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On sets with rank one in simple homogeneous structures. (English) Zbl 1360.03073

By now, the structures that are countable, homogeneous, and stable are well understood. The authors extend these results to more general simple structures. They give a series of technical lemmas and theorems. For instance, the main result reads:
Theorem 5.1. Let \({\mathcal M}\) be countable, binary, homogeneous and simple with trivial dependence. Suppose that \(G\subseteq M^{\text{eq}}\) is \(A\)-definable, where \(A\subseteq M\) is finite, only finitely many sorts are represented in \(G\), \(\mathrm{SU}(a/A)= 1\) and \(\operatorname{acl}(\{a\}\cup A)\cap G=\{a\}\) for every \(a\in G\). Let \({\mathcal G}\) denote the canonically embedded structure in \({\mathcal M}^{\mathrm{eq}}\) over \(A\) with universe \(G\). Then \({\mathcal G}\) is a reduct of a binary random structure.
The authors give a detailed introduction and preliminaries to give the background material. But they assume “a working knowledge” of simplicity theory.

MSC:

03C50 Models with special properties (saturated, rigid, etc.)
03C45 Classification theory, stability, and related concepts in model theory
03C15 Model theory of denumerable and separable structures
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References:

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