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BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Model theory of generalized Urysohn spaces Conant, Gabriel


Many well known examples of homogeneous metric spaces and graphs can be viewed as analogs of the rational Urysohn space (for example, the random graph as the Urysohn space with distances {0,1,2}). In 2007, Delhomme, Laflamme, Pouzet, and Sauer characterized the countable subsets S of nonnegative reals for which an ``S-Urysohn space" exists. Sauer later showed that, under mild closure assumptions on S, the existence of the S-Urysohn space is equivalent to associativity of a natural binary operation on S induced by usual addition of real numbers. In this talk, I consider the R-Urysohn space, where R is an arbitrary ordered commutative monoid. I will first construct an extension R* of R, such that any model of the theory of the R-Urysohn space (in a discrete relational language) can be given the structure of an R*-metric space. I will then characterize quantifier elimination in this theory by continuity of addition in R*. Finally, I will characterize various model theoretic properties of the R-Urysohn space using natural algebraic properties of R.

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