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

Miroslav Ploscica; Friedrich Wehrung

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arXiv · 2206.11099

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

Miroslav Ploscica·Friedrich Wehrung

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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.

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BibTeX RIS

Miroslav Ploscica, Friedrich Wehrung. 2022-06-20. Spectral subspaces of spectra of Abelian lattice-ordered groups in size aleph one. https://arxiv.org/abs/2206.11099

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