By a partial partition of $X$, I mean a collection of non-empty subsets of $X$ that are disjoint. Given partial partitions $\mathcal{A}$ and $\mathcal{B}$ of $X$, define $\mathcal{A} \leq \mathcal{B}$ if and only if the following properties hold:
- Covering Property. $\bigcup \mathcal{A} \subseteq \bigcup \mathcal{B}$.
- Inclusion-Exclusion Property For all $A \in \mathcal{A}$ and $B \in \mathcal{B}$, either $A \supseteq B$, or $A \cap B = \emptyset$.
I'm just wondering whether this is a partial order on partial partitions? If so, the least element of this poset is clearly $\emptyset$, and the greatest element is clearly the discrete partition. If this really is a partial order, I'd also be interested to know whether it has meets and joins, too.