Applied Categorical Structures

, Volume 26, Issue 2, pp 369–400 | Cite as

Convex Spaces, Affine Spaces, and Commutants for Algebraic Theories

  • Rory B. B. Lucyshyn-Wright


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.


Convex space Convex module Affine space Affine module Commutant Centralizer clone Commutation Algebraic theory Lawvere theory Universal algebra Ring Rig Semiring Preordered ring Ordered ring Module Monad Commutative theory Semilattice Matrix Kronecker product 

Mathematics Subject Classification

18C10 18C15 18C20 18C05 08A62 08B99 08C05 52A01 52A05 51N10 16B50 16B70 16D10 16D90 16W80 13J25 13C99 15A27 15A69 15A99 15B51 06A11 06A12 06F25 


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© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Mount Allison UniversitySackvilleCanada

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