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A trichotomy theorem for o-minimal structures

Authors

Ya’acov Peterzil, Sergei Starchenko
Type: Article
Publication Date: 1998-11-01
Citations: 75
DOI: https://doi.org/10.1112/s0024611598000549

Abstract

Let M = 〈M, <, …〉 be alinearly ordered structure. We define M to be o-minimal if every definable subset of M is a finite union of intervals. Classical examples are ordered divisible abelian groups and real closed fields. We prove a trichotomy theorem for the structure that an arbitraryo-minimal M can induce on a neighbourhood of any a in M. Roughly said, one of the following holds: (i) a is trivial (technical term), or (ii) a has a convex neighbourhood on which M induces the structure of an ordered vector space, or (iii) a is contained in an open interval on which M induces the structure of an expansion of a real closed field. The proof uses 'geometric calculus' which allows one to recover a differentiable structure by purely geometric methods. 1991 Mathematics Subject Classification: primary 03C45; secondary 03C52, 12J15, 14P10.

Locations

  • Proceedings of the London Mathematical Society
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Quasi-o-minimal structures

2000-09-01
Oleg Belegradek , Ya’acov Peterzil , Frank Olaf Wagner
Abstract A structure ( M , &lt;, …) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets … Abstract A structure ( M , &lt;, …) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal; one of them is the ordered group of integers. We develop a technique to investigate quasi-o-minimality and use it to study quasi-o-minimal ordered groups (possibly with extra structure). Main results: any quasi-o-minimal ordered group is abelian; any quasi-o-minimal ordered ring is a real closed field, or has zero multiplication; every quasi-o-minimal divisible ordered group is o-minimal; every quasi-o-minimal archimedian densely ordered group is divisible. We show that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U -rank 1.

Notes on o-Minimality and Variations

2000-01-01
Dugald Macpherson
The article surveys some topics related to o-minimality, and is based on three lectures. The emphasis is on o-minimality as an analogue of strong minimality, rather than as a setting … The article surveys some topics related to o-minimality, and is based on three lectures. The emphasis is on o-minimality as an analogue of strong minimality, rather than as a setting for the model theory of expansions of the reals. Section 2 gives some basics (the Monotonicity and Cell Decomposition Theorems) together with a discussion of dimension. Section 3 concerns the Peterzil–Starchenko Trichotomy Theorem (an o-minimal analogue of Zil’ber Trichotomy). There follows some material on definable groups, with powerful applications of the Trichotomy Theorem in work by Peterzil, Pillay and Starchenko. The final section introduces weak o-minimality, P -minimality, and C-minimality. These are analogues of o-minimality intended as settings for certain henselian valued fields with extra structure.

Almost o-minimal structures and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="fraktur">X</mml:mi></mml:math>-structures

2022-06-01
Masato Fujita
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A note on noninterpretability in o-minimal structures

1998-01-01
Ricardo Bianconi
We prove that if M is an o-minimal structure whose underlying order is dense then Th(M) does not interpret the theory of an infinite discretely ordered structure. We also make … We prove that if M is an o-minimal structure whose underlying order is dense then Th(M) does not interpret the theory of an infinite discretely ordered structure. We also make a conjecture concerning the class of the theory of an infinite discretely ordered o-minimal structure. Introduction. In [9], Świerczkowski proves that Th(〈ω,<〉) is not interpretable (with parameters) in RCF (the theory of real closed fields) by showing that a pre-ordering with successors is not definable in R. (We recall that a pre-ordering with successors is a reflexive and transitive binary relation , satisfying ∀x∀y (x y ∨ y x) and ∀x∃y Succ(x, y), where Succ(x, y)⇔ x y ∧ x 6≈ y ∧ ∀z (x z y → z ≈ x ∨ z ≈ y), and x ≈ y means x y ∧ y x.) Recall that a structure (M,<,Ri)i∈I is said to be o-minimal if < is a total ordering on M and every definable (with parameters) subset of M is a finite union of points in M and open intervals (a, b), where a ∈M ∪ {−∞} and b ∈M ∪ {∞}. Recall also that if M is o-minimal, then all N |= Th(M) are o-minimal, where Th(M) is the theory of M (see [1]). Certain properties of RCF are used in the proof of the main result of [9], such as o-minimality and definable Skolem functions. We show that this result remains true in the more general setting of o-minimal densely ordered structures. Noninterpretability results. We show the following: Theorem. Let M be an o-minimal structure whose underlying order is dense. Then Th(M) does not interpret the theory of a preordered structure with successors. 1991 Mathematics Subject Classification: 03C40, 03C45, 06F99.

