Cardinalities in finite monoids of $G$-equivariant functions
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912261042339840 |
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| author | Ruiz-Medina, Ramón H. |
| author_facet | Ruiz-Medina, Ramón H. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03772 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cardinalities in finite monoids of $G$-equivariant functions Ruiz-Medina, Ramón H. Group Theory 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. |
| title | Cardinalities in finite monoids of $G$-equivariant functions |
| topic | Group Theory |
| url | https://arxiv.org/abs/2503.03772 |