×

zbMATH — the first resource for mathematics

Definable sets in ordered structures. (English) Zbl 0542.03016
In this research announcement the notion of an o-minimal structure is introduced and several results stated. The totally ordered structure M is o-minimal if every definable (with parameters) subset X of M is a finite union of intervals and points. The theory T is o-minimal if every model of T is. Among the results stated are the following - an ordered group G is o-minimal iff G is abelian and divisible; an ordered ring R is o- minimal iff R is a real closed field; if \(f:M\to M\) is a definable function on the o-minimal structure M then f is piecewise continuous; if T is o-minimal then over every set A there is a unique prime model up to A-isomorphism. We also prove in our model-theoretic context results on the (topological) structure of semi-algebraic sets.

MSC:
03C45 Classification theory, stability and related concepts in model theory
06F15 Ordered groups
06F25 Ordered rings, algebras, modules
03C60 Model-theoretic algebra
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] P. Erdös, L. Gillman, and M. Henriksen, An isomorphism theorem for real-closed fields, Ann. of Math. (2) 61 (1955), 542 – 554. · Zbl 0065.02305 · doi:10.2307/1969812 · doi.org
[2] John N. Mather, Stratifications and mappings, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 195 – 232.
[3] Alfred Tarski, A Decision Method for Elementary Algebra and Geometry, RAND Corporation, Santa Monica, Calif., 1948. · Zbl 0035.00602
[4] L. van den Dries, Remarks on Tarski’s problem concerning (R, +,., exp), Proc. Peano Conf. (Florence, 1982) (to appear).
[5] Hassler Whitney, Elementary structure of real algebraic varieties, Ann. of Math. (2) 66 (1957), 545 – 556. · Zbl 0078.13403 · doi:10.2307/1969908 · doi.org
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.