Burnside Semiring

The Burnside semiring or Burnside rig of a Group $G$ is a Semiring on the set of Isomorphism Classes of the Elementary Topos of Finite G-Sets $\mathsf{FinSet}^{G}$, where addition is given by Coproducts in $\mathsf{FinSet}^{G}$ and multiplication by Products.

More generally, the Burnside semiring of a Distributive Category $\mathcal{C}$ is the semiring formed by taking Isomorphism Classes of $\mathcal{C}$.

Properties

• The Burnside semiring of a group is an Exponential Semiring, where exponentiation is given by Cartesian Closure of $\mathsf{FinSet}^{G}$.