Abstract
For a group G acting on a set X, let \(\textrm{End}_G(X)\) be the monoid of all G-equivariant transformations, or G-endomorphisms, of X, and let \(\textrm{Aut}_G(X)\) be its group of units. After discussing few basic results in a general setting, we focus on the case when G and X are both finite in order to determine the smallest cardinality of a set \(W \subseteq \textrm{End}_G(X)\) such that \(W \cup \textrm{Aut}_G(X)\) generates \(\textrm{End}_G(X)\); this is known in semigroup theory as the relative rank of \(\textrm{End}_G(X)\) modulo \(\textrm{Aut}_G(X)\).
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Acknowledgements
The first and second author were supported by a CONACYT Basic Science Grant (No. A1-S-8013) and a PhD CONACYT National Scholarship, respectively. Both authors sincerely thank Csaba Schneider for his kind and enriching replies to our questions on the Imprimitive Wreath Product Embedding Theorem. We also thank the anonymous referee of this paper for his insightful comments.
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Communicated by Jorge Almeida.
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Castillo-Ramirez, A., Ruiz-Medina, R.H. The relative rank of the endomorphism monoid of a finite G-set. Semigroup Forum 106, 51–66 (2023). https://doi.org/10.1007/s00233-023-10340-7
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DOI: https://doi.org/10.1007/s00233-023-10340-7