Definable Sets in Ordered Structures. II

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

doi:10.2307/2000053

Cited by 52 publications

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“…In particular, each v ∈ Vert(W ) is a fixed point. 14) That proof uses the same routing property for a simplical division S of .…”

Section: Claim 1 the Set Smentioning

confidence: 97%

“…W ∈ W cl(K) is a (k − 1)-dimensional g-simplex, then either (I) W lies on boundary ∂(K) = cl(K)\K and W is a face of some k-dimensional g-simplex W ∈ W cl(K) , or (II) W is a common face of two k-dimensional g-simplexes W , W ∈ W cl(K) . Then, we can follow exactly the lines of a standard proof for Sperner's Lemma 14) . …”

Section: Lemma (Generalized Sperner's Lemma) Let W Be a G-simplical mentioning

confidence: 98%

“…3) Cf. Pillay and Steinhorn [18], Knight et al [14], and van den Dries [7]. 4) Standard notions in the first order predicate logic can be found in Enderton [10], and Chang and Keisler [4].…”

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confidence: 99%

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A fixed point theorem for o‐minimal structures

Wong

2003

Mathematical Logic Qtrly

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We prove a definable analogue to Brouwer's Fixed Point Theorem for o-minimal structures of real closed field expansions: A continuous definable function mapping from the unit simplex into itself admits a fixed point, even though the underlying space is not necessarily topologically complete. Our proof is direct and elementary; it uses a triangulation technique for o-minimal functions, with an application of Sperner's Lemma. This paper operates in the definable context of o-minimal structures as developed in the series of papers of van den Dries [6], Pillay and Steinhorn [18], Knight et al. [14], Wilkie [21], and others. We prove a definable analogue to Brouwer's Fixed Point Theorem for o-minimal structures of real closed field expansions: Every continuous definable function mapping from the unit simplex into itself admits a fixed point, even though the underlying space is not necessarily topologically complete 1) . Our proof applies a triangulation technique for o-minimal functions (cf. van den Dries [7]), with an application of Sperner's Lemma 2) .Definable analogues to Brouwer's Theorem, and its generalization, the Lefschetz Fixed Point Theorem, have been recently obtained by Edmundo [8,9] and Berarducci and Otero [1]. These papers use o-minimal homology developed by Woerheide [22] and others. In contrast, our proof is direct and elementary and does not use o-minimal homology. We begin by reviewing some notations about o-minimal structures 3) . Recall that for a structure M = (M, . . . ), a set X M n is parametrically definable if there is a first order formula ϕ(x 1 , . . . , x n ) ∈ L(M) (the first order predicate language 4) of M together with names for the elements of M ), such that X is the set of all b ∈ M n satisfying ϕ( · ) in M, i. e. X = {b ∈ M n : M ϕ[b ]}. For brevity, we drop mention of parameters, and refer simply to "definable". A function isAn o-minimal structure is an ordered structure M = (M, <, . . . ) in which every definable (with parameters) subset of M is a finite union of intervals 5) .Our results apply to those structures M that are expansions of real closed ordered fields 6) , so M = (M, <, +, · , 0, 1, . . . ) 7) . As usual, we endow M with the interval topology, i. e. intervals (a, b) form a basis; and M n is given the product topology. We use to denote the closed unit simplex in M n , i. e.,

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“…In particular, each v ∈ Vert(W ) is a fixed point. 14) That proof uses the same routing property for a simplical division S of .…”

Section: Claim 1 the Set Smentioning

confidence: 97%

Section: Lemma (Generalized Sperner's Lemma) Let W Be a G-simplical mentioning

confidence: 98%

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A fixed point theorem for o‐minimal structures

Wong

2003

Mathematical Logic Qtrly

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“…In this paper when we say that some subset or function is definable, we mean it is first-order definable (possibly with parameters) in the sense of the structure M. A general reference for first-order logic is [9]. All the notions related to o-minimality can be found in [17,12,18], see also [19] for a nice overview. We start with the definition of an o-minimal structure: The field of reals R, <, +, ·, 0, 1 , the group of rationals Q, <, +, 0 , the field of reals with exponential function, the field of reals expanded by restricted pfaffian functions and the exponential function are o-minimal structures.…”

Section: O-minimality and Definabilitymentioning

confidence: 99%

“…The o-minimal structures [17,12,18,19] enjoy very nice finiteness properties. Regarding the above discussion it seemed interesting to define "o-minimal transition systems".…”

Section: Introductionmentioning

confidence: 99%

A note on the undecidability of the reachability problem for o-minimal dynamical systems

Brihaye

2006

Math. Log. Quart.

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On the strong cell decomposition property for weakly o-minimal structures

Wencel

2013

Math. Log. Quart.

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We consider a class of weakly o-minimal structures admitting an o-minimal style cell decomposition, for which one can construct certain canonical o-minimal extension. The paper contains several fundamental facts concerning the structures in question. Among other things, it is proved that the strong cell decomposition property is preserved under elementary equivalences. We also investigate fiberwise properties (of definable sets and definable functions), definable equivalence relations, and conditions implying elimination of imaginaries.

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Definable Sets in Ordered Structures. II

Cited by 52 publications

References 0 publications

A fixed point theorem for o‐minimal structures

A fixed point theorem for o‐minimal structures

A note on the undecidability of the reachability problem for o-minimal dynamical systems

On the strong cell decomposition property for weakly o-minimal structures

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