Notes on planar semimodular lattices. VIII. Congruence lattices of SPS lattices

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.