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Normal subsemigroups of finite transformation semigroups

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Abstract

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.

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  1. Department of Mathematics, University of Mary Washington, Fredericksburg, VA, 22401, USA

    Janusz Konieczny

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  1. Janusz Konieczny
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Correspondence to Janusz Konieczny.

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Communicated by Ganna Kudryavtseva.

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Konieczny, J. Normal subsemigroups of finite transformation semigroups. Semigroup Forum 107, 680–691 (2023). https://doi.org/10.1007/s00233-023-10383-w

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