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

Abstract

Denote by Id$_c G$ the lattice of all principal $\ell$-ideals of an Abelian $\ell$-group $G$. Our main result is the following. Theorem. For every countable Abelian $\ell$-group $G$, every countable completely normal distributive 0-lattice $L,$ and every closed 0-lattice homomorphism $\varphi : {\rm Id}_c G \to L$, there are a countable Abelian $\ell$-group $H$, an $\ell$-homomorphism $f: G \to H$, and a lattice isomorphism $\iota: {\rm Id}_c H \to L$ such that $\varphi = \iota \circ {\rm Id}_c f$. We record the following consequences of that result: (1) A 0-lattice homomorphism $\varphi: K \to L$, between countable completely normal distributive 0-lattices, can be represented, with respect to the functor Id$_c$, by an $\ell$-homomorphism of Abelian $\ell$-groups iff it is closed. (2) A distributive 0-lattice $D$ of cardinality at most $\aleph_1$ is isomorphic to some Id$_c G$ iff $D$ is completely normal and for all $a,b \in D$ the set $\{x\in D | a \leq b \vee x$ has a countable coinitial subset. This solves Mundici's MV-spectrum Problem for cardinalities up to $\aleph_1$. The bound $\aleph_1$ 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.