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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References
Balcar, B., Simon, P.: Disjoint refinement. In: Monk, J.D., Bonnet, R. (eds.) Handbook of Boolean Algebras, vol. 2, pp. 333–388. North-Holland, Amsterdam (1989)
Cherlin, G.: Homogeneous directed graphs. The imprimitive case. In: Berline, Ch., Bouscaren, E., Dickmann, M., Krivine, J.-L., Lascar, D., Parigot, M., Pelz, E., Sabbagh, G. (eds.) Logic colloquium ’85 (Orsay, 1985), Studies in Logic and the Foundations of Mathematics, vol. 122, pp. 67–88. North-Holland, Amsterdam (1987)
Cherlin, G.: The classification of countable homogeneous directed graphs and countable homogeneous n-tournaments, vol. 131. Memories of the American Mathematical Society no. 621, American Mathematical Society (1998)
Fraïssé R.: Theory of relations, revised edn. With an appendix by Norbert Sauer, Studies in Logic and the Foundations of Mathematics, 145, North-Holland, Amsterdam (2000)
Hernández-Hernández F.: Distributivity of quotients of countable products of Boolean algebras. Rend. Istit. Mat. Univ. Trieste 41, 27–33 (2009)
Kurilić M.S.: From A 1 to D 5: towards a forcing-related classification of relational structures. J. Symb. Log. 79(1), 279–295 (2014)
Kurilić M.S.: Maximally embeddable components. Arch. Math. Logic 52(7), 793–808 (2013)
Kurilić, M.S.: Forcing with copies of countable ordinals. Proc. Amer. Math. Soc. (in press)
Kurilić M.S.: Posets of copies of countable scattered linear orders. Ann. Pure Appl. Logic 165, 895–912 (2014)
Kurilić M.S., Todorčević S.: Forcing by non-scattered sets. Ann. Pure Appl. Logic 163, 1299–1308 (2012)
Lachlan A.H.: Countable homogeneous tournaments. Trans. Am. Math. Soc. 284, 431–461 (1984)
Lachlan A.H., Woodrow R.E.: Countable ultrahomogeneous undirected graphs. Trans. Am. Math. Soc. 262(1), 51–94 (1980)
Macpherson D.: A survey of homogeneous structures. Discrete Math. 311(15), 1599–1634 (2011)
Schmerl J.: Countable homogeneous partially ordered sets. Alg. Univ. 9, 317–321 (1970)
Shelah S., Spinas O.: The distributivity numbers of finite products of P(ω)/fin. Fund. Math. 158(1), 81–93 (1998)
Szymański, A., Xua, Z.H.: The behaviour of ω 2 * under some consequences of Martin’s axiom, General topology and its relations to modern analysis and algebra, V (Prague, 1981), vol. 3, pp. 577–584, Sigma Series in Pure Mathematics, Heldermann, Berlin (1983)
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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