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

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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.

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Notes

  1. Strictly speaking we should set the Heyting implication \(\rightarrow \) apart from the lattice signature, thus for example stating that “every finite distributive lattice expands to a unique Heyting algebra”. The shorter formulation, which we shall keep for the sake of simplicity, reflects a standard abuse of terminology that should create no confusion here.

  2. “Right” and “left” appear to have been unfortunately mixed up at various places in [18], particularly on pages 12 and 13. Since this is mostly a matter of choosing sides, that paper’s results are unaffected. We nonetheless attempt to fix this here.

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Funding

The first author was supported by Slovak VEGA grant 1/0152/22.

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Authors and Affiliations

  1. Faculty of Natural Sciences, Šafárik’s University, Jesenná 5, Košice, 04154, Slovakia

    Miroslav Ploščica

  2. Normandie Université, UNICAEN, CNRS UMR 6139, LMNO, 14000, Caen, France

    Friedrich Wehrung

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  1. Miroslav Ploščica
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  2. Friedrich Wehrung
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Correspondence to Friedrich Wehrung.

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Ploščica, M., Wehrung, F. Spectral subspaces of spectra of Abelian lattice-ordered groups in size aleph one. Acta Sci. Math. (Szeged) 89, 339–356 (2023). https://doi.org/10.1007/s44146-023-00080-z

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