Definable groups in models of Presburger Arithmetic and G^{00}

Authors

Alf Onshuus, Mariana Vicaría

Type: Preprint

Publication Date: 2016-12-29

Citations: 0

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Abstract

This paper is devoted to understand groups definable in Presburger arithmetic. We prove the following theorems: Theorem 1. Every group definable in a model of Presburger Arithmetic is abelian-by-finite. Theorem 2. Every bounded group definable in a model (Z,+,<) of Presburger Arithmetic is definably isomorphic to (Z, +)^{n} mod out by a lattice.

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arXiv (Cornell University)
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Definable groups in models of Presburger Arithmetic and G^{00}

2016-01-01

Alf Onshuus , Mariana Vicaría

This paper is devoted to understand groups definable in Presburger arithmetic. We prove the following theorems: Theorem 1. Every group definable in a model of Presburger Arithmetic is abelian-by-finite. Theorem … This paper is devoted to understand groups definable in Presburger arithmetic. We prove the following theorems: Theorem 1. Every group definable in a model of Presburger Arithmetic is abelian-by-finite. Theorem 2. Every bounded group definable in a model (Z,+,<) of Presburger Arithmetic is definably isomorphic to (Z, +)^{n} mod out by a lattice.

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Definable groups in models of Presburger Arithmetic

2020-03-09

Alf Onshuus , Mariana Vicaría

Groups definable in Presburger arithmetic

2024-08-10

Juan Pablo Acosta López

Here we give a complete list of the groups definable in Presburger arithmetic up to a finite index subgroup. Here we give a complete list of the groups definable in Presburger arithmetic up to a finite index subgroup.

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Linear Orders in Presburger Arithmetic

2022-01-01

Fedor Pakhomov , Alexander Zapryagaev

We prove that any linear order definable in the standard model (Z, <, +) of Presburger arithmetic is (Z, <, +)-definably embeddable into the lexicographic ordering on Z^n, for some … We prove that any linear order definable in the standard model (Z, <, +) of Presburger arithmetic is (Z, <, +)-definably embeddable into the lexicographic ordering on Z^n, for some n.

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Groups definable in Presburger arithmetic

2019-01-01

Juan Pablo Acosta López

We determine all groups definable in Presburger arithmetic, up to a finite index subgroup. We determine all groups definable in Presburger arithmetic, up to a finite index subgroup.

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Groups definable in Presburger arithmetic

2019-03-31

Juan Pablo Acosta López

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Definable sets up to definable bijections in Presburger groups

2018-01-04

Raf Cluckers , Immanuel Halupczok

We entirely classify definable sets up to definable bijections in Z-groups, where the language is the one of ordered abelian groups. From this, we deduce, among others, a classification of … We entirely classify definable sets up to definable bijections in Z-groups, where the language is the one of ordered abelian groups. From this, we deduce, among others, a classification of definable families of bounded definable sets.

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There are no intermediate structures between the group of integers and Presburger arithmetic

2016-03-01

Gabriel Conant

We show that if a first-order structure $\mathcal{M}$, with universe $\mathbb{Z}$, is an expansion of $(\mathbb{Z},+,0)$ and a reduct of $(\mathbb{Z},+,<,0)$, then $\mathcal{M}$ must be interdefinable with $(\mathbb{Z},+,0)$ or $(\mathbb{Z},+,<,0)$. We show that if a first-order structure $\mathcal{M}$, with universe $\mathbb{Z}$, is an expansion of $(\mathbb{Z},+,0)$ and a reduct of $(\mathbb{Z},+,<,0)$, then $\mathcal{M}$ must be interdefinable with $(\mathbb{Z},+,0)$ or $(\mathbb{Z},+,<,0)$.

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There are no intermediate structures between the group of integers and Presburger arithmetic

2016-01-01

Gabriel Conant

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THERE ARE NO INTERMEDIATE STRUCTURES BETWEEN THE GROUP OF INTEGERS AND PRESBURGER ARITHMETIC

2018-03-01

Gabriel Conant

Abstract We show that if a first-order structure ${\cal M}$ , with universe ℤ, is an expansion of (ℤ,+,0) and a reduct of (ℤ,+,<,0), then ${\cal M}$ must be interdefinable … Abstract We show that if a first-order structure ${\cal M}$ , with universe ℤ, is an expansion of (ℤ,+,0) and a reduct of (ℤ,+,<,0), then ${\cal M}$ must be interdefinable with (ℤ ,+,0) or (ℤ ,+,<,0).

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Presburger sets and p-minimal fields

2003-03-01

Raf Cluckers

Abstract We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full … Abstract We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable bijection. We also exhibit a tight connection between the definable sets in an arbitrary p-minimal field and Presburger sets in its value group. We give a negative result about expansions of Presburger structures and prove uniform elimination of imaginaries for Presburger structures within the Presburger language.

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Presburger sets and p-minimal fields

2002-01-01

Raf Cluckers

We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification … We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable bijection. We also exhibit a tight connection between the definable sets in an arbitrary p-minimal field and Presburger sets in its value group. We give a negative result about expansions of Presburger structures and prove uniform elimination of imaginaries for Presburger structures within the Presburger language.

