Semimodular lattices and geometric shapes

1. Semimodular lattices and geometric shapes

2. Prelimineries • G. Grätzer andE. Knapppublished in the Acta Sci. Math.a sequence of papers on planar semimodular lattices. • Their proved that the slim planar arethe cover-preserving join- homomorphic images of the direct product of two chains. • In G. Czédli and E. T. Schmidt wegeneralized this result,we proved the following three theorems:

3. Czédli-Schmidt: Acta Math. Hungar. 2008. • Theorem 1.Everysemimodular lattice is thecover-preserving join-homomorphic image of the direct product of chains. • The number of the chains is: w((J(L)) where J(L) is the poset of join-irreducible elements ofL, [w((J(L)) is the width of J(L)] . The direct product of chains can be considered as coordinate system (grid), that means the semimodular lattices are coordinatisable. (if i.e:w((J(L)) = 2 hight and width).

4. J(L):the poset of join-irreducibleelements • Theorem 2. The semimodular latticeLis the cover-preserving join-homomorphic image of the distributive latticeD=H(J(L)). H(P) is the poset of all order ideals ofP

5. The cover-preserving join-congruence can be charakterisedby the restriction on the covering squars(Czédli- Schmidt)Theorem 3.

6. The draughting of our aim • From Theorem 1. it followsthat we can derive every semimodular latticefom a distributive lattice applaying e good tool, this is the cover-preserving join-homomorphism. • The finite distributice lattices can be considered as geometric shapes, the boolean algebra 23is a cube. We get theplanar disztributivelatticesby gluing of rectangulars.This is thrue in higher dimension. • This approachcan be applied by semimodular lattices.

7. Thesourceelement of a cover-preserving join-congruence Q (yello sircle).s is a source element, if there exists an lowercovert, ofssuch thats and t are congruent but a is not congruent with b if s/t perspektív down toa/b.

8. A cover-preserving join-congruence inducet by a source element

9. A source elementsis congruent with every upper cover of s.These upper covers generate a boolean lattice, the restriction ofQ has only one non-trivial congruence class ths“baret”

10. The independence fof the source elements

11. The source and the associated matrix in the w(J(L))= 2 case

12. The (0,1) matrix • 0 1 0 0 0 0 • 0 0 0 0 0 1 • 0 0 1 0 0 0 • 0 0 0 0 0 0 • 0 0 0 0 1 0

13. Boole lattice • In a Boole lattice means that the source elements form an antichain. • Every geometric latticeGcan be characterised as a pair (B,S) , whereBis a Boole lattice andSis a antichain. Example: Fano plane B=27 and 7 source elements

14. An aplication • In joint paper with G. Czédli we determine the numberof all slim semimodular lattices of given length.

16. P=J(L) the poset of join-irreducible elements. Vertikalandhorizontaledges.If we delate the two horisontal edgesit ramains two disjunkt chains.These determine the direct product (big rectangular). The horizontál edges determine the two, small blue rectangulars.

17. Order join-homomorphism: u congruentv.We getL (grey) as the Hall-Dilworth gluing of two rectangular lattices.If we extend L with the blue rectanulars (gadgets) the result is a big rectangular“envelop”.

18. The order P=J(L), w(P)= 3 with a horisontal edge:uv,(on the next figure the corresponding distributívelattice, we cut out a rectangle

19. The horisontal edge determines a rectagle

20. Rectangular lattices • DefinitionThesemimodular lattice L is called rectangular if J(L) the cardinal sum of chains.

21. A backyard (non degenerate source element)a source element s,the corresponding factor is M7. (generalisation: Boole lattice with basket)

23. Remark • For semimodular lattices of w(J(L))>2 we can define geometric conceptsface, and so on.

24. The third power of the three-element chainwith two souce elements, on the next figure you ca see the“result” M3[C3].

25. M3[C4] contains three planes.