2. The representation theorem for finite distributive lattices

2.1 Theorem. Let $L$ be a finite distributive lattice, and let $P =$   JI$(L)$, the set of join-irreducibles of $L$ with the partial order inherited from $L$. Then $L \cong$   Downsets$(P)$.

Corollary. Every finite distributive lattice is isomorphic to a sublattice of some power set, namely    Pow$(X)$ where $X$ is the set of its join-irreducibles.

(Actually, every infinite distributive lattice is likewise isomorphic to a lattice of sets, but in a more subtle way. This is the Stone Representation Theorem.)

Remark. In a finite distributive lattice, every element is uniquely an irredundant join of join-irreducibles. (Contrast this with the case of $M _ 3$.)

An example of a lattice of downsets is shown in Figure .

2.2 Corollary. Let $L$ be a finite distributive lattice, and let $Q =$   JI$(L) ^ \cup$, the dual of JI$(L)$. Then $L \cong$   2$^ Q$, the lattice of isotone functions from $Q$ to 2.

Proof. Observe that for a partially ordered set $Q$,    2$^ Q \cong$   Downsets$(P ^ \cup)$. In fact, each isotone function $f: Q\rightarrow$   2 gives a decomposition of $Q$ into $f ^ {-1}(0)$, which is a downset, and $f ^ {-1}(1)$, which is the complementary upset, and in the other direction each downset gives a complementary upset and isotone function. This correspondence reverses order since $f \leq g$ when $g$ ``has more 1's'' than $f$, or equivalently ``fewer 0's'', so that $f ^ {-1}(0) \supseteq f ^ {-1}(1)$.

Note. As we shall see later, infinite distributive lattices can also be represented as lattices of subsets.