Isomorphic and Strongly Connected Components

Abstract

We study the partial orderings of the form $⟨\mathbb{P}(\mathbb{X}),\subset ⟩$, 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 $⟨\mathbb{P}(\mathbb{X}),\subset ⟩$ 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.

arXiv comment16 pages, 1 figure