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Strong semimodular lattices and Frankl’s conjecture. (English) Zbl 1013.06008

Frankl’s conjecture says that in every nontrivial finite lattice \(L\) there is a join-irreducible element \(a\) such that the principal filter \(F(a)\) has at most \(|L|/2\) elements. It is known that this conjecture holds for lower semimodular or section-complemented lattices, but it is unknown whether it holds for upper semimodular lattices. The author introduces the concept of a strong semimodular lattice and proves Frankl’s conjecture for these lattices.

MSC:

06C10 Semimodular lattices, geometric lattices
06C15 Complemented lattices, orthocomplemented lattices and posets
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