An inequality for the sizes of prime filters of finite distributive lattices
The results in this paper were inspired by an old problem which became known as Frankl's conjecture, i.e., for every (nonempty) finite union-closed family of finite sets, there is an element contained in at least half of them. There is an equivalent lattice-theoretic statement: every finite lattice \(L\) contains a join-irreducible element \(a\) such that the number \(u(a)\) of elements in the prime filter \(\{b \in L: a\leq b\}\) is \(\leq |L|/2\). The main result (Theorem 1) in the paper shows that for any finite distributive lattice \(L\), \[ \sum_{a\in J(L)}4^{u(a)}\geq j(L)4^{|L|/2}, \] where \(J(L)\) denote the set of all join-irreducible elements of \(L\) and \(j(L)=|J(L)|\). It should be noted that the result implies Frankl's conjecture for finite distributive lattices. The authors prove that every finite non-Boolean lattice \(L\) contains either a prime filter of size at most \(|L|/3\) or a prime filter of size at least \(2|L|/3\).
scientific article; zbMATH DE number 1334540