Maximally embeddable components

Abstract

We investigate the partial orderings of the form ${\langle \mathbb{P}(\mathbb{X}), \subset \rangle}$ , where ${\mathbb{X} =\langle X, \rho \rangle }$ is a countable binary relational structure and ${\mathbb{P} (\mathbb{X})}$ the set of the domains of its isomorphic substructures and show that if the components of ${\mathbb{X}}$ are maximally embeddable and satisfy an additional condition related to connectivity, then the poset ${\langle \mathbb{P} (\mathbb{X}), \subset \rangle }$ is forcing equivalent to a finite power of (P(ω)/ Fin)⁺, or to the poset (P(ω × ω)/(Fin × Fin))⁺, or to the product ${(P(\Delta )/\fancyscript{E}\fancyscript{D}_{\rm fin})^+ \times ((P(\omega )/{\rm Fin})^+)^n}$ , 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.