Points of joint continuity for the~semigroup of ultrafilters on an~Abelian group

I. V. Protasov

Sbornik. Mathematics, Tome 187 (1996) no. 2, pp. 287-296

Résumé

The Stone-Cech compactification $\beta G$ of a discrete Abelian group $G$ is identified with the set of all ultrafilters on $G$. The operation of addition on $G$ can be extended naturally to a semigroup operation on $\beta G$. A pair of ultrafilters $(p,q)$ on $G$ is a point of joint continuity for the semigroup $\beta G$ if and only if the family of subsets $\{P+Q:P\in p,\ Q\in q\}$ forms an ultrafilter base. The main result of the present paper can be stated as follow: if $G$ is countable group with finitely many elements of order 2 and $(p,q)$ is a point of joint continuity for $\beta G$, then at least one of the ultrafilters $p$ of $q$ must be principal. Examples demonstrating that the restrictions imposed on $G$ are essential are constructed under some further assumptions additional to the standard axioms of $ZFC$ set theory.