Isomorphic and strongly connected components

Abstract

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.

Different similarities

“…| and, by (10) and (11), 14) is true. By (14) there are bijections ϕ j : X j → Y F (j) ; let ϕ = j∈J ϕ j : κ → κ.…”

“…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].…”

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

“…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).…”