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A Characterisation of the Relations Definable in Presburger Arithmetic

2008-04-29

Mathias Barra

DEFINABLE SETS IN WEAK PRESBURGER ARITHMETIC

2007-09-01

Christian Choffrut , Achille Frigeri

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Expansion of Presburger arithmetic with the exchange property

2018-01-01

Nathanaël Mariaule

Let $G$ be a model of Presburger arithmetic. Let $\mathcal{L}$ be an expansion of the language of Presburger $\mathcal{L}_{Pres}$. In this paper we prove that the $\mathcal{L}$-theory of $G$ is … Let $G$ be a model of Presburger arithmetic. Let $\mathcal{L}$ be an expansion of the language of Presburger $\mathcal{L}_{Pres}$. In this paper we prove that the $\mathcal{L}$-theory of $G$ is $\mathcal{L}_{Pres}$-minimal iff it has the exchange property and any bounded definable set has a maximum.

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Model theory of finite-by-Presburger Abelian groups and finite extensions of $p$-adic fields

2016-01-01

Jamshid Derakhshan , Angus Macintyre

We define a class of pre-ordered abelian groups that we call finite-by-Presburger groups, and prove that their theory is model-complete. We show that certain quotients of the multiplicative group of … We define a class of pre-ordered abelian groups that we call finite-by-Presburger groups, and prove that their theory is model-complete. We show that certain quotients of the multiplicative group of a local field of characteristic zero are finite-by-Presburger and interpret the higher residue rings of the local field. We apply these results to give a new proof of the model completeness in the ring language of a local field of characteristic zero (a result that follows also from work of Prestel-Roquette).

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Model theory of the field of $p$-adic numbers expanded by a multiplicative subgroup

2018-01-01

Nathanaël Mariaule

Let $G$ be a multiplicative subgroup of $\mathbb{Q}_p$. In this paper, we describe the theory of the pair $(\mathbb{Q}_p, G)$ under the condition that $G$ satisfies Mann property and is … Let $G$ be a multiplicative subgroup of $\mathbb{Q}_p$. In this paper, we describe the theory of the pair $(\mathbb{Q}_p, G)$ under the condition that $G$ satisfies Mann property and is small as subset of a first-order structure. First, we give an axiomatisation of the first-order theory of this structure. This includes an axiomatisation of the theory of the group $G$ as valued group (with the valuation induced on $G$ by the $p$-adic valuation). If the subgroups $G^{[n]}$ of $G$ have finite index for all $n$, we describe the definable sets in this theory and prove that it is NIP. Finally, we extend some of our results to the subanalytic setting.

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Expansions of Presburger arithmetic with the exchange property

2021-10-14

Nathanaël Mariaule

Abstract Let G be a model of Presburger arithmetic. Let be an expansion of the language of Presburger . In this paper, we prove that the ‐theory of G is … Abstract Let G be a model of Presburger arithmetic. Let be an expansion of the language of Presburger . In this paper, we prove that the ‐theory of G is ‐minimal iff it has the exchange property and is definably complete (i.e., any bounded definable set has a maximum). If the ‐theory of G has the exchange property but is not definably complete, there is a proper definable convex subgroup H . Assuming that the induced theories on H and are definable complete and o ‐minimal respectively, we prove that any definable set of G is ‐definable.

Stable reducts of elementary extensions of Presburger arithmetic

2024-12-13

Eran Alouf , Antongiulio Fornasiero , Itay Kaplan

Suppose $N$ is elementarily equivalent to an archimedean ordered abelian group $(G,+,<)$ with small quotients (for all $1 \leq n < \omega$, $[G: nG]$ is finite). Then every stable reduct … Suppose $N$ is elementarily equivalent to an archimedean ordered abelian group $(G,+,<)$ with small quotients (for all $1 \leq n < \omega$, $[G: nG]$ is finite). Then every stable reduct of $N$ which expands $(G,+)$ (equivalently every reduct that does not add new unary definable sets) is interdefinable with $(G,+)$. This extends previous results on stable reducts of $(\mathbb{Z}, +, <)$ to (stable) reducts of elementary extensions of $\mathbb{Z}$. In particular this holds for $G = \mathbb{Z}$ and $G = \mathbb{Q}$. As a result we answer a question of Conant from 2018.

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Quantifier elimination of the products of ordered abelian groups(Study of definability in nonstandard models of arithmetic)

2006-02-01

Hiroshi Tanaka , Hirokazu Yokoyama

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.

Quasi-o-minimal structures

2000-09-01

Oleg Belegradek , Ya’acov Peterzil , Frank Olaf Wagner

Abstract A structure ( M , <, …) 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 , <, …) 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.

Forking in VC-minimal theories

2012-11-02

Sarah Cotter , Sergei Starchenko

Abstract We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable … Abstract We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable over M , generalizing a result of Dolich on o-minimal theories in [4].

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The Banach-Tarski paradox

1985-01-01

Stanley Wagon

The Banach—Tarski paradox is one of the most shocking results of mathematics. In this chapter we show how tilings of the hyperbolic plane can help us visualize the paradox. The … The Banach—Tarski paradox is one of the most shocking results of mathematics. In this chapter we show how tilings of the hyperbolic plane can help us visualize the paradox. The images shown here display three congruent subsets of the hyperbolic plane. In the left image, the congruence is evident. The right image changes the viewpoint a little and changes the green to a blue shade; it is evident that the red set is congruent to its complement. Thus these sets are, simultaneously, one half and one third of the hyperbolic plane.

Characterizing rosy theories

2007-09-01

Clifton Ealy , Alf Onshuus

Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences … Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.

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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On NIP and invariant measures

2011-05-13

Ehud Hrushovski , Anand Pillay

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are … We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = \mathrm{tp}(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over \mathrm{bdd}(A) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G^{000} = G^{00} for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o -minimal expansions of real closed fields.

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