Posets of copies of countable ultrahomogeneous tournaments

The poset of copies of a relational structure $X$ is the partial order $P\left(X\right):=〈\{Y\subset X:Y\cong X\},\subset 〉$ and each similarity of such posets (e.g. isomorphism, forcing equivalence = isomorphism of Boolean completions, $B_X:=\mathrm{ro}\mathrm{sq}P\left(X\right)$) determines a classification of structures. Here we consider the structures from Lachlan's list of countable ultrahomogeneous tournaments: $Q$ (the rational line), $S\left(2\right)$ (the circular tournament), and $T^{\infty }$ (the countable homogeneous universal tournament); as well as the ultrahomogeneous digraphs $S\left(3\right)$, $Q\left[I_n\right]$, $S\left(2\right)\left[I_n\right]$ and $T^{\infty }\left[I_n\right]$ from Cherlin's list.

If $G_{\mathrm{Rado}}$ (resp. $Q_n$) denotes the countable homogeneous universal graph (resp. n-labeled linear order), it turns out that $P\left(T^{\infty }\right)\cong P\left(G_{\mathrm{Rado}}\right)$ and that $P\left(Q_n\right)$ densely embeds in $P\left(S\right(n\left)\right)$, for $n\in \{2,3\}$.

Consequently, $B_X\cong \mathrm{ro}(S⁎\pi )$, where $S$ is the poset of perfect subsets of $R$ and π an $S$-name such that $1_S⊩\text{“}\pi$ is a separative, atomless and σ-closed forcing” (thus $1_S⊩\text{“}\pi {\equiv }_{forc}{\left(P\right(\omega )/\mathrm{Fin})}^{+}$”, under CH), whenever $X$ is a countable structure equimorphic with $Q$, $Q_n$, $S\left(2\right)$, $S\left(3\right)$, $Q\left[I_n\right]$ or $S\left(2\right)\left[I_n\right]$.

Also, $B_X\cong \mathrm{ro}(S⁎\pi )$, where $1_S⊩\text{“}\pi$ is an ω-distributive forcing”, whenever $X$ is a countable graph containing a copy of $G_{\mathrm{Rado}}$, or a countable tournament containing a copy of $T^{\infty }$, or $X=T^{\infty }\left[I_n\right]$.