Maximally Embeddable Components

Abstract

We investigate the partial orderings of the form (P(X),\subset), where X 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),\subset) is forcing equivalent to a finite power of (P(\omega)/Fin)^+, or to (P(\omega \times \omega)/(Fin \times Fin))^+, or to the direct product (P(\Delta)/ED_fin)^+ \times ((P(\omega)/Fin)^+)^n, for some n \in \omega. In particular we obtain forcing equivalents of the posets of copies of countable equivalence relations, disconnected ultrahomogeneous graphs and some partial orderings.