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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:
[1] Omitting types in -minimal theories 51 pp 63– (1986)
[2] DOI: 10.1090/S0002-9947-1986-0833698-1 · doi:10.1090/S0002-9947-1986-0833698-1
[3] DOI: 10.1090/S0273-0979-1986-15468-6 · Zbl 0612.03008 · doi:10.1090/S0273-0979-1986-15468-6
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[7] DOI: 10.1016/0168-0072(87)90004-2 · doi:10.1016/0168-0072(87)90004-2
[8] DOI: 10.1090/S0002-9947-1986-0833697-X · doi:10.1090/S0002-9947-1986-0833697-X
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