Christian - we have a system for numbering everyone's puzzles, so yours is CW1.

I like it! I like it so much I'm gonna try it. A partition of \$$X\$$ is a subset \$$P \subseteq P(X)\$$, or if you want to show off, an element \$$P \in P(P(X)) \$$. But the first description is better for characterizing partitions. Any subset \$$S \subseteq P(X) \$$ has a meet \$$\bigvee S\$$ which is the union of all the sets in \$$S\$$, and for a partition this union must be all of \$$X\$$, so we need

$$\bigvee P = X$$

I don't see an equally terse way of saying that the sets in \$$P\$$ are pairwise disjoint and nonempty. Clearly this says that the 1-fold intersections of sets \$$P\$$ are nonempty while the 2-fold and higher intersections are empty.