Distributive lattices arising from a collection of sets closed under intersection.

Hello everyone,

It's well known that every collection of subsets of a set X which is closed both under intersection and reunion is also a distributive lattice (the order relation being sets inclusion of course). However, if I suppose the collection to be intersection closed only, the result is no longer true; but the collection is still a lattice. Does anybody know when such a lattice is distributive ?

Thank you,

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I don't think there is a satisfactory answer. If $L$ is a lattice and $x\in L$, let $V_x = \lbrace y\in L : y\leq x\rbrace$. Then $L$ is isomorphic to the set of $V_x$'s ordered by inclusion, and meet in $L$ corresponds to intersection of the $V_x$'s. Thus the question is equivalent to: given a lattice $L$, when is it distributive? –  Richard Stanley Feb 23 '11 at 19:08
Also, the collection closed under intersection is a semilattice. (It can be made into a lattice by adjoining at most one element, but sometimes at least one element is needed.) Gerhard "Ask Me About System Design" Paseman, 2011.02.23 –  Gerhard Paseman Feb 23 '11 at 23:58
Oh, ok, sure thank you. Do you know how I may accept your answers ? I'am searching for the "check mark" but I don't find it ... –  Selim Mar 2 '11 at 20:52
I'll post an answer for you to accept. Gerhard "Ask Me About System Design" Paseman, 2011.03.02 –  Gerhard Paseman Mar 2 '11 at 21:01

From Richard Stanley:

I don't think there is a satisfactory answer. If $L$ is a lattice and $x\in L$, let $V_x = \lbrace y\in L : y\leq x\rbrace$. Then $L$ is isomorphic to the set of $V_x$'s ordered by inclusion, and meet in $L$ corresponds to intersection of the $V_x$'s. Thus the question is equivalent to: given a lattice $L$, when is it distributive?

From Gerhard Paseman:

Also, the collection closed under intersection is a semilattice. (It can be made into a lattice by adjoining at most one element, but sometimes at least one element is needed.)