We study the partial orderings of the form $\langle { \mathbb P} ({ \mathbb X}), \subset \rangle $, where ${ \mathbb X}$ is a binary relational structure with the connectivity components isomorphic to a strongly connected structure ${ \mathbb Y}$ and ${ \mathbb P} ({ \mathbb X})$ is the set of (domains of) substructures of ${ \mathbb X}$ isomorphic to ${ \mathbb X}$. We show that, for example, for a countable ${ \mathbb X}$, the poset $\langle { \mathbb P} ({ \mathbb X}), \subset \rangle $ is either isomorphic to a finite power of ${ \mathbb P} ({ \mathbb Y})$ or forcing equivalent to a separative atomless $\sigma$-closed poset and, consistently, to $P(\omega )/$Fin. In particular, this holds for each ultrahomogeneous structure ${ \mathbb X}$ such that ${ \mathbb X}$ or ${ \mathbb X} ^c$ is a disconnected structure and in this case ${ \mathbb Y}$ can be replaced by an ultrahomogeneous connected digraph.
Categories: math.LO