The nullary forms of distributivity follow automatically:
Any lattice that satisfies one of the two binary distributivity laws must also satisfy the other; isn't that nice? This convenience does not extend to infintary distributivity, however.
Conversely, the notion of coherent category may be understood as a categorification of the notion of distributive lattices.
The completely distributive algebraic lattices (the frames of opens of Alexandroff locales ) form a reflective subcategory of that of all distributive lattices. The reflector is called canonical extension.