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.