Definable groups in models of Presburger arithmetic. (English) Zbl 1481.03025
Summary: 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 abelian group definable in a model \((\mathbb{Z}, +, <)\) of Presburger Arithmetic is definably isomorphic to \((\mathbb{Z},+)^n\) mod out by a lattice.
Theorem 1. Every group definable in a model of Presburger Arithmetic is abelian-by-finite.
Theorem 2. Every bounded abelian group definable in a model \((\mathbb{Z}, +, <)\) of Presburger Arithmetic is definably isomorphic to \((\mathbb{Z},+)^n\) mod out by a lattice.
MSC:
| 03C64 | Model theory of ordered structures; o-minimality |
| 03C60 | Model-theoretic algebra |
| 03C45 | Classification theory, stability, and related concepts in model theory |
| 03C10 | Quantifier elimination, model completeness, and related topics |
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