# Isomorphism between Order Ideals and Distributive Lattices

by I<3Gauss
Tags: distributive, ideals, isomorphism, lattices, order
 P: 14 The poset on the set of order ideals of a poset p, denoted L(p), is a distributive lattice, and it is pretty clear why this is since the supremum of two order ideals and the infimum of 2 order ideals are just union and intersection respectively, and we know that union and intersection are distributive operations. However, Richard Stanley, in his book Enumerative Combinatorics, states that for any distributive lattice L, there exists a poset P such that L(P) is isomorphic to L. I was wondering what the proof is for this particular statement is since I have been trying to prove this to no avail. I appreciate the responses.
 Mentor P: 4,198 At first glance, it seems like it should be isomorphic to its own set of order ideals. I feel like that's too easy of an answer though, and I'm making an assumption that's not true
 P: 14 That's an interesting idea but however, i also do not have the guts yet or the know how to make such an assumption.
Mentor
P: 4,198

## Isomorphism between Order Ideals and Distributive Lattices

$$x\mapsto I_x$$ where $$I_x$$ is the order ideal of x.