On slim rectangular lattices

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 rectangular, if there are complementary $$a, b \in I$$ such is to left b. claim rectangular slim also lattice. will present some applications, including recent result G. Czédli. In paper with E. Knapp about dozen years ago, we introduced natural diagrams for lattices. Five later, Czédli $${\mathcal {C}}_1$$ -diagrams. prove they same.