Isomorphic and Strongly Connected Components

Abstract

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.