I am interested in atomistic lattices satisfying the following property:
If $x \leq y$ and for each atom $a \leq y$ the pair $(a,x)$ is modular then there exists $z$ such that for each atom $b$ the pair $(b,z)$ is modular and $z \land y=x$.
An example of a lattice satisfying such a condition is the lattice of convex sets in any Euclidean space with $A \land B :=A \cap B$, $A \lor B := co(A \cup B)$. It is also satisfied by any matroid lattice.
Is it known in the literature? Can anyone come up with equivalent or slightly stronger condition that would have simpler formulation?