Maximally embeddable components

Abstract

We investigate the partial orderings of the form 〈 ℙ(X), ⊂ 〉, where X = 〈X, ρ〉 is a countable binary relational structure and ℙ(X) the set of the domains of its isomorphic substructures and show that if the components of X are maximally embeddable and satisfy an additional condition related to connectivity, then the poset 〈ℙ(X), ⊂〉 is forcing equivalent to a finite power of (P(ω)/ Fin)+, or to the poset (P(ω × ω)/(Fin × Fin))+, or to the product, for some n ∈ ω. In particular we obtain forcing equivalents of the posets of copies of countable equivalence relations, disconnected ultrahomogeneous graphs and some partial orderings. © 2013 Springer-Verlag Berlin Heidelberg.