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Points of joint continuity for the semigroup of ultrafilters on an Abelian group
I. V. Protasov National Taras Shevchenko University of Kyiv
Abstract:
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.
UDC:
512.536
MSC: Primary
22A15; Secondary
20K45,
20M99,
54D80 Received: 24.11.1994
DOI:
10.4213/sm112
Fulltext:
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