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.(adsbygoogle = window.adsbygoogle || []).push({});

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# Isomorphism between Order Ideals and Distributive Lattices

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