Posets of Copies of Countable Ultrahomogeneous Tournaments

Abstract

The poset of copies of a relational structure $\mathbb{X}$ is the partial order $\mathbb{P}(\mathbb{X}):=⟨\{Y\subset X:\mathbb{Y}\cong \mathbb{X}\},\subset ⟩$ 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 $1_{\mathbb{S}}⊩"\pi$ 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 $1_{\mathbb{S}}⊩"\pi$ 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 $\mathbb{X}={\mathbb{T}}^{\infty }[{\mathbb{I}}_n]$.

14 pages