An Identity of Distributive Lattices

Abstract

In a finite distributive lattice $Ł$ we define two functions $s(\alpha )=|\{\delta \in \mathcal{L}|\delta \le \alpha \}|$ and $l(\alpha )=|\{\delta \in \mathcal{L}|\delta \ge \alpha \}|$. In this present article we prove that the sum of these two functions over a finite distributive lattice are equal. Using this identity we give a formula for the number of non-comparable pairs of elements in a finite distributive lattice.