The Burnside ring of simple 𝒞-sets
Abstract
The Burnside ring of a finite category, introduced by Webb, generalizes the classical Burnside ring of a finite group. However, unlike the classical case, the Burnside ring of a finite category has finite rank if and only if the category is equivalent to a groupoid. In this article, we introduce a new invariant associated with a finite category 𝒞, called the \emph{simple Burnside ring} of 𝒞 and denoted by BS(𝒞). This construction is obtained from simple 𝒞-sets and generalizes the classical Burnside ring of a finite group. Moreover, the ring BS(𝒞) always has finite rank. We develop the basic theory of simple 𝒞-sets and study several structural properties of the ring BS(𝒞). In particular, we determine all ring homomorphisms from BS(𝒞) to ℤ, describe its prime spectrum, and obtain a decomposition theorem expressing BS(𝒞) as a product of simple Burnside rings of strongly connected subcategories.
Related
- A simple construction of a simple nil ring
- Automorphism groups of Cayley graphs on almost simple groups with normal connection sets
- A Simple Explicit Construction of Smaller Strong Blocking Sets
- Crossed Burnside rings for groupoids
- Mazurov's question on fixed points of automorphisms of free Burnside groups
BibTeX
@misc{leon2026burnside,
title = {{The Burnside ring of simple $\mathcalC$-sets}},
author = {José Miguel Calderón León and Alberto G. Raggi-Cárdenas and Itzel Rosas and Ramón H. Ruiz-Medina},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.25036},
eprint = {2607.25036},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.25036},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Burnside ring of simple 𝒞-sets
AU - José Miguel Calderón León
AU - Alberto G. Raggi-Cárdenas
AU - Itzel Rosas
AU - Ramón H. Ruiz-Medina
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2607.25036
UR - https://arxiv.org/abs/2607.25036
AB - The Burnside ring of a finite category, introduced by Webb, generalizes the classical Burnside ring of a finite group. However, unlike the classical case, the Burnside ring of a finite category has finite rank if and only if the category is equivalent to a groupoid. In this article, we introduce a new invariant associated with a finite category 𝒞, called the \emph{simple Burnside ring} of 𝒞 and denoted by B^S(𝒞). This construction is obtained from simple 𝒞-sets and generalizes the classical Burnside ring of a finite group. Moreover, the ring B^S(𝒞) always has finite rank. We develop the basic theory of simple 𝒞-sets and study several structural properties of the ring B^S(𝒞). In particular, we determine all ring homomorphisms from B^S(𝒞) to ℤ, describe its prime spectrum, and obtain a decomposition theorem expressing B^S(𝒞) as a product of simple Burnside rings of strongly connected subcategories.
ER -