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Groups, measures, and the NIP

2007-02-02
Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating … We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.
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Definable Compactness and Definable Subgroups of o-Minimal Groups

1999-06-01
Ya’acov Peterzil , Charles Steinhorn
The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an … The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

A trichotomy theorem for o-minimal structures

1998-11-01
Ya’acov Peterzil , Sergei Starchenko
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 … 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.

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.

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.

Definable homomorphisms of abelian groups in o-minimal structures

1999-11-01
Ya’acov Peterzil , Sergei Starchenko

On Groups and Rings Definable In O-Minimal Expansions of Real Closed Fields

1996-01-01
Margarita Otero , Ya’acov Peterzil , Anand Pillay
Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in … Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in M. We show that every M–definable ring without zero divisors is definably isomorphic to R, R(√(−l)) or the ring of quaternions over R. One corollary is that no model of Texp is interpretable in a model of Tan.

MODEL THEORY OF DIFFERENCE FIELDS, II: PERIODIC IDEALS AND THE TRICHOTOMY IN ALL CHARACTERISTICS

2002-08-02
Zoé Chatzidakis , Ehud Hrushovski , Ya’acov Peterzil
We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the … We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the automorphism by its nth power. We show that these reducts admit a successively smoother theory as n becomes large; and we succeed in defining a limit structure to these reducts, or rather to the structure they induce on one-dimensional sets. This limit structure is shown to be a Zariski geometry in (roughly) the sense of Hrushovski and Zil'ber. The trichotomy is thus obtained for the limit structure as a consequence of a general theorem, and then shown to be inherited by the original theory. 2000 Mathematical Subject Classification: 03C60; (primary) 03C45, 03C98, 08A35, 12H10 (secondary)

Definability of restricted theta functions and families of abelian varieties

2013-03-15
Ya’acov Peterzil , Sergei Starchenko
We consider some classical maps from the theory of abelian varieties and their moduli spaces, and we prove their definability on restricted domains in the o-minimal structure Ran,exp. In particular, … We consider some classical maps from the theory of abelian varieties and their moduli spaces, and we prove their definability on restricted domains in the o-minimal structure Ran,exp. In particular, we prove that the projective embedding of the moduli space of the principally polarized abelian variety Sp(2g,Z)\Hg is definable in Ran,exp when restricted to Siegel's fundamental set Fg. We also prove the definability on appropriate domains of embeddings of families of abelian varieties into projective spaces.
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Linear Groups Definable in o-Minimal Structures

2002-01-01
Ya’acov Peterzil , Anthony L. Pillay , Sergei Starchenko

Linear O-minimal structures

1993-02-01
James Loveys , Ya’acov Peterzil

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.

Additive reducts of real closed fields

1992-03-01
David Marker , Ya’acov Peterzil , Anand Pillay
In [MP] Marker and Pillay showed that if X ⊂ C n is constructible but ( C , +, X ) is not locally modular, then multiplication is definable in … In [MP] Marker and Pillay showed that if X ⊂ C n is constructible but ( C , +, X ) is not locally modular, then multiplication is definable in the structure ( C , +, X ). That result extended earlier results of Martin [M] and Rabinovich and Zil'ber [RZ]. Here we will examine additive reducts of R and Q p . Definition. A subset X of R n is called semialgebraic if it is definable in the structure ( R , +,·). A subset X of R n is called semilinear if it is definable in the structure ( R , +, &lt;,λ r ) r∈ b , where λ r is the function x ↦ rx [scalar multiplication by r ]. Every semilinear set is a Boolean combination of sets of the form { : p ( ) = 0} and { : q ( ) &gt; 0}, where p ( ) and q ( ) are linear polynomials. Van den Dries asked the following question: if X is semialgebraic but not semilinear, can we define multiplication in ( R , +, &lt;, X )? This was answered negatively by Pillay, Scowcroft and Steinhorn. Theorem 1.1 [PSS]. Suppose X ⊂ R n is semialgebraic and X ⊂ I n for some bounded interval I. Then multiplication is not definable in ( R , +, &lt;, X ,λ r ) r ∈ R . In particular if X = · ∣ [0, l ] 2 , the graph of multiplication restricted to the unit interval, then X is not semilinear so we have a negative answer to van den Dries' question. Peterzil showed that this is the only restriction.

A descending chain condition for groups definable in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>o</mml:mi></mml:math>-minimal structures

2005-03-22
Alessandro Berarducci , Margarita Otero , Ya’acov Peterzil +1 more

Complex analytic geometry and analytic-geometric categories

2009-01-01
Ya’acov Peterzil , Sergei Starchenko
The notion of an analytic-geometric category was introduced by v. d. Dries and Miller in [Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. … The notion of an analytic-geometric category was introduced by v. d. Dries and Miller in [Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540.]. It is a category of subsets of real analytic manifolds which extends the category of subanalytic sets. This paper discusses connections between the subanalytic category, or more generally analytic-geometric categories, and complex analytic geometry. The questions are of the following nature: We start with a subset A of a complex analytic manifold M and assume that A is an object of an analytic-geometric category (by viewing M as a real analytic manifold of double dimension). We then formulate conditions under which A, its closure or its image under a holomorphic map is a complex analytic set.

Expansions of algebraically closed fields in o-minimal structures

2001-09-01
Ya’acov Peterzil , Sergei Starchenko

Generic sets in definably compact groups

2007-01-01
Ya’acov Peterzil , Anand Pillay
A subset $X$ of a group $G$ is called <em>left generic</em> if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably … A subset $X$ of a group $G$ is called <em>left generic</em> if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably compact group in an o-minimal structure and a definable $X\subseteq G$ is not right generic then i
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On central extensions and definably compact groups in o-minimal structures

2010-11-17
Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay
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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.

A structure theorem for semibounded sets in the reals

1992-09-01
Ya’acov Peterzil
In [MPP] it was shown that in every reduct of = ‹ ℝ, +, ·, &lt;› that properly expands ℳ = ‹ℝ, +, &lt;, λ a › a ∈ℝ , … In [MPP] it was shown that in every reduct of = ‹ ℝ, +, ·, &lt;› that properly expands ℳ = ‹ℝ, +, &lt;, λ a › a ∈ℝ , all the bounded semi-algebraic (that is, -definable) sets are definable. Said differently, every such is an expansion of = ‹ℝ, +, &lt;, λ a , B i › a ∈ℝ, i ∈ I where { B i } i ∈ I is the collection of all bounded semialgebraic sets and the λ a 's are scalar multiplication by a . In [PSS] (see Theorem 1.2 below) it was shown that the structure is a proper reduct of ; that is, one cannot define in it all the semialgebraic sets. In [Pe] we show that is the only reduct properly between ℳ and . As a first step towards this result, we investigate in this paper the definable sets in reducts such as . (We point out that ‘definable’ will always mean ‘definable with parameters’.) Definition 1.1. Let X ⊆ ℝ n . X is called semi-bounded if it is definable in the structure ‹ℝ, +, &lt;, λ a , B 1 , …, B k › a ∈ℝ , where the B i 's are bounded subsets of ℝ n . The main result of this paper (see Theorem 3.1) shows roughly that, in Ominimal expansions of that satisfy the partition condition (see Definition 2.3), every semibounded set can be partitioned into finitely many sets, each of which is of a form similar to a cylinder. Namely, these sets are obtained through the “stretching” of a bounded cell by finitely many linear vectors. As a corollary (see Theorem 1.4), we get different characterizations of semibounded sets, either in terms of their structure or in terms of their definability power. The following result, by A. Pillay, P. Scowcroft and C. Steinhorn, was the main motivation for this paper. The theorem is formulated here in a slightly stronger form than originally, but the proof itself is essentially the original one. A short version of the proof is included in §4.

Uniform definability of the Weierstrass $\wp$ functions and generalized tori of dimension one

2005-03-30
Ya’acov Peterzil , Sergei Starchenko

Scope Dominance with Upward Monotone Quantifiers

2005-10-01
Alon Altman , Ya’acov Peterzil , Yoad Winter

On torsion-free groups in o-minimal structures

2005-10-01
Ya’acov Peterzil , Sergei Starchenko
We consider groups definable in the structure Ran and certain o-minimal expansions of it.We prove: If G = G, * is a definable abelian torsion-free group, then G is definably … We consider groups definable in the structure Ran and certain o-minimal expansions of it.We prove: If G = G, * is a definable abelian torsion-free group, then G is definably isomorphic to a direct sum of R, + k and R >0 , • m , for some k, m 0. Futhermore, this isomorphism is definable in the structure R, +, •, G .In particular, if G is semialgebraic, then the isomorphism is semialgebraic.We show how to use the above result to give an "o-minimal proof" to the classical Chevalley theorem for abelian algebraic groups over algebraically closed fields of characteristic zero.We also prove: Let M be an arbitrary o-minimal expansion of a real closed field R and G a definable group of dimension n.The group G is torsion-free if and only if G, as a definable group-manifold, is definably diffeomorphic to R n .
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Reducts of some structures over the reals

1993-09-01
Ya’acov Peterzil
Abstract We consider reducts of the structure ℛ = 〈ℝ, +, ·, &lt;〉 and other real closed fields. We compete the proof that there exists a unique reduct between 〈ℝ, … Abstract We consider reducts of the structure ℛ = 〈ℝ, +, ·, &lt;〉 and other real closed fields. We compete the proof that there exists a unique reduct between 〈ℝ, +, &lt;,λ a 〉 a ∈ ℝ and ℛ, and we demonstrate how to recover the definition of multiplication in more general contexts than the semialgebraic one. We then conclude a similar result for reducts between 〈ℝ, ·, &lt;〉 and ℛ and for general real closed fields.

