Cover preserving embedding of modular lattices into partition lattices

Tight embedding of modular lattices into partition lattices: progress and program

“…Huhn's paper sparked the groundbreaking paper of Jónsson and Nation [7] which showed that for each field k each 2-distributive modular lattice L with |L| < |k| tightly embeds into the subspace lattice L(k n ) (n = d(L)). In turn [7] triggered [5], which triggered [15] and whence the present article.…”

“…Progress initially sprang from the author's effort to present the key concepts and results of [15] in crisper ways. All of this harks back to a result of Huhn [6] that characterizes 2-distributive modular lattices in terms of certain forbidden sublattices.…”

Modular and semimodular lattices

“…On the other hand, if L is a semimodular lattice then its height function is a pseudorank function, and the isometrical embedding of L preserves the height of each element, moreover it also preserves the covering relation under some necessary conditions. In the first section, we recall a proof of Marcel Wild [89], which shows that every finite semimodular lattice has a cover-preserving embedding into a geometric lattice. This argument is a motivation for the second section, where we prove a generalization of an embedding result of George Grätzer and Emil W. Kiss [43], see [78].…”

“…Wild [89] matroidokkal történő fedésőrző beágyazása található véges féligmoduláris hálókra, ami motivációt ad azáltalános eset bizonyításához, amit a fejezet második részében közlök [78].…”

“…The construction is due to Wild [89], who noticed that the technique used by Finkbeiner and Dilworth is actually matroid theory. He also noticed that their construction gives a different proof for the corollary of Grätzer and Kiss' theorem.…”