Some more algebra on ultrafilters in metric spaces

TL;DR

This paper extends the algebraic study of ultrafilters in metric spaces, introducing metric analogs of prime and cancellable ultrafilters and exploring their properties using the concept of parallelity.

Contribution

It defines and analyzes metric counterparts of prime, strongly prime, and right cancellable ultrafilters, advancing the algebraic understanding of ultrafilters in metric spaces.

Findings

Introduces metric versions of prime and cancellable ultrafilters.

Establishes properties of ultrafilters using parallelity in metric spaces.

Extends algebraic structures from discrete groups to metric spaces.

Abstract

We continue algebraization of the set of ultrafilters on a metric spaces initiated in [6]. In particular, we define and study metric counterparts of prime, strongly prime and right cancellable ultrafilters from the Stone-$\check{C}$ech compactification of a discrete group as a right topological semigroup [3]. Our approach is based on the concept of parallelity introduced in the context of balleans in [4].