Complex analytic geometry in a nonstandard setting

2008-05-22
Ya’acov Peterzil , Sergei Starchenko
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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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EXPANSIONS OF ALGEBRAICALLY CLOSED FIELDS II: FUNCTIONS OF SEVERAL VARIABLES

2003-05-01
Ya’acov Peterzil , Sergei Starchenko
Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field … Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field [Formula: see text]. We develop the basic theory of such K-differentiability for definable functions of several variables, proving theorems on removable singularities as well as analogues of the Weierstrass preparation and division theorems for definable functions. We consider also definably meromorphic functions and prove that every definable function which is meromorphic on K n is necessarily a rational function. We finally discuss definable analogues of complex analytic manifolds, with possible connections to the model theoretic work on compact complex manifolds, and present two examples of "nonstandard manifolds" in our setting.

Tame Complex Analysis and <font>o</font>-minimality

2011-04-15
Ya’acov Peterzil , Sergei Starchenko

THE CENTRAL PATH IN SMOOTH CONVEX SEMIDEFINITE PROGRAMS

2002-04-01
L. M. Graña Drummond , Ya’acov Peterzil
Abstract In this paper we study the welldefinedness of the central path associated to a nonlinear convex semidefinite programming problem with smooth objective and constraint functions. Under standard assumptions, we … Abstract In this paper we study the welldefinedness of the central path associated to a nonlinear convex semidefinite programming problem with smooth objective and constraint functions. Under standard assumptions, we prove that the existence of the central path is equivalent to the nonemptiness and boundedness of the optimal set. Other equivalent conditions are given, such as the existence of a strictly dual feasible point or the existence of a single central point. The monotonic behavior of the primal and dual logarithmic barriers and of the primal and dual objective functions along the trajectory is also discussed. The existence and optimality of cluster points is established and finally, under the additional assumption of analyticity of the data functions, the convergence of the primal-dual trajectory is proved.

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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Pillay's conjecture and its solution—a survey

2010-06-07
Ya’acov Peterzil
Introduction. These notes were originally written for a tutorial I gave in a Modnet Summer meeting which took place in Oxford 2006. I later gave a similar tutorial in the … Introduction. These notes were originally written for a tutorial I gave in a Modnet Summer meeting which took place in Oxford 2006. I later gave a similar tutorial in the Wroclaw Logic colloquium 2007. The goal was to survey recent work in model theory of o-minimal structures, centered around the solution to a beautiful conjecture of Pillay on definable groups in o-minimal structures. The conjecture (which is now a theorem in most interesting cases) suggested a connection between arbitrary definable groups in o-minimal structures and compact real Lie groups.

Definable quotients of locally definable groups

2012-03-23
Pantelis E. Eleftheriou , Ya’acov Peterzil
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Constructing a group-interval in o-minimal structures

1994-06-01
Ya’acov Peterzil

Topological groups, $\mu$-types and their stabilizers

2017-09-18
Ya’acov Peterzil , Sergei Starchenko
We consider an arbitrary topological group G definable in a structure \mathcal M , such that some basis for the topology of G consists of sets definable in \mathcal M … We consider an arbitrary topological group G definable in a structure \mathcal M , such that some basis for the topology of G consists of sets definable in \mathcal M . To each such group G we associate a compact G -space of partial types, S^{\mu}_G(M)=\{p_{\mu}\colon p\in S_G(M)\} which is the quotient of the usual type space S_G(M) by the relation of two types being "infinitesimally close to each other". In the o-minimal setting, if p is a definable type then it has a corresponding definable subgroup Stab ^{\mu}(p) , which is the stabilizer of p_{\mu} . This group is nontrivial when p is unbounded in the sense of \mathcal M ; in fact it is a torsion-free solvable group. Along the way, we analyze the general construction of S^{\mu}_G(M) and its connection to the Samuel compactification of topological groups.
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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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A note on stable sets, groups, and theories with NIP

2007-05-18
Alf Onshuus , Ya’acov Peterzil
Abstract Let M be an arbitrary structure. Then we say that an M ‐formula φ ( x ) defines a stable set in M if every formula φ ( x … Abstract Let M be an arbitrary structure. Then we say that an M ‐formula φ ( x ) defines a stable set in M if every formula φ ( x ) ∧ α ( x , y ) is stable. We prove: If G is an M ‐definable group and every definable stable subset of G has U ‐rank at most n (the same n for all sets), then G has a maximal connected stable normal subgroup H such that G / H is purely unstable. The assumptions hold for example if M is interpretable in an o‐minimal structure. More generally, an M ‐definable set X is weakly stable if the M ‐induced structure on X is stable. We observe that, by results of Shelah, every weakly stable set in theories with NIP is stable. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Computing o-minimal topological invariants using differential topology

2006-10-24
Ya’acov Peterzil , Sergei Starchenko
We work in an o-minimal expansion of a real closed field. Using piecewise smoothness of definable functions we define the topological degree for definable continuous functions. Using this notion of … We work in an o-minimal expansion of a real closed field. Using piecewise smoothness of definable functions we define the topological degree for definable continuous functions. Using this notion of the degree we obtain a new proof for the existence of torsion points in a definably compact group, and also a new proof of an o-minimal analogue of the Brouwer fixed point theorem.

G-linear sets and torsion points in definably compact groups

2009-04-15
Margarita Otero , Ya’acov Peterzil
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Geometry, Calculus and Zil'ber's Conjecture

1996-03-01
Ya’acov Peterzil , Sergei Starchenko
§1. Introduction . By and large, definitions of a differentiable structure on a set involve two ingredients, topology and algebra. However, in some cases, partial information on one or both … §1. Introduction . By and large, definitions of a differentiable structure on a set involve two ingredients, topology and algebra. However, in some cases, partial information on one or both of these is sufficient. A very simple example is that of the field ℝ (or any real closed field) where algebra alone determines the ordering and hence the topology of the field: In the case of the field ℂ, the algebraic structure is insufficient to determine the Euclidean topology; another topology, Zariski, is associated with the ield but this will be too coarse to give a diferentiable structure. A celebrated example of how partial algebraic and topological data ( G a locally euclidean group) determines a differentiable structure ( G is a Lie group) is Hilbert's 5th problem and its solution by Montgomery-Zippin and Gleason. The main result which we discuss here (see [13] for the full version) is of a similar flavor: we recover an algebraic and later differentiable structure from a topological data. We begin with a linearly ordered set ⟨ M , &lt;⟩, equipped with the order topology, and its cartesian products with the product topologies. We then consider the collection of definable subsets of M n , n = 1, 2, …, in some first order expansion ℳ of ⟨ M , &lt;⟩.

On central extensions and definably compact groups in o-minimal structures

2008-01-01
Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay
We prove several structural results on definably compact groups G in o-minimal expansions of real closed fields, such as (i) G is definably an almost direct product of a semisimple … We prove several structural results on definably compact groups G in o-minimal expansions of real closed fields, such as (i) G is definably an almost direct product of a semisimple group and a commutative group, and (ii) the group (G, .) is elementarily equivalent to (G/G^00, .). We also prove results on the internality of finite covers of G in an o-minimal environment, as well as the full compact domination conjecture. These results depend on key theorems about the interpretability of central and finite extensions of definable groups, in the o-minimal context. These methods and others also yield interpretability results for universal covers of arbitrary definable real Lie groups, from which we can deduce the semialgebraicity of finite covers of Lie groups such as SL(2,R).

