distributive lattice

A distributive lattice is a lattice in which meet and join interact as cleanly as intersection and union of sets do: each spreads over the other. This is the most ‘set-like’ kind of lattice, and indeed the punchline of the theory is that every distributive lattice can be realized as an honest lattice of sets, so distributivity is precisely the abstract fingerprint of set operations.

Formally, a lattice (L, ∧, ∨) is distributive if it satisfies the distributive identity a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) for all a, b, c. By a standard lemma this single law is equivalent to its dual a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c), so one law suffices. Distributive lattices form a subvariety of all lattices, cut out by adding this identity to the lattice axioms.

There is a clean forbidden-sublattice test: a lattice is distributive if and only if it contains neither the diamond M_3 (the five-element lattice of three pairwise-incomparable middle elements) nor the pentagon N_5 as a sublattice. Birkhoff's representation theorem sharpens the set-realization to the finite case: every finite distributive lattice is isomorphic to the lattice of down-sets (order ideals) of the poset of its join-irreducible elements — a perfect duality between finite distributive lattices and finite posets.