Maximally Embeddable Components

TL;DR

This paper explores the structure of certain posets derived from countable binary relational structures, showing they are forcing equivalent to well-known posets under specific conditions related to embeddability and connectivity.

Contribution

It characterizes the forcing equivalence classes of posets of isomorphic substructure domains for maximally embeddable components with connectivity conditions.

Findings

Posets are forcing equivalent to finite powers of (P(ω)/Fin)^+

Posets are equivalent to (P(ω×ω)/(Fin×Fin))^+

Includes posets of countable equivalence relations and ultrahomogeneous graphs

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.