Posets of Copies of Countable Non-Scattered Labeled Linear Orders

Abstract

We show that the poset of copies ℙ(ℚn) = 〈 { f[X] : f∈ Emb (ℚn) } , ⊂ 〉 of the countable homogeneous universal n-labeled linear order, ℚn, is forcing equivalent to the poset S∗ π, where S is the Sacks perfect set forcing and 1 S⊢ “π is an atomless separative σ-closed forcing”. Under CH (or under some weaker assumptions) 1 S⊢ “π is forcing equivalent to P(ω)/Fin”. In addition, these statements hold for each countable non-scattered n-labeled linear order L and we have rosq ℙ(L) ≅ rosq ℙ(ℚn) ≅ rosq (S∗ π).