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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
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##### References:
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