Distributive lattices with sufficiently many chain intervals (Q1057292)

A lattice is called reductive (a generalization of weakly atomic) if given \(a<b\), there exist c,d with \(a\leq c<d\leq b\) such that the interval [c,d] is a chain. The paper begins with sufficient conditions for a modular lattice to be reductive (e.g. if it is complete and contains no complete sublattice isomorphic to the power set lattice of the integers). Next it is shown that a distributive lattice is reductive if and only if it has an essential extension which is a product of chains. Finally it is shown how to decompose a distributive lattice subdirectly into a reductive part and a totally irreductive part.