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

2016-12-29

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.

Definable groups in models of Presburger Arithmetic

2020-03-09

Alf Onshuus , Mariana Vicaría

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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

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 $(\mathbb{Z},+,<,0)$.

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

2016-01-01

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

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.

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.

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).

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.

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

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

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