Forcing by non-scattered sets

Kurilić, Miloš S.; Todorčević, Stevo

doi:10.1016/j.apal.2012.02.004

Cited by 16 publications

(21 citation statements)

References 3 publications

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“…For example, under CH all countable linear orders are partitioned in only two classes. Namely, by [5], CH implies that for a non-scattered countable linear order L the poset P(L), ⊂ is forcing equivalent to the iteration S * (P (ω)/ Fin) + , where S is the Sacks forcing. Otherwise, for scattered orders, by [7] we have Theorem 1.1 For each countable scattered linear order L the separative quotient of the poset P(L), ⊂ is ω 1 -closed and atomless.…”

Section: Introductionmentioning

confidence: 99%

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Forcing with copies of countable ordinals

Kurilić¹

2014

Proc. Amer. Math. Soc.

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Let α be a countable ordinal and P(α) the collection of its subsets isomorphic to α. We show that the separative quotient of the poset P(α), ⊂ is isomorphic to a forcing product of iterated reduced products of Boolean algebras of the form P (ω γ )/I ω γ , where γ ∈ Lim ∪{1} and I ω γ is the corresponding ordinal ideal. Moreover, the poset P(α), ⊂ is forcing equivalent to a twostep iteration of the form (P (ω)/ Fin) + * π, where [ω] "π is an ω 1 -closed separative pre-order" and, if h = ω 1 , to (P (ω)/ Fin) + . Also we analyze the quotients over ordinal ideals P (ω δ )/I ω δ and the corresponding cardinal invariants h ω δ and t ω δ .

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Section: Introductionmentioning

confidence: 99%

“…By (ii) of Claim 3.3 we have A ∈ (Fin ×I ω β ) + and (5) is proved. So (3) is true.By(5) we have I L = Fin ×I ω β . Since ω β+1 , ∈ ∼ = L, by Fact 2.1(a) we have I ω β+1 ∼ = I L and, hence,…”

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confidence: 99%

Forcing with copies of countable ordinals

Kurilić¹

2014

Proc. Amer. Math. Soc.

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“…First, antichains in the poset of copies of the random (Rado) graph were analyzed in [12]. Second, forcing-related properties of the posets of copies of ultrahomogeneous structures were investigated in [13,14,15]. Third, in [7,8,9] a classification of relational structures with respect to the properties of posets P(X), ⊂ is given.…”

Section: Theorem 12 (Schmerl) Each Countable Ultrahomogeneous Partial...mentioning

confidence: 99%

Antichains of Copies of Ultrahomogeneous Structures

Kurilić¹,

Kuzeljević²

2019

Preprint

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We investigate possible cardinalities of maximal antichains in the poset of copies P(X), ⊂ of a countable ultrahomogeneous relational structure X. It turns out that if the age of X has the strong amalgamation property, then, defining a copy of X to be large iff it has infinite intersection with each orbit of X, the structure X can be partitioned into countably many large copies, there are almost disjoint families of large copies of size continuum and, hence, there are (maximal) antichains of size continuum in the poset P(X). Finally, we show that the posets of copies of all countable ultrahomogeneous partial orders contain maximal antichains of cardinality continuum and determine which of them contain countable maximal antichains. That holds, in particular, for the random (universal ultrahomogeneous) poset.

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“…Case 1: L is non-scattered. By [3], for each non-scattered linear order L the poset P(L), ⊂ is forcing equivalent to the two-step iteration S * π, where S is the Sacks forcing and 1 S "π is a σ-closed forcing". If the equality sh(S) = ℵ 1 or PFA holds in the ground model, then the second iterand is forcing equivalent to the poset (P (ω)/ Fin) + of the Sacks extension.…”

Section: Linear Ordersmentioning

confidence: 99%

From A1 to D5: Towards a Forcing-Related Classification of Relational Structures

Kurilić¹

2014

J. symb. log.

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We investigate the partial orderings of the form P(X), ⊂ , where X is a relational structure and P(X) the set of the domains of its isomorphic substructures. A rough classification of countable binary structures corresponding to the forcing-related properties of the posets of their copies is obtained. Theorem 6.1 For a countable binary relational structure X = ω, ρ the following conditions are equivalent:ρ is isomorphic to one of the following relational structures: 1 The empty relation, ω, ∅ , 2 The complete graph, ω, ω 2 \ ∆ ω , 3 The natural strict linear order on ω, ω, < , 4 The inverse of the natural strict linear order on ω, ω, < −1 , 5 The diagonal relation, ω, ∆ ω , 6 The full relation, ω, ω 2 , 7 The natural linear order on ω, ω, ≤ , 8 The inverse of the natural linear order on ω, ω, ≤ −1 ; (d) P(X) is a somewhere dense set in [ω] ω , ⊂ ; (e) I X = Fin. Then the poset sq P(X), ⊂ = (P (ω)/ Fin) + is atomless and σ-closed. Proof. The implication (a) ⇒ (b) is trivial and it is easy to check (c) ⇒ (a). (b) ⇒ (c). Let P(X) be a dense set in [ω] ω , ⊂ . Claim 1. The relation ρ is reflexive or irreflexive.Proof of Claim 1. If R = {x ∈ ω : xρx} ∈ [ω] ω , then there is C ⊂ R such that ω, ρ ∼ = C, ρ C and, since ρ C is reflexive, ρ is reflexive as well. Otherwise we have I = {x ∈ ω : ¬xρx} ∈ [ω] ω and, similarly, ρ must be irreflexive.

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Forcing by non-scattered sets

Cited by 16 publications

References 3 publications

Forcing with copies of countable ordinals

Forcing with copies of countable ordinals

Antichains of Copies of Ultrahomogeneous Structures

From A1 to D5: Towards a Forcing-Related Classification of Relational Structures

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