  Let \alpha be a countable ordinal and \P(\alpha) the collection of its
subsets isomorphic to \alpha. We show that the separative quotient of the set
\P (\alpha) ordered by the inclusion is isomorphic to a forcing product of
iterated reduced products of Boolean algebras of the form P(\omega
^\gamma)/I(\omega ^\gamma), where \gamma is a limit ordinal or 1 and I(\omega
^\gamma) the corresponding ordinal ideal. Moreover, the poset \P(\alpha) is
forcing equivalent to a two-step iteration P(\omega)/Fin * \pi, where \pi is an
\omega_1-closed separative pre-order in each extension by P(\omega)/Fin and, if
the distributivity number is equal to\omega_1, to P(\omega)/Fin. Also we
analyze the quotients over ordinal ideals P(\omega ^\delta)/I(\omega ^\delta)
and their distributivity and tower numbers.
