Isomorphism between Order Ideals and Distributive Lattices

[I<3Gauss]

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.

 

[Office_Shredder]

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

 

[I<3Gauss]

That's an interesting idea but however, i also do not have the guts yet or the know how to make such an assumption.

 

[Office_Shredder]

Think about the map

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

 

[I<3Gauss]

thanks for the tip, i think i kind of see why this is now. I guess if a poset P was the join irreducible set of some Lattice L, and this particular poset P is isomorphic to the join-irreducibles of L(P), which is the set of all Ix, then L would be isomorphic to L(P)?

The tip that you gave would easily prove that a poset P would be isomorphic to the join-irreducibles of J(P), but I guess what I still don't understand is the following:

How does proving P is the subposet of join-irreducibles of L is isomorphic to the join-irreducibles of L(P) help us in proving that L is indeed isomorphic to L(P)?