Countably compact group topologies on the free Abelian group of size continuum (and a Wallace semigroup) from a selective ultrafilter

TL;DR

This paper demonstrates that the existence of a selective ultrafilter leads to the construction of a countably compact Hausdorff group topology on the free Abelian group of size continuum and a Wallace semigroup, highlighting deep connections between ultrafilters and topological algebra.

Contribution

It establishes a new link between selective ultrafilters and the existence of specific topological structures on free Abelian groups and semigroups.

Findings

Existence of a selective ultrafilter implies a countably compact Hausdorff group topology.

Constructs a Wallace semigroup from a selective ultrafilter.

Shows the existence of these structures is consistent with certain set-theoretic assumptions.

Abstract

We prove that the existence of a selective ultrafilter implies the existence of a countably compact Hausdorff group topology on the free Abelian group of size continuum. As a consequence, we show that the existence of a selective ultrafilter implies the existence of a Wallace semigroup (i.e., a countably compact both-sided cancellative topological semigroup which is not a topological group).