The Burnside ring of simple C-sets

2026/07/27 by José Miguel Calderón León, Alberto G. Raggi-Cárdenas, Itzel Rosas +1
#math.CT

paper · pdf

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 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.

Citations

• A Burnside ring for monoids

Related

• The extended Burnside ring and module categories
• Simplicial Burnside ring
• A note on the lambda-structure on the Burnside ring
• Explicit Generators for the Unit Group of the Burnside ring
• Generalised Burnside Rings, G-categories and Module Categories