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Posets of copies of countable non-scattered labeled linear orders. (English) Zbl 07204024
Summary: We show that the poset of copies \(\mathbb{P} (\mathbb{Q}_n )=\langle \{ f[X]: f\in \text{Emb} (\mathbb{Q}_n ) \},\subset \rangle\) of the countable homogeneous universal \(n\)-labeled linear order, \( \mathbb{Q}_n \), is forcing equivalent to the poset \(\mathbb{S} \ast \pi \), where \(\mathbb{S}\) is the Sacks perfect set forcing and \(1_{\mathbb{S}} \Vdash\) “\(\pi\) is an atomless separative \(\sigma \)-closed forcing”. Under CH (or under some weaker assumptions) \(1_{\mathbb{S}} \Vdash `` \pi\) is forcing equivalent to \(P( \omega )\)/Fin”. In addition, these statements hold for each countable non-scattered \(n\)-labeled linear order \(\mathbb{L}\) and we have \(\text{ro } \text{sq} \mathbb{P} (\mathbb{L} )\cong \text{ro } \text{sq} \mathbb{P} (\mathbb{Q}_n )\cong \text{ro } \text{sq} (\mathbb{S} \ast \pi )\).
MSC:
06-XX Order, lattices, ordered algebraic structures
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