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 $M$?

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 $M$?