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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
2. Conrad, P. F., Some structure theorems for lattice-ordered groups, Trans. Amer. Math. Soc. 99 (1961), 212–240.Google Scholar
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