Tame Topology and O-minimal Structures

1998-05-07
L. P. D. van den Dries
Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment … Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment of the theory of o-minimal structures from a geometric and topological viewpoint, assuming only rudimentary algebra and analysis. The book starts with an introduction and overview of the subject. Later chapters cover the monotonicity theorem, cell decomposition, and the Euler characteristic in the o-minimal setting and show how these notions are easier to handle than in ordinary topology. The remarkable combinatorial property of o-minimal structures, the Vapnik-Chervonenkis property, is also covered. This book should be of interest to model theorists, analytic geometers and topologists.

Vaught's conjecture for o-minimal theories

1988-03-01
Laura L. Mayer
The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is … The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is a model of T then M is linearly ordered and every definable subset of the universe of M consists of finitely many intervals and points. The theory of real closed fields is an example of an o-minimal theory. We examine the structure of the countable models for T, T an arbitrary o-minimal theory (in a countable language). We completely characterize these models, provided that T does not have 2 ω countable models. This proviso (viz. that T has fewer than 2 ω countable models) is in the tradition of classification theory: given a cardinal α , if T has the maximum possible number of models of size α , i.e. 2 α , then no structure theorem is expected (cf. [Sh1]). O-minimality is introduced in §1. §1 also contains conventions and definitions, including the definitions of cut and noncut. Cuts and noncuts constitute the nonisolated types over a set. In §2 we study a notion of independence for sets of nonisolated types and the corresponding notion of dimension. In §3 we define what it means for a nonisolated type to be simple. Such types generalize the so-called “components” in Pillay and Steinhorn's analysis of ω -categorical o-minimal theories [PS]. We show that if there is a nonisolated type which is not simple then T has 2 ω countable models.

Vaught's conjecture for o-minimal theories

1988-03-01
Laura L. Mayer
The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is … The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is a model of T then M is linearly ordered and every definable subset of the universe of M consists of finitely many intervals and points. The theory of real closed fields is an example of an o-minimal theory. We examine the structure of the countable models for T, T an arbitrary o-minimal theory (in a countable language). We completely characterize these models, provided that T does not have 2 ω countable models. This proviso (viz. that T has fewer than 2 ω countable models) is in the tradition of classification theory: given a cardinal α , if T has the maximum possible number of models of size α , i.e. 2 α , then no structure theorem is expected (cf. [Sh1]). O-minimality is introduced in §1. §1 also contains conventions and definitions, including the definitions of cut and noncut. Cuts and noncuts constitute the nonisolated types over a set. In §2 we study a notion of independence for sets of nonisolated types and the corresponding notion of dimension. In §3 we define what it means for a nonisolated type to be simple. Such types generalize the so-called “components” in Pillay and Steinhorn's analysis of ω -categorical o-minimal theories [PS]. We show that if there is a nonisolated type which is not simple then T has 2 ω countable models.

Definable Open Sets As Finite Unions of Definable Open Cells

2010-04-01
Simon Andrews
We introduce CE-cell decomposition, a modified version of the usual o-minimal cell decomposition. We show that if an o-minimal structure ℛ admits CE-cell decomposition then any definable open set in … We introduce CE-cell decomposition, a modified version of the usual o-minimal cell decomposition. We show that if an o-minimal structure ℛ admits CE-cell decomposition then any definable open set in ℛ may be expressed as a finite union of definable open cells. The dense linear ordering and linear o-minimal expansions of ordered abelian groups are examples of such structures.
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Definable structures in o-minimal theories: One dimensional types

2010-10-31
Assaf Hasson , Alf Onshuus , Ya’acov Peterzil
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An introduction to o-minimal structures

2000-01-01
Mário J. Edmundo
The first papers on o-minimal structures appeared in the mid 1980s, since then the subject has grown into a wide ranging generalisation of semialgebraic, subanalytic and subpfaffian geometry. In these … The first papers on o-minimal structures appeared in the mid 1980s, since then the subject has grown into a wide ranging generalisation of semialgebraic, subanalytic and subpfaffian geometry. In these notes we try to show that this is in fact the case by presenting several examples of o-minimal structures and by listing some geometric properties of sets and maps definable in o-minimal structures. We omit here any reference to the pure model theory of o-minimal structures and to the theory of groups and rings definable in o-minimal structures.

