We investigate the partial orderings of the form 〈P(X),⊂〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle \mathbb{P}(\mathbb{X}), \subset \rangle}$$\end{document}, where X=〈X,ρ〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X} =\langle X, \rho \rangle }$$\end{document} is a countable binary relational structure and P(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{P} (\mathbb{X})}$$\end{document} the set of the domains of its isomorphic substructures and show that if the components of X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document} (...) are maximally embeddable and satisfy an additional condition related to connectivity, then the poset 〈P(X),⊂〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle \mathbb{P} (\mathbb{X}), \subset \rangle }$$\end{document} 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(Δ)/EDfin)+×((P(ω)/Fin)+)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(P(\Delta )/\fancyscript{e}\fancyscript{d}_{\rm fin})^+ \times ((P(\omega )/{\rm Fin})^+)^n}$$\end{document}, for some n∈ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \in \omega}$$\end{document}. In particular we obtain forcing equivalents of the posets of copies of countable equivalence relations, disconnected ultrahomogeneous graphs and some partial orderings. (shrink)

We show that the separative quotient of the poset 〈P,⊂〉 of isomorphic suborders of a countable scattered linear order L is σ-closed and atomless. So, under the CH, all these posets are forcing-equivalent /Fin)+).