J Austral Math Soc Ser A 46 pp8--21, 1989.

Constructions of S-Groups, Relatively Free S-Groups

Don Brunker and Denis Higgs

Abstract

A S-group is an abelian group on which 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 S-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 S-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 S-group, the values of the infinite sums may be determined solely from a knowledge of which series are summable. Such a S-group is said to be relatively free and it is shown that all metrizable abelian groups are relatively free.

1980 AMS Subject Classification (1985 Revision): 20K99

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Authors

Don Brunker

Bureau of Industry Economies, Canberra, A.C.T. 2600, Australia.

Denis Higgs

Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada.