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Semimodularity in interval lattices. (Russian. English summary) Zbl 0664.06007
For a lattice L let Int(L) be the lattice of all intervals of L (including the empty set). The author proves the following results. Int(L) is lower semimodular if and only if for each pair of incomparable elements x and y in L there exist \(u,v\in L\) such that \(x\wedge y<u<x<v<x\vee y\). Int(L) is upper semimodular iff card \(L\leq 2\). Let \(L_ 1\) be a lattice satisfying either the descending chain condition or the ascending chain condition; then \(Int(L_ 1)\) is lower semimodular iff \(L_ 1\) is a chain.
Reviewer: J.Jakubik

MSC:
06C10 Semimodular lattices, geometric lattices
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