Distance structures for generalized metric spaces

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• 作者：Gabriel Conant ^{gconant@nd.edu}
• 关键词：03C10 ; 03C15 ; 54E35 ; 20N02
• 刊名：Annals of Pure and Applied Logic
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：168
• 期：3
• 页码：622-650
• 全文大小：665 K
• 卷排序：168

文摘

Let R=(R,⊕,≤,0)R=(R,⊕,≤,0) be an algebraic structure, where ⊕ is a commutative binary operation with identity 0, and ≤ is a translation-invariant total order with least element 0. Given a distinguished subset S⊆RS⊆R, we define the natural notion of a “generalized” RR-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 S⁎S⁎ on the space of quantifier-free 2-types consistent with the axioms of RR-metric spaces with distances in S, and show that, if A   is an RR-metric space with distances in S  , then any model of Th(A)Th(A) logically inherits a canonical S⁎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 RR, with least element 0, and consider URUR, the countable Urysohn space over RR. We show that quantifier elimination for Th(UR)Th(UR) is characterized by continuity of addition in R⁎R⁎, which can be expressed as a first-order sentence of RR in the language of ordered monoids. Finally, we analyze an example of Casanovas and Wagner in this context.