×

\(T\)-convexity and tame extensions. (English) Zbl 0856.03028

Let \(T\) be a complete o-minimal theory extending the theory of real closed fields. The authors first characterize those convex subrings \(V\) of a model \(\mathfrak R\) of \(T\) which are the convex hull of some elementary substructure of \(\mathfrak R\). They then study the theory \(T\)(convex) of pairs \(({\mathfrak R}, V)\) where \(\mathfrak R\) is a model of \(T\) and \(V\neq {\mathfrak R}\) is the convex hull of some elementary substructures of \(\mathfrak R\). Among others they prove that \(T\)(convex) admits quantifier elimination if \(T\) does and is universal. Moreover, they show that \(T\)(convex) is complete, and is model complete in case \(T\) is model complete. The authors next concentrate on ‘tame’ extensions \({\mathcal R}'\subset {\mathcal R}\), where \({\mathcal R}'\) is an elementary substructure of a model \(\mathfrak R\) of \(T\). This extension is called tame if for every \({\mathfrak R}'\)-bounded \(r\in {\mathfrak R}\) there exists a (necessarily unique) \(r'\in {\mathfrak R}'\) such that \(|r- r'|< \varepsilon\) for all positive \(\varepsilon\) in \({\mathfrak R}'\). The element \(r'\) is then called the ‘standard part’ of \(r\) \((r'= \text{st}(r))\). The author denotes by \(T\)(tame) the theory of such triples \(({\mathfrak R}, {\mathfrak R}', \text{st})\) with \({\mathfrak R}'\neq {\mathfrak R}\) and proves similar results as for \(T\)(convex).

MSC:

03C52 Properties of classes of models
12L12 Model theory of fields
03C60 Model-theoretic algebra
03C10 Quantifier elimination, model completeness, and related topics
03C35 Categoricity and completeness of theories
12J10 Valued fields
12J15 Ordered fields
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Einführung in die Mathematische Logik und Modelltheorie (1986) · Zbl 0616.03001
[2] Proceedings of the American Mathematical Society
[3] On the elementary theory of restricted elementary functions 53 pp 796– (1988)
[4] DOI: 10.1090/S0273-0979-1986-15468-6 · Zbl 0612.03008 · doi:10.1090/S0273-0979-1986-15468-6
[5] Algebraic theories with definable Skolem functions 49 pp 625– (1984)
[6] DOI: 10.2307/1971463 · Zbl 0693.14012 · doi:10.2307/1971463
[7] DOI: 10.1016/0168-0072(83)90019-2 · Zbl 0538.03028 · doi:10.1016/0168-0072(83)90019-2
[8] Real algebraic geometry, proceedings (Rennes, 1991) pp 145– (1992)
[9] DOI: 10.1090/conm/155/01374 · doi:10.1090/conm/155/01374
[10] DOI: 10.1016/0168-0072(91)90028-K · Zbl 0756.03018 · doi:10.1016/0168-0072(91)90028-K
[11] DOI: 10.1090/S0002-9947-1986-0833697-X · doi:10.1090/S0002-9947-1986-0833697-X
[12] DOI: 10.1016/0168-0072(94)90048-5 · Zbl 0823.03018 · doi:10.1016/0168-0072(94)90048-5
[13] Definable types in o-minimal theories 59 pp 185– (1994)
[14] DOI: 10.1090/S0002-9947-1986-0833698-1 · doi:10.1090/S0002-9947-1986-0833698-1
[15] DOI: 10.1007/BF02758635 · Zbl 0823.03017 · doi:10.1007/BF02758635
[16] Annals of Mathematics 85 pp 19– (1994)
[17] Journal of the American Mathematical Society
[18] On Dedekind complete O-minimal structures 52 pp 156– (1987)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.