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Generalization of Hölder's Theorem to Ordered Modules

Published online by Cambridge University Press:  20 November 2018

T. M. Viswanathan
T. M. Viswanathan*
Affiliation:
Queen's University, Kingston, Ontario The University of Western Ontario, London, Ontario
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Hölder's theorem on archimedean groups states:

An ordered (abelian) group G is order isomorphic to an ordered subgroup of the ordered group R of real numbers if and only if it is archimedean.

We comprehend this theorem in the following setting: G is a Z-module and Ris the completion with respect to the open interval topology of the ordered field Q; Qitself is the ordered quotient field of the ordered domain Z.

Rephrasing the situation, we raise the following question: We start with a fully ordered domain A,let Kbe its ordered quotient field. We endow Kwith the open interval topology and consider , the topological completion of K. Is it possible to impose a compatible order structure on and if this can be done, when can we say that an ordered A-module Mis order isomorphic to an ordered A-submodule of ? In Theorem 3.1, we obtain a set of necessary and sufficient conditions for this isomorphism to hold.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 149 - 157
Copyright
Copyright © Canadian Mathematical Society 1969

References

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