  A numbering of a countable family $S$ is a surjective map from the set of
natural numbers $\omega$ onto $S$. The paper studies Rogers semilattices, i.e.
upper semilattices induced by the reducibility between numberings, for families
$S\subset P(\omega)$. Working in set theory ZF+DC+PD, we obtain the following
results on families from various levels of the analytical hierarchy.
  For a non-zero number $n$, by $E^1_n$ we denote $\Pi^1_n$ if $n$ is odd, and
$\Sigma^1_n$ if $n$ is even. We show that for a finite family $S$ of $E^1_n$
sets, its Rogers $E^1_n$-semilattice has the greatest element if and only if
$S$ contains the least element under set-theoretic inclusion. Furthermore, if
$S$ does not have the $\subseteq$-least element, then the corresponding Rogers
$E^1_n$-semilattice is upwards dense.
