  The rankable and compressible sets have been studied for more than a quarter
of a century, ever since Allender [1] and Goldberg and Sipser [6] introduced
the formal study of polynomial-time ranking. Yet even after all that time,
whether the rankable and compressible sets are closed under the most important
boolean and other operations remains essentially unexplored. The present paper
studies these questions for both polynomial-time and recursion-theoretic
compression and ranking, and for almost every case arrives at a Closed, a
Not-Closed, or a Closed-Iff-Well-Known-Complexity-Classes-Collapse result for
the given operation. Even though compression and ranking classes are capturing
something quite natural about the structure of sets, it turns out that they are
quite fragile with respect to closure properties, and many fail to possess even
the most basic of closure properties. For example, we show that with respect to
the join (aka disjoint union) operation: the P-rankable sets are not closed,
whether the semistrongly P-rankable sets are closed is closely linked to
whether P = UP $\cap$ coUP, and the strongly P-rankable sets are closed.