Algebraic and o-minimal flows on complex and real tori

2018-06-05
Ya’acov Peterzil , Sergei Starchenko
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Mild manifolds and a non-standard Riemann existence theorem

2008-12-30
Ya’acov Peterzil , Sergei Starchenko
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Lattices in Locally Definable Subgroups of 〈Rn,+〉

2013-01-01
Pantelis E. Eleftheriou , Ya’acov Peterzil
Let M be an o-minimal expansion of a real closed field R. We define the notion of a lattice in a locally definable group and then prove that every connected, … Let M be an o-minimal expansion of a real closed field R. We define the notion of a lattice in a locally definable group and then prove that every connected, definably generated subgroup of 〈Rn,+〉 contains a definable generic set and therefore admits a lattice.
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Definable one dimensional structures in o-minimal theories

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

GROUP COVERS, o-MINIMALITY, AND CATEGORICITY

2010-12-01
Alessandro Berarducci , Ya’acov Peterzil , Anthony L. Pillay
We study the model theory of "covers" of groups H definable in an o-minimal structure M. We pose the question of whether any finite central extension G of H is … We study the model theory of "covers" of groups H definable in an o-minimal structure M. We pose the question of whether any finite central extension G of H is interpretable in M, proving some cases (such as when H is abelian) as well as stating various equivalences. When M is an o-minimal expansion of the reals (so H is a definable Lie group) this is related to Milnor's conjecture [15], and many cases are known. We also prove a strong relative Lω1, ω-categoricity theorem for universal covers of definable Lie groups, and point out some notable differences with the case of covers of complex algebraic groups (studied by Zilber and his students).
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Groups, measures, and the NIP

2006-01-01
Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating … We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce a new notion "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.

A THEORY OF PAIRS FOR NON-VALUATIONAL STRUCTURES

2019-01-25
ELITZUR BAR-YEHUDA , Assaf Hasson , Ya’acov Peterzil
Given a weakly o-minimal structure $\mathcal M$ and its o-minimal completion $\bar {\mathcal M}$, we first associate to $\bar {\mathcal M}$ a canonical language and then prove that $Th(\mathcal M)$ … Given a weakly o-minimal structure $\mathcal M$ and its o-minimal completion $\bar {\mathcal M}$, we first associate to $\bar {\mathcal M}$ a canonical language and then prove that $Th(\mathcal M)$ determines $Th(\bar {\mathcal M})$. We then investigate the theory of the pair $\mathcal M^P=(\bar {\mathcal M};M)$ in the spirit of the theory of dense pairs of o-minimal structures, and prove, among other results, that it is near model complete, and every $\mathcal M^P$-definable open subset of $\bar M^n$ is already definable in $\bar {\mathcal M}$. We give an example of a weakly o-minimal structure which interprets $\bar {\mathcal M}$ and show that it is not elementarily equivalent to any reduct of an o-minimal trace.
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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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Simple groups definable in O-minimal structures

1998-01-01
Ya’acov Peterzil , Anand Pillay , Sergei Starchenko
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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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The infinitesimal subgroup of interpretable groups in some dp-minimal valued fields

2025-02-12
Yatir Halevi , Assaf Hasson , Ya’acov Peterzil
We continue our local analysis of groups interpretable in various dp-minimal valued fields, as introduced in [8]. We associate with every infinite group $G$ interpretable in those fields an infinite … We continue our local analysis of groups interpretable in various dp-minimal valued fields, as introduced in [8]. We associate with every infinite group $G$ interpretable in those fields an infinite type-definable infinitesimal subgroup $\nu(G)$, generated by the four infinitesimal subgroups $\nu_D(G)$ associated with the distinguished sorts $K$, $\textbf{k}$, $\Gamma$ and $K/\mathcal{O}$. To show that $\nu(G)$ is type-definable, we show that the resulting subgroups $\nu_D(G)$ commute with each other as $D$ ranges over the four distinguished sorts. We then study the basic properties of $\nu(G)$. Among others, we show that $\nu(G_1\times G_2)=\nu(G_1)\times \nu(G_2)$ and that if $G_1\le G$ is a definable subgroup then $\nu(G_1)$ is relatively definable in $\nu(G)$. We also discuss possible connections between $\mathrm{dp\text{-}rk}(\nu(G))$ and elimination of imaginaries.
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Limits of definable families and dilations in nilmanifolds

2024-06-27
Ya’acov Peterzil , Sergei Starchenko
Let $G$ be a unipotent group and $\mathcal F=\{F_t:t\in (0,\infty)\}$ a family of subsets of $G$, with $\mathcal F$ definable in an o-minimal expansion of the real field. Given a … Let $G$ be a unipotent group and $\mathcal F=\{F_t:t\in (0,\infty)\}$ a family of subsets of $G$, with $\mathcal F$ definable in an o-minimal expansion of the real field. Given a lattice $\Gamma\subseteq G$, we study the possible Hausdorff limits of $\pi(\mathcal F)$ in $G/\Gamma$ as $t$ tends to $\infty$ (here $\pi:G\to G/\Gamma$ is the canonical projection). Towards a solution, we associate to $\mathcal F$ finitely many real algebraic subgroups $L\subseteq G$, and, uniformly in $\Gamma$, determine if the only Hausdorff limit at $\infty$ is $G/\Gamma$, depending on whether $L^\Gamma=G$ or not. The special case of polynomial dilations of a definable set is treated in details.
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On groups interpretable in various valued fields

2024-06-19
Yatir Halevi , Assaf Hasson , Ya’acov Peterzil

Model-theoretic Elekes–Szabó for stable and o-minimal hypergraphs

2024-02-15
Artem Chernikov , Ya’acov Peterzil , Sergei Starchenko
A theorem of Elekes and Szabó recognizes algebraic groups among certain complex algebraic varieties with maximal size intersections with finite grids. We establish a generalization to relations of any arity … A theorem of Elekes and Szabó recognizes algebraic groups among certain complex algebraic varieties with maximal size intersections with finite grids. We establish a generalization to relations of any arity and dimension definable in: (1) stable structures with distal expansions (includes algebraically and differentially closed fields of characteristic 0) and (2) o-minimal expansions of groups. Our methods provide explicit bounds on the power-saving exponent in the nongroup case. Ingredients of the proof include a higher arity generalization of the abelian group configuration theorem in stable structures (along with a purely combinatorial variant characterizing Latin hypercubes that arise from abelian groups) and Zarankiewicz-style bounds for hypergraphs definable in distal structures.
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ON DEFINABLE GROUPS AND D-GROUPS IN CERTAIN FIELDS WITH A GENERIC DERIVATION

2024-01-15
Ya’acov Peterzil , Anand Pillay , Françoise Point
We continue our study from Peterzil et al. (2022, Preprint, arXiv:2208.08293) of finite-dimensional definable groups in models of the theory $T_{\partial }$, the model companion of an o-minimal ${\mathcal {L}}$-theory … We continue our study from Peterzil et al. (2022, Preprint, arXiv:2208.08293) of finite-dimensional definable groups in models of the theory $T_{\partial }$, the model companion of an o-minimal ${\mathcal {L}}$-theory T expanded by a generic derivation $\partial $ as in Fornasiero and Kaplan (2021, Journal of Mathematical Logic 21, 2150007).We generalize Buium's notion of an algebraic D-group to ${\mathcal {L}}$-definable D-groups, namely $(G,s)$, where G is an ${\mathcal {L}}$-definable group in a model of T, and $s:G\to \tau (G)$ is an ${\mathcal {L}}$-definable group section. Our main theorem says that every definable group of finite dimension in a model of $T_\partial $ is definably isomorphic to a group of the form $$ \begin{align*}(G,s)^\partial=\{g\in G:s(g)=\nabla g\},\end{align*} $$for some ${\mathcal {L}}$-definable D-group $(G,s)$ (where $\nabla (g)=(g,\partial g)$).We obtain analogous results when T is either the theory of p-adically closed fields or the theory of pseudo-finite fields of characteristic $0$.
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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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On definable groups and D-group in certain fields with a generic derivation

2023-01-01
Ya’acov Peterzil , Anand Pillay , Françoise Point
We continue our earlier study of finite dimensional definable groups in models of the the model companion of an o-minimal L-theory T expanded by a generic derivation as in [F-K]. … We continue our earlier study of finite dimensional definable groups in models of the the model companion of an o-minimal L-theory T expanded by a generic derivation as in [F-K]. We generalize Buium's notion of an algebraic D-group to L-definable D-groups, namely (G,s), where G is a L-definable group in a model of T, and s is an L-definable group section into the prolongation of G. Our main theorem says that every definable group of finite dimension in a model of the theory is definably isomorphic to the ``sharp'' points of an L-definable D-group. We obtain analogous results when T is either the theory of p-adically closed fields or the theory of pseudo-finite fields of characteristic zero.
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Semisimple groups interpretable in various valued fields

2023-01-01
Yatir Halevi , Assaf Hasson , Ya’acov Peterzil
We study infinite groups interpretable in power bounded $T$-convex, $V$-minimal or $p$-adically closed fields. We show that if $G$ is an interpretable definably semisimple group (i.e., has no definable infinite … We study infinite groups interpretable in power bounded $T$-convex, $V$-minimal or $p$-adically closed fields. We show that if $G$ is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups) then, up to a finite index subgroup, it is definably isogenous to a group $G_1\times G_2$, where $G_1$ is a $K$-linear group and $G_2$ is a $\mathbf{k}$-linear group. The analysis is carried out by studying the interaction of $G$ with four distinguished sorts: the valued field $K$, the residue field $\mathbf{k}$, the value group $\Gamma$, and the closed $0$-balls $K/\mathcal{O}$.
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Interpretable fields in various valued fields

