Abstract
In this note, I find a new property of the congruence lattice, \({{\,\mathrm{Con}\,}}L\), of an SPS lattice L (slim, planar, semimodular, where “slim” is the absence of \({\mathsf {M}}_3\) sublattices) with more than 2 elements: there are at least two dual atoms in\({{\,\mathrm{Con}\,}}L\). So the three-element chain cannot be represented as the congruence lattice of an SPS lattice, supplementing a result of G. Czédli.
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Grätzer, G. Notes on planar semimodular lattices. VIII. Congruence lattices of SPS lattices. Algebra Univers. 81, 15 (2020). https://doi.org/10.1007/s00012-020-0641-1
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DOI: https://doi.org/10.1007/s00012-020-0641-1