Hausdorff measure on o-minimal structures

2010-11-07
Antongiulio Fornasiero , Elisa Vasquez Rifo
We introduce the Hausdorff measure for definable sets in an o-minimal structure, and prove the Cauchy-Crofton and co-area formulae for the o-minimal Hausdorff measure. We also prove that every definable … We introduce the Hausdorff measure for definable sets in an o-minimal structure, and prove the Cauchy-Crofton and co-area formulae for the o-minimal Hausdorff measure. We also prove that every definable set can be partitioned into rectifiable sets, and that the Whitney arc property holds for basic rectifiable sets.

Almost o-minimal structures and $\mathfrak X$-structures

2021-04-03
Masato Fujita
We propose new structures called almost o-minimal structures and $\mathfrak X$-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a … We propose new structures called almost o-minimal structures and $\mathfrak X$-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a definable set with a bounded open interval is a finite union of points and open intervals. The latter is a variant of van den Dries and Miller's analytic geometric categories and Shiota's $\mathfrak X$-sets and $\mathfrak Y$-sets. In them, the family of definable sets are closed only under proper projections unlike first-order structures. We demonstrate that an $\mathfrak X$-expansion of an ordered divisible abelian group always contains an o-minimal expansion of an ordered group such that all bounded $\mathfrak X$-definable sets are definable in the structure. Another contribution of this paper is a uniform local definable cell decomposition theorem for almost o-minimal expansions of ordered groups $\mathcal M=(M,<,0,+,\ldots)$. Let $\{A_\lambda\}_{\lambda\in\Lambda}$ be a finite family of definable subsets of $M^{m+n}$. Take an arbitrary positive element $R \in M$ and set $B=]-R,R[^n$. Then, there exists a finite partition into definable sets \begin{equation*} M^m \times B = X_1 \cup \ldots \cup X_k \end{equation*} such that $B=(X_1)_b \cup \ldots \cup (X_k)_b$ is a definable cell decomposition of $B$ for any $b \in M^m$ and either $X_i \cap A_\lambda = \emptyset$ or $X_i \subseteq A_\lambda$ for any $1 \leq i \leq k$ and $\lambda \in \Lambda$. Here, the notation $S_b$ denotes the fiber of a definable subset $S$ of $M^{m+n}$ at $b \in M^m$. We introduce the notion of multi-cells and demonstrate that any definable set is a finite union of multi-cells in the course of the proof of the above theorem.

Structure theorems in tame expansions of o-minimal structures by a dense set

2015-10-12
Pantelis E. Eleftheriou , Ayhan Günaydın , Philipp Hieronymi
We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\cal M$ … We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\cal M$ by a dense set $P$, such that three tameness conditions hold. We prove a structure theorem for definable sets and functions in analogy with the influential cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of $\mathcal {\widetilde M}$, as it achieves a decomposition of definable sets into \emph{unions} of `cones', instead of only boolean combinations of them. We also develop the right dimension theory in the tame setting. Applications include: (i) the dimension of a definable set coincides with a suitable pregeometric dimension, and it is invariant under definable bijections, (ii) every definable map is given by an $\cal L$-definable map off a subset of its domain of smaller dimension, and (iii) around generic elements of a definable group, the group operation is given by an $\cal L$-definable map.

Structure theorems in tame expansions of o-minimal structures by a dense set

2015-01-01
Pantelis E. Eleftheriou , Ayhan Günaydın , Philipp Hieronymi
We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\cal M$ … We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\cal M$ by a dense set $P$, such that three tameness conditions hold. We prove a structure theorem for definable sets and functions in analogy with the influential cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of $\mathcal {\widetilde M}$, as it achieves a decomposition of definable sets into \emph{unions} of `cones', instead of only boolean combinations of them. We also develop the right dimension theory in the tame setting. Applications include: (i) the dimension of a definable set coincides with a suitable pregeometric dimension, and it is invariant under definable bijections, (ii) every definable map is given by an $\cal L$-definable map off a subset of its domain of smaller dimension, and (iii) around generic elements of a definable group, the group operation is given by an $\cal L$-definable map.

