The poset of copies of a relational structure ${ \mathbb X}$ is the partial order ${ \mathbb P} ({ \mathbb X} ) := \langle \{ Y \subset X: { \mathbb Y} \cong { \mathbb X}\}, \subset \rangle$ and each similarity of such posets (e.g. isomorphism, forcing equivalence) determines a classification of structures. We consider the countable ultrahomogeneous tournaments: ${ \mathbb Q} $ (the rational line), ${ \mathbb S} (2)$ (the circular tournament), and ${ \mathbb T} ^\infty$ (the random tournament); as well as the ultrahomogeneous digraphs ${ \mathbb S} (3)$, ${ \mathbb Q} [{ \mathbb I}_n]$, ${ \mathbb S} (2)[{ \mathbb I}_n]$ and ${ \mathbb T} ^\infty [{ \mathbb I}_n]$ from Cherlin's list. If ${ \mathbb G} _{ { \mathrm{ Rado}}}$ (resp. ${ \mathbb Q} _n$) denotes the countable homogeneous universal graph (resp. $n$-labeled linear order), it turns out that ${ \mathbb P} ({ \mathbb T} ^\infty)\cong { \mathbb P} ({ \mathbb G}_{ { \mathrm{ Rado}}})$ and that ${ \mathbb P} ({ \mathbb Q} _n)$ densely embeds in ${ \mathbb P} ({ \mathbb S} (n))$, for $n\in\{ 2,3\}$. Consequently, ${ \mathbb B} _{ \mathbb X} \cong { \mathrm{ ro}}\, ({ \mathbb S} \ast \pi)$, where ${ \mathbb S}$ is the Sacks forcing and $1_{ \mathbb S} \Vdash "\pi $ is a separative, atomless and $\sigma$-closed forcing", whenever ${ \mathbb X}$ is a countable structure equimorphic with ${ \mathbb Q}$, ${ \mathbb Q} _n$, ${ \mathbb S} (2)$, ${ \mathbb S} (3)$, ${ \mathbb Q} [{ \mathbb I}_n]$ or ${ \mathbb S} (2)[{ \mathbb I}_n]$. Also, ${ \mathbb B} _{ \mathbb X} \cong { \mathrm{ ro}}\, ({ \mathbb S} \ast \pi)$, where $1_{ \mathbb S} \Vdash "\pi $ is an $\omega$-distributive forcing", whenever ${ \mathbb X}$ is a countable graph embedding ${ \mathbb G} _{ { \mathrm{ Rado}}}$, or a countable tournament embedding ${ \mathbb T} ^\infty$, or ${ \mathbb X} ={ \mathbb T} ^\infty [{ \mathbb I}_n]$.
Categories: math.LO