Proposal: Partitioning a lattice element

I’ve given two different definitions for partitioning an element of a complete lattice (generalizing partitioning of a set). I called them weak partitioning and strong partitioning.

The problem is whether these two definitions are equivalent for all complete lattices, or if are not then under which additional conditions these are equivalent. (I suspect these may be equivalent under the additional condition that our lattice is distributive.)

I may seem greedy producing now already second or third (dependently on how to count) polymath proposal about my problems, but I just want to present all of my proposal for (hopefully) fair judges Terence Tao and Tim Gowers. The more the better.