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Neostability in countable homogeneous metric spaces. (arXiv:1504.02427v4 [math.LO] UPDATED)
来源于:arXiv
Given a countable, totally ordered commutative monoid
$\mathcal{R}=(R,\oplus,\leq,0)$, with least element $0$, there is a countable,
universal and ultrahomogeneous metric space $\mathcal{U}_\mathcal{R}$ with
distances in $\mathcal{R}$. We refer to this space as the $\mathcal{R}$-Urysohn
space, and consider the theory of $\mathcal{U}_\mathcal{R}$ in a binary
relational language of distance inequalities. This setting encompasses many
classical structures of varying model theoretic complexity, including the
rational Urysohn space, the free $n^{\text{th}}$ roots of the complete graph
(e.g. the random graph when $n=2$), and theories of refining equivalence
relations (viewed as ultrametric spaces). We characterize model theoretic
properties of $\text{Th}(\mathcal{U}_\mathcal{R})$ by algebraic properties of
$\mathcal{R}$, many of which are first-order in the language of ordered
monoids. This includes stability, simplicity, and Shelah's SOP$_n$-hierarchy.
Using the submonoid of idempotents in 查看全文>>