Abstract
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 (Adv Math 225:2455–2463, 2010), 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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He, P., Wang, Xp. A Characterization of a Semimodular Lattice. Stud Logica 106, 691–698 (2018). https://doi.org/10.1007/s11225-017-9761-9
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DOI: https://doi.org/10.1007/s11225-017-9761-9