2022-04-28
Yatir Halevi , Assaf Hasson , Ya’acov Peterzil
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Additive reducts of real closed fields and strongly bounded structures

2022-01-01
Hind Abu Saleh , Ya’acov Peterzil
Given a real closed field $R$, we identify exactly four proper reducts of $R$ which expand the underlying (unordered) $R$-vector space structure. Towards this theorem we introduce a new notion, … Given a real closed field $R$, we identify exactly four proper reducts of $R$ which expand the underlying (unordered) $R$-vector space structure. Towards this theorem we introduce a new notion, of strongly bounded reducts of linearly ordered structures: A reduct $\mathcal M$ of a linearly ordered structure $\langle R;<,\cdots\rangle $ is called \emph{strongly bounded} if every $\mathcal M$-definable subset of $R$ is either bounded or co-bounded in $R$. We investigate strongly bounded additive reducts of o-minimal structures and as a corollary prove the above theorem on additive reducts of real closed fields.
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On groups interpretable in various valued fields

2022-01-01
Yatir Halevi , Assaf Hasson , Ya’acov Peterzil
We study infinite groups interpretable in three families of valued fields: $V$-minimal, power bounded $T$-convex, and $p$-adically closed fields. We show that every such group $G$ has unbounded exponent and … We study infinite groups interpretable in three families of valued fields: $V$-minimal, power bounded $T$-convex, and $p$-adically closed fields. We show that every such group $G$ has unbounded exponent and that if $G$ is dp-minimal then it is abelian-by-finite. Along the way, we associate with any infinite interpretable group an infinite type-definable subgroup which is definably isomorphic to a group in one of four distinguished sorts: the underlying valued field $K$, its residue field $\mathbf{k}$ (when infinite), its value group $\Gamma$, or $K/\mathcal{O}$, where $\mathcal{O}$ is the valuation ring. Our work uses and extends techniques developed in [11] to circumvent elimination of imaginaries.
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On definable groups in real closed fields with a generic derivation, and related structures

2022-01-01
Ya’acov Peterzil , Anand Pillay , Françoise Point
We study finite-dimensional groups definable in models of the theory of real closed fields with a generic derivation (also known as CODF). We prove that any such group definably embeds … We study finite-dimensional groups definable in models of the theory of real closed fields with a generic derivation (also known as CODF). We prove that any such group definably embeds in a semialgebraic group. We extend the results to several more general contexts; strongly model complete theories of large geometric fields with a generic derivation, model complete o-minimal expansions of RCF with a generic derivation, open theories of topological fields with a generic derivation. We also give a general theorem on recovering a definable group from generic data in the context of geometric structures.
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Definably semisimple groups interpretable in $p$-adically closed fields

2022-01-01
Yatir Halevi , Assaf Hasson , Ya’acov Peterzil
Let $K$ be a $p$-adically closed field and $G$ a group interpretable in $K$. We show that if $G$ is definably semisimple (i.e. $G$ has no definable infinite normal abelian … Let $K$ be a $p$-adically closed field and $G$ a group interpretable in $K$. We show that if $G$ is definably semisimple (i.e. $G$ has no definable infinite normal abelian subgroups) then there exists a finite normal subgroup $H$ such that $G/H$ is definably isomorphic to a $K$-linear group. The result remains true in models of $\mathrm{Th}(\mathbb{Q}_p^{an})$.
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O-minimal flows on nilmanifolds

2021-11-22
Ya’acov Peterzil , Sergei Starchenko
Let G be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of UT(n,R), and let Γ be a lattice in G, with π:G→G∕Γ the quotient … Let G be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of UT(n,R), and let Γ be a lattice in G, with π:G→G∕Γ the quotient map. For a semialgebraic X⊆G, and more generally a definable set in an o-minimal structure on the real field, we consider the topological closure of π(X) in the compact nilmanifold G∕Γ. Our theorem describes cl(π(X)) in terms of finitely many families of cosets of real algebraic subgroups of G. The underlying families are extracted from X, independently of Γ. We also prove an equidistribution result in the case of curves.
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Strongly minimal groups in o-minimal structures

2021-05-28
Pantelis E. Eleftheriou , Assaf Hasson , Ya’acov Peterzil
We prove Zilber’s Trichotomy Conjecture for strongly minimal expansions of 2-dimensional groups, definable in o-minimal structures: Theorem. Let \mathcal{M} be an o-minimal expansion of a real closed field, \langle G;+\rangle … We prove Zilber’s Trichotomy Conjecture for strongly minimal expansions of 2-dimensional groups, definable in o-minimal structures: Theorem. Let \mathcal{M} be an o-minimal expansion of a real closed field, \langle G;+\rangle a 2-dimensional group definable in \mathcal{M} , and \mathcal{D}=\langle G;+,\ldots\rangle a strongly minimal structure, all of whose atomic relations are definable in \mathcal{M} . If \mathcal{D} is not locally modular, then an algebraically closed field K is interpretable in \mathcal{D} , and the group G , with all its induced \mathcal{D} -structure, is definably isomorphic in \mathcal{D} to an algebraic K -group with all its induced K -structure.
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Fields interpretable in $P$-minimal fields

2021-03-28
Yatir Halevi , Assaf Hasson , Ya’acov Peterzil
We prove that an infinite field interpretable in a $p$-adically closed field $K$ is definably isomorphic to a finite extension of $K$. The result remains true in any $P$-minimal field … We prove that an infinite field interpretable in a $p$-adically closed field $K$ is definably isomorphic to a finite extension of $K$. The result remains true in any $P$-minimal field where definable functions are generically differentiable.

Interpretable fields in real closed valued fields and some expansions

2021-01-01
Assaf Hasson , Ya’acov Peterzil
Let $\mathcal M=\langle K;O\rangle$ be a real closed valued field and let $k$ be its residue field. We prove that every interpretable field in $\mathcal M$ is definably isomorphic to … Let $\mathcal M=\langle K;O\rangle$ be a real closed valued field and let $k$ be its residue field. We prove that every interpretable field in $\mathcal M$ is definably isomorphic to either $K$, $K(\sqrt{-1})$, $k$, or $k(\sqrt{-1})$. The same result holds when $K$ is a model of $T$, for $T$ an o-minimal power bounded expansion of a real closed field, and $O$ is a $T$-convex subring. The proof is direct and does not make use of known results about elimination of imaginaries in valued fields.

Interpretable Fields in Various Valued Fields

2021-01-01
Yatir Halevi , Assaf Hasson , Ya’acov Peterzil
Let $\mathcal{K}=(K,v,\ldots)$ be a dp-minimal expansion of a non-trivially valued field of characteristic $0$ and $\mathcal{F}$ an infinite field interpretable in $\mathcal{K}$. Assume that $\mathcal{K}$ is one of the following: … Let $\mathcal{K}=(K,v,\ldots)$ be a dp-minimal expansion of a non-trivially valued field of characteristic $0$ and $\mathcal{F}$ an infinite field interpretable in $\mathcal{K}$. Assume that $\mathcal{K}$ is one of the following: (i) $V$-minimal, (ii) power bounded $T$-convex, or (iii) $P$-minimal (assuming additionally in (iii) generic differentiability of definable functions). Then $\mathcal{F}$ is definably isomorphic to a finite extension $K$ or, in cases (i) and (ii), its residue field. In particular, every infinite field interpretable in $\mathbb{Q}_p$ is definably isomorphic to a finite extension of $\mathbb{Q}_p$, answering a question of Pillay's. Using Johnson's work on dp-minimal fields and the machinery developed here, we conclude that if $\mathcal{K}$ is an infinite dp-minimal pure field then every field definable in $\mathcal{K}$ is definably isomorphic to a finite extension of $K$. The proof avoids elimination of imaginaries in $\mathcal{K}$ replacing it with a reduction of the problem to certain distinguished quotients of $K$.
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Fields interpretable in $P$-minimal fields

2021-01-01
Yatir Halevi , Assaf Hasson , Ya’acov Peterzil
We prove that an infinite field interpretable in a $p$-adically closed field $K$ is definably isomorphic to a finite extension of $K$. The result remains true in any $P$-minimal field … We prove that an infinite field interpretable in a $p$-adically closed field $K$ is definably isomorphic to a finite extension of $K$. The result remains true in any $P$-minimal field where definable functions are generically differentiable.
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Model-theoretic Elekes-Szabó for stable and o-minimal hypergraphs

