scholarly journals INTERPRETABLE SETS IN DENSE O-MINIMAL STRUCTURES

2018 ◽  
Vol 83 (04) ◽  
pp. 1477-1500
Author(s):  
WILL JOHNSON
Keyword(s):  
Negative Answer ◽  
Connected Components ◽  
Lower Dimension ◽  
Definable Set ◽  
Definable Subset ◽  
Minimal Structure ◽  
Piecewise Continuous ◽  
Definable Functions

AbstractWe give an example of a dense o-minimal structure in which there is a definable quotient that cannot be eliminated, even after naming parameters. Equivalently, there is an interpretable set which cannot be put in parametrically definable bijection with any definable set. This gives a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Additionally, we show that interpretable sets in dense o-minimal structures admit definable topologies which are “tame” in several ways: (a) they are Hausdorff, (b) every point has a neighborhood which is definably homeomorphic to a definable set, (c) definable functions are piecewise continuous, (d) definable subsets have finitely many definably connected components, and (e) the frontier of a definable subset has lower dimension than the subset itself.

Sheaves of continuous definable functions

1988 ◽  
Vol 53 (4) ◽  
pp. 1165-1169 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Real Closed Field ◽  
Spectral Space ◽  
Real Spectrum ◽  
Closed Field ◽  
Elementary Extension ◽  
Definable Set ◽  
Definable Subset ◽  
Skolem Functions ◽  
Minimal Structure ◽  
Definable Functions

Let M be an o-minimal structure or a p-adically closed field. Let be the space of complete n-types over M equipped with the following topology: The basic open sets of are of the form Ũ = {p ∈ Sn (M): U ∈ p} for U an open definable subset of Mn. is a spectral space. (For M = K a real closed field, is precisely the real spectrum of K[X1, …, Xn]; see [CR].) We will equip with a sheaf of LM-structures (where LM is a suitable language). Again for M a real closed field this corresponds to the structure sheaf on (see [S]). Our main point is that when Th(M) has definable Skolem functions, then if p ∈ , it follows that M(p), the definable ultrapower of M at p, can be factored through Mp, the stalk at p with respect to the above sheaf. This depends on the observation that if M ≺ N, a ∈ Nn and f is an M-definable (partial) function defined at a, then there is an open M-definable set U ⊂ Nn with a ∈ U, and a continuous M-definable function g:U → N such that g(a) = f(a).In the case that M is an o-minimal expansion of a real closed field (or M is a p-adically closed field), it turns out that M(p) can be recovered as the unique quotient of Mp which is an elementary extension of M.

The main theorem

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Projective Variety ◽  
Unit Vector ◽  
Definable Set ◽  
Valued Field ◽  
Definable Subset ◽  
Deformation Retraction ◽  
Relative Curve

This chapter introduces the main theorem, which states: Let V be a quasi-projective variety over a valued field F and let X be a definable subset of V x Γ‎superscript Script Small l subscript infinity over some base set V ⊂ VF ∪ Γ‎, with F = VF(A). Then there exists an A-definable deformation retraction h : I × unit vector X → unit vector X with image an iso-definable subset definably homeomorphic to a definable subset of Γ‎superscript w subscript Infinity, for some finite A-definable set w. The chapter presents several preliminary reductions to essentially reduce to a curve fibration. It then constructs a relative curve homotopy and a liftable base homotopy, along with a purely combinatorial homotopy in the Γ‎-world. It also constructs the homotopy retraction by concatenating the previous three homotopies together with an inflation homotopy. Finally, it describes a uniform version of the main theorem with respect to parameters.

Γ‎-internal spaces

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Algebraic Variety ◽  
Topological Structure ◽  
Unit Vector ◽  
Definable Set ◽  
Valued Field ◽  
Definable Subset

This chapter describes the topological structure of Γ‎-internal spaces. Let V be an algebraic variety over a valued field. An iso-definable subset X of unit vector V is said to be Γ‎-internal if it is in pro-definable bijection with a definable set which is Γ‎-internal. A number of delicate issues arise here. A pro-definable subset X of unit vector V is Γ‎-parameterized if there exists a definable subset Y of Γ‎ⁿ, for some n, and a pro-definable map g : Y → unit vector V with image X. The chapter presents an example showing that there exists Γ‎-parameterized subsets of unit vector V which are not iso-definable, whence not Γ‎-internal. It also presents the main results about the topological structure of Γ‎-internal spaces.

scholarly journals CELL DECOMPOSITION AND CLASSIFICATION OF DEFINABLE SETS INp-OPTIMAL FIELDS

2017 ◽  
Vol 82 (1) ◽  
pp. 120-136 ◽  
Author(s):  
LUCK DARNIÈRE ◽  
IMMANUEL HALPUCZOK
Keyword(s):  
Decomposition Theorem ◽  
Cell Decomposition ◽  
Extreme Value ◽  
Property A ◽  
Definable Subset ◽  
Skolem Functions ◽  
Definable Sets ◽  
A Cell ◽  
Preparation Theorem ◽  
Definable Functions

AbstractWe prove that forp-optimal fields (a very large subclass ofp-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef’s paper [7]. We derive from it the existence of definable Skolem functions and strongp-minimality. Then we turn to stronglyp-minimal fields satisfying the Extreme Value Property—a property which in particular holds in fields which are elementarily equivalent to ap-adic one. For such fieldsK, we prove that every definable subset ofK×Kdwhose fibers overKare inverse images by the valuation of subsets of the value group is semialgebraic. Combining the two we get a preparation theorem for definable functions onp-optimal fields satisfying the Extreme Value Property, from which it follows that infinite sets definable over such fields are in definable bijection iff they have the same dimension.

