## 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:

 [1] Birkhoff, G., Lattice theory, (1967), Amer. Math. Soc New York · Zbl 0126.03801 [2] Faigle, U., Geometries on partially ordered sets, J. combin. theory ser. B, 28, 26-51, (1980) · Zbl 0359.05018 [3] Faigle, U.; Richter, G.; Stern, M., Geometric exchange properties in lattices of finite length, Algebra univ., 19, 355-365, (1984) · Zbl 0548.06003 [4] Stern, M., On complemented modular lattices of finite length, () [5] Stern, M., Strongness in semimodular lattices, Discrete math., 80, 79-88, (1990) · Zbl 0724.06006
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