2021-01-01
Artem Chernikov , Ya’acov Peterzil , Sergei Starchenko
A theorem of Elekes and Szab\'{o} recognizes algebraic groups among certain complex algebraic varieties with maximal size intersections with finite grids. We establish a generalization to relations of any arity … A theorem of Elekes and Szab\'{o} recognizes algebraic groups among certain complex algebraic varieties with maximal size intersections with finite grids. We establish a generalization to relations of any arity and dimension, definable in: 1) stable structures with distal expansions (includes algebraically and differentially closed fields of characteristic $0$); and 2) $o$-minimal expansions of groups. Our methods provide explicit bounds on the power saving exponent in the non-group case. Ingredients of the proof include: a higher arity generalization of the abelian group configuration theorem in stable structures, along with a purely combinatorial variant characterizing Latin hypercubes that arise from abelian groups; and Zarankiewicz-style bounds for hypergraphs definable in distal structures.
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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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Locally definable subgroups of semialgebraic groups

2019-10-07
Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil
We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let [Formula: … We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let [Formula: see text] be an abelian semialgebraic group over a real closed field [Formula: see text] and let [Formula: see text] be a semialgebraic subset of [Formula: see text]. Then the group generated by [Formula: see text] contains a generic set and, if connected, it is divisible. More generally, the same result holds when [Formula: see text] is definable in any o-minimal expansion of [Formula: see text] which is elementarily equivalent to [Formula: see text]. We observe that the above statement is equivalent to saying: there exists an [Formula: see text] such that [Formula: see text] is an approximate subgroup of [Formula: see text].
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Definable one-dimensional topologies in O-minimal structures

2019-05-31
Ya’acov Peterzil , Ayala Rosel
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A THEORY OF PAIRS FOR NON-VALUATIONAL STRUCTURES

2019-01-25
ELITZUR BAR-YEHUDA , Assaf Hasson , Ya’acov Peterzil
Given a weakly o-minimal structure $\mathcal M$ and its o-minimal completion $\bar {\mathcal M}$, we first associate to $\bar {\mathcal M}$ a canonical language and then prove that $Th(\mathcal M)$ … Given a weakly o-minimal structure $\mathcal M$ and its o-minimal completion $\bar {\mathcal M}$, we first associate to $\bar {\mathcal M}$ a canonical language and then prove that $Th(\mathcal M)$ determines $Th(\bar {\mathcal M})$. We then investigate the theory of the pair $\mathcal M^P=(\bar {\mathcal M};M)$ in the spirit of the theory of dense pairs of o-minimal structures, and prove, among other results, that it is near model complete, and every $\mathcal M^P$-definable open subset of $\bar M^n$ is already definable in $\bar {\mathcal M}$. We give an example of a weakly o-minimal structure which interprets $\bar {\mathcal M}$ and show that it is not elementarily equivalent to any reduct of an o-minimal trace.
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An o-minimalist view of the group configuration

2019-01-01
Ya’acov Peterzil
The group configuration in o-minimal structures gives rise, just like in the stable case, to a transitive action of a type-definable group on a partial type. Because $acl=dcl$ the o-minimal … The group configuration in o-minimal structures gives rise, just like in the stable case, to a transitive action of a type-definable group on a partial type. Because $acl=dcl$ the o-minimal proof is significantly simpler than Hrushovski's original argument. Several equivalent versions, which are more suitable to the o-minimal setting, are formulated, in functional language and also in terms of a certain $4$-ary relation. In addition, the following question is considered: Can every definably connected type-definable group be definably embedded into a definable group of the same dimension? Two simple cases with a positive answer are given.

Locally definable and approximate subgroups of semialgebraic groups

2018-12-27
Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil
We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic … We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and in particular is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $\Sigma_{i=1}^m(X-X)$ is an approximate subgroup of $G$.

Locally definable subgroups of semialgebraic groups

2018-12-27
Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil
We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic … We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and, if connected, it is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $\Sigma_{i=1}^m(X-X)$ is an approximate subgroup of $G$.

Strongly minimal groups in o-minimal structures

2018-10-03
Pantelis E. Eleftheriou , Assaf Hasson , Ya’acov Peterzil
We prove Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups, definable in o-minimal structures: Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a … We prove Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups, definable in o-minimal structures: Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a 2-dimensional group definable in M, and D = (G;+,...) a strongly minimal structure, all of whose atomic relations are definable in M. If D is not locally modular, then an algebraically closed field K is interpretable in D, and the group G, with all its induced D-structure, is definably isomorphic in D to an algebraic K-group with all its induced K-structure.

O-minimal flows on nilmanifolds

2018-09-14
Ya’acov Peterzil , Sergei Starchenko
Let $G$ be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of $\mathrm{UT}(n,\mathbb{R})$, and let $\Gamma$ be a lattice in $G$, with $\pi:G\to G/\Gamma$ the … Let $G$ be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of $\mathrm{UT}(n,\mathbb{R})$, and let $\Gamma$ be a lattice in $G$, with $\pi:G\to G/\Gamma$ the quotient map. For a semi-algebraic $X\subseteq G$, and more generally a definable set in an o-minimal structure on the real field, we consider the topological closure of $\pi(X)$ in the compact nilmanifold $G/\Gamma$. Our theorem describes $\mathrm{cl}(\pi(X))$ in terms of finitely many families of cosets of real algebraic subgroups of $G$. The underlying families are extracted from $X$, independently of $\Gamma$. We also prove an equidistribution result in the case of curves.

A note on o-minimal flows and the Ax–Lindemann–Weierstrass theorem for semi-abelian varieties over $\mathbb C$

2018-09-03
Ya’acov Peterzil , Sergei Starchenko
In this short note we present an elementary proof of eorem 1.2 from [UY2], and also the Ax–Lindemann–Weierstrass theorem for abelian and semi-abelian varieties. The proof uses ideas of Pila, … In this short note we present an elementary proof of eorem 1.2 from [UY2], and also the Ax–Lindemann–Weierstrass theorem for abelian and semi-abelian varieties. The proof uses ideas of Pila, Ullmo, Yafaev, Zannier (see, e.g., [PZ]) and is based on basic properties of sets definable in o-minimal structures. It does not use the Pila–Wilkie counting theorem.

Definable one dimensional topologies in o-minimal structures

2018-07-21
Ya’acov Peterzil , Ayala Rosel
We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable … We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable space (namely, a definable subset of $M^{n}$ with the induced subspace topology). One of the main results says that it is sufficient for $X$ to be regular and decompose into finitely many definably connected components.

Algebraic and o-minimal flows on complex and real tori

2018-06-05
Ya’acov Peterzil , Sergei Starchenko
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Locally definable subgroups of semialgebraic groups

2018-01-01
Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil
We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic … We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and, if connected, it is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $Σ_{i=1}^m(X-X)$ is an approximate subgroup of $G$.
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Strongly minimal groups in o-minimal structures

2018-01-01
Pantelis E. Eleftheriou , Assaf Hasson , Ya’acov Peterzil
We prove Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups, definable in o-minimal structures: Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a … We prove Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups, definable in o-minimal structures: Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a 2-dimensional group definable in M, and D = (G;+,...) a strongly minimal structure, all of whose atomic relations are definable in M. If D is not locally modular, then an algebraically closed field K is interpretable in D, and the group G, with all its induced D-structure, is definably isomorphic in D to an algebraic K-group with all its induced K-structure.

O-minimal flows on nilmanifolds

2018-01-01
Ya’acov Peterzil , Sergei Starchenko
Let $G$ be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of $\mathrm{UT}(n,\mathbb{R})$, and let $\Gamma$ be a lattice in $G$, with $\pi:G\to G/\Gamma$ the … Let $G$ be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of $\mathrm{UT}(n,\mathbb{R})$, and let $\Gamma$ be a lattice in $G$, with $\pi:G\to G/\Gamma$ the quotient map. For a semi-algebraic $X\subseteq G$, and more generally a definable set in an o-minimal structure on the real field, we consider the topological closure of $\pi(X)$ in the compact nilmanifold $G/\Gamma$. Our theorem describes $\mathrm{cl}(\pi(X))$ in terms of finitely many families of cosets of real algebraic subgroups of $G$. The underlying families are extracted from $X$, independently of $\Gamma$. We also prove an equidistribution result in the case of curves.
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Definable one dimensional topologies in o-minimal structures

2018-01-01
Ya’acov Peterzil , Ayala Rosel
We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable … We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable space (namely, a definable subset of $M^{n}$ with the induced subspace topology). One of the main results says that it is sufficient for $X$ to be regular and decompose into finitely many definably connected components.
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Euler characteristic of imaginaries in o‐minimal structures

2017-11-29
Sofya Kamenkovich , Ya’acov Peterzil
Abstract We define the notion of Euler characteristic for definable quotients in an arbitrary o‐minimal structure and prove some fundamental properties. Abstract We define the notion of Euler characteristic for definable quotients in an arbitrary o‐minimal structure and prove some fundamental properties.
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Topological groups, $\mu$-types and their stabilizers

