The nullary forms of distributivity hold in any lattice:
Any lattice that satisfies one of the two binary distributivity laws must also satisfy the other; isn't that nice? (This may safely be left as an exercise.) This convenience does not extend to infinitary distributivity, however.
As mentioned above, the theory of distributive lattices is self-dual, something that is proved in almost any account (or left as an exercise), but which is not manifestly obvious from the standard definition which chooses one of the two distributivity laws and goes from there. In this section we provide some other characterizations or axiomatizations which are manifestly self-dual.
Here is one such characterization:
A lattice is distributive if and only if the identity
Again this may be left as a (somewhat mechanical) exercise.
Perhaps more useful in practice is the characterization in terms of “forbidden sublattices” due to Birkhoff. Namely, introduce the “pentagon” as the 5-element lattice where and is incomparable to , so that and . Introduce the “thick diamond” as the 5-element lattice with pairwise incomparable. Both and are self-dual. Birkhoff’s characterization is the following (manifestly self-dual) criterion.
A lattice is distributive if and only if there is no embedding of or into that preserves binary meets and binary joins.
This can be useful for determining distributivity or its failure, especially in cases where one can visualize a lattice via its Hasse diagram.
The necessity of the forbidden sublattice condition is clear in view of the fact that the cancellation law stated in the next result fails in and . This result gives another self-dual axiomatization of distributive lattices.
A lattice is distributive if and only if the cancellation law holds: for all , we have whenever and (for some ).
“Only if”: if and , then
which implies , and similarly we have .
and so by cancellation of the 's, we conclude . Similarly (dually), for , we have . Hence is modular.
Now we show is distributive. Let and consider the three elements
where the non-definitional equalities follow from modularity. Using the first expressions, we compute
where the third and fourth lines use modularity. By symmetry in the letters , we also have . Now the second expressions are dual to the first, so by duality we compute
Now by cancellation of the 's, we may conclude , but in that case we obtain
so that is distributive by Proposition 1.
While the expressions for in the preceding proof may look as though they come out of thin air, the underlying idea is that the sublattice of generated by is the image of a lattice map out of the free modular lattice on three elements. The only obstruction to distributivity in is the presence of an -sublattice appearing in the center of its Hasse diagram. The middle elements of that sublattice correspond to the formal expressions for given above, and the proof shows that under the cancellation law, we have in , making the thick diamond collapse to a point in and removing the obstruction to distributivity.
From Proposition 2, it is not very hard to deduce Birkhoff’s theorem. The presence of a copy of or in a non-distributive lattice is deduced from a failure of the cancellation law where we have three elements with , , and . If are comparable, say , then the set forms an . If are incomparable, then we have either , or , or both and ; in the first two cases we get an (e.g., for the first case), and in the third case the set forms an .
Any linear order is a distributive lattice.
An integral domain is a Prüfer domain? iff its lattice of ideals is distributive. The classical example is ; equivalently, the (opposite of the) multiplicative monoid ordered by divisibility, with at the bottom and at the top.
The lattice of Young diagrams ordered by inclusion is distributive.
A distributive lattice that is complete (not necessarily completely distributive) may be infinitely distributive or said to satisfiy the infinite distributive law :
Note that this property does not imply the dual co-infinitely distributive property:
Instead this dual gives the lattice co-Heyting structure where the co-implication or “subtraction” () is
Since a finite distributive lattice is completely distributive it is a bi-Heyting lattice, as shown above.
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 hom-functor
where is the 2-element distributive lattice, and in the other direction by
where is the 2-element poset with .
This baby form of Birkhoff duality is (in one form or another) mentioned in many places; the formulation in terms of hom-functors may be found 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.