Distance structures for generalized metric spaces

TL;DR

This paper develops a framework for generalized metric spaces over algebraic structures, characterizes their model-theoretic properties, and applies these to universal, homogeneous spaces like the Urysohn space, including conditions for quantifier elimination.

Contribution

It introduces a new approach to generalized metric spaces over ordered algebraic structures and characterizes their model-theoretic properties, extending the theory of universal homogeneous metric spaces.

Findings

Constructed an ordered additive structure on types of generalized metric spaces.

Characterized the existence of universal, homogeneous metric spaces over these structures.

Identified conditions for quantifier elimination related to the continuity of addition in the algebraic structure.

Abstract

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 Delhomm\'{e}, 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.