Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-09T09:47:52.755Z Has data issue: false hasContentIssue false

Cauchy completion of Abelian tight Riesz groups

Published online by Cambridge University Press:  09 April 2009

B. F. Sherman
Affiliation:
Department of Mathematics, Monash University, Clayton, 3168, Australia
Rights & Permissions [Opens in a new window]

Extract

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.

This paper concerns the completions of partially ordered groups introduced by Fuchs (1965a) and the author (to appear); the p.o. groups under consideration are, generally, abelian tight Riesz groups, and so, throughout, the word “group” will refer to an abelian group.

In section 3 we meet the cornerstone of the work, the central product theorem, by means of which we can interpret the Cauchy completion of a tight Riesz group in terms of the completion of any of its o-ideals; one particularly important case arises when the group has a minimal o-ideal. Such a minimal o-ideal is o-simple, and in section 6 the completion of an isolated o-simple tight Riesz group is shown to be a tight Riesz real vector space.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1975

References

Dieudonné, J. (1941), ‘Sur la Théorie de la Divisibilité,’ Bull. Soc. Math. Fr. 69, 133144.CrossRefGoogle Scholar
Fuchs, L. (1963), Partially Ordered Algebraic Systems, (Pergamon Press, 1963).Google Scholar
Fuchs, L. (1965), ‘Riesz Groups’, Ann. Scuela Norm. Sup. Pisa (3) 19, 134.Google Scholar
Fuchs, L. (1965a), ‘Approximation of Lattice Ordered Groups’, Ann. Univ. Sci. Budapest Eötvös, Sect. Math. 8 187203.Google Scholar
Grätzer, G. (1968), Universal Algebra, (Van Nostrand, 1968).Google Scholar
Lorenzen, P. (1939), ‘Abstrakte Begründung der Multiplikativen Ideal theorie’, Math. Zeit 45 433553.CrossRefGoogle Scholar
Loy, R. and Miller, J. (1972), ‘Tight Riesz Groups,’ J. Austral. Math. Soc. 13 (2) (1972), 224–40.CrossRefGoogle Scholar
Reilly, N. (1971), Compatible Tight Riesz Orders and Prime Subgroups. (Preprint, Simon Fraser University, 1971).Google Scholar
Reilly, N. (1972), ‘Permutational Products of Lattice Ordered Groups’, J. Math. Austral. Soc. 13 2534.CrossRefGoogle Scholar
Ribenboim, P. (1959), Théorie des Groups Ordonnés, (Bahia Blanca, 1959.)Google Scholar
Schaefer, H. (1966), Topological Vector Spaces, (Springer-Verlag, 1966.)Google Scholar
Sherman, B. ‘Cauchy Completion of Partially Ordered Groups’. J. Austral. Math. Soc. (to appear).Google Scholar