Frankl's Conjecture for Large Semimodular and Planar Semimodular

A lattice L is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a join-irreducible element f ∈ L such that at most half of the elements x of L satisfy f ≤ x. Frankl’s conjecture, also called as union-closed sets conjecture, is well-known in combinatorics, and it is equivalent to the statement that every finite lattice satisfies Frankl’s conjecture. Let m denote the number of nonzero join-irreducible elements of L .I t is well-known that L consists of at most 2 m elements. Let us say that L is large if it has more than 5 · 2 m�3 elements. It is shown that every large semimodular lattice satisfies Frankl’s conjecture. The second result states that every finite semimodular planar lattice L satisfies Frankl’s conjecture. If, in addition, L has at least four elements and its largest element is joinreducible then there are at least two choices for the above-mentioned f .