Slim Semimodular Lattices. II. A Description by Patchwork Systems

Rectangular lattices are special planar semimodular lattices introduced by G. Grätzer and E. Knapp in 2009. By a patch lattice we mean a rectangular lattice whose weak corners are coatoms. As a sort of gluings, 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 “sewn together”; see Figure 3 for a first impression. For a modular planar lattice, our patchwork system coincides with the S-glued system introduced by C. Herrmann in 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 the Hall-Dilworth gluing over chains; this fact gives another structure theorem for planar semimodular lattices since patch lattices are obtained from the four-element non-chain lattice by adding forks, introduced in our preceding