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Strongness in semimodular lattices. (English) Zbl 0724.06006

The author gives another characterization of strongness in semimodular lattices \({\mathcal L}\) of finite length [cf. also: the author, Wiss. Z., Martin-Luther-Univ. Halle-Wittenberg, Math.-Naturwiss. Reihe 38, No.5, 21-23 (1989; Zbl 0704.06004)]: \({\mathcal L}\) is strong iff it does not contain a certain hexagon sublattice (Theorem 2) iff \(a'=a_+\) for any \(a\in {\mathcal L}\) (Theorem 5); here \(a'\) denotes the join of all lower covers \(v'\) of join-irreducible elements v with \(v\leq a\), and \(a_+\) means the meet of all lower covers of a.

MSC:

06C10 Semimodular lattices, geometric lattices

Citations:

Zbl 0704.06004
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References:

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