Structure theorems in tame expansions of o-minimal structures by a dense set

2020-08-01
Pantelis E. Eleftheriou , Ayhan Günaydın , Philipp Hieronymi
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A pathological o-minimal quotient

2014-01-01
Will Johnson
We give an example of a definable quotient in an o-minimal structure which cannot be eliminated over any set of parameters, giving a negative answer to a question of Eleftheriou, … We give an example of a definable quotient in an o-minimal structure which cannot be eliminated over any set of parameters, giving a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Equivalently, there is an o-minimal structure M whose elementary diagram does not eliminate imaginaries. We also give a positive answer to a related question, showing that any imaginary in an o-minimal structure is interdefinable over an independent set of parameters with a tuple of real elements. This can be interpreted as saying that interpretable sets look "locally" like definable sets, in a sense which can be made precise.

Definable compactness in o-minimal structures

2025-03-05
Pablo Andújar Guerrero

Tameness Results for Expansions of the Real Field by Groups

2013-01-01
Michael Tychonievich
Expanding on the ideas of o-minimality, we study three kinds of expansions of the real field and discuss certain tameness properties that they possess or lack. In Chapter 1, we … Expanding on the ideas of o-minimality, we study three kinds of expansions of the real field and discuss certain tameness properties that they possess or lack. In Chapter 1, we introduce some basic logical concepts and theorems of o-minimality. In Chapter 2, we prove that the ring of integers is definable in the expansion of the real field by an infinite convex subset of a finite-rank additive subgroup of the reals. We give a few applications of this result. The main theorem of Chapter 3 is a structure theorem for expansions of the real field by families of restricted complex power functions. We apply it to classify expansions of the real field by families of locally closed trajectories of linear vector fields. Chapter 4 deals with polynomially bounded o-minimal structures over the real field expanded by multiplicative subgroups of the reals. The main result is that any nonempty, bounded, definable d-dimensional submanifold has finite d-dimensional Hausdorff measure if and only if the dimension of its frontier is less than d.

Zil’ber’s Trichotomy and o-minimal Structures

1998-01-01
Ya’acov Peterzil
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O-minimal structures

2007-11-01
A. J. Wilkie
The notion of an o-minimal expansion of the ordered field of real numbers was invented by L van den Dries [vdD1] as a framework for investigating the model theory of … The notion of an o-minimal expansion of the ordered field of real numbers was invented by L van den Dries [vdD1] as a framework for investigating the model theory of the real exponential function exp : R ! R : x ! ex, and thereby settle an old problem of Tarski. More on this later, but for the moment it is best motivated as being a candidate for Grothendieck’s idea of “tame topology” as expounded in his Esquisse d’un Programme [Gr]. It seems to me that such a candidate should satisfy (at least) the following criteria. (A) It should be a framework that is flexible enough to carry out many geometrical and topological constructions on real functions and on subsets of real euclidean spaces. (B) But at the same time it should have built in restrictions so that we are a priori guaranteed that pathological phenomena can never arise. In particular, there should be a meaningful notion of dimension for all sets under consideration and any that can be constructed from these by use of the operations allowed under (A). (C) One must be able to prove finiteness theorems that are uniform over fibred collections. None of the standard restrictions on functions that arise in elementary real analysis satisfy both (A) and (B). For example, there exists a continuous function G : (0, 1) ! (0, 1)2 which is surjective, thereby destroying any hope of a dimension theory for a framework that admits all continuous functions. Restricting to the smooth (i.e. C1) 985–02 environment fares no better. For every closed subset of any euclidean space, in particular the subset graph(G) of R3, is the set of zeros of some smooth function. So by the use of a few simple constructions that we would certainly wish to allow under (A), we soon arrive at dimension-destroying phenomena. The same is even true (though this is harder to prove) if we start from just those smooth functions that are everywhere real analytic (i.e. equal the sum of their Taylor series on a neighbourhood of every point), although, as we shall see, this class of functions is locally well-behaved and as such can serve as a model for the three criteria above. Rather than enumerate analytic conditions on sets and functions sufficient to guarantee the criteria (A), (B) and (C) however, we shall give one succinct axiom, the o-minimality axiom, which implies them. Of course, this is a rather open-ended (and currently flourishing) project because of the large number of questions that one can ask under (C). One must also provide concrete examples of collections of sets and functions that satisfy the axiom and this too is an active area of research. In this talk I shall survey both aspects of the theory. Our formulation of the o-minimality axiom makes use of definability theory from mathematical logic. We begin with a collection F of real valued functions of real variables (not necessarily all of the same number of arguments). We consider the ordered field structure on R augmented by the functions in F. This gives us a first-order structure (or model ) RF := hR;+, ·,−,<,Fi, and we denote the corresponding firstorder logical language by L(F). We then call the structure RF o-minimal if whenever (x) is an L(F)-formula (with parameters) then the subset of R defined by (x) is a finite union of open intervals and points (i.e. it is the union of finitely many connected sets). I shall elucidate what is meant by an L(F)-formula and by the subset of R (and, more generally, of Rn) defined by such a formula in the next two sections. However, I should emphasize at this stage that such a formula not only defines a subset , denoted (RF), of Rn, but also a subset (R) of Rn where R is any ordered ring augmented by a collection of functions, F say, such that F and F are in correspondence via a bijection that preserves the number of places (arity) of the functions. One can, and should, define the notion o-minimality for such structures hR;Fi and it was at (rather more than) this level of generality that the true foundations of the subject were laid by Pillay and Steinhorn in [P-S], shortly after van den Dries’ work on the real field. Indeed, it turned out that the solution to Tarski’s problem on the real exponential function (the case F = {exp} in the above notation) relied heavily on the Pillay-Steinhorn theory of o-minimality for structures based on ordered fields other than the reals. This having been said, I shall concentrate in this lecture on the real case, alluding only occasionally to the more general situation, and leave the reader to adapt the definitions and theorems to the setting of o-minimal expansions of arbitrary ordered fields.

