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Dual spaces of totally ordered rings. (English) Zbl 0671.06010

This paper continues and relies on earlier work of the author [Dual spaces of totally ordered abelian groups, ibid. 37, 613-627 (1987; Zbl 0645.06007)]. There, he assigns a dual space to every totally ordered abelian group in such a way that the evaluation map into the second dual is one-to-one and order preserving. In this paper he investigates the question whether, for totally ordered rings, convolution may be used to define a multiplication on the second dual in such a way that the evaluation map also preserves the multiplication. The author succeeds for a class of rings that include all lexicographically ordered power series rings with real coefficients on totally ordered cancellative semigroups. The precise statements of the results are too involved to be recorded here.
Reviewer: K.Keimel

MSC:

06F25 Ordered rings, algebras, modules

Citations:

Zbl 0645.06007
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References:

[1] A. Bigard K. Keimel S. Wolfenstein: Groupes et Anneaux Réticulés. Lecture Notes in Mathematics 608, Springer-Verlag, Berlin, 1977. · Zbl 0384.06022
[2] L. Fuchs: Partially Ordered Algebraic Systems. Pergamon Press, Oxford, 1963. · Zbl 0137.02001
[3] E. Hewitt K. A. Ross: Abstract Harmonic Analysis, Volume I. Springer-Verlag, Berlin, 1963. · Zbl 0115.10603
[4] J. L. Kelley: General Topology. Graduate Texts in Mathematics 27, Springer-Verlag, New York, reprint of D. Van Nostrand Co. edition, 1955. · Zbl 0066.16604
[5] B. H. Neumann: On ordered division rings. Trans. Amer. Math. Soc. 66 (1949), 202-252. · Zbl 0035.30401 · doi:10.2307/1990552
[6] R. H. Redfield: Dual spaces of totally ordered abelian groups. Czechoslovak Math. J. 37(112), (1987), 613-627. · Zbl 0645.06007
[7] R. H. Redfield: Embeddings into power series rings. Manuscripta Math. 56 (1986), 247-268. · Zbl 0613.06011 · doi:10.1007/BF01180767
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