  Statistical discoveries are often obtained through multiple hypothesis
testing. A variety of procedures exists to evaluate multiple hypotheses, for
instance the ones of Benjamini-Hochberg, Bonferroni, Holm or Sidak. We are
particularly interested in multiple testing procedures with two desired
properties: (solely) monotonic and well-behaved procedures. This article
investigates to which extent the classes of (monotonic or well-behaved)
multiple testing procedures, in particular the subclasses of so-called step-up
and step-down procedures, are closed under basic set operations, specifically
the union, intersection, difference and the complement of sets of rejected or
non-rejected hypotheses. The present article proves two main results: First,
taking the union or intersection of arbitrary (monotonic or well-behaved)
multiple testing procedures results in new procedures which are monotonic but
not well-behaved, whereas the complement or difference generally preserves
neither property. Second, the two classes of (solely monotonic or well-behaved)
step-up and step-down procedures are closed under taking the union or
intersection, but not the complement or difference.