On semibounded expansions of ordered groups

2023-01-01
Pantelis E. Eleftheriou , Alex Savatovsky
We explore <em>semibounded</em> expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We show that if $\mathcal R=\langle \mathbb R, \lt , … We explore <em>semibounded</em> expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We show that if $\mathcal R=\langle \mathbb R, \lt , +, \ldots \rangle $ is a semibounded o-minimal structure and
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On NIP and invariant measures

2007-01-01
Ehud Hrushovski , Anand Pillay
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of $NIP$ (not the independence property), continuing aspects of math.LO/0607442. Among key results are: (i) if … We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of $NIP$ (not the independence property), continuing aspects of math.LO/0607442. Among key results are: (i) if $p = tp(b/A)$ does not fork over $A$ then the Lascar strong type of $b$ over $A$ coincides with the compact strong type of $b$ over $A$ and any global nonforking extension of $p$ is Borel definable over $bdd(A)$ (ii) analogous statements for Keisler measures and definable groups, including the fact that $G^{000} = G^{00}$ for $G$ definably amenable, (iii) definitions, characterizations and properties of "generically stable" types and groups (iv) uniqueness of translation invariant Keisler measures on groups with finitely satisfiable generics (vi) A proof of the compact domination conjecture for definably compact commutative groups in $o$-minimal expansions of real closed fields.

Zarankiewicz's problem for semilinear hypergraphs

2020-09-07
Abdul Basit , Artem Chernikov , Sergei Starchenko +2 more
A bipartite graph $H = \left(V_1, V_2; E \right)$ with $|V_1| + |V_2| = n$ is semilinear if $V_i \subseteq \mathbb{R}^{d_i}$ for some $d_i$ and the edge relation $E$ consists … A bipartite graph $H = \left(V_1, V_2; E \right)$ with $|V_1| + |V_2| = n$ is semilinear if $V_i \subseteq \mathbb{R}^{d_i}$ for some $d_i$ and the edge relation $E$ consists of the pairs of points $(x_1, x_2) \in V_1 \times V_2$ satisfying a fixed Boolean combination of $s$ linear equalities and inequalities in $d_1 + d_2$ variables for some $s$. We show that for a fixed $k$, the number of edges in a $K_{k,k}$-free semilinear $H$ is almost linear in $n$, namely $|E| = O_{s,k,\varepsilon}(n^{1+\varepsilon})$ for any $\varepsilon > 0$; and more generally, $|E| = O_{s,k,r,\varepsilon}(n^{r-1 + \varepsilon})$ for a $K_{k, \ldots,k}$-free semilinear $r$-partite $r$-uniform hypergraph. As an application, we obtain the following incidence bound: given $n_1$ points and $n_2$ open boxes with axis parallel sides in $\mathbb{R}^d$ such that their incidence graph is $K_{k,k}$-free, there can be at most $O_{k,\varepsilon}(n^{1+\varepsilon})$ incidences. The same bound holds if instead of boxes one takes polytopes cut out by the translates of an arbitrary fixed finite set of halfspaces. We also obtain matching upper and (superlinear) lower bounds in the case of dyadic boxes on the plane, and point out some connections to the model-theoretic trichotomy in $o$-minimal structures (showing that the failure of an almost linear bound for some definable graph allows one to recover the field operations from that graph in a definable manner).

