The countable lifting property for Riesz space surjections

Abstract

A surjective homomorphism A↠φB of Riesz spaces (real vector lattices) has the “countable lifting property” CLP if: For each countable pairwise disjoint {bn} in B, there are disjoint{an} in A with φ(an)=bn for each n. Previous thoughts on this are due to Topping (1965), Conrad (1968), and in considerable depth, Moore (1970), (and little subsequent, to our knowledge). Here, we consider the issue mostly (not entirely) for Riesz spaces resembling C(X)’s. We show (inter alia): A↠φB will have CLP if (a) B is laterally σ-complete; or if (b) B=C(Y) for Y locally compact and σ-compact; or if (c) A is an f-algebra with identity, which is archimedean and uniformly complete, and B is (merely) archimedean (e.g., A=C(X) and B=C(Y), for any X, Y). The main technical device is the notion: b is a weak supremum of {bn} if b=⋁λnbn for some {λn}⊆(0,+∞).