Theorem 1. Let L be a finite lattice. Let δ be an equivalence relation on L with intervals as equivalence classes. Then δ is a congruence relation if the following
condition and its dual hold:
(C+) If x is covered by y, z ∈ L and x ≡ y (mod δ), then z ≡ y + z (mod δ).
There are lots of important, and complicated, things (Burke).