Construction d'un groupe dans les structures <i>C</i>-minimales

2008-09-01
Fares Maalouf
Abstract We will study some aspects of the local structure of models of certain C -minimal theories. We will prove (theorem 19) that, in a sufficiently saturated C -minimal structure … Abstract We will study some aspects of the local structure of models of certain C -minimal theories. We will prove (theorem 19) that, in a sufficiently saturated C -minimal structure in which the algebraic closure has the exchange property and which is locally modular, we can construct an infinite type-definable group around any non trivial point (a term to be defined later).

The prospects for mathematical logic in the twenty-first century

2002-01-01
Samuel R. Buss , Alexander S. Kechris , Anand Pillay +1 more
The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model … The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

Analytic and pseudo-analytic structures (a survey)

2004-01-01
Boris Zilber
The paper is an extended version of the talk in the Logic Colloquium-2000 at Paris. We discuss a series of results and problems around Hrushovski's construction of counter-examples to the … The paper is an extended version of the talk in the Logic Colloquium-2000 at Paris. We discuss a series of results and problems around Hrushovski's construction of counter-examples to the Trichotomy conjecture.

Definability in the group of infinitesimals of a compact Lie group

2020-03-09
Martin Bays , Ya’acov Peterzil
We show that for G a simple compact Lie group, the infinitesimal subgroup G 00 is bi-interpretable with a real closed convexly valued field. We deduce that for G an … We show that for G a simple compact Lie group, the infinitesimal subgroup G 00 is bi-interpretable with a real closed convexly valued field. We deduce that for G an infinite definably compact group definable in an o-minimal expansion of a field, G 00 is bi-interpretable with the disjoint union of a (possibly trivial) ℚ-vector space and finitely many (possibly zero) real closed valued fields. We also describe the isomorphisms between such infinitesimal subgroups, and along the way prove that every definable field in a real closed convexly valued field R is definably isomorphic to R.
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Interpretable groups are definable

2014-03-18
Pantelis E. Eleftheriou , Ya’acov Peterzil , Janak Ramakrishnan
We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product … We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product of one-dimensional definable group-intervals (or one-dimensional definable groups). We discuss the general open question of elimination of imaginaries in an o-minimal structure.
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Definable quotients of locally definable groups

2012-03-23
Pantelis E. Eleftheriou , Ya’acov Peterzil
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Simple algebraic and semialgebraic groups over real closed fields

2000-06-13
Ya’acov Peterzil , Anand Pillay , Sergei Starchenko
We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic … We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

Definably simple groups in o-minimal structures

2000-02-24
Ya’acov Peterzil , Anthony L. Pillay , Sergei Starchenko
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}=\langle G, \cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a group definable in an o-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle G,\cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (while <italic>definable</italic> means definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). Assume <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable proper subgroup of finite index. In this paper we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no nontrivial abelian normal subgroup, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the direct product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable subgroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1 comma ellipsis comma upper H Subscript k Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">H_1,\ldots ,H_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript i"> <mml:semantics> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">H_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin’s conjecture.

A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007-03-01
Ehud Hrushovski , Ya’acov Peterzil
Abstract We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions … Abstract We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.
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Simple groups definable in O-minimal structures

2017-03-02
Ya’acov Peterzil , Anand Pillay , Sergei Starchenko
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Definable groups as homomorphic images of semi-linear and field-definable groups

2012-03-23
Pantelis E. Eleftheriou , Ya’acov Peterzil
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Returning to semi-bounded sets

2009-06-01
Ya’acov Peterzil
Abstract An o-minimal expansion of an ordered group is called semi-bounded if there is no definable bijection between a bounded and an unbounded interval in it (equivalently, it is an … Abstract An o-minimal expansion of an ordered group is called semi-bounded if there is no definable bijection between a bounded and an unbounded interval in it (equivalently, it is an expansion of the group by bounded predicates and group automorphisms). It is shown that every such structure has an elementary extension N such that either N is a reduct of an ordered vector space, or there is an o-minimal structure , with the same universe but of different language from N , with (i) Every definable set in N is definable in , and (ii) has an elementary substructure in which every bounded interval admits a definable real closed field. As a result certain questions about definably compact groups can be reduced to either ordered vector spaces or expansions of real closed fields. Using the known results in these two settings, the number of torsion points in definably compact abelian groups in expansions of ordered groups is given. Pillay's Conjecture for such groups follows.

