On distributive sublattices of modular lattices
Let \(L\) be a modular lattice and let \(K\) and \(X\) be distributive sublattices of \(L\). \textit{B. Jónsson} [Proc. Am. Math. Soc. 6, 682-688 (1955; Zbl 0066.28301)] gave necessary and sufficient condition for a sublattice of \(L\) generated by \(K\cup X\) to be distributive. The author presents (under some additional conditions on \(K\) and \(X\)) another sufficient condition ensuring distributivity of a sublattice generated by \(K\cup X\). Theorem. Let \(K\) and \(X\) be distributive sublattices of a modular lattice \(L\) such that (i) for all \(x\in X\) there are \(u\), \(v\in K\cap X\) with \(u\leq x\leq v,\) (ii) for all \(x\), \(y\in X\) and \(u\in K\), if \(x\leq u\leq y\) then \(u\in X\). Then the sublattice of \(L\) generated by \(K\cup X\) is distributive.
distributive sublattice of a modular lattice
scientific article; zbMATH DE number 1226564