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 σ-closed poset and, consistently, to P(ω)/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.
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Kurilić, M.S. Isomorphic and strongly connected components. Arch. Math. Logic 54, 35–48 (2015). https://doi.org/10.1007/s00153-014-0399-2
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DOI: https://doi.org/10.1007/s00153-014-0399-2