广义度量空间的距离结构
Distance structures for generalized metric spaces
摘要：
设$\mathcal{R}=(R,\oplus,\leq,0)$是一个代数结构，其中$\oplus$是具有单位元$0$的交换二元运算，$\leq$是具有最少元$0$的交换不变全序.给定一个杰出的子集$S\subseteq R$，我们定义了一个“广义”的$\mathcal{R}$ -度量空间的自然概念，在$S$的距离。我们研究这样的度量空间的一阶结构的关系语言组成的距离不等式的$S$的每个元素。我们首先构造了一个有序的可加结构$\mathcal{S}^*$的空间上的量词自由$2$ -类型符合公理的$\mathcal{R}$ -度量空间的距离在$S$，并表明，如果$A$是一个$\mathcal{R}$ -度量空间的距离在$S$，那么任何模型的$\text{Th}(A)$逻辑继承了一个典型的$\mathcal{S}^*$ -度量。我们的主要应用这个框架涉及可数的，普遍的，和均匀的度量空间，作为推广的理性Urysohn空间。我们适应以前的工作Delhomme，Laflamme，Pouzet，和绍尔充分刻画存在这样的空间。然后我们确定了一个可数的全序交换幺半群$\mathcal{R}$，它具有最少元$0$，并考虑$\mathcal{R}$上的可数Urysohn空间$\mathcal{U}_\mathcal{R}$.我们证明了$\text{Th}(\mathcal{U}_\mathcal{R})$的量词消去的特征在于$\mathcal{R}^*$中加法的连续性，它可以表示为有序幺半群语言中$\mathcal{R}$的一阶句子.最后，我们分析了卡萨诺瓦和瓦格纳在这种情况下的一个例子。
英文摘要：
Let $\mathcal{R}=(R,\oplus,\leq,0)$ be an algebraic structure, where $\oplus$ is a commutative binary operation with identity $0$, and $\leq$ is a translation-invariant total order with least element $0$. Given a distinguished subset $S\subseteq R$, we define the natural notion of a "generalized" $\mathcal{R}$-metric space, with distances in $S$. We study such metric spaces as first-order structures in a relational language consisting of a distance inequality for each element of $S$. We first construct an ordered additive structure $\mathcal{S}^*$ on the space of quantifier-free $2$-types consistent with the axioms of $\mathcal{R}$-metric spaces with distances in $S$, and show that, if $A$ is an $\mathcal{R}$-metric space with distances in $S$, then any model of $\text{Th}(A)$ logically inherits a canonical $\mathcal{S}^*$-metric. Our primary application of this framework concerns countable, universal, and homogeneous metric spaces, obtained as generalizations of the rational Urysohn space. We adapt previous work of Delhommé, Laflamme, Pouzet, and Sauer to fully characterize the existence of such spaces. We then fix a countable totally ordered commutative monoid $\mathcal{R}$, with least element $0$, and consider $\mathcal{U}_\mathcal{R}$, the countable Urysohn space over $\mathcal{R}$. We show that quantifier elimination for $\text{Th}(\mathcal{U}_\mathcal{R})$ is characterized by continuity of addition in $\mathcal{R}^*$, which can be expressed as a first-order sentence of $\mathcal{R}$ in the language of ordered monoids. Finally, we analyze an example of Casanovas and Wagner in this context.
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