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Implications between Conditions on l-Groups

  • Franklin D. Pedersen (a1)

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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.

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References

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1. Birkhoff, G., Lattice theory, rev. éd. Amer. Math. Soc. Colloq. Publ. Vol. 25 (Amer. Math. Soc, Providence, R.I., 1948).
2. Conrad, P. F., Some structure theorems for lattice-ordered groups, Trans. Amer. Math. Soc. 99 (1961), 212240.
3. Conrad, P. F., The lattice of all convex l-subgroups of a lattice-ordered group, Czechoslovak Math. J. 15 (90) (1965), 101123.
4. Conrad, P. F., Harvey, J., and Holland, C., The Hahn embedding theorem for abelian lattice-ordered groups, Trans. Amer. Math. Soc. 108 (1963), 143169.
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.
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Implications between Conditions on l-Groups

  • Franklin D. Pedersen (a1)

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