Spectral spaces of countable abelian lattice-ordered groups
Abstract
A 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 -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 -group (we say that D is -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--representable bounded distributive lattice, of cardinality 1 , with an -representable countable L∞, ω-elementary sublattice. In particular, there is no characterization, of the class of all -representable distributive lattices, in arbitrary cardinality, by any class of L∞, ω sentences.