On two types of ultrafilter extensions of binary relations

TL;DR

This paper compares two distinct ultrafilter extension types of binary relations, showing their inclusion relationship, interaction with algebraic operations, and topological characterizations, including a continuous mapping between ultrafilters and filters.

Contribution

It clarifies the relationship between two ultrafilter extension types, providing topological insights and mappings that unify their understanding across different mathematical contexts.

Findings

The extension from model theory is properly included in the universal algebra extension.

Both extensions are characterized topologically using the Vietoris topology.

A continuous map from ultrafilters to filters is established.

Abstract

There exist two distinct types of ultrafilter extensions of binary relations, one discovered in universal algebra and modal logic, and another, in model theory and algebra of ultrafilters. We show that the extension of the latter type is properly included in the extension of the former type, and describe their interaction with the relation algebra operations. Then we provide topological characterizations of both extensions and show that the larger extension continuously maps the space of ultrafilters into the space of filters endowed with the Vietoris topology.