Diagrams and rectangular extensions of planar semimodular lattices

Abstract

In 2009, G. Grätzer 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. These diagrams are unique in a strong sense; we also explore many of their additional properties. We demonstrate the power of our new classes of diagrams in two ways. First, we prove a simplified version of our earlier Trajectory Coloring Theorem, which describes the inclusion con${(\mathfrak{p}) \supseteq}$ con${(\mathfrak{q})}$ for prime intervals ${\mathfrak{p}}$ and ${\mathfrak{q}}$ in slim rectangular lattices. Second, we prove G. Grätzer’s Swing Lemma for the same class of lattices, which describes the same inclusion more simply.