Finite semimodular lattices
description
Transcript of Finite semimodular lattices
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])