1. Finite semimodular lattices Presentation by pictures November 2012

2. 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

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

4. A planar distributive lattice

5. The smallest “building stones” of planar distributive lattices are the following three lattices, the planar distributive pigeonholes. We can get all planar distributive lattice using a special gluing: the patchwork.

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

7. 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).

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

9. Patchwork irreducible planar lattices and pigeonholes, antislimming M n

10. The patching in the 3-dimensional case

11. 3D patchwork of distributive lattices

12. 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 M 3 (it is diamond- free).

13. The smallest semimodular but not modular planar lattice

14. 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 S 7 from C 3 x C 3 : C 3 x C 3 / 

15. Nesting S 7 and “inside” a fork (red)

16. The extension of the fork

17. We make a 2D pigeonhole.

18. Patchwork of slim semimodular lattices (pigeonholes)

19. 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.

20. Higher dimension Rectanular lattice: J(L) is the disjoint sum of chains.

21. 3D patchwork

22. The beret  on B 3 (the factor is M 3 )

23. The source lattice S 3 (inside the 3-fork)

24. Rectangular lattices The Edelman-Jaison lattice

25. (C 2 ) 4 /   is the beret)

26. Modularity, M 3 – free areas

27. A modular 3D rectangular lattice as patchwork (M 3 [C 3 ])