  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.

Logic