A Characterization of a Semimodular Lattice

Studia Logica 106 (4):691-698 (2018)

A geometric lattice is the lattice of closed subsets of a closure operator on a set which is zero-closure, algebraic, atomistic and which has the so-called exchange property. There are many profound results about this type of lattices, the most recent one of which, due to Czédli and Schimdt, says that a lattice L of finite length is semimodular if and only if L has a cover-preserving embedding into a geometric lattice G of the same length. The goal of our paper is to offer the following result: a lattice of finite length is semimodular if and only if every cell in L is a 4-element Boolean lattice and the 7-element non-distributive atomistic lattice having 3 atoms is not a cover-preserving sublattice of L.
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DOI 10.1007/s11225-017-9761-9
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Lattice Theory.Garrett Birkhoff - 1940 - Journal of Symbolic Logic 5 (4):155-157.

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