Spectral spaces of countable abelian lattice-ordered groups

International audienceA compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X, closed under finite intersections. Theorem. A topological space X is homeomorphic to the spectrum of some countable Abelian ℓ-group with unit (resp., MV-algebra) iff X is spectral, has a countable basis of open sets, and for any points x and y in the closure of a singleton {z}, either x is in the closure of {y} or y is in the closure of {x}. We establish this result by proving that a countable distributive lattice D with zero is isomorphic to the lattice of all principal ideals of an Abelian ℓ-group (we say that D is ℓ-representable) iff for all a, b ∈ D there are x, y ∈ D such that a ∨ b = a ∨ y = b ∨ x and x ∧ y = 0. On the other hand, we construct a non-ℓ-representable bounded distributive lattice, of cardinality ℵ 1 , with an ℓ-representable countable L∞,ω-elementary sublattice. In particular, there is no characterization, of the class of all ℓ-representable distributive lattices, in arbitrary cardinality, by any class of L∞,ω sentences