Hostname: page-component-cb9f654ff-9b74x Total loading time: 0 Render date: 2025-08-31T11:29:20.190Z Has data issue: false hasContentIssue false
  • English
  • Français

l-simple lattice-ordered groups

Published online by Cambridge University Press:  20 January 2009

A. M. W. Glass
A. M. W. Glass
Affiliation:
Bowling Green State University, Bowling Green, Ohio 43403, U.S.A.
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.

Let G be a lattice-ordered group (l-group) and H a subgroup of G. H is said to be an l-subgroup of G if it is a sublattice of G. H is said to be convex if h1, h2 ∈ H and h2 ≦ g ≦ h2 imply g ∈ H. The normal convex l-subgroups (l-ideals) of an l-group play the same role in the study of lattice-ordered groups as do normal subgroups in the investigation of groups. For this reason, an l-group is said to be l-simple if it has no non-trivial l-ideals. As in group theory, a central task in the examination of lattice-ordered groups is to characterise those l-groups which are l-simple.

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 2 , September 1974 , pp. 133 - 138
Copyright
Copyright © Edinburgh Mathematical Society 1974

References

REFERENCES

(1) Chang, C. C. and Keisler, H. J., Model Theory (North Holland, 1973).Google Scholar
(2) Frayne, T. E., Morel, A. C. and Scott, D. S., Reduced direct products, Fund. Math. 51 (1962), 195–228.CrossRefGoogle Scholar
(3) Holland, W. C., The lattice-ordered groups of automorphisms of an ordered set, Michigan Math. J. 10 (1963), 399–408.CrossRefGoogle Scholar
(4) Holland, W. C., Transitive lattice-ordered permutation groups, Math. Z. 87 (1965), 420–433.CrossRefGoogle Scholar
(5) Holland, W. C., A class of simple lattice-ordered groups, Proc. Amer. Math. Soc. 16 (1965), 326–329.CrossRefGoogle Scholar
(6) Holland, W. C., Ordered permutation groups, Permutations (actes du colloque Paris, juillet 1972), (Gauthier-Villars, 1974).Google Scholar
(7) Mccleary, S. H., o-primitive ordered permutation groups, Pacific J. Math. 40 (1972), 349–372.CrossRefGoogle Scholar
(8) Mccleary, S. H., o-2 transitive ordered permutation groups, Pacific J. Math. 49 (1973), 425–430.CrossRefGoogle Scholar
(9) Mccleary, S. H., The lattice-ordered group of automorphisms of an α-set Pacific J. Math. 49 (1973), 417–424.CrossRefGoogle Scholar