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

TL;DR

This paper explores how nonstandard methods can characterize extensions of arbitrary relations to ultrafilters, expanding beyond congruences to include weak, strong, and mixed extensions.

Contribution

It demonstrates that nonstandard techniques used for congruences can be applied to a broader class of relations and their interactions in ultrafilter spaces.

Findings

Nonstandard methods effectively characterize relation extensions to ultrafilters.

Extensions of arbitrary relations can be analyzed similarly to congruences.

The interplay of different types of relation extensions is clarified.

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.