On the minimality of powers of minimal bounded abelian groups
Authors:
Dikran Dikranjan and Alberto Tonolo
Journal:
Proc. Amer. Math. Soc. 127 (1999), 31013110
MSC (1991):
Primary 54F45, 54D05, 54D30; Secondary 22A05, 22D05, 54D25
Published electronically:
April 23, 1999
MathSciNet review:
1600132
Fulltext PDF Free Access
Abstract 
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Abstract: We describe the structure of totally disconnected minimal  bounded abelian groups by reducing the description to the case of those of them which are subgroups of powers of the adic integers . 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), 775790) when arbitrary powers of minimal bounded abelian groups are minimal. We prove that the positive answer to this question is equivalent to nonexistence of measurable cardinals.
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 D. Dikranjan, Compactness and connectedness in topological groups, Topology Appl., 84 (1998), 227252. CMP 98:10
 [D4]
 D. Dikranjan, The structure of minimal countably compact abelian groups, Preprint.
 [D5]
 D. Dikranjan, Recent advances in minimal topological groups, Topology Appl., 85 (1998), 5391. CMP 98:11
 [D6]
 D. Dikranjan, adic numbers and minimality of powers, Preprint.
 [DPS]
 D. Dikranjan, I. Prodanov and L. Stoyanov, Topological groups: characters, dualities and minimal group topologies, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 130, Marcel Dekker Inc., New YorkBasel, 1990. MR 91e:22001
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 [T1]
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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/S0002993999048340
PII:
S 00029939(99)048340
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
