A solution to the MV-spectrum problem in size aleph one

Abstract

Denote by IdcG the lattice of all principal ℓ-ideals of an Abelian ℓ-group G. Our main result is the following. TheoremFor every countable Abelian ℓ-group G, every countable completely normal distributive 0-lattice L, and every closed 0-lattice homomorphismφ:IdcG→L, there are a countable Abelian ℓ-group H, an ℓ-homomorphismf:G→H, and a lattice isomorphismι:IdcH→Lsuch thatφ=ι∘Idcf. We record the following consequences of that result:(1)A 0-lattice homomorphism φ:K→L, between countable completely normal distributive 0-lattices, can be represented, with respect to the functor Idc, by an ℓ-homomorphism of Abelian ℓ-groups iff it is closed.(2)A distributive 0-lattice D of cardinality at most ℵ1 is isomorphic to some IdcG iff D is completely normal and for all a,b∈D the set {x∈D|a≤b∨x} has a countable coinitial subset. This solves Mundici's MV-spectrum Problem for cardinalities up to ℵ1. The bound ℵ1 is sharp.Item (1) is extended to commutative diagrams indexed by forests in which every node has countable height. All our results are stated in terms of vector lattices over any countable totally ordered division ring.