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A Note on Commutative l-Groups

Published online by Cambridge University Press:  09 April 2009

T. P. Speed  and
E. Strzelecki
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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

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