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Lifting Disjoint Sets in Vector Lattices

Published online by Cambridge University Press:  20 November 2018

Paul Conrad
Paul Conrad*
Affiliation:
Tulane University, New Orleans, Louisiana
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A subset {s α| α ϵ △} of a lattice-ordered group (l-group) is disjoint if S α Λ S β = 0 for all α≠ β in △. An l-group G has the lifting property if for each l-ideal S of G and each countable disjoint subset X1 , X2, … of G/S one can choose elements 0 ≦ xi ϵ Xi so that x1, x2, … is a disjoint subset of G. In (2) Topping showed by an example, that uncountable sets of disjoint elements cannot necessarily be lifted and asserted (Theorem 8) that each vector lattice has the lifting property. His proof is valid for finite disjoint subsets of G/S, but we show by an example that this is, in general, all that one can establish.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 1362 - 1364
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Conrad, P., Harvey, J., and Holland, C., The Hahn embedding theorem for abelian latticeordered groups, Trans. Amer. Math. Soc. 108 (1963), 143–169.Google Scholar
2. Topping, D. M., Some homological pathology in vector lattices, Can. J. Math. 17 (1965), 411–428.Google Scholar