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The ordered set of principal congruences of a countable lattice

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Abstract

For a lattice L, let Princ(L) denote the ordered set of principal congruences of L. In a pioneering paper, G. Grätzer characterized the ordered set Princ(L) of a finite lattice L; here we do the same for a countable lattice. He also showed that every bounded ordered set H is isomorphic to Princ(L) of a bounded lattice L. We prove a related statement: if an ordered set H with a least element is the union of a chain of principal ideals (equivalently, if 0 \({\in}\) H and H has a cofinal chain), then H is isomorphic to Princ(L) of some lattice L.

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Authors and Affiliations

  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, 6720, Hungary

    Gábor Czédli

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  1. Gábor Czédli
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Correspondence to Gábor Czédli.

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Presented by F. Wehrung.

To the memory of András P. Huhn

This research was supported by the European Union and co-funded by the European Social Fund under the project “Telemedicine-focused research activities on the field of Mathematics, Informatics and Medical sciences” of project number “TÁMOP-4.2.2.A-11/1/KONV-2012-0073”, and by NFSR of Hungary (OTKA), grant number K83219.

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Czédli, G. The ordered set of principal congruences of a countable lattice. Algebra Univers. 75, 351–380 (2016). https://doi.org/10.1007/s00012-016-0376-1

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