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Countable vector lattices

Published online by Cambridge University Press:  17 April 2009

Paul F. Conrad
Paul F. Conrad
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, Kansas, USA.
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In his paper “On the structure of ordered real vector spaces” (Publ. Math. Debrecen 4 (1955–56), 334–343), Erdös shows that a totally ordered real vector space of countable dimension is order isomorphic to a lexicographic direct sum of copies of the group of real numbers. Brown, in “Valued vector spaces of countable dimension” (Publ. Math. Debrecen 18 (1971), 149–151), extends the result to a valued vector space of countable dimension and greatly simplifies the proof. In this note it is shown that a finite valued vector lattice of countable dimension is order isomorphic to a direct sum of o–simple totally ordered vector spaces. One obtains as corollaries the result of Erdös and the applications that Brown makes to totally ordered spaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 10 , Issue 3 , June 1974 , pp. 371 - 376
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Brown, Ron, “Valued vector spaces of countable dimension”, Publ. Math. Debrecen 18 (1971), 149–151.Google Scholar
[2]Conrad, Paul, “Methods of ordering a vector space”, J. Indian Math. Soc. 22 (1958), 1–25.Google Scholar
[3]Conrad, P., Lattice ordered groups (Lecture Notes, Tulane University, Louisiana, 1970).Google Scholar
[4]Erdös, J., “On the structure of ordered real vector spaces”, Publ. Math. Debrecen 4 (1955–1956), 334–343.Google Scholar
[5]Fuchs, L., Partially ordered algebraic systems (Pergamon Press, Oxford, London, New York, Paris, 1963).Google Scholar