Isomorphic and strongly connected components

Abstract

© 2014, Springer-Verlag Berlin Heidelberg. We study the partial orderings of the form $${\langle \mathbb{P} (\mathbb {X}), \subset\rangle}$$⟨P(X),⊂⟩, where $${\mathbb{X}}$$X is a binary relational structure with the connectivity components isomorphic to a strongly connected structure $${\mathbb{Y}}$$Y and $${\mathbb{P} (\mathbb{X})}$$P(X) is the set of (domains of) substructures of $${\mathbb {X}}$$X isomorphic to $${\mathbb{X}}$$X. We show that, for example, for a countable $${\mathbb{X}}$$X, the poset $${\langle \mathbb {P} (\mathbb{X}), \subset\rangle}$$⟨P(X),⊂⟩ is either isomorphic to a finite power of $${\mathbb{P} (\mathbb{Y})}$$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 $${\mathbb{X}}$$X such that $${\mathbb{X}}$$X or $${\mathbb{X}^{c}}$$Xc is a disconnected structure and in this case $${\mathbb{Y}}$$Y can be replaced by an ultrahomogeneous connected digraph.