On slim rectangular lattices

Abstract

Let L be a slim, planar, semimodular lattice (slim means that it does not contain an $${{\textsf{M}}}_3$$ -sublattice). We call the interval $$I = [o, i]$$ of L rectangular, if there are complementary $$a, b \in I$$ such that a is to the left of b. We claim that a rectangular interval of a slim rectangular lattice is also a slim rectangular lattice. We will present some applications, including a recent result of G. Czédli. In a paper with E. Knapp about a dozen years ago, we introduced natural diagrams for slim rectangular lattices. Five years later, G. Czédli introduced $${\mathcal {C}}_1$$ -diagrams. We prove that they are the same.