Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-30T12:15:04.558Z Has data issue: false hasContentIssue false
  • English
  • Français

Constructions of Σ-groups, relatively free Σ-groups

Part of: Abelian groups

Published online by Cambridge University Press:  09 April 2009

Don Brunker  and
Denis Higgs
Don Brunker
Affiliation:
Bureau of Industry Economics Canberra, A.C.T. 2600, Australia
Denis Higgs
Affiliation:
Department of Pure Mathematics, University of Waterloo Waterloo, OntarioCanada
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A Σ-group is an abelian group on which is given a family of infinite sums having properties suggested by, but weaker than, those which hold for absolutely convergent series of real or complex numbers. Two closely related questions are considered. The first concerns the construction of a Σ-group from an arbitrary abelian group on which certain series are given to be summable, certain of these series being required to sum to zero. This leads to a Σ-theoretic construction of R from Q and in general of the completion of an arbitrary metrizable abelian group (with the associated unconditional sums) from that group. The second question is whether, in a given Σ-group, the values of the infinite sums may be determined solely from a knowledge of which series are summable. Such a Σ-group is said to be relatively free and it is shown that all metrizable abelian groups are relatively free.

MSC classification

Secondary: 20K99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 1 , February 1989 , pp. 8 - 21
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Bourbaki, N., Elements of mathematics III. General topology (Hermann, Paris, and Addison-Wesley, Reading, Mass., 1966).Google Scholar
[2]Brunker, D. M. S., Topics in the algebra of axiomatic infinite sums, (Ph.D. Dissertation, University of Waterloo, 10 1980).Google Scholar
[3]Fleischer, I., ‘Completion and regularization of Σ-groups’, preprint.Google Scholar
[4]Higgs, D., ‘Axiomatic infinite sums—an algebraic approach to integration theory’, Proceedings of the conference on integration, topology, and geometry in linear spaces, (Contemporary Mathematics, Vol. 2, pp. 205–212, American Mathematical Society, Providence, R.I., 1980).CrossRefGoogle Scholar
[5]Rado, R., ‘The cardinal module and some theorems on families of sets’, Ann. Mat. Pura Appl. (4) 102 (1975), 135–154.Google Scholar
[6]Wylie, S., ‘Intercept-finite cell complexes’, Algebraic geometry and topology, a symposium in honor of S. Lefschetz (Princeton Mathematical Series, No. 12, pp. 389–399, Princeton University Press, Princeton, N.J., 1957).Google Scholar