An inequality for the sizes of prime filters of finite distributive lattices

Abstract

Let L be a finite distributive lattice, and let J( L) denote the set of all join-irreducible elements of L. Set J( L) = | J( L)|. For each a ∈ J( L), let u( a) denote the number of elements in the prime filter { x ∈ L: x⩾ a} Our main theorem is Theorem 1. For any finite distributive lattice L, ∑ a ∈J(L) 4 u(a)⩾j(L)4 |L| 2 . The base 4 here can most likely be replaced by a smaller number, but it cannot be replaced by any number strictly between 1 and 1.6159. We also make a few other observations about prime filters and the numbers u( a), a ∈ J( L), among which is: every finite distributive non-Boolean lattice L contains a prime filter of size at most |L|/3 or at least 2|L|/3. The above inequality is certainly not true for all finite lattices. However, we give another inequality, equivalent to the above for distributive lattices, which might hold for all finite lattices. If so, this would give an immediate proof of a conjecture known as Frankl's conjecture.