Definable sets in ordered structures. II

Knight, Julia F.; Pillay, Anand; Steinhorn, Charles

doi:10.1090/s0002-9947-1986-0833698-1

Cited by 134 publications

(52 citation statements)

References 6 publications

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“…We now require a deeper asymptotic analysis and I must assume that the reader is familiar with the basic general properties of 0-minimal structures. These can be found in the foundational papers [10] and [7]. (See also [15] for more recent developments.…”

Section: Smooth 0-minimal Theoriesmentioning

confidence: 99%

Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function

Wilkie¹

1996

J. Amer. Math. Soc.

323

155

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Section: Smooth 0-minimal Theoriesmentioning

confidence: 99%

Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function

Wilkie¹

1996

J. Amer. Math. Soc.

323

155

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“…Then for any formula tp(x,y) E L(M) there is N <uj such that, for any b C xM, (^(Xjb) A x(x))M is a union of at most N intervals and points. Moreover the definable (in M) subsets of (x(x)M)n satisfy all the results of [1] We will prove PROPOSITION 2.3. For any L-formula tp{x,y), there is tpl(jc,y) which is a finite Boolean combination of formulae ip(x) and formulae x(V) sucn that M 1= <p(x,y) «-» tpl(x,y).…”

Section: Mixed O-minimalmentioning

confidence: 73%

“…We will say that x(x) 's o-minimal in M if every definable (in M) subset X of \M is a finite union of points and intervals (with endpoints). Then the proofs in [1] and §1 of this paper give: FACT 2.1. Let \(x) be o-minimal in M with (xM, <) a dense ordering without endpoints.…”

Section: Mixed O-minimalmentioning

confidence: 97%

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Definable sets in ordered structures. III

Pillay¹,

Steinhorn²

1988

Trans. Amer. Math. Soc.

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ABSTRACT. We show that any o-minimal structure has a strongly o-minimal theory. Introduction.In this paper we prove that an arbitrary o-minimal structure M is strongly o-minimal. This was proved in [1] in the case when the ordering on M is dense.In §1 we show that for discrete M, o-minimal implies strongly o-minimal. This is, of course, a result on uniform finite bounds. The proof has some interesting differences with the dense case, partly because here one has to prove uniform bound results for functions defined on finite intervals. In [3] it was shown that strongly ominimal discrete structures are "trivial", i.e. there are no definable functions other than translations in one variable. Thus, with the results of §1, discrete o-minimal structures lose their interest.In §2 we show that the discrete and dense parts of an arbitrary o-minimal structure are "orthogonal", from which our main result follows.Recall that the structure (M, <,...) is said to be o-minimal if

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“…The reader may wish to consult the lecture notes from the recent thematic program on o-minimality at the Fields Institute for a more recent account or Wilkie's Bourbaki notes [48] for a fuller survey. The foundational papers by Pillay and Steinhorn [32,33] and together with Knight [15] remain vital. Definition 3.1.…”

Section: Introduction To O-minimalitymentioning

confidence: 99%