Definable one dimensional structures in o-minimal theories

2010-10-31
Assaf Hasson , Alf Onshuus , Ya’acov Peterzil

Additive reducts of real closed fields and strongly bounded structures

2023-10-21
Hind Abu Saleh , Ya’acov Peterzil
Additive reducts of real closed fields and strongly bounded structures Additive reducts of real closed fields and strongly bounded structures
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Definable structures in o-minimal theories: One dimensional types

2010-10-31
Assaf Hasson , Alf Onshuus , Ya’acov Peterzil
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Structure theorems for o-minimal expansions of groups

2000-03-01
Mário J. Edmundo

Solvable groups definable in o-minimal structures

2003-10-09
Mário J. Edmundo
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Thorn independence in the field of real numbers with a small multiplicative group

2007-11-20
Alexander Berenstein , Clifton Ealy , Ayhan Günaydın
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Model theory of special subvarieties and Schanuel-type conjectures

2016-05-11
Boris Zilber
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On lovely pairs of geometric structures

2009-11-29
Alexander Berenstein , Evgueni Vassiliev

Almost o-minimal structures and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="fraktur">X</mml:mi></mml:math>-structures

2022-06-01
Masato Fujita
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Functions continuous on curves in o-minimal structures

2009-01-01
Janak Ramakrishnan
We give necessary and sufficient conditions on a non-oscillatory curve in an o-minimal field such that, for any bounded definable function, the germ of the function on an initial segment … We give necessary and sufficient conditions on a non-oscillatory curve in an o-minimal field such that, for any bounded definable function, the germ of the function on an initial segment of the curve can be continuously extended to a closed definable set. This situation is translated into a question about types: What are the conditions on an $n$-type such that, for any bounded definable function, there is a definable set containing the type on which the function is continuous, and can be extended continuously to the set's closure? All such types are definable, and we give the precise conditions that are equivalent to existence of a desired definable set.

Higher Dimensional Trichotomy Conjectures in Model Theory

2021-01-01
Benjamin Castle
Author(s): Castle, Benjamin | Advisor(s): Scanlon, Thomas | Abstract: In this thesis we study the Restricted Trichotomy Conjectures for algebraically closed and o-minimal fields. These conjectures predict a classification of … Author(s): Castle, Benjamin | Advisor(s): Scanlon, Thomas | Abstract: In this thesis we study the Restricted Trichotomy Conjectures for algebraically closed and o-minimal fields. These conjectures predict a classification of all sufficiently complex, that is, non-locally modular, strongly minimal structures which can be interpreted from such fields. Such problems have been historically divided into `lower dimensional' and `higher dimensional' cases; this thesis is devoted to a number of partial results in the higher dimensional cases. In particular, in ACF_0 and over o-minimal fields, we prove that all higher dimensional strongly minimal structures whose definable sets satisfy certain geometric restrictions are locally modular. We also make progress toward verifying these geometric restrictions in any counterexample. Finally, in the last chapter we give a full proof of local modularity for strongly minimal expansions of higher dimensional groups in ACF_0.

Some remarks on CM-triviality

2009-02-01
Ikuo Yoneda
We show that any rosy CM-trivial theory has weak canonical bases, and CM-triviality in the real sort is equivalent to CM-triviality with geometric elimination of imaginaries. We also show that … We show that any rosy CM-trivial theory has weak canonical bases, and CM-triviality in the real sort is equivalent to CM-triviality with geometric elimination of imaginaries. We also show that CM-triviality is equivalent to the modularity in O-minimal theories with elimination of imaginaries.
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Incidence systems on Cartesian powers of algebraic curves

2017-01-01
Assaf Hasson , Dmitry Sustretov
We show that a reduct of the Zariski structure of an algebraic curve which is not locally modular interprets a field, answering a question of Zilber's. We show that a reduct of the Zariski structure of an algebraic curve which is not locally modular interprets a field, answering a question of Zilber's.
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Definable Open Sets As Finite Unions of Definable Open Cells

2010-04-01
Simon Andrews
We introduce CE-cell decomposition, a modified version of the usual o-minimal cell decomposition. We show that if an o-minimal structure ℛ admits CE-cell decomposition then any definable open set in … We introduce CE-cell decomposition, a modified version of the usual o-minimal cell decomposition. We show that if an o-minimal structure ℛ admits CE-cell decomposition then any definable open set in ℛ may be expressed as a finite union of definable open cells. The dense linear ordering and linear o-minimal expansions of ordered abelian groups are examples of such structures.
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Combinatorial Bounds in Distal Structures

