An Identity of Distributive Lattices

In a finite distributive lattice $Ł$ we define two functions $s(α)=|\{δ\in \mathcal{L} | δ\leq α\}|$ and $l(α)=|\{δ\in \mathcal{L} | δ\geq α\}|$. 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.

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