Birkhoff's representation theorem

Remarks

Every element $a$ of a finite lattice is the join of the join-irreducible elements below it, so the map is injective and order-preserving in both directions. Distributivity is used for surjectivity: if $j \le x \vee y$ with $j$ join-irreducible, then $j = (j \wedge x) \vee (j \wedge y)$, so $j \le x$ or $j \le y$. This shows that the map sends joins to unions, and so every lower set $D$ is the image of $\bigvee D$.

Example: the divisors of $12$ ordered by divisibility form a distributive lattice. Its join-irreducible elements are the prime powers $2$, $4$ and $3$, with $2 \le 4$ and $3$ incomparable to both. This poset has six lower sets, matching the six divisors $1, 2, 3, 4, 6, 12$; for example $6$ corresponds to $\set{2, 3}$.

Distributivity is necessary: in the diamond lattice M3 all three middle elements are join-irreducible and pairwise incomparable, so the lower sets of $J(M_3)$ number $8$, while $M_3$ has $5$ elements.

Going deeper

For a student

The theorem gives a correspondence between finite posets and finite distributive lattices: a poset $P$ gives the lattice of its lower sets, and a lattice $L$ gives the poset $J(L)$. This correspondence extends to a duality between the category of finite posets with order-preserving maps and the category of finite distributive lattices with bounded lattice homomorphisms. For infinite distributive lattices, the analogue needs topology, as in Priestley duality. In the infinite case, prime filters take the place of join-irreducible elements, and the prime ideal theorem for distributive lattices guarantees that there are enough of them.