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Implications between Conditions on l-Groups

Published online by Cambridge University Press:  20 November 2018

Franklin D. Pedersen
Franklin D. Pedersen*
Affiliation:
Southern Illinois University, Carbondale, Illinois
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The study of lattice-ordered groups (l-groups) in recent years has yielded several publications concerning the lattice of all convex l-subgroups (denoted by Γ(G)). As for example, in [3], Conrad has shown a direct connection between one possible structure of Γ(G) as a lattice and certain finite conditions on the elements of the group G. In the process of developing variations of this approach, there has been imposed several different conditions on G and Γ(G). For instance, it was proposed by Conrad that if all the minimal prime subgroups have non-zero polars, then the group elements would all be finite-valued. In this paper we show that in general this is not true.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970 , pp. 1 - 6
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Birkhoff, G., Lattice theory, rev. éd. Amer. Math. Soc. Colloq. Publ. Vol. 25 (Amer. Math. Soc, Providence, R.I., 1948).Google Scholar
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3. Conrad, P. F., The lattice of all convex l-subgroups of a lattice-ordered group, Czechoslovak Math. J. 15 (90) (1965), 101–123.Google Scholar
4. Conrad, P. F., Harvey, J., and Holland, C., The Hahn embedding theorem for abelian lattice-ordered groups, Trans. Amer. Math. Soc. 108 (1963), 143–169.Google Scholar
5. Pedersen, F., Contributions to the theory of regular subgroups and prime subgroups of a lattice ordered group, Doctoral Dissertation, Tulane University, New Orleans, Louisiana, 1967.Google Scholar