Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Convex spaces, affine spaces, and commutants for algebraic theories   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.

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The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
18
The categories of this paper are: econ
1
The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol