Posets of copies of countable scattered linear orders

Kurilić, Miloš S.

doi:10.1016/j.apal.2013.11.005

Cited by 13 publications

(18 citation statements)

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“…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. Under CH, it is forcing equivalent to the poset (P (ω)/ Fin) + .…”

Section: Introductionmentioning

confidence: 99%

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%

Forcing with copies of countable ordinals

Kurilić¹

2014

Proc. Amer. Math. Soc.

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“…So, for example, for the structures from column D of Diagram 1 of [6] the corresponding posets are forcing equivalent to an atomless ω 1closed poset and, consistently, to P (ω)/Fin. This class of structures includes all scattered linear orders [9] (in particular, all countable ordinals [8]), all structures with maximally embeddable components [7] (in particular, all countable equivalence relations and all disjoint unions of countable ordinals) and in this paper we show that it contains a large class of ultrahomogeneous structures.…”

Section: Introductionmentioning

confidence: 56%

Isomorphic and strongly connected components

Kurilić

2014

Arch. Math. Logic

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We study the partial orderings of the form P(X), ⊂ , where X is a binary relational structure with the connectivity components isomorphic to a strongly connected structure Y and P(X) is the set of (domains of) substructures of X isomorphic to X. We show that, for example, for a countable X, the poset P(X), ⊂ is either isomorphic to a finite power of P(Y) or forcing equivalent to a separative atomless σ -closed poset and, consistently, to P(ω)/Fin. In particular, this holds for each ultrahomogeneous structure X such that X or X c is a disconnected structure and in this case Y can be replaced by an ultrahomogeneous connected digraph.

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“…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. Fourth, the order types of the maximal chains in the posets of copies of countable ultrahomogeneous graphs and countable ultrahomogeneous partial orders are described in [10,11].…”

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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Posets of copies of countable scattered linear orders

Abstract: We show that the separative quotient of the poset P(L), ⊂ of isomorphic suborders of a countable scattered linear order L is σ-closed and atomless. So, under the CH, all these posets are forcing-equivalent (to (P (ω)/ Fin) + ). 1 1 2010 MSC: 06A05, 06A06, 03C15, 03E40, 03E35.

Cited by 13 publications

References 5 publications

Forcing with copies of countable ordinals

Forcing with copies of countable ordinals

Isomorphic and strongly connected components

Antichains of Copies of Ultrahomogeneous Structures

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