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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the minimality of powers
of minimal $\omega $-bounded abelian groups


Authors: Dikran Dikranjan and Alberto Tonolo
Journal: Proc. Amer. Math. Soc. 127 (1999), 3101-3110
MSC (1991): Primary 54F45, 54D05, 54D30; Secondary 22A05, 22D05, 54D25
Published electronically: April 23, 1999
MathSciNet review: 1600132
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Abstract | References | Similar Articles | Additional Information

Abstract: We describe the structure of totally disconnected minimal $\omega$-
bounded abelian groups by reducing the description to the case of those of them which are subgroups of powers of the $p$-adic integers $\mathbb{Z}_{p}$. In this case the description is obtained by means of a functorial correspondence, based on Pontryagin duality, between topological and linearly topologized groups introduced by Tonolo. As an application we answer the question (posed in Pseudocompact and countably compact abelian groups: Cartesian products and minimality, Trans. Amer. Math. Soc. 335 (1993), 775-790) when arbitrary powers of minimal $\omega $-bounded abelian groups are minimal. We prove that the positive answer to this question is equivalent to non-existence of measurable cardinals.


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Additional Information

Dikran Dikranjan
Affiliation: Dipartimento di Matematica e Informatica, Udine University, via delle Scienze 206 (località Rizzi), 33100 Udine, Italy
Email: dikranja@dimi.uniud.it

Alberto Tonolo
Affiliation: Dipartimento di Matematica Pura ed Applicata, Padova University, Via Belzoni 7, 35131 Padova, Italy
Email: tonolo@math.unipd.it

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04834-0
PII: S 0002-9939(99)04834-0
Keywords: Totally disconnected group, connected group, countably compact group, $\omega $-bounded group, minimal group, measurable cardinal
Received by editor(s): March 25, 1997
Received by editor(s) in revised form: December 13, 1997
Published electronically: April 23, 1999
Communicated by: Alan Dow
Article copyright: © Copyright 1999 American Mathematical Society