四种概念，以及部分变换的半群\(P_n\) 、完全变换的半群\(T_n\)和\({\mathcal {I}}_n\ ）具有n 个元素的有限集上的部分单射变换。我们针对四种共轭关系中的每一个 描述了\(P_n\)、\(T_n\)和\({\mathcal {I}}_n\) 的正规子半群。