The poset of all copies of the random graph has the 2-localization property

Kurilić, Miloš S.; Todorčević, Stevo

doi:10.1016/j.apal.2016.04.001

Cited by 10 publications

(7 citation statements)

References 13 publications

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“…Results on the theme of copies of an homogeneous structure have been developped in a series of papers by Kurilić, Kuzeljević and Todorcević (e.g. [9,10,11,12]). Some of the results presented here have been the subject of a lecture by the third author in March 2019 at the FG1 Seminar of the Technical University of Vienna [18].…”

Section: An Outline Of the Resultsmentioning

confidence: 99%

The poset of copies for automorphism groups of countable relational structures

Laflamme¹,

Pouzet

Sauer

et al. 2020

Preprint

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Let G be a subgroup of the symmetric group S(U ) of all permutations of a countable set U . Let G be the topological closure of G in the function topology on U U . We initiate the study of the poset G[U ] ∶= {f [U ] f ∈ G} of images of the functions in G, being ordered under inclusion. This set G[U ] of subsets of the set U will be called the poset of copies for the group G. A denomination being justified by the fact that for every subgroup G of the symmetric group S(U ) there exists a homogeneous relational structure R on U such that G is the set of embeddings of the homogeneous structure R into itself and G[U ] is the set of copies of R in R and that the set of bijections G ∩ S(U ) of U to U forms the group of automorphisms of R.

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Section: An Outline Of the Resultsmentioning

confidence: 99%

The poset of copies for automorphism groups of countable relational structures

Laflamme¹,

Pouzet

Sauer

et al. 2020

Preprint

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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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“…Example 3. 13 The implications i and j can not be reversed. Let X = (0, 1) Q , ≤ and Y = (0, 1] Q , ≤ be suborders of the rational line, Q.…”

Section: Example 311mentioning

confidence: 99%

“…Writing P(X) instead of P(X), ⊂ , some coarser classifications of structures are obtained if the equality is replaced by the following weaker conditions: P(X) ∼ = P(Y) (implied by Emb(X) ∼ = Emb(Y)), sq P(X) ∼ = sq P(Y) (where sq P denotes the separative quotient of a poset P), and P(X) ≡ P(Y) (the forcing equivalence of posets of copies). Concerning the last (and the coarsest non-trivial) similarity relation we note that the forcing related properties of posets of copies was investigated for countable structures in general in [6], for equivalence relations and similar structures in [7], for ordinals in [8], for scattered and non-scattered linear orders in [9] and [11], and for several ultrahomogeneous structures in [10], [11], [12], and [13].…”

Section: Introductionmentioning

confidence: 99%

Different similarities

Kurilić

2015

Arch. Math. Logic

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We establish the hierarchy among twelve equivalence relations (similarities) on the class of relational structures: the equality, the isomorphism, the equimorphism, the full relation, four similarities of structures induced by similarities of their self-embedding monoids and intersections of these equivalence relations. In particular, fixing a language L and a cardinal κ, we consider the interplay between the restrictions of these similarities to the class Mod L (κ) of all L-structures of size κ. It turns out that, concerning the number of different similarities and the shape of the corresponding Hasse diagram, the class of all structures naturally splits into three parts: finite structures, infinite structures of unary languages, and infinite structures of non-unary languages (where all these similarities are different).

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The poset of all copies of the random graph has the 2-localization property

Cited by 10 publications

References 13 publications

The poset of copies for automorphism groups of countable relational structures

The poset of copies for automorphism groups of countable relational structures

Antichains of Copies of Ultrahomogeneous Structures

Different similarities

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