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Dually atomistic lattices. (English) Zbl 0759.06006

A lattice \(L\) of finite length is called strong if, for all join- irreducible elements \(u\in L\) and for all \(x\in L\), \(u\leq x\lor u'\) implies \(u\leq x\), where \(u'\) denotes the unique lower cover of \(u\). The main results of the paper are the following: (i) A lattice of finite length is dually atomistic if and only if each atom of each upper interval has a complement in this interval. (ii) A lattice of finite length whose greatest element is a join of atoms, is dually atomistic if and only if its dual lattice is strong. These results are applied to semimodular lattices yielding some generalizations of well-known results on modular and geometric lattices.

MSC:

06B05 Structure theory of lattices
06C10 Semimodular lattices, geometric lattices
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References:

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