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 infinitary distributivity, however.
Any linear order is a distributive lattice.
Let be the category of finite distributive lattices and lattice homomorphisms, and let be the category of finite posets and order-preserving functions. These are contravariantly equivalent, thanks to the presence of an ambimorphic object:
This equivalence is given by the functor
where is the 2-element distributive lattice, and
where is the 2-element poset with .
This is mentioned in
Every distributive lattice, regarded as a category (a (0,1)-category), is a coherent category. Conversely, the notion of coherent category may be understood as a categorification of the notion of distributive lattice. A different categorification is the notion of distributive category.
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.