Frankl's conjecture and the dual covering property (Q1043790)

Frankl's conjecture states that for any union-closed family of subsets \(F\) of a finite set \(X\) with at least two members there exists an element \(x\) of \(X\) such that at least half of the members of \(F\) contain \(x\). B. Poonen reformulated this for lattices as follows: Let \(L\) be a finite lattice with at least two elements. Then there is a join-irreducible element \(j\) such that the number of elements in the principal filter generated by \(j\) does not exceed half of the size of \(L\). The present authors generalize this for posets where they define that an element \(j\) of a poset is join-irreducible if \(j\) covers a unique element. It was already proved that Frankl's conjecture for lattices is valid for some large classes of lattices including modular lattices, relatively complemented lattices or lower semimodular lattices. The main result here is: Frankl's conjecture for posets is true for posets satisfying the dual covering property. Such posets are treated in the paper.