Let $L$ be an atomistic ortholattice (i.e. every element can be written as a join of atoms) with top and bottom elements 0 and 1, and let $M$ be a distributive atomic sub-ortholattice of $L$.


Is $M$ generated by its atoms, in the sense that every element in $M$ can be written as a join of the atoms in $M$?



distributive sublattices of atomistic ortholattices
No. Let $L$ be the power set of ${\bf N}$ ordered by inclusion. It is an atomic Boolean algebra. Let a subset of ${\bf N}$ belong to $M$ if its characteristic function is periodic. Then $M$ is a nonatomic Boolean subalgebra of $L$.


