Convex spaces, affine spaces, and commutants for algebraic theories

TL;DR

This paper explores the algebraic structures of affine and convex spaces over rings and preordered rings, establishing dualities and commutant relationships among their theories, with applications to real numbers and probability measures.

Contribution

It characterizes the theories of affine and convex spaces as commutants of module theories, revealing dualities and generalizing classical notions in algebraic theories.

Findings

The theory of left R-affine spaces is a commutant of pointed right R-modules.

For many rigs R, these theories are mutual commutants in the finitary theory of R.

The duality between convex spaces and R_+-modules over the reals generalizes the integral representation of probability measures.

Abstract

Certain axiomatic notions of $\textit{affine space}$ over a ring and $\textit{convex space}$ over a preordered ring are examples of the notion of $\mathcal{T}$-algebra for an algebraic theory $\mathcal{T}$ in the sense of Lawvere. Herein we study the notion of $\textit{commutant}$ for Lawvere theories that was defined by Wraith and generalizes the notion of $\textit{centralizer clone}$. We focus on the Lawvere theory of $\textit{left $R$-affine spaces}$ for a ring or rig $R$, proving that this theory can be described as a commutant of the theory of pointed right $R$-modules. Further, we show that for a wide class of rigs $R$ that includes all rings, these theories are commutants of one another in the full finitary theory of $R$ in the category of sets. We define $\textit{left $R$-convex spaces}$ for a preordered ring $R$ as left affine spaces over the positive part $R_+$ of $R$. We show that for any $\textit{firmly archimedean}$ preordered algebra $R$ over the dyadic rationals, the theories of left $R$-convex spaces and pointed right $R_+$-modules are commutants of one another within the full finitary theory of $R_+$ in the category of sets. Applied to the ring of real numbers $\mathbb{R}$, this result shows that the connection between convex spaces and pointed $\mathbb{R}_+$-modules that is implicit in the integral representation of probability measures is a perfect `duality' of algebraic theories.