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Conjugacy in abstract semigroups, transformation and diagram monoids, and conjugacy growth
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Orientador(es)
Resumo(s)
We study conjugacy relations on semigroups and monoids, focusing on the relation a∼nb, defined by the existence of g,h∈S1 such that ag=gb, bh=ha, hag=b, and gbh=a. This notion emerged as one that yields particularly elegant results. The interplay between ∼n and other standard conjugacy relations is analyzed, and some results on special classes of abstract semigroups are established. We then specialize to the case of transformation semigroups. A complete classification of ∼n-classes is obtained for the full transformation monoid Tn, the symmetric inverse monoid In, and the endomorphism monoid of G-sets, among others. We also investigate the natural conjugacy in diagram semigroups, including the partition monoid, the Brauer monoid, and the partial Brauer monoid. Finally, we investigate the conjugacy growth function in polycyclic monoids and obtain a precise asymptotic estimate. The paper concludes with some open problems.
Descrição
Publisher Copyright: © 2026 The Author(s)
Palavras-chave
Conjugacy Conjugacy growth series Partition monoids Polycyclic monoids Transformation semigroups Algebra and Number Theory