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Definable types in $${\mathcal O}$$-minimal theories. (English) Zbl 0801.03026
Generalizing van den Dries’s work on real closed fields, the authors prove the following. Assume that the theory $$T$$ is complete and $$O$$- minimal. (So, the language $$L$$ has a binary relation $$<$$, $$T$$ insists it to be a linear order, and in any model $$M$$ of $$T$$ a definable set with parameters from $$M$$ is a finite union of intervals.) Main result: An $$n$$- type over $$M$$, $$p(\overline{v})$$, is definable exactly when for any $$a$$ that realizes $$p(\overline{v})$$ in some elementary extension of $$M$$, $$M$$ is Dedekind complete in the prime model over $$M\cup \{\overline{a}\}$$. Here, by definition, $$p(\overline{v})$$ is definable if for any $$L$$- formula $$\theta( \overline{v}, \overline{w})$$ there is an $$L_ M$$- formula $$d\theta(\overline{w})$$ so that $$\theta( \overline{v}, \overline{a})\in p(\overline{v})$$ iff $$M\models d\theta(\overline{a})$$ for any $$\overline{a}\in M$$. A 1-type $$p(v)$$ over $$M$$ is said to be a cut of $$M$$ if there are non-empty disjoint $$C_ 0,C_ 1\subseteq M$$ such that $$C_ 0\cup C_ 1= M$$, $$C_ 0[C_ 1]$$ has no greatest [least] element, and for each $$c\in C_ 0 [c\in C_ 1]$$ “$$c<v$$” [“$$v<c$$”] belong to $$p(v)$$. The definition in the paper has a misprint. $$M$$ is Dedekind complete in $$N\supseteq M$$, the definition dictates, if no cut of $$M$$ is realized in $$N$$. The authors use this main result in the determination of the behaviour of $$f$$ on $$M$$, where $$f$$ is a definable function on $$N\succ M$$, $$M$$ and $$N$$ are $$O$$-minimal and expansions of divisible ordered abelian groups, and $$M$$ is Dedekind complete in $$N$$. They also characterize a definable type in a densely ordered $$O$$-minimal structure $$M$$ to be the one that has a unique coheir over any $$N\succ M$$.

##### MSC:
 03C50 Models with special properties (saturated, rigid, etc.) 03C45 Classification theory, stability and related concepts in model theory
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##### References:
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