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

Advances in Mathematics

Todorčević

2017

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Let R, ∼ be the Rado graph, Emb(R) the monoid of its self-embeddings, P(R) = {f (R) : f ∈ Emb(R)} the set of copies of R contained in R, and I R the ideal of subsets of R which do not contain a copy of R. We consider the poset P(R), ⊂ , the algebra P (R)/I R , and the inverse of the right Green's pre-order on Emb(R), and show that these pre-orders are forcing equivalent to a two step iteration of the form P * π, where the poset P is similar to the Sacks perfect set forcing: adds a generic real, has the ℵ 0 -covering property and, hence, preserves ω 1 , has the Sacks property and does not produce splitting reals, while π codes an ω-distributive forcing. Consequently, the Boolean completions of these four posets are isomorphic and the same holds for each countable graph containing a copy of the Rado graph. 2010 MSC: 05C80, 03C15, 03C50, 03E40, 06A06, 20M20. Theorem 1.2 For each countable non-scattered graph G, ∼ and, in particular, for the Rado graph, the poset P(G) is forcing equivalent to the two-step iteration P * π, where 1 P "π is an ω-distributive forcing" and the poset P is similar to the Sacks forcing: adds a generic real, has the ℵ 0 -covering property (thus preserves ω 1 ), has the Sacks property and does not produce splitting reals. In addition, ro sq P(G) ∼ = ro(P (R)/I R ) + ∼ = ro(P * π) and these complete Boolean algebras are weakly distributive 3 .In fact, if G, ∼ is a countable graph containing a copy of the Rado graph, then these two structures are equimorphic and, by [11], forcing equivalent. So it is sufficient to prove the previous theorem assuming that G, ∼ is the Rado graph.Finally we note that the results of this paper are related to the investigation of the monoids of self-embeddings. We recall that the right Green's pre-order R on a monoid M, ·, 1 is defined by x R y iff x · z = y, for some z. It is easy to check (see [12]) that the poset of copies P(X) of a structure X is isomorphic to the antisymmetric quotient of the pre-order Emb(X), ( R ) −1 and, consequently, these pre-orders are forcing equivalent. Thus, by Theorem 1.2, for the Rado graph we have Emb(R), ( R ) −1 ≡ (P * π) and the Boolean completion of the preorder Emb(R), ( R ) −1 is a weakly distributive complete Boolean algebra. PreliminariesFirst we introduce a convenient notation. If G, ∼ is a graph (namely, if ∼ is a symmetric and irreflexive binary relation on the set G) and K ⊂ H ∈ [G] <ω , letThe object of our study is the Rado graph (the Erdős-Rényi graph, the countable random graph) introduced independently by Erdős and Rényi [2] and Rado [17]. 3 A complete Boolean algebra B is called weakly distributive (or (ω, ·, <ω)-distributive) iff for each cardinal κ and each matrix [bnα : n, α ∈ ω × κ] of elements of B we have n∈ω α∈κ bnα = s:ω→[κ] <ω n∈ω α∈s(n) bnα.

“…it is a graph-isomorphism, by (iii) satisfies (7) and, by (iv), satisfies (8) (7) and (8). Then there is a…”

Section: Theorem 42 For Each Maximal Antichain a In The Poset P(r) Amentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Copies of the random graph

Advances in Mathematics

Todorčević

2017

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“…Let κ = j∈J X j and κ = k∈K Y k be the partitions determined by the relations ≈ ρ and ≈ σ respectively. By (6)…”

Section: Infinite Unary Structuresmentioning

confidence: 99%

“…If X i = X i , ρ i , i ∈ I, are connected L b -structures and X i ∩ X j = ∅, for different i, j ∈ I, then the structure i∈I X i = i∈I X i , i∈I ρ i is the disjoint union of the structures X i , i ∈ I, and the structures X i , i ∈ I, are its components. Fact 3.16 ( [6]) If X is an L b -structure, then at least one of the structures X and X c is connected.…”

Section: Proof Of Theorem 37mentioning

confidence: 99%

Different similarities

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).

“…Thus, the intended classification is in fact the classification of the posets of the form P(X), ⊂ determined by their forcing-related properties. Clearly, this classification induces a coarse classification of structures as well (see [4,5,6,7,8], for countable relational structures).…”

Section: Introductionmentioning

confidence: 99%

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

Annals of Pure and Applied Logic

Todorčević

2016

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Let G be a countable graph containing a copy of the countable random graph (Erdős-Rényi graph, Rado graph), Emb(G) the monoid of its selfembeddings, P(G) = {f [G] : f ∈ Emb(G)} the set of copies of G contained in G, and I G the ideal of subsets of G which do not contain a copy of G. We show that the poset P(G), ⊂ , the algebra P (G)/I G , and the inverse of the right Green's pre-order Emb(G), R have the 2-localization property. The Boolean completions of these pre-orders are isomorphic and satisfy the following law: for each double sequencewhere Bt( <ω ω) denotes the set of all binary subtrees of the tree <ω ω. 2010 MSC: 03C15, 03C50, 03E40, 05C80, 06A06, 20M20.