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Let \alpha be a countable ordinal and \P(\alpha) the collection of itssubsets isomorphic to \alpha. We show that the separative quotient of the set\P (\alpha) ordered by the inclusion is isomorphic to a forcing product ofiterated 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) isforcing 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, ifthe distributivity number is equal to\omega_1, to P(\omega)/Fin. Also weanalyze the quotients over ordinal ideals P(\omega ^\delta)/I(\omega ^\delta)and their distributivity and tower numbers.

Forcing With Copies of Countable Ordinals