2017-09-18
Ya’acov Peterzil , Sergei Starchenko
We consider an arbitrary topological group G definable in a structure \mathcal M , such that some basis for the topology of G consists of sets definable in \mathcal M … We consider an arbitrary topological group G definable in a structure \mathcal M , such that some basis for the topology of G consists of sets definable in \mathcal M . To each such group G we associate a compact G -space of partial types, S^{\mu}_G(M)=\{p_{\mu}\colon p\in S_G(M)\} which is the quotient of the usual type space S_G(M) by the relation of two types being "infinitesimally close to each other". In the o-minimal setting, if p is a definable type then it has a corresponding definable subgroup Stab ^{\mu}(p) , which is the stabilizer of p_{\mu} . This group is nontrivial when p is unbounded in the sense of \mathcal M ; in fact it is a torsion-free solvable group. Along the way, we analyze the general construction of S^{\mu}_G(M) and its connection to the Samuel compactification of topological groups.
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Adding Multiplication to an O-minimal Expansion of the Additive Group of Real Numbers

2017-03-29
Ya’acov Peterzil , Patrick Speissegger , Sergei Starchenko

Zil'ber's Trichotomy and o-minimal Structures

2017-03-02
Ya’acov Peterzil
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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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Algebraic and o-minimal flows on complex and real tori

2017-01-31
Ya’acov Peterzil , Sergei Starchenko
We consider the covering map $\pi:\mathbb{C}^n\to \mathbb{T}$ of a compact complex torus. Given an algebraic variety $X\subseteq \mathbb{C}^n$ we describe the topological closure of $\pi(X)$ in $\mathbb T$. We obtain … We consider the covering map $\pi:\mathbb{C}^n\to \mathbb{T}$ of a compact complex torus. Given an algebraic variety $X\subseteq \mathbb{C}^n$ we describe the topological closure of $\pi(X)$ in $\mathbb T$. We obtain a similar description when $\mathbb{T}$ is a real torus and $X\subseteq \mathbb{R}^n$ is a set definable in an o-minimal structure over the reals.

Algebraic and o-minimal flows on complex and real tori

2017-01-01
Ya’acov Peterzil , Sergei Starchenko
We consider the covering map $\pi:\mathbb{C}^n\to \mathbb{T}$ of a compact complex torus. Given an algebraic variety $X\subseteq \mathbb{C}^n$ we describe the topological closure of $\pi(X)$ in $\mathbb T$. We obtain … We consider the covering map $\pi:\mathbb{C}^n\to \mathbb{T}$ of a compact complex torus. Given an algebraic variety $X\subseteq \mathbb{C}^n$ we describe the topological closure of $\pi(X)$ in $\mathbb T$. We obtain a similar description when $\mathbb{T}$ is a real torus and $X\subseteq \mathbb{R}^n$ is a set definable in an o-minimal structure over the reals.

A note on o-minimal flows and the Ax--Lindemann--Weierstrass theorem for abelian varieties over $\mathbb{C}$

2016-01-01
Ya’acov Peterzil , Sergei Starchenko
In this short note we present an elementary proof of the Ax-Lindemann-Weierstrass theorem for abelian and semi-abelian varieties. The proof uses ideas of Pila, Ullmo, Yafaev, Zannier and is based … In this short note we present an elementary proof of the Ax-Lindemann-Weierstrass theorem for abelian and semi-abelian varieties. The proof uses ideas of Pila, Ullmo, Yafaev, Zannier and is based on basic properties of sets definable in o-minimal structures. It does not use the Pila-Wilkie counting theorem.

Topological groups, \mu-types and their stabilizers

2014-09-18
Ya’acov Peterzil , Sergei Starchenko
We consider an arbitrary topological group $G$ definable in a structure $\mathcal M$, such that some basis for the topology of $G$ consists of sets definable in $\mathcal M$. To … We consider an arbitrary topological group $G$ definable in a structure $\mathcal M$, such that some basis for the topology of $G$ consists of sets definable in $\mathcal M$. To each such group $G$ we associate a compact $G$-space of partial types $S^\mu_G(M)=\{p_\mu:p\in S_G(M)\}$ which is the quotient of the usual type space $S_G(M)$ by the relation of two types being infinitesimally close to each other. In the o-minimal setting, if $p$ is a definable type then it has a corresponding definable subgroup $Stab_\mu(p)$, which is the stabilizer of $p_\mu$. This group is nontrivial when $p$ is unbounded in the sense of $\mathcal M$; in fact it is a torsion-free solvable group. Along the way, we analyze the general construction of $S^\mu_G(M)$ and its connection to the Samuel compactification of topological groups.

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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Topological groups, μ-types and their stabilizers

2014-01-01
Ya’acov Peterzil , Sergei Starchenko
We consider an arbitrary topological group $G$ definable in a structure $\mathcal M$, such that some basis for the topology of $G$ consists of sets definable in $\mathcal M$. To … We consider an arbitrary topological group $G$ definable in a structure $\mathcal M$, such that some basis for the topology of $G$ consists of sets definable in $\mathcal M$. To each such group $G$ we associate a compact $G$-space of partial types $S^\mu_G(M)=\{p_\mu:p\in S_G(M)\}$ which is the quotient of the usual type space $S_G(M)$ by the relation of two types being "infinitesimally close to each other". In the o-minimal setting, if $p$ is a definable type then it has a corresponding definable subgroup $Stab_\mu(p)$, which is the stabilizer of $p_\mu$. This group is nontrivial when $p$ is unbounded in the sense of $\mathcal M$; in fact it is a torsion-free solvable group. Along the way, we analyze the general construction of $S^\mu_G(M)$ and its connection to the Samuel compactification of topological groups.
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Definability of restricted theta functions and families of abelian varieties

2013-03-15
Ya’acov Peterzil , Sergei Starchenko
We consider some classical maps from the theory of abelian varieties and their moduli spaces, and we prove their definability on restricted domains in the o-minimal structure Ran,exp. In particular, … We consider some classical maps from the theory of abelian varieties and their moduli spaces, and we prove their definability on restricted domains in the o-minimal structure Ran,exp. In particular, we prove that the projective embedding of the moduli space of the principally polarized abelian variety Sp(2g,Z)\Hg is definable in Ran,exp when restricted to Siegel's fundamental set Fg. We also prove the definability on appropriate domains of embeddings of families of abelian varieties into projective spaces.
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Lattices in Locally Definable Subgroups of 〈Rn,+〉

2013-01-01
Pantelis E. Eleftheriou , Ya’acov Peterzil
Let M be an o-minimal expansion of a real closed field R. We define the notion of a lattice in a locally definable group and then prove that every connected, … Let M be an o-minimal expansion of a real closed field R. We define the notion of a lattice in a locally definable group and then prove that every connected, definably generated subgroup of 〈Rn,+〉 contains a definable generic set and therefore admits a lattice.
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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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Coauthor Papers Together
Sergei Starchenko 34
Pantelis E. Eleftheriou 17
Assaf Hasson 17
Anand Pillay 16
Yatir Halevi 9
Ehud Hrushovski 7
Alf Onshuus 4
Elías Baro 4
Alessandro Berarducci 4
Margarita Otero 4
Françoise Point 3
Anthony L. Pillay 3
Ayala Rosel 3
Janak Ramakrishnan 3
Frank Olaf Wagner 2
Oleg Belegradek 2
Hind Abu Saleh 2
Artem Chernikov 2
Yoad Winter 1
David Marker 1
Martin Bays 1
Sofya Kamenkovich 1
Alon Altman 1
Charles Steinhorn 1
ELITZUR BAR-YEHUDA 1
L. M. Graña Drummond 1
Patrick Speissegger 1
Zoé Chatzidakis 1
James Loveys 1

On groups and fields definable in o-minimal structures

1988-09-01
Anand Pillay

Definable Compactness and Definable Subgroups of o-Minimal Groups

1999-06-01
Ya’acov Peterzil , Charles Steinhorn
The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an … The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

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.

DEFINABLY COMPACT ABELIAN GROUPS

2004-12-01
Mário J. Edmundo , Margarita Otero
Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the … Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the o-minimal fundamental group of G is isomorphic to ℤ n ; for each k&gt;0, the k-torsion subgroup of G is isomorphic to (ℤ/kℤ) n , and the o-minimal cohomology algebra over ℚ of G is isomorphic to the exterior algebra over ℚ with n generators of degree one.
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On Groups and Rings Definable In O-Minimal Expansions of Real Closed Fields

1996-01-01
Margarita Otero , Ya’acov Peterzil , Anand Pillay
Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in … Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in M. We show that every M–definable ring without zero divisors is definably isomorphic to R, R(√(−l)) or the ring of quaternions over R. One corollary is that no model of Texp is interpretable in a model of Tan.

Euler characteristic in semialgebraic and other o-minimal groups

1994-10-01
Adam Strzeboński

A trichotomy theorem for o-minimal structures

1998-11-01
Ya’acov Peterzil , Sergei Starchenko
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 … 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.

