代数理论的凸空间、仿射空间和交换子
Convex spaces, affine spaces, and commutants for algebraic theories
摘要：
环上的$\textit{affine space}$和预序环上的$\textit{convex space}$的某些公理化概念是Lawvere意义上的代数理论$\mathcal{T}$的$\mathcal{T}$-代数概念的例子。在这里，我们研究了由Wraith定义的Lawvere理论的$\textit{commutant}$的概念，并推广了$\textit{centralizer clone}$的概念。本文研究了环或环上$R$的$\textit{left $ R $-affine spaces}$的Lawvere理论，证明了该理论可以描述为点右$R$-模理论的交换子.此外，我们表明，对于广泛的一类钻机$R$，其中包括所有的环，这些理论是交换的另一个在全有限理论的$R$的类别集。定义了一个预序环$R$的$\textit{left $ R $-convex spaces}$为$R$的正部分$R_+$上的左仿射空间.证明了对并元有理数上的任意$\textit{firmly archimedean}$预序代数$R$，左$R$-凸空间和右点$R_+$-模的理论在集合范畴中的$R_+$的全有限理论中是彼此的交换子.应用于实数环$\mathbb{R}$，这一结果表明，凸空间和点$\mathbb{R}_+$-模之间的联系，是隐含在概率测度的积分表示是一个完美的"对偶"的代数理论。
英文摘要：
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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