# Spectral spaces of countable abelian lattice-ordered groups

## TL;DR

This paper characterizes spectral spaces that correspond to countable abelian lattice-ordered groups and explores the lattice-theoretic properties related to their spectra, revealing limitations in their logical definability.

## Contribution

It provides a topological characterization of spectra of countable abelian ll-groups and shows the non-definability of ll-representability in arbitrary cardinalities.

## Key findings

- Spectral spaces correspond to spectra of countable abelian ll-groups.
- A lattice-theoretic condition characterizes ll-representability.
- There are limitations in characterizing ll-representability via inite logic.

## 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 {\ell}-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 {\ell}-group (we say that D is {\ell}-representable) iff for all a, b $\in$ D there are x, y $\in$ D such that a $\lor$ b = a $\lor$ y = b $\lor$ x and x $\land$ y = 0. On the other hand, we construct a non-{\ell}-representable bounded distributive lattice, of cardinality $\aleph$ 1 , with an {\ell}-representable countable L$\infty, \omega$-elementary sublattice. In particular, there is no characterization, of the class of all {\ell}-representable distributive lattices, in arbitrary cardinality, by any class of L$\infty, \omega$ sentences.