2014/12/15 by Gábor Czédli, Czédli, Gábor
Computer Science · Mathematics · #06C10 #Advanced Algebra and Logic #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.RA #msc:06C10 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1412.4453
48 pages, 14 figures
arxiv created 2014/12/15 · openalex publication_date 2014/12/15 · arxiv updated 2014/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. 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)⊇\con(q) for prime intervals p and q in slim rectangular lattices. Second, we prove G. Grätzer's Swing Lemma for the same lattices, which describes the same inclusion more simply.