Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-29T11:48:47.236Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on Commutative l-Groups

Published online by Cambridge University Press:  09 April 2009

T. P. Speed  and
E. Strzelecki
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 commutative lattice ordered group. Theorem 1 gives necessary and sufficient conditions under which a⊥ with a∈G is a maximal l-ideal. A wide family of, l-groups G having the property that the orthogonal complement of each atom is a maximal l-ideal is described. Conditionally σ-complete and hence conditionally complete vector lattices belong to the family.It follows immediately that if a is an atom in a conditionally complete vector lattice then a⊥ is a maximal vector lattice ideal. This theorem has been proved in [7] by Yamamuro. Theorem 2 generalizes another result contained in [7]. Namely we prove that if M is a closed maximal l-ideal of an archimedean l-group G then there exists an atom a ∈ G such that M = a⊥.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 1 , February 1971 , pp. 69 - 74
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Birkhoff, G., Lattice Theory (American Mathematical Society, Providence, Rhose Island 1967). Colloquium Publication, 25.Google Scholar
[2]Conrad, P. F. and McAllister, D., ‘The completion of a lattice ordered group’, J. Aust. Math. Soc. 9 (1969), 182–208.CrossRefGoogle Scholar
[3]Johnson, D. and Kist, J., ‘Complemented ideals and extremally disconnected spaces’, Arch. Math. 12 (1961), 349–354.CrossRefGoogle Scholar
[4]Masterson, J. J., ‘Structure spaces of a vector lattice and its Dedekind completion’, Proc. Kon. Ned. Akad. v. Wet. 71 (1968), 468–478.Google Scholar
[5]Speed, T. P., ‘On commutative l-groups’ (to appear).Google Scholar
[6]Vulikh, B. Z., Introduction to the theory of partially ordered spaces (Wolters-Noordhoff Scientific Publications, Groningen, 1967).Google Scholar
[7]Yamamuro, S., ‘A note on vector lattices’, J. Aust. Math. Soc. 7 (1967) 32–38.CrossRefGoogle Scholar