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Semimodular lattices. (Semimodulare Verbände.) (German) Zbl 0654.06006

The author gives some characterizations of “semimodularity” using join- irreducible elements of a lattice \(L\) of finite height. Some older results are included to clarify the connection with the new results: \(L\) is semimodular iff for join-irreducible elements \(u, v\) and the unique lower neighbor \(u'<u\) of \(u\)there holds \(v\leq b\vee u\), \(v\nleq b\vee u'\) imply \(u\leq b\vee v\vee u')\) iff \(u\wedge b=u'\) implies \(u\vee b\) covers \(b\) iff \(u\wedge b=u'\) implies \((u,b)M^*\) for \(b\in L\). The condition \((u,b)M^*\) is as in F. Maeda and S. Maeda’s book “Theory of symmetric lattices.” Berlin etc.: Springer-Verlag (1970; Zbl 0219.06002), where other characterizations of “semimodular” are given in theorems 7.6, 7.10. They are consequences of the above mentioned result.
Further characterizations can be found in G. Birkhoff’s book “Lattice theory” 3rd ed. Providence, R.I.: AMS (1967; Zbl 0153.02501), and I. Rival’s paper [Algebra Univers. 6, 99–103 (1976; Zbl 0356.06018)].

MSC:

06C10 Semimodular lattices, geometric lattices
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