Spectral spaces of countable Abelian lattice-ordered groups
Friedrich Wehrung · Transactions of the American Mathematical Society · 2018
It is well known that the $\ell$-spectrum of an Abelian $\ell$-group, defined as the set of all its prime $\ell$-ideals with the hull-kernel topology, is a completely normal generalized spectral space. We establish the following converse of this result. Theorem. Every second countable, completely normal generalized spectral space is homeomorphic to the $\ell$-spectrum of some Abelian $\ell$-group. We obtain this result by proving that a countable distributive lattice $D$ with zero is isomorphic to the Stone dual of some $\ell$-spectrum (we say that $D$ is $\ell$-representable) iff for all $a,b\in D$ there are $x,y\in D$ such that $a\vee b=a\vee y=b\vee x$ and $x\wedge y=0$. On the other hand, we construct a non-$\ell$-representable bounded distributive lattice, of cardinality $\aleph _1$, with an $\ell$-representable countable $\mathscr {L}_{\infty ,\omega }$-elementary sublattice. In particular, there is no characterization, of the class of all $\ell$-representable distributive lattices, by any class of $\mathscr {L}_{\infty ,\omega }$ sentences.