Geometric categories and o-minimal structures

1996-08-01
Lou van den Dries , Chris Miller

Groups definable in local fields and pseudo-finite fields

1994-02-01
Ehud Hrushovski , Anand Pillay

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.

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.

Definable homomorphisms of abelian groups in o-minimal structures

1999-11-01
Ya’acov Peterzil , Sergei Starchenko

Groups, measures, and the NIP

2007-02-02
Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating … We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.
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Expansions of algebraically closed fields in o-minimal structures

2001-09-01
Ya’acov Peterzil , Sergei Starchenko

Locally definable groups in o-minimal structures

2005-05-18
Mário J. Edmundo
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The universal covering homomorphism in o‐minimal expansions of groups

2007-09-26
Mário J. Edmundo , Pantelis E. Eleftheriou
Abstract Suppose G is a definably connected, definable group in an o‐minimal expansion of an ordered group. We show that the o‐minimal universal covering homomorphism $ \tilde p $ : … Abstract Suppose G is a definably connected, definable group in an o‐minimal expansion of an ordered group. We show that the o‐minimal universal covering homomorphism $ \tilde p $ : $ \tilde G $ → G is a locally definable covering homomorphism and π 1 ( G ) is isomorphic to the o‐minimal fundamental group π ( G ) of G defined using locally definable covering homomorphisms. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)
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TYPE-DEFINABILITY, COMPACT LIE GROUPS, AND o-MINIMALITY

2004-12-01
Anand Pillay
We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient … We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient structure is o-minimal. The gist is that in this o-minimal case, any definable group G should have a smallest type-definable subgroup of bounded index, and that the quotient, when equipped with the logic topology, should be a compact Lie group of the "right" dimension. I give positive answers to the conjectures in the special cases when G is 1-dimensional, and when G is definably simple.

Linear Groups Definable in o-Minimal Structures

2002-01-01
Ya’acov Peterzil , Anthony L. Pillay , Sergei Starchenko

Some model theory of compact Lie groups

1991-01-01
Ali Nesin , Anand Pillay
We consider questions of first order definability in a compact Lie group <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>. Our main result is that … We consider questions of first order definability in a compact Lie group <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>. Our main result is that if such <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 simple (and centerless) then the Lie group structure 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 first order definable from the abstract group structure. Along the way we also show (i) if <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 non-Abelian and connected then a copy of the field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is interpretable. in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper G comma dot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(G, \cdot )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and (ii) <italic>any</italic> "<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional" field interpretable in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-struck upper R comma plus comma dot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {R}, +, \cdot )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>definably</italic> (i.e., semialgebraically) isomorphic to the ground field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
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Linear O-minimal structures

1993-02-01
James Loveys , Ya’acov Peterzil

Definable sets in ordered structures. II

1986-01-01
Julia F. Knight , Anand Pillay , Charles Steinhorn
It is proved that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> … It is proved that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (in which the underlying order is dense) is strongly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal (namely, every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> elementarily equivalent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</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="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal). It is simultaneously proved that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</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="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal, then every definable set of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-tuples of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has finitely many "definably connected components."

TAME TOPOLOGY AND O-MINIMAL STRUCTURES (London Mathematical Society Lecture Note Series 248)

2000-05-01
A. J. Wilkie
By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998). By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998).

Constructing a group-interval in o-minimal structures

1994-06-01
Ya’acov Peterzil

On the real exponential field with restricted analytic functions

1994-02-01
Lou van den Dries , Chris Miller

Intersection theory for o-minimal manifolds

2001-01-01
Alessandro Berarducci , Margarita Otero

Forking and independence in o-minimal theories

2004-03-01
Alfred Dolich
In the following we try to answer a simple question, “what does forking look like in an o-minimal theory”, or more generally, “what kinds of notions of independence with what … In the following we try to answer a simple question, “what does forking look like in an o-minimal theory”, or more generally, “what kinds of notions of independence with what kinds of properties are admissible in an o-minimal theory?” The motivation of these question begin with the study of simple theories and generalizations of simple theories. In [3] Kim and Pillay prove that the class of simple theories may be described exactly as those theories bearing a notion of independence satisfying various axioms. Thus it is natural to ask, if we weaken the assumptions as to which axioms must hold, what kind of theories do we get? Another source of motivation, also stemming from the study of simple theories, comes from the work of Shelah in [8] and [7]. Here Shelah addresses a “classification” type problem for class of models of a theory, showing that a theory will have the appropriate “structure” type property if one can construct a partially ordered set, satisfying various properties, of models of the theory. Using this criterion Shelah shows that the class of simple theories has this “structure” property, yet also that several non-simple examples do as well (though it should be pointed out that o-minimal theories can not be among these since any theory with the strict order property will have the corresponding “non-structure” property [8]). Thus one is lead to ask, what are the non-simple theories meeting this criterion, and one is once again led to study the types of independence relation a theory might bear. Finally, Shelah in [6] provides some possible definitions of what axioms for a notion of independence one should possibly look for in order to hope that theories bearing such a notion of independence should be amenable closer analysis. In studying all of the above mentioned situations it readily becomes clear that dividing and forking play a central role in all of them, even though we are no longer dealing with the simple case where we know that dividing and forking are very well behaved. All of these considerations lead one to look for classes of non-simple theories of which something is known where one can construct interesting notions of independence and consequently also say something about the nature of forking and dividing in these contexts. Given this one is naturally lead to one of the most well behaved classes of non-simple theories, namely the o-minimal theories.

Groups definable in ordered vector spaces over ordered division rings

2007-12-01
Pantelis E. Eleftheriou , Sergei Starchenko
Abstract Let M = 〈 M , +, &lt;, 0, {λ} λЄ D 〉 be an ordered vector space over an ordered division ring D , and G = 〈 … Abstract Let M = 〈 M , +, &lt;, 0, {λ} λЄ D 〉 be an ordered vector space over an ordered division ring D , and G = 〈 G , ⊕, e G 〉 an n -dimensional group definable in M . We show that if G is definably compact and definably connected with respect to the t -topology, then it is definably isomorphic to a ‘definable quotient group’ U/L , for some convex V -definable subgroup U of 〈 M n , +〉 and a lattice L of rank n . As two consequences, we derive Pillay's conjecture for a saturated M as above and we show that the o-minimal fundamental group of G is isomorphic to L .

A descending chain condition for groups definable in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>o</mml:mi></mml:math>-minimal structures

2005-03-22
Alessandro Berarducci , Margarita Otero , Ya’acov Peterzil +1 more

Generic sets in definably compact groups

2007-01-01
Ya’acov Peterzil , Anand Pillay
A subset $X$ of a group $G$ is called <em>left generic</em> if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably … A subset $X$ of a group $G$ is called <em>left generic</em> if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably compact group in an o-minimal structure and a definable $X\subseteq G$ is not right generic then i
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Type-definable and invariant groups in o-minimal structures

2007-03-01
Jana Maříková
Abstract Let M be a big o-minimal structure and G a type-definable group in M n . We show that G is a type-definable subset of a definable manifold in … Abstract Let M be a big o-minimal structure and G a type-definable group in M n . We show that G is a type-definable subset of a definable manifold in M n 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 M k with the topology induced by M k . Part of this result holds for the wider class of so-called invariant groups: each invariant group G in M n has a unique topology making it a topological group and inducing the same topology on a large invariant subset of the group as M n .

On fields definable inQ p

1989-03-01
Anand Pillay

Solvable groups definable in o-minimal structures

2003-10-09
Mário J. Edmundo
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An additive measure in o-minimal expansions of fields

2004-12-01
Alessandro Berarducci
Journal Article An additive measure in o-minimal expansions of fields Get access Alessandro Berarducci, Alessandro Berarducci Search for other works by this author on: Oxford Academic Google Scholar Margarita Otero … Journal Article An additive measure in o-minimal expansions of fields Get access Alessandro Berarducci, Alessandro Berarducci Search for other works by this author on: Oxford Academic Google Scholar Margarita Otero Margarita Otero Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 55, Issue 4, December 2004, Pages 411–419, https://doi.org/10.1093/qmath/hah010 Published: 01 December 2004

On central extensions and definably compact groups in o-minimal structures

2010-11-17
Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay
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O-minimal spectra, infinitesimal subgroups and cohomology

2007-12-01
Alessandro Berarducci
Abstract By recent work on some conjectures of Pillay, each definably compact group G in a saturated o-minimal expansion of an ordered field has a normal “infinitesimal subgroup” G 00 … Abstract By recent work on some conjectures of Pillay, each definably compact group G in a saturated o-minimal expansion of an ordered field has a normal “infinitesimal subgroup” G 00 such that the quotient G/G 00 , equipped with the “logic topology”, is a compact (real) Lie group. Our first result is that the functor G ↦ G/G 00 sends exact sequences of definably compact groups into exact sequences of Lie groups. We then study the connections between the Lie group G/G 00 and the o-minimal spectrum of G . We prove that G/G 00 is a topological quotient of . We thus obtain a natural homomorphism Ψ* from the cohomology of G/G 00 to the (Čech-)cohomology of . We show that if G 00 satisfies a suitable contractibility conjecture then is acyclic in Čech cohomology and Ψ is an isomorphism. Finally we prove the conjecture in some special cases.
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Geometric Stability Theory

1996-09-12
Anand Pillay
Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine … Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine structure of models of stable theories, using the geometry of forking; this often achieves global results relevant to classification theory. Topics range from Zilber-Cherlin classification of infinite locally finite homogenous geometries, to regular types, their geometries, and their role in superstable theories. The structure and existence of definable groups is featured prominently, as is work by Hrushovski. The book is unique in the range and depth of material covered and will be invaluable to anyone interested in modern model theory.

