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?