Slim Semimodular Lattices. I. A Visual Approach

Abstract

Abstract. 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 paper.

Patch extensions and trajectory colorings of slim rectangular lattices

“…Planar semimodular lattices. It is proved in G. Czédli and E. T. Schmidt [11,Lemmas 5 and 6], or in G. Czédli and E. T. Schmidt [12,Proposition 5], that slim lattices are planar; for slim semimodular lattices this was proved earlier in G. Grätzer and E. Knapp [21]. In this paper, a lattice diagram is a planar Hasse diagram of a finite lattice.…”

“…;H] is the (single) fork extension introduced in G. Czédli and E. T. Schmidt [12]. For P = S (n) 7 , we call D[S (n) 7 ;H] the n-fold fork extension of D at the 4-cell H; we speak of multi-fork extensions if n is not specified.…”

“…The particular case of (single) fork extensions is proved in G. Czédli and E. T. Schmidt [12,Theorem 11]. We know from G. Czédli and E. T. Schmidt [14,Theorem 3.4] that each slim patch diagram P is obtained from a single 4-cell by a sequence of fork extensions.…”

“…Hence, having an S 7 sublattice, D is not distributive. Therefore, we can choose an element t ∈ D such that ↓t is not distributive but ↓t is distributive for all t < t. The combination of [12,Lemma 15] and [12, Proof of Lemma 22] contains the statement that t is the top of a a cover-preserving S 7 sublattice and also the top of a strong fork; this concept is defined in [12] but we do not need it. In our terminology, this statement says that there is a rectangular diagram D containing t and a 4-cell H of D with top t such that D is obtained from D by a (single) fork extension at H .…”

“…In the paper, our first goal is to generalize the fork extensions of slim semimodular lattices, introduced by G. Czédli and E. T. Schmidt [12], to patch extensions and in particular, multi-fork extensions. Multi-fork extensions lead to a new structural description of slim rectangular lattices, see Theorem 3.7.…”