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

1996-01-01
Andrew O.M. Wilkie
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A growth dichotomy for o-minimal expansions of ordered groups

1998-01-01
Chris Miller , Sergei Starchenko
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an o-minimal expansion of a divisible ordered abelian group … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an o-minimal expansion of a divisible ordered abelian group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper R comma greater-than comma plus comma 0 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(R,&gt;,+,0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with a distinguished positive element <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Then the following dichotomy holds: Either there is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable binary operation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dot"> <mml:semantics> <mml:mo>⋅</mml:mo> <mml:annotation encoding="application/x-tex">\cdot</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper R comma greater-than comma plus comma dot comma 0 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(R,&gt;,+,\cdot ,0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an ordered real closed field; or, for every definable function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon upper R right-arrow upper R"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f:R\to R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there exists a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda element-of StartSet 0 EndSet union upper A u t left-parenthesis upper R comma plus right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>∪</mml:mo> <mml:mi>Aut</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda \in \{0\}\cup \operatorname {Aut}(R,+)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript x right-arrow plus normal infinity Endscripts left-bracket f left-parenthesis x right-parenthesis minus lamda left-parenthesis x right-parenthesis right-bracket element-of upper R"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:munder> <mml:mo stretchy="false">[</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>−</mml:mo> <mml:mi>λ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\lim _{x\to +\infty }[f(x)-\lambda (x)]\in R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper M colon equals left-parenthesis upper M comma greater-than comma ellipsis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">M</mml:mi> </mml:mrow> <mml:mo>:=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {M}:=(M,&gt;,\dots )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable groups with underlying set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
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Complex analytic geometry and analytic-geometric categories

2009-01-01
Ya’acov Peterzil , Sergei Starchenko
The notion of an analytic-geometric category was introduced by v. d. Dries and Miller in [Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. … The notion of an analytic-geometric category was introduced by v. d. Dries and Miller in [Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540.]. It is a category of subsets of real analytic manifolds which extends the category of subanalytic sets. This paper discusses connections between the subanalytic category, or more generally analytic-geometric categories, and complex analytic geometry. The questions are of the following nature: We start with a subset A of a complex analytic manifold M and assume that A is an object of an analytic-geometric category (by viewing M as a real analytic manifold of double dimension). We then formulate conditions under which A, its closure or its image under a holomorphic map is a complex analytic set.

Locally definable homotopy

2009-05-08
Elías Baro , Margarita Otero
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Additive reducts of real closed fields

1992-03-01
David Marker , Ya’acov Peterzil , Anand Pillay
In [MP] Marker and Pillay showed that if X ⊂ C n is constructible but ( C , +, X ) is not locally modular, then multiplication is definable in … In [MP] Marker and Pillay showed that if X ⊂ C n is constructible but ( C , +, X ) is not locally modular, then multiplication is definable in the structure ( C , +, X ). That result extended earlier results of Martin [M] and Rabinovich and Zil'ber [RZ]. Here we will examine additive reducts of R and Q p . Definition. A subset X of R n is called semialgebraic if it is definable in the structure ( R , +,·). A subset X of R n is called semilinear if it is definable in the structure ( R , +, &lt;,λ r ) r∈ b , where λ r is the function x ↦ rx [scalar multiplication by r ]. Every semilinear set is a Boolean combination of sets of the form { : p ( ) = 0} and { : q ( ) &gt; 0}, where p ( ) and q ( ) are linear polynomials. Van den Dries asked the following question: if X is semialgebraic but not semilinear, can we define multiplication in ( R , +, &lt;, X )? This was answered negatively by Pillay, Scowcroft and Steinhorn. Theorem 1.1 [PSS]. Suppose X ⊂ R n is semialgebraic and X ⊂ I n for some bounded interval I. Then multiplication is not definable in ( R , +, &lt;, X ,λ r ) r ∈ R . In particular if X = · ∣ [0, l ] 2 , the graph of multiplication restricted to the unit interval, then X is not semilinear so we have a negative answer to van den Dries' question. Peterzil showed that this is the only restriction.

Strongly minimal expansions of algebraically closed fields

1992-10-01
Ehud Hrushovski

A new strongly minimal set

1993-07-01
Ehud Hrushovski

Valued fields, metastable groups

2019-07-23
Ehud Hrushovski , Silvain Rideau
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Local analysis for semi-bounded groups

2012-01-01
Pantelis E. Eleftheriou
An o-minimal expansion $\mathcal{M}=\langle M, <, +, 0, \dots\rangle$ of an ordered group is called <i>semi-bounded</i> if it does not expand a real closed field. Possibly, it defines a real … An o-minimal expansion $\mathcal{M}=\langle M, <, +, 0, \dots\rangle$ of an ordered group is called <i>semi-bounded</i> if it does not expand a real closed field. Possibly, it defines a real closed field with bounded domain $I\subseteq M$. Let us call
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Imaginaries in real closed valued fields

2005-06-14
T. Mellor

Some remarks on definable equivalence relations in O-minimal structures

1986-09-01
Anand Pillay
Let M be an O-minimal structure. We use our understanding, acquired in [KPS], of the structure of definable sets of n -tuples in M , to study definable (in M … Let M be an O-minimal structure. We use our understanding, acquired in [KPS], of the structure of definable sets of n -tuples in M , to study definable (in M ) equivalence relations on M n . In particular, we show that if E is an A -definable equivalence relation on M n ( A ⊂ M ) then E has only finitely many classes with nonempty interior in M n , each such class being moreover also A -definable. As a consequence, we are able to give some conditions under which an O -minimal theory T eliminates imaginaries (in the sense of Poizat [P]). If L is a first order language and M an L -structure, then by a definable set in M , we mean something of the form X ⊂ M n , n ≥ 1, where X = {( a 1 …, a n ) ∈ M n : M ⊨ ϕ (ā)} for some formula ∈ L ( M ). (Here L ( M ) means L together with names for the elements of M .) If the parameters from come from a subset A of M , we say that X is A-definable . M is said to be O-minimal if M = ( M , &lt;,…), where &lt; is a dense linear order with no first or last element, and every definable set X ⊂ M is a finite union of points, and intervals ( a, b ) (where a, b ∈ M ∪ {± ∞}). (This notion is as in [PS] except here we demand the underlying order be dense.) The complete theory T is said to be O-minimal if every model of T is O-minimal. (Note that in [KPS] it is proved that if M is O-minimal, then T = Th( M ) is O-minimal.) In the remainder of this section and in §2, M will denote a fixed but arbitrary O-minimal structure. A,B,C,… will denote subsets of M .

The Elementary Theory of Restricted Analytic Fields with Exponentiation

1994-07-01
Lou van den Dries , Angus Macintyre , David Marker
numbers with exponentiation is model complete. When we combine this with Hovanskii's finiteness theorem [9], it follows that the real exponential field is o-minimal. In o-minimal expansions of the real … numbers with exponentiation is model complete. When we combine this with Hovanskii's finiteness theorem [9], it follows that the real exponential field is o-minimal. In o-minimal expansions of the real field the definable subsets of R' share many of the nice structural properties of semialgebraic sets. For example, definable subsets have only finitely many connected components, definable sets can be stratified and triangulated, and continuous definable maps are piecewise trivial (see [5]). In this paper we will prove a quantifier elimination result for the real field augmented by exponentiation and all restricted analytic functions, and use this result to obtain o-minimality. We were led to this while studying work of Ressayre [13] and several of his ideas emerge here in simplified form. However, our treatment is formally independent of the results of [16], [17], [9], and [13].

Weakly o-minimal structures and real closed fields

2000-04-13
Dugald Macpherson , David Marker , Charles Steinhorn
A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures … A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that every weakly o-minimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.
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A survey on groups definable in o-minimal structures

2008-05-22
Margarita Otero
Groups definable in o-minimal structures have been studied for the last twenty years. The starting point of all the development is Pillay's theorem that a definable group is a definable … Groups definable in o-minimal structures have been studied for the last twenty years. The starting point of all the development is Pillay's theorem that a definable group is a definable group manifold (see Section 2). This implies that when the group has the order type of the reals, we have a real Lie group. The main lines of research in the subject so far have been the following:
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