Lattice formulation of the union-closed sets conjecture

Prove that every finite lattice L with more than one element contains a join-irreducible element j such that the principal filter of j, (↑j)L = {x ∈ L : j ≤L x}, has cardinality at most |L|/2.

Background

The paper studies the lattice-theoretic formulation of the union-closed sets conjecture, also known as Frankl’s conjecture. In this formulation, finite union-closed set families correspond, via the intersection-closed dual, to finite lattices. The conjecture asserts that every finite lattice with more than one element has a join-irreducible element lying below at most half of the lattice’s elements.

The paper does not resolve this conjecture; instead, it derives necessary structural conditions that any minimum-size counterexample would have to satisfy, including restrictions on irreducible elements and on the sizes of principal filters.