Countably compact group topologies on the free abelian group of size continuum (and a Wallace semigroup) from a selective ultrafilter (Q2338576)

It is well known that a non-trivial free abelian group does not admit a compact Hausdorff group topology. On the other hand, it was shown that the free abelian group generated by \(\mathfrak{c}\) elements can be endowed with a countably compact Hausdorff group topology under some set-theoretic assumptions. \textit{R. E. Madariaga-Garcia} and \textit{A. H. Tomita} [Topology Appl. 154, No. 7, 1470--1480 (2007; Zbl 1116.54004)] also obtained such a group assuming the existence of \(\mathfrak{c}\) many pairwise incomparable selective ultrafilters and asked whether the existence of one selective ultrafilter implies the existence of a countably compact group topology on the free abelian group of size \(\mathfrak{c}\).\par It was known that compact both-sided cancellative semigroups are topological groups. In the 1950's, Wallace asked whether every countably compact topological semigroup with both-sided cancellation is a topological group. A counterexample to Wallace's question has been called a Wallace semigroup. The main example in the above cited paper of \textit{R. E. Madariaga-Garcia} and \textit{A. H. Tomita} yields a Wallace semigroup from the existence of \(\mathfrak{c}\) selective ultrafilters. \par In this paper, the authors 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 \(\mathfrak{c}\) answering the above first question, and the existence of a Wallace semigroup.