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

2022/06/20 by Miroslav Ploščica, Ploscica, Miroslav, Friedrich Wehrung +1
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2206.11099

openalex publication_date 2022/06/20 · openalex created_date 2022/06/22 · openalex updated_date 2026/07/28

Abstract

It is well known that the lattice Idc G of all principal ℓ-ideals of any Abelian ℓ-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 ℵ 2. We prove that every completely normal distributive 0-lattice with at most ℵ 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 ℵ 1 compact open sets, is homeomorphic to a spectral subspace of the ℓ-spectrum of some Abelian ℓ-group.

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