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An exposition on the supersimplicity of certain expansions of the additive group of the integers

Amador Martin-Pizarro, Daniel Palacín

TL;DR

This note develops a general model-theoretic framework for proving that certain expansions of the additive group $\mathbb{Z}$ by a unary predicate are supersimple of rank $1$. By reducing formulas to basic $\mathcal{L}_P^+$-formulae and leveraging an additively random predicate, the authors establish quantifier elimination up to these basic formulas and show that algebraic closure coincides with the pure closure of the generated subgroup. They provide a back-and-forth argument (via a Separation Lemma) to prove supersimplicity and analyze dividing to ensure non-algebraic formulas do not divide, thereby encompassing known results for primes (under Dickson's conjecture) and square-free predicates, and relating to Chatzidakis–Pillay's generic predicate framework. The work offers a unifying, model-theoretic approach to expansions of $(\mathbb{Z},+)$ with number-theoretic predicates and suggests avenues for extending the framework to additional predicates (e.g., CP98 variants).

Abstract

In this short note, we present a self-contained exposition of the supersimplicity of certain expansions of the additive group of the integers, such as adding a generic predicate (due to Chatzidakis and Pillay), a predicate for the square-free integers (due to Bhardwaj and Tran) or a predicate for the prime integers (due to Kaplan and Shelah, assuming Dickson's conjecture).

An exposition on the supersimplicity of certain expansions of the additive group of the integers

TL;DR

This note develops a general model-theoretic framework for proving that certain expansions of the additive group by a unary predicate are supersimple of rank . By reducing formulas to basic -formulae and leveraging an additively random predicate, the authors establish quantifier elimination up to these basic formulas and show that algebraic closure coincides with the pure closure of the generated subgroup. They provide a back-and-forth argument (via a Separation Lemma) to prove supersimplicity and analyze dividing to ensure non-algebraic formulas do not divide, thereby encompassing known results for primes (under Dickson's conjecture) and square-free predicates, and relating to Chatzidakis–Pillay's generic predicate framework. The work offers a unifying, model-theoretic approach to expansions of with number-theoretic predicates and suggests avenues for extending the framework to additional predicates (e.g., CP98 variants).

Abstract

In this short note, we present a self-contained exposition of the supersimplicity of certain expansions of the additive group of the integers, such as adding a generic predicate (due to Chatzidakis and Pillay), a predicate for the square-free integers (due to Bhardwaj and Tran) or a predicate for the prime integers (due to Kaplan and Shelah, assuming Dickson's conjecture).

Paper Structure

This paper contains 3 sections, 4 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. The setting
  3. Quantifier elimination and supersimplicity

Key Result

Lemma 2.6

(cf. KS17) Assume that the substructure $A$ is pure in $\mathbb{U}$ and consider a basic $\mathcal{L}_P^+$-formula $\varphi(x, \bar{a})$ in $\operatorname{tp}(b/A)$, where $b$ is a singleton. Then there are a primary $\mathcal{L}_P$-formula $\psi(x, \bar{a}_1)$ over $A$, a non-constant $\mathcal{L

Theorems & Definitions (15)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Lemma 2.8
  • proof
  • ...and 5 more