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arXiv · 2503.03772

Cardinalities in finite monoids of $G$-equivariant functions

Ramón H. Ruiz-Medina
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Abstract

A set with a group action is referred to as a $G$-set, and the set of functions that commute with this action forms a monoid under function composition. This paper examines the case where the $G$-set is finite, which implies that the monoid of $G$-equivariant functions is also finite. The document provides formulas for calculating the cardinality of this monoid, its group of units, and explores special cases of $G$-equivariant functions, known as fixing elementary collapsings. All of these results are expressed in terms of specific properties of the $G$-set, including the number of orbits and certain indices of the subgroups acting as stabilizers.

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Ramón H. Ruiz-Medina. 2025-03-04. Cardinalities in finite monoids of $G$-equivariant functions. https://arxiv.org/abs/2503.03772

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