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I look for an elegant way to notate a partition $\mathcal{Q}$ based on another partition $\mathcal{P}$. Two elements who are already in the same partition in $\mathcal{P}$ also belong to the same partition in $\mathcal{Q}$ only if a function $\delta$ maps to elements that are both member of the same (but possibly another partion of $\mathcal{P}$) the same for both elements on all inputs.

So two items are in the same partion of $\mathcal{Q}$ if: $\left(\exists P_1\in \mathcal{P}:p_1\in P_1\wedge p_2\in P_1\right)\wedge\left(\forall i\in I:\exists P_2\in\mathcal{P}:\delta\left(p_1,i\right)\in P_2\wedge \delta\left(p_2,i\right)\in P_2\right)$

Is there an elegant notation for this?

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It might be easier to talk about the associated equivalence relations. If $\mathcal{P}$ is a partition (of whatever underlying set is involved here), write $p_1\stackrel{\mathcal{P}}\equiv p_2$ iff $p_1$ and $p_2$ belong to the same ‘piece’ of $\mathcal{P}$. Now you can define

$$p_1\stackrel{\mathcal{Q}}\equiv p_2\text{ iff }p_1\stackrel{\mathcal{P}}\equiv p_2\text{ and }\delta(p_1,i)\stackrel{\mathcal{P}}\equiv\delta(p_2,i)\text{ for all }i\in I\;.\tag{1}$$

However, I know of no name or standard notation for this notion.

Alternatively, each $i\in I$ induces a partition $\mathcal{P}_i$ by $p_1\stackrel{\mathcal{P}_i}\equiv p_2$ iff $\delta(p_1,i)\stackrel{\mathcal{P}}\equiv\delta(p_2,i)$, and $\mathcal{Q}$ can then be described as the coarsest partition refining $\mathcal{P}$ and all $\mathcal{P}_i$ for $i\in I$. In terms of equivalence relations, writing $E_\mathcal{P}$ instead of $\stackrel{\mathcal{P}}\equiv$ for greater readability, this is

$$E_\mathcal{Q}=E_\mathcal{P}\cap\bigcap_{i\in I}E_{\mathcal{P}_i}\;,$$

which is compact, but not necessarily more readable than $(1)$.

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