Spectral subspaces of spectra of Abelian lattice-ordered groups in size aleph one

TL;DR

This paper characterizes which completely normal distributive 0-lattices of size at most 1 are realizable as spectra of Abelian ll-groups, showing a positive result for lattices of size 1.

Contribution

It proves that all such lattices with size at most 1 are homomorphic images of principal ll-ideals of Abelian ll-groups, extending previous counterexamples.

Findings

Every completely normal distributive 0-lattice with 1 elements is a homomorphic image of some Idc G.

Spectral spaces with at most 1 compact open sets are homeomorphic to spectral subspaces of Abelian ll-group spectra.

Counterexamples exist at size 2, showing the size restriction is sharp.

Abstract

It is well known that the lattice Idc G of all principal {\ell}-ideals of any Abelian {\ell}-group G is a completely normal distributive 0-lattice, and that not every completely normal distributive 0-lattice is a homomorphic image of some Idc G, via a counterexample of cardinality $\aleph 2. We prove that every completely normal distributive 0-lattice with at most $\aleph 1 elements is a homomorphic image of some Idc G. By Stone duality, this means that every completely normal generalized spectral space, with at most $\aleph 1 compact open sets, is homeomorphic to a spectral subspace of the {\ell}-spectrum of some Abelian {\ell}-group.