← Notes on Rieg Theory

Older Versions__2023_07_23Ver__abstract.tex

\begin{abstract}
In ring theory classes, we are told that \dq{natural numbers do not have the structure of a ring, but only a semiring (or a \emph{rig}).} In fact, it does not have the structure of a ring like $2-3=-1 \notin \N$, but it does have a rich algebraic structure in another direction, exponentials $2^3=8$.

In combinatorics, when considering the action of a finite group on a finite set, one sometimes constructs a Burnside rig of the isomorphism classes. However, even in this case, if we do not extend it to a ring, it has an exponential structure.

In this article, we investigate the properties of such semirings with exponentials, which we call \emph{riegs}, and observe their connections with bicartesian closed categories and toposes as categorifications.

Let us spoil some of the wonders of rieg theory. In contrast to field theory, there are only a finite number of possible \dq{positive characteristics} of a rieg! They are $1,2,6,42$, and $1806$.
\end{abstract}