Applying the Czédli-Schmidt Sequences to Congruence Properties of Planar Semimodular Lattices

Abstract

Abstract Following Grätzer and Knapp, 2009, a planar semimodular lattice L is rectangular, if the left boundary chain has exactly one doubly-irreducible element, cl, and the right boundary chain has exactly one doubly-irreducible element, cr, and these elements are complementary. The Czédli-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 M3 sublattice) rectangular lattice L, the congruence lattice Con L has exactly length[cl, 1] + length[cr, 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.