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.