Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-99-04834-0
Published electronically: April 23, 1999
MathSciNet review: 1600132
Full-text PDF

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.


References [Enhancements On Off] (What's this?)

  • [B] B.Banaschewski, Minimal topological algebras, Math. Ann. 211 (1974), 107-114. MR 50:9988
  • [CG] W. W. Comfort and D. Grant, Cardinal invariants, pseudocompactness and minimality: Some recent advances in the topological theory of topological groups, Topology Proc. 6 (1981), 227-265. MR 84b:54009
  • [CR] W. Comfort and K. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483-496. MR 34:7699
  • [D1] D. Dikranjan, Sur la minimalité des produits de groupes topologiques abéliens, C. R. Acad. Sci. Paris 299 Série I (1984), 303-306. MR 86a:22001
  • [D2] D. Dikranjan, Zero-dimensionality of some pseudocompact groups, Proc. Amer. Math. Soc. 120 n. 4 (1994), 1299-1308. MR 94f:54077
  • [D3] D. Dikranjan, Compactness and connectedness in topological groups, Topology Appl., 84 (1998), 227-252. 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), 53-91. CMP 98:11
  • [D6] D. Dikranjan, $p$-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 York-Basel, 1990. MR 91e:22001
  • [DS1] D. Dikranjan and D. Shakhmatov, Products of minimal abelian groups, Math. Zeit. 204 (1990), 583-603. MR 91i:22002
  • [DS2] D. Dikranjan and D. Shakhmatov, Compact-like totally dense subgroups of compact groups, Proc. Amer. Math. Soc. 114 (1992), 1119-1129. MR 92g:22009
  • [DS3] D. Dikranjan and D. Shakhmatov, Pseudocompact and countably compact abelian groups: Cartesian products and minimality, Trans. Amer. Math. Soc. 335 (1993), 775-790. MR 93d:22001
  • [E] R. Engelking, General Topology, 2nd edition, Heldermann Verlag, Berlin, 1989. MR 91c:54001
  • [HR] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. 1, 2nd edition, Springer-Verlag, 1979. MR 81k:43001
  • [J] T. Jech, Set Theory, Academic Press, New York, 1978. MR 80a:03062
  • [P] I. Prodanov, Precompact minimal group topologies and $p$-adic numbers, Annuaire Univ. Sofia Fac. Math. Méc. 66 (1971/72), 249-266. MR 54:449
  • [PS] I. Prodanov and L.Stoyanov, Every minimal abelian group is precompact, C. R. Acad. Bulgare Sci. 37 (1984), 23-26. MR 85k:22007
  • [Ste] R. M. Stephenson, Jr., Minimal topological groups, Math. Ann. 192 (1971), 193-195. MR 44:4141
  • [St] L. Stoyanov, Weak periodicity and minimality of topological groups, Annuaire Univ. Sofia Fac. Math. Méc. 73 (1978/79), 155-167. MR 89a:22003
  • [T1] A. Tonolo, On the existence of a finest equivalent linear topology, Comm. Algebra 20(2) (1992), 437-455, Erratum ibid. 22(6), (1994), 2317. CMP 94:10
  • [T2] A. Tonolo, On a class of minimal topological modules, in: A. Facchini and C. Menini ed., Abelian Groups and Modules, Proc. of the Padova Conference, Padova, Italy, June 23-July 1, 1994, Mathematics and its Applications, 343 Kluwer Academic Publishers, Dordrecht-Boston-London, The Netherlands (1995), 459-466. MR 97a:16090

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 54F45, 54D05, 54D30, 22A05, 22D05, 54D25

Retrieve articles in all journals with MSC (1991): 54F45, 54D05, 54D30, 22A05, 22D05, 54D25


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: https://doi.org/10.1090/S0002-9939-99-04834-0
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

American Mathematical Society