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.
A maximal almost disjoint (mad) family ⊆ [ω]ω is Cohen-stable if and only if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-unstable. It is shown that a mad family. .is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the sets G[A]. A ∈ are nowhere dense. An ℵ0-mad family, . is a mad family with the property that given any countable family ℬ ⊂ [ω]ω such that each element of ℬ meets infinitely many elements of in an infinite set there is an element of meeting each element of ℬ in an infinite set. It is shown that Cohen-stable mad families exist if and only if there exist ℵ0-mad families. Either of the conditions b = c or a < cov() implies that there exist Cohen-stable mad families. Similar results are obtained for splitting families. For example, a splitting family. . is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the boundaries of the sets G[S], S ∈ are nowhere dense. Also. Cohen-stable splitting families of cardinality ≤ κ exist if and only if ℵ0-splitting families of cardinality ≤ κ exist.
A relational structure X is called reversible iff each bijective homomorphism from X onto X is an isomorphism, and linear orders are prototypical examples of such structures. One way to detect new reversible structures of a given relational language L is to notice that the maximal or minimal elements of isomorphism-invariant sets of interpretations of the language L on a fixed domain X determine reversible structures. We isolate certain syntactical conditions providing that a consistent L ∞ω -theory defines a class of interpretations having extreme elements on a fixed domain and detect several classes of reversible structures. In particular, we characterize the reversible countable ultrahomogeneous graphs. 2010 MSC: 03C30, 03C52, 03C98, 05C63, 05C20,
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