Diagrams and rectangular extensions of planar semimodular lattices

Abstract

In 2009, G. Gr\"atzer and E. Knapp proved that every planar semimodular lattice has a rectangular extension. We prove that, under reasonable additional conditions, this extension is unique. This theorem naturally leads to a hierarchy of special diagrams of planar semimodular lattices. Besides that these diagrams are unique in a strong sense, we explore many of their further properties. Finally, we demonstrate the power of our new diagrams in two ways. First, we prove a simplified version of our earlier Trajectory Coloring Theorem, which describes the inclusion con(p)\supseteq\con(q) for prime intervals p and q in slim rectangular lattices. Second, we prove G. Gr\"atzer's Swing Lemma for the same lattices, which describes the same inclusion more simply.