Convex Spaces, Affine Spaces, and Commutants for Algebraic Theories

Abstract

Certain axiomatic notions of affine space over a ring and convex space over a preordered ring are examples of the notion of $$\mathscr {T}$$ -algebra for an algebraic theory $$\mathscr {T}$$ in the sense of Lawvere. Herein we study the notion of commutant for Lawvere theories that was defined by Wraith and generalizes the notion of centralizer clone. We focus on the Lawvere theory of 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 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 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.