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Retractors in local positive logic

Arturo Rodriguez Fanlo, Ori Segel

TL;DR

This work extends Hrushovski–Lascar style analysis to local positive logic by developing a robust theory of type spaces and saturation (retractors) in a multi-sorted, locality-aware setting. It introduces local positive types and bounded satisfiability, proves a Universality Lemma for local positive topologies, and establishes that local positive compactness is equivalent to the existence of retractors, while also analyzing automorphism groups of retractors and their topological/dynamical properties. The results generalize several non-local positive-logic phenomena to the local, multi-sorted framework and lay groundwork for applications to definability patterns and rough/subgroup-type structures. Together, these contributions provide a cohesive toolkit for studying saturation, completeness, and symmetry in local positive logics.

Abstract

We study type spaces and saturation for local positive logic.

Retractors in local positive logic

TL;DR

This work extends Hrushovski–Lascar style analysis to local positive logic by developing a robust theory of type spaces and saturation (retractors) in a multi-sorted, locality-aware setting. It introduces local positive types and bounded satisfiability, proves a Universality Lemma for local positive topologies, and establishes that local positive compactness is equivalent to the existence of retractors, while also analyzing automorphism groups of retractors and their topological/dynamical properties. The results generalize several non-local positive-logic phenomena to the local, multi-sorted framework and lay groundwork for applications to definability patterns and rough/subgroup-type structures. Together, these contributions provide a cohesive toolkit for studying saturation, completeness, and symmetry in local positive logics.

Abstract

We study type spaces and saturation for local positive logic.
Paper Structure (15 sections, 60 theorems, 18 equations)

This paper contains 15 sections, 60 theorems, 18 equations.

Table of Contents

  1. Preliminaries in local positive logic
  2. Local languages
  3. Local structures
  4. Positive closedness
  5. Completeness
  6. Denials
  7. Local positive types
  8. Local positive types over parameters
  9. Spaces of types
  10. Completeness via types
  11. Retractors
  12. Local positive logic topologies
  13. Local positive compactness
  14. Local retractors and non-local universal models
  15. Automorphisms of retractors

Key Result

Lemma 2.3

Let $\Gamma\subseteq{\mathrm{LFor}}^x_+({\mathtt{L}})$ be a subset of . Then, $\Gamma$ is a partial of ${\mathtt{T}}$ if and only if there are $M\models^{\mathrm{pc}}_{\mathtt{L}}{\mathtt{T}}$ and $a\in M^x$ such that $M\models_{\mathtt{L}}\Gamma(a)$.

Theorems & Definitions (153)

  • Remark 1.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Example 2.6
  • Corollary 2.7
  • ...and 143 more