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

TL;DR

This paper investigates the structure of congruence lattices of SPS lattices, revealing that such lattices with more than two elements always have at least two dual atoms, limiting certain lattice representations.

Contribution

It establishes a new property of congruence lattices of SPS lattices, showing the necessity of at least two dual atoms for lattices with more than two elements.

Findings

Cong$L$ of an SPS lattice with >2 elements has at least two dual atoms.

Three-element chain cannot be the congruence lattice of an SPS lattice.

Supports and extends previous results on congruence lattice representations.

Abstract

In this note, I find a new property of the congruence lattice, 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: \emph{there are at least two dual atoms in Con$L$}. So the three-element chain cannot be represented as the congruence lattice of an SPS lattice, supplementing a~result of G. Cz\'edli.