Lifting Disjoint Sets in Vector Lattices
Paul F. Conrad · Canadian Journal of Mathematics · 1968
A subset {s α| α ϵ △} of a lattice-ordered group (l-group) is disjoint if Sα Λ Sβ= 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.