Let $[q]=\{1,\dots,q\}$. We obtain a set hierarchy $H$ (aka rooted tree) on $[q]$ using partition refinements. This means that the vertices $v\in V(H)$ of $H$ are subsets $v\subseteq[q]$ of $[q]$, and $V(H)$ is a family of these subsets. I'm interested in the set $R$ of all such families $V(H)$, meaning the set of all refinement schemes of $[q]$.
QUESTION: Does this family $R$ have a name?
I looked for families of sets, but either I'm blind or none of the usual suspects is a match. In case my description was ambiguous, here's a formal recursive definition.
Let $\mathcal P(S)$ be the power set of a set $S$ and $\mathcal P_\circ(S)=\mathcal P(S)\setminus\{\emptyset\}$. Also, let $S^{\underline 2}=\{x\in S^2:x_1\neq x_2\}$. For $S\in\mathcal P_\circ([q])$ let $P^{(3)}(S)=\{P^{(2)}\in\mathcal P(\mathcal P_\circ(S)):\forall P\in(P^{(2)})^{\underline 2}\,P_1\cap P_2=\emptyset,\bigcup_{P\in P^{(2)}}P=S\}\in\mathcal P(\mathcal P(\mathcal P_\circ([q])))$ be the set of partitions of $S$. For more general $S^{(2)}\in\mathcal P(\mathcal P_\circ([q]))$ let $I^{(2)}(S^{(2)})=\{S\in S^{(2)}:\exists T\in S^{(2)}\,T\subsetneqq S\}$ be the interior vertices (in the hierarchy) and $L^{(2)}(S^{(2)})=S^{(2)}\setminus I^{(2)}(S^{(2)})$ the leaves.
Now, let $R_0=\{\{[q]\}\}$, for $n\in\mathbb Z_{>0}$ let $R_n=\{V^{(2)}\cup P^{(2)}:V^{(2)}\in R_{n-1},L\in L^{(2)}(V^{(2)}),P^{(2)}\in P^{(3)}(L)\}$, and let $R=\bigcup_n R_n$. This means, in each step we take a refinement scheme, take one of the smallest sets, and add a partition of that set to the refinement scheme. This doesn't change anything if the partition was trivial (the set itself), otherwise we also consider a "split" of that set. It's easy to see that the (normalized) hierarchy can be recovered from the refinement scheme using the inclusion relation.