Let $U$ is a complete lattice with least element 0.

*Weak partitioning* is a collection $S$ of nonempty subsets of $U$ such that $\forall x\in S: x\cap\bigcup(S\setminus\{x\})=0$.

*Strong partitioning* is a collection $S$ of nonempty subsets of $U$ such that $\forall A,B\in PS:(A\cap B=\emptyset \Rightarrow \bigcup A\cap\bigcup B=0)$.

Easy to show that every strong partitioning is weak partitioning.

Is weak and strong partitioning the same?

If not, under which additional conditions these are the same?