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Weak, strong and mixed extensions of relations to spaces of ultrafilters

Leonardo Raffaello Maximilian Gasparro, Lorenzo Luperi Baglini

Abstract

The use of nonstandard methods to characterize properties of weak, strong and mixed extensions of congruences to ultrafilters has been the main topic of several recent papers. We show that similar methods can be used to characterize the extensions of arbitrary realtions and their interplay.

Weak, strong and mixed extensions of relations to spaces of ultrafilters

Abstract

The use of nonstandard methods to characterize properties of weak, strong and mixed extensions of congruences to ultrafilters has been the main topic of several recent papers. We show that similar methods can be used to characterize the extensions of arbitrary realtions and their interplay.

Paper Structure

This paper contains 8 sections, 15 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Funding
  3. Nonstandard preliminaries
  4. Weak and strong extensions of relations
  5. Nonstandard characterizations
  6. Weak and strong extensions of equivalence and pre-order relations
  7. Mixed extensions of relations
  8. Open questions

Key Result

Theorem 1

For an equivalence relation $R \subseteq I^2$, the following facts are equivalent.

Theorems & Definitions (43)

  • Definition 1.1
  • Definition 1.2
  • Definition
  • Theorem : A
  • Theorem : B
  • Remark 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • ...and 33 more