Slim Semimodular Lattices. II. A Description by Patchwork Systems

Abstract

Rectangular lattices are special planar semimodular lattices introduced by G. Gratzer and E. Knapp in Acta Sci Math 75:29–48, 2009. A patch lattice is a rectangular lattice whose weak corners are coatoms. As a variant of gluing, we introduce the concept of a patchwork system. We prove that every glued sum indecomposable, planar, semimodular lattice is a patchwork of its maximal patch lattice intervals. For a planar modular lattice, our patchwork system is the same as the S-glued system introduced by C. Herrmann in Math Z 130:255–274, 1973. Among planar semimodular lattices, patch lattices are characterized as the patchwork-irreducible ones. They are also characterized as the indecomposable ones with respect to gluing over chains; this gives another structure theorem.