The Burnside ring of simple $\mathcal{C}$-sets
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 $\mathcal{C}$, called the \emph{simple Burnside ring} of $\mathcal{C}$ and denoted by $B^S(\mathcal{C})$. This construction is obtained from simple $\mathcal{C}$-sets and generalizes the classical Burnside ring of a finite group. Moreover, the ring $B^S(\mathcal{C})$ always has finite rank. We develop the basic theory of simple $\mathcal{C}$-sets and study several structural properties of the ring $B^S(\mathcal{C})$. In particular, we determine all ring homomorphisms from $B^S(\mathcal{C})$ to $\mathbb{Z}$, describe its prime spectrum, and obtain a decomposition theorem expressing $B^S(\mathcal{C})$ as a product of simple Burnside rings of strongly connected subcategories.