Applying the Cz\'edli-Schmidt Sequences to congruence properties of planar semimodular lattices

TL;DR

This paper uses Czédli-Schmidt Sequences to analyze the structure of congruence lattices in rectangular semimodular lattices, revealing precise counts of dual atoms and prime ideals.

Contribution

It applies Czédli-Schmidt Sequences to establish new structure theorems for congruence lattices of slim rectangular lattices.

Findings

The congruence lattice of a slim rectangular lattice has a specific number of dual atoms.

Each dual atom in the congruence lattice corresponds to a congruence with exactly two classes.

Prime ideals in slim rectangular lattices are explicitly described.

Abstract

Following G.~Gr\"atzer and E.~Knapp, 2009, a planar semimodular lattice $L$ is \emph{rectangular}, if~the left boundary chain has exactly one doubly-irreducible element, $c_l$, and the right boundary chain has exactly one doubly-irreducible element, $c_r$, and these elements are complementary. The Cz\'edli-Schmidt Sequences, introduced in 2012, construct rectangular lattices. We use them to prove some structure theorems. In particular, we prove that for a slim (no $\mathsf{M}_3$ sublattice) rectangular lattice~$L$, the congruence lattice $\Con L$ has exactly $\length[c_l,1] + \length[c_r,1]$ dual atoms and a dual atom in $\Con L$ is a congruence with exactly two classes. We also describe the prime ideals in a slim rectangular lattice.