Isomorphic and strongly connected components

Abstract

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 ?-closed poset and, consistently, to P(?)/Fin. In particular, this holds for each ultrahomogeneous structure $${\mathbb{X}}$$ X such that $${\mathbb{X}}$$ X or $${\mathbb{X}^{c}}$$ X c is a disconnected structure and in this case $${\mathbb{Y}}$$ Y can be replaced by an ultrahomogeneous connected digraph.