INTERPRETING GROUPS INSIDE MODULAR STRONGLY MINIMAL HOMOGENEOUS MODELS

2003 ◽  
Vol 03 (01) ◽  
pp. 127-142 ◽  
Author(s):  
TAPANI HYTTINEN
Keyword(s):  
Saturated Model ◽  
Closure Operation ◽  
Definable Subset ◽  
Minimal Structure ◽  
Homogeneous Models

A large homogeneous (not necessarily saturated) model M is strongly minimal, if any definable subset is either bounded or has bounded complement. In this case (M, bcl) is a pregeometry, where bcl denotes the bounded closure operation. In this paper, we show that if M is a large homogeneous strongly minimal structure and (M, bcl) is non-trivial and locally modular, then M interprets a group. In addition, we give a description of such groups.

Type-definable and invariant groups in o-minimal structures

2007 ◽  
Vol 72 (1) ◽  
pp. 67-80 ◽  
Author(s):  
Jana Maříková
Keyword(s):  
Topological Group ◽  
Real Closed Field ◽  
Closed Field ◽  
Group Topology ◽  
Invariant Subset ◽  
Definable Subset ◽  
Minimal Structure ◽  
Invariant Group

AbstractLet M be a big o-minimal structure and G a type-definable group in Mn. We show that G is a type-definable subset of a definable manifold in Mn that induces on G a group topology. If M is an o-minimal expansion of a real closed field, then G with this group topology is even definably isomorphic to a type-definable group in some Mk with the topology induced by Mk. Part of this result holds for the wider class of so-called invariant groups: each invariant group G in Mn has a unique topology making it a topological group and inducing the same topology on a large invariant subset of the group as Mn.

scholarly journals Defining integer-valued functions in rings of continuous definable functions over a topological field

2020 ◽  
Vol 20 (03) ◽  
pp. 2050014
Author(s):  
Luck Darnière ◽  
Marcus Tressl
Keyword(s):  
Definable Set ◽  
Ordered Field ◽  
Dimension Formula ◽  
Valued Field ◽  
Pure Dimension ◽  
Isolated Points ◽  
Topological Field ◽  
Structure Formula ◽  
Definable Functions ◽  
Ring Of Functions

Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, or an expansion of a local field, we prove that the ring of integers [Formula: see text] is interpretable in [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] is definably connected of pure dimension [Formula: see text], then [Formula: see text] defines the subring [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] has no isolated points, then there is a discrete ring [Formula: see text] contained in [Formula: see text] and naturally isomorphic to [Formula: see text], such that the ring of functions [Formula: see text] which take values in [Formula: see text] is definable in [Formula: see text].

scholarly journals Lipschitz–Killing curvatures and polar images

2019 ◽  
Vol 19 (2) ◽  
pp. 205-230
Author(s):  
Nicolas Dutertre
Keyword(s):  
Definable Set ◽  
Smooth Submanifolds ◽  
Minimal Structure

Abstract We relate the Lipschitz–Killing measures of a definable set X ⊂ ℝn in an o-minimal structure to the volumes of generic polar images. For smooth submanifolds of ℝn, such results were established by Langevin and Shifrin. Then we give infinitesimal versions of these results. As a corollary, we obtain a relation between the polar invariants of Comte and Merle and the densities of generic polar images.

scholarly journals ŁOJASIEWICZ-TYPE INEQUALITIES AND GLOBAL ERROR BOUNDS FOR NONSMOOTH DEFINABLE FUNCTIONS IN O-MINIMAL STRUCTURES

2015 ◽  
Vol 93 (1) ◽  
pp. 99-112
Author(s):  
DŨNG PHI HOÀNG
Keyword(s):  
Error Bound ◽  
Global Error ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Global Error Bound ◽  
Minimal Structure ◽  
Palais Smale Condition ◽  
Necessary And Sufficient ◽  
Definable Functions ◽  
The Relationship

In this paper, we give some Łojasiewicz-type inequalities for continuous definable functions in an o-minimal structure. We also give a necessary and sufficient condition for the existence of a global error bound and the relationship between the Palais–Smale condition and this global error bound. Moreover, we give a Łojasiewicz nonsmooth gradient inequality at infinity near the fibre for continuous definable functions in an o-minimal structure.

Definable sets in boolean ordered o-minimal structures. II

2003 ◽  
Vol 68 (1) ◽  
pp. 35-51 ◽  
Author(s):  
Roman Wencel
Keyword(s):  
Pairwise Disjoint ◽  
Closed Subset ◽  
Boolean Algebras ◽  
Ordered Structures ◽  
Definable Set ◽  
Minimal Theory ◽  
Boolean Subalgebra ◽  
Minimal Structure ◽  
Definable Sets ◽  
Simple Nature

AbstractLet (M, ≤,…) denote a Boolean ordered o-minimal structure. We prove that a Boolean subalgebra of M determined by an algebraically closed subset contains no dense atoms. We show that Boolean algebras with finitely many atoms do not admit proper expansions with o-minimal theory. The proof involves decomposition of any definable set into finitely many pairwise disjoint cells, i.e., definable sets of an especially simple nature. This leads to the conclusion that Boolean ordered structures with o-minimal theories are essentially bidefinable with Boolean algebras with finitely many atoms, expanded by naming constants. We also discuss the problem of existence of proper o-minimal expansions of Boolean algebras.

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