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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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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.) Gerhard "Ask Me About System Design" Paseman, 2011.03.02 |
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