Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Notes on ultrafilter extensions of almost bounded structures

Zalán Molnár

TL;DR

The paper extends the study of ultrafilter extensions from bounded to almost bounded relational structures with a single binary relation, showing that key model-theoretic properties—such as being an elementary substructure and lifting elementary embeddings—continue to hold under suitable conditions. By decomposing the edge relation into parts and introducing a transfer framework, it demonstrates that the ultrafilter extension $oldsymbol{A}^ ext{ue}$ can align closely with ultrapowers $oldsymbol{A}^*$, particularly in the countable case where isomorphism is attainable for regular ultrapowers. It further analyzes the modal logic implications, showing that almost boundedness can yield $oldsymbol{A}^ ext{ue}$ and ultrapowers sharing modal theories under specific hypotheses, while also admitting cases where their modal theories diverge. The results deepen the connection between ultrafilter extensions, ultrapowers, and modal canonicity, and identify precise structural and combinatorial conditions under which these relationships persist. The discussion outlines open questions about broader classes (e.g., locally finite or path-bounded), inviting further exploration of how ultrafilter extensions behave beyond the currently established boundaries.

Abstract

We extend some of our earlier results on the interconnection between ultrafilter extensions, and ultrapowers. Throughout we restrict ourselves to relational structures with one binary relation. Recently it was shown that for bounded structures, where a universal finite bound on the maximal in-and out-degree is given, ultrafilter extensions are elementary extensions of the original structures. Comparing the constructions, it seems that the real challenge is when the degree has no global finite bound, or there are elements with infinite degree. This is a first step towards this direction by slightly relaxing the notion of boundedness, called almost bounded structures. Among others, we show that members of this class are still elementary substructures of their extensions, elementary embeddings can be lifted up to the extensions, moreover for the countable case, the extensions are isomorphic to certain ultrapowers. We also comment on the modal logics they generate.

Notes on ultrafilter extensions of almost bounded structures

TL;DR

The paper extends the study of ultrafilter extensions from bounded to almost bounded relational structures with a single binary relation, showing that key model-theoretic properties—such as being an elementary substructure and lifting elementary embeddings—continue to hold under suitable conditions. By decomposing the edge relation into parts and introducing a transfer framework, it demonstrates that the ultrafilter extension can align closely with ultrapowers , particularly in the countable case where isomorphism is attainable for regular ultrapowers. It further analyzes the modal logic implications, showing that almost boundedness can yield and ultrapowers sharing modal theories under specific hypotheses, while also admitting cases where their modal theories diverge. The results deepen the connection between ultrafilter extensions, ultrapowers, and modal canonicity, and identify precise structural and combinatorial conditions under which these relationships persist. The discussion outlines open questions about broader classes (e.g., locally finite or path-bounded), inviting further exploration of how ultrafilter extensions behave beyond the currently established boundaries.

Abstract

We extend some of our earlier results on the interconnection between ultrafilter extensions, and ultrapowers. Throughout we restrict ourselves to relational structures with one binary relation. Recently it was shown that for bounded structures, where a universal finite bound on the maximal in-and out-degree is given, ultrafilter extensions are elementary extensions of the original structures. Comparing the constructions, it seems that the real challenge is when the degree has no global finite bound, or there are elements with infinite degree. This is a first step towards this direction by slightly relaxing the notion of boundedness, called almost bounded structures. Among others, we show that members of this class are still elementary substructures of their extensions, elementary embeddings can be lifted up to the extensions, moreover for the countable case, the extensions are isomorphic to certain ultrapowers. We also comment on the modal logics they generate.

Paper Structure

This paper contains 13 sections, 4 theorems, 40 equations.

Table of Contents

  1. Overview
  2. Background and notation
  3. Background.
  4. Notation.
  5. Results of the paper
  6. Almost bounded structures
  7. Coherent decomposition
  8. A Ł oś's Lemma type property
  9. Comparing the constructions
  10. On elementary embeddings
  11. Comparing modal logics
  12. Further structural and combinatorial properties
  13. Discussion

Key Result

Theorem \ref{thm:almost_bounded}

Let $\mathcal{A}$ be an almost bounded structure with $|A|=\lambda$. If the ultrapower $\mathcal{A}^*$ is at least $2^\lambda$-regular , then $\mathcal{A}\preceq\mathcal{A}^{\mathfrak{ue}}\preceq\mathcal{A}^*$.

Theorems & Definitions (20)

  • Theorem \ref{thm:almost_bounded}
  • Theorem \ref{cor:elementary}
  • Theorem \ref{thm:almost_modal}
  • Theorem \ref{sat}
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 10 more