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Generalizing Galvin and Jónsson’s classification to \(N_5\)

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Abstract

The problem of determining (up to lattice isomorphism) the lattices that are sublattices of free lattices is in general an extremely difficult and an unsolved problem. A notable result towards solving this problem was established by Galvin and Jónsson when they classified (up to lattice isomorphism) all of the distributive sublattices of free lattices in 1959. In this paper, we weaken the requirement that a sublattice of a free lattice be distributive to requiring that a such a lattice belongs in the variety of lattices generated by the pentagon \(N_5\). Specifically, we use McKenzie’s list of join-irreducible covers of the variety generated by \(N_5\) to extend Galvin and Jónsson’s results by proving that all sublattices of a free lattice that belong to the variety generated by \(N_5\) satisfy three structural properties. Afterwards, we explain how the results in this paper can be partially extended to lattices from seven known infinite sequences of semidistributive lattice varieties.

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Acknowledgements

The author would like to thank Claude Laflamme and Robert Woodrow for their support during the writing of an earlier draft of this paper, and the author would like to thank Stephanie van Willigenburg for her support during the writing of the current paper. Furthermore, the author would like to thank an anonymous reviewer for introducing the author to splitting equations, for revising Lemma 3.1 of an earlier draft of this paper and its proof, and for insightful feedback which led the author to write the first part of Section 3.

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  1. Mathematics Department, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Brian T. Chan

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  1. Brian T. Chan
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Correspondence to Brian T. Chan.

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Presented by R. Freese.

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The author was supported in part by the Natural Sciences and Engineering Research Council of Canada [funding reference number PGSD2 - 519022 - 2018].

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Chan, B.T. Generalizing Galvin and Jónsson’s classification to \(N_5\). Algebra Univers. 81, 43 (2020). https://doi.org/10.1007/s00012-020-00674-6

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  • DOI: https://doi.org/10.1007/s00012-020-00674-6

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