An extremal problem for finite topologies and distributive lattices

Abstract

Let ( r 1, r 2, …) be a sequence of non-negative integers summing to n. We determine under what conditions there exists a finite distributive lattice L of rank n with r i join-irreducibles of rank i, for all i = 1, 2, …. When L exists, we give explicit expressions for the greatest number of elements L can have of any given rank, and for the greatest total number of elements L can have. The problem is also formulated in terms of finite topological spaces.