  It is known that for any class C closed under union and intersection, the
Boolean closure of C, the Boolean hierarchy over C, and the symmetric
difference hierarchy over C all are equal. We prove that these equalities hold
for any complexity class closed under intersection; in particular, they thus
hold for unambiguous polynomial time (UP). In contrast to the NP case, we prove
that the Hausdorff hierarchy and the nested difference hierarchy over UP both
fail to capture the Boolean closure of UP in some relativized worlds.
  Karp and Lipton proved that if nondeterministic polynomial time has sparse
Turing-complete sets, then the polynomial hierarchy collapses. We establish the
first consequences from the assumption that unambiguous polynomial time has
sparse Turing-complete sets: (a) UP is in Low_2, where Low_2 is the second
level of the low hierarchy, and (b) each level of the unambiguous polynomial
hierarchy is contained one level lower in the promise unambiguous polynomial
hierarchy than is otherwise known to be the case.
