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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,

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Finite semimodular lattices

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  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 latticesare 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 • Mn

  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 M3 (it is diamond-free).

  13. The smallest semimodular but not modular planar lattice

  14. The beretD 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 / D

  15. NestingS7 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 Don B3 (the factor is M3)

  23. The source lattice S3 (inside the 3-fork)

  24. Rectangular latticesThe Edelman-Jaison lattice

  25. (C2)4/D(D is the beret)

  26. Modularity,M3 – free areas

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

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