2023-10-13
Aaron Anderson
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10th Asian Logic Conference

2009-06-01
Toshiyasu Arai
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Weakly one-based geometric theories

2012-04-04
Alexander Berenstein , Evgueni Vassiliev
Abstract We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find … Abstract We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω -categorical geometric theory interprets an infinite vector space over a finite field.
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Non-standard lattices and o-minimal groups

2013-03-01
Pantelis E. Eleftheriou
Abstract We describe a recent program from the study of definable groups in certain o-minimal structures. A central notion of this program is that of a (geometric) lattice . We … Abstract We describe a recent program from the study of definable groups in certain o-minimal structures. A central notion of this program is that of a (geometric) lattice . We propose a definition of a lattice in an arbitrary first-order structure. We then use it to describe, uniformly, various structure theorems for o-minimal groups, each time recovering a lattice that captures some significant invariant of the group at hand. The analysis first goes through a local level, where a pertinent notion of pregeometry and generic elements is each time introduced.

Almost o-minimal structures and $\mathfrak X$-structures

2021-04-03
Masato Fujita
We propose new structures called almost o-minimal structures and $\mathfrak X$-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a … We propose new structures called almost o-minimal structures and $\mathfrak X$-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a definable set with a bounded open interval is a finite union of points and open intervals. The latter is a variant of van den Dries and Miller's analytic geometric categories and Shiota's $\mathfrak X$-sets and $\mathfrak Y$-sets. In them, the family of definable sets are closed only under proper projections unlike first-order structures. We demonstrate that an $\mathfrak X$-expansion of an ordered divisible abelian group always contains an o-minimal expansion of an ordered group such that all bounded $\mathfrak X$-definable sets are definable in the structure. Another contribution of this paper is a uniform local definable cell decomposition theorem for almost o-minimal expansions of ordered groups $\mathcal M=(M,<,0,+,\ldots)$. Let $\{A_\lambda\}_{\lambda\in\Lambda}$ be a finite family of definable subsets of $M^{m+n}$. Take an arbitrary positive element $R \in M$ and set $B=]-R,R[^n$. Then, there exists a finite partition into definable sets \begin{equation*} M^m \times B = X_1 \cup \ldots \cup X_k \end{equation*} such that $B=(X_1)_b \cup \ldots \cup (X_k)_b$ is a definable cell decomposition of $B$ for any $b \in M^m$ and either $X_i \cap A_\lambda = \emptyset$ or $X_i \subseteq A_\lambda$ for any $1 \leq i \leq k$ and $\lambda \in \Lambda$. Here, the notation $S_b$ denotes the fiber of a definable subset $S$ of $M^{m+n}$ at $b \in M^m$. We introduce the notion of multi-cells and demonstrate that any definable set is a finite union of multi-cells in the course of the proof of the above theorem.

Book Review: Tame topology and o-minimal structures

2000-03-02
David Marker
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Uniform bounds on growth in o‐minimal structures

2010-07-12
Janak Ramakrishnan
Abstract We prove that a function definable with parameters in an o‐minimal structure is bounded away from ∞ as its argument goes to ∞ by a function definable without parameters, … Abstract We prove that a function definable with parameters in an o‐minimal structure is bounded away from ∞ as its argument goes to ∞ by a function definable without parameters, and that this new function can be chosen independently of the parameters in the original function. This generalizes a result in [1]. Moreover, this remains true if the argument is taken to approach any element of the structure (or ±∞), and the function has limit any element of the structure (or ±∞) (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)
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A tetrachotomy for expansions of the real ordered additive group

2021-06-19
Philipp Hieronymi , Erik Walsberg
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One-basedness and groups of the form G/G 00

2011-07-15
Davide Penazzi
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Coverings by open cells

2014-01-28
Mário J. Edmundo , Pantelis E. Eleftheriou , Luca Prelli

Model theory of special subvarieties and Schanuel-type conjectures

2015-01-14
Boris Zilber
We use the language and tools available in model theory to redefine and clarify the rather involved notion of a {\em special subvariety} known from the theory of Shimura varieties … We use the language and tools available in model theory to redefine and clarify the rather involved notion of a {\em special subvariety} known from the theory of Shimura varieties (mixed and pure).