Diagrams and rectangular extensions of planar semimodular lattices

TL;DR

This paper establishes the uniqueness of rectangular extensions of planar semimodular lattices under certain conditions, introduces a hierarchy of special diagrams, and demonstrates their utility in simplifying key lattice theorems.

Contribution

It proves the uniqueness of rectangular extensions under additional conditions and develops a hierarchy of special diagrams with strong uniqueness properties.

Findings

Unique diagrams for planar semimodular lattices are constructed.

Simplified proof of the Trajectory Coloring Theorem using new diagrams.

Proof of G. Grätzer's Swing Lemma for slim rectangular 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.