Rieg theory: the research programme

The aim is to develop the theory of semirings with exponentiation across four subjects: logic, finite and profinite arithmetic, enumerative combinatorics, and category theory.

The central object is a rieg: a commutative semiring with an exponentiation operation satisfying the exponent laws, with the convention $0^0=1$.

The concrete questions in this project are:

• Rational subrieg of the nonnegative reals: determine the relations introduced by exponentiation.
• Free rieg elements as games: find a game interpretation compatible with the rieg operations and evaluation at zero.
• Unary functions represented by terms: characterize the functions on natural numbers obtained from a free rieg.
• Connected compact Hausdorff riegs: answered positively by the unit interval with $\max$, $\min$ and $x^y=\max(x,1-y)$; by the compact no-go theorem every example has noncancellative addition.
• Finite-valued presheaf categories: identify when categorical counting has the topos structure motivating Burnside riegs.

The target is a single mathematical account that states the definitions and examples precisely and separates established results from these remaining questions. The proposed title is Notes on Rieg theory: semiring with exponentials in logic, profinite arithmetic, enumerative combinatorics, and category theory.