Finite semimodular lattices

Presentation by pictures

November 2012

Introduction

• We present here some new structure theorems for finite semimodular lattices which is a geometric approach.

• We introduce some new constructions:

--- a special gluing, the patchwork,

--- the nesting,

and spacial lattices:

--- source lattices,

--- pigeonhole lattices

Planar distributive lattices

How does it look like a finite planar distributive lattice ? On the following picture we have a typical example:

A planar distributive lattice

The smallest “building stones” of planar distributive latticesare the following three lattices, the planar distributive

pigeonholes.We can get all planar distributive lattice using a special

gluing: the patchwork.

Here is a special case of the Hall-Dilworth gluig: patching of two squares along the edges

Dimension

• Dim(L) the Kuros-Ore dimension is is the minimal number of join-irreducibles to span the unit element of L,

• dim(L) is the width of J(L).

The same lattice with colored covering squeres, this is a patchwork

Patchwork irreducible planar lattices and pigeonholes,antislimming

• Mn

The patching in the 3-dimensional case

3D patchwork of distributive lattices

Planar semimodular lattices

• A planar semimodular lattices L is called slim if no three join-irreducible elements form an antichain. This is equvivalent to: L does not contain M3 (it is diamond-free).

The smallest semimodular but not modular planar lattice

The beret of a lattice L is the set of all dual atoms and 1. This is a cover-preserving join-

congruence where the beret is the only one non-trivial congruence class. We get S7 from C3 x C3 :

C3 x C3 /

NestingS7 and “inside” a fork (red)

The extension of the fork

We make a 2D pigeonhole.

Patchwork of slim semimodular lattices (pigeonholes)

Slim semimodular lattices

• Theorem. (Czédli-Schmidt) Every slim semimodular lattice is the patchwork of pigeonholes.

• Corollary. Every planar semimodular lattice is the antislimming of a patchwork of pigeonholes.

Higher dimension

• Rectanular lattice: J(L) is the disjoint sum of chains.

3D patchwork

The beret on B3 (the factor is M3)

The source lattice S3 (inside the 3-fork)

Rectangular latticesThe Edelman-Jaison lattice

(C2)4/is the beret)

Modularity,M3 – free areas

A modular 3D rectangular lattice as patchwork (M3[C3])