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

Abstract

It is well known that the lattice $${{\,\mathrm{Id_c}\,}}{G}$$ of all principal $$\ell $$ -ideals of any Abelian $$\ell $$ -group G is a completely normal distributive 0-lattice; yet not every completely normal distributive 0-lattice is a homomorphic image of some $${{\,\mathrm{Id_c}\,}}{G}$$ , 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 $${{\,\mathrm{Id_c}\,}}{G}$$ . 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.