Is there accepted notation and/or terminology for the smallest cover of $S$ with cells from $P$?

Question

Let $X$ denote a set. Then for $S \subseteq X$ and $P$ a partitioning of $X$, define $P \diamond S$ as the smallest cover of $S$ with cells from $P$. Explicitly:

$$P \diamond S = \bigcup\{Q \in P \mid Q\cap S \ne \emptyset\}$$

So $P \diamond S$ is kind of like a blurred-out version of $S$.

Question. Is there accepted notation and/or terminology for the operation $\diamond$?

Motivation. Consider a function $f : X \rightarrow Y$. We obtain two corresponding functions $f_\mathrm{ker}$ and $f_\mathrm{im}$ as follows.

$$f_\mathrm{ker} : \mathcal{P}(X) \rightarrow \mathcal{P}(X) \qquad f_\mathrm{im} : \mathcal{P}(Y) \rightarrow \mathcal{P}(Y)$$

$$f_\mathrm{ker}(A) = f^{-1}(f(A)) \qquad f_\mathrm{im}(B) = f(f^{-1}(B))$$

Now let $\mathrm{im}(f)$ denote the image of $f$. Then the latter map has a rather simple description:

$$f_\mathrm{im}(B) = \mathrm{im}(f) \cap B$$

Let us furthermore write $\mathrm{ker}(f)$ for the partitioning induced on $X$ by $f$. Then:

$$f_\mathrm{ker}(A) = \mathrm{ker}(f) \diamond A$$

Answer 1

On page (2) of this article, they essentially make the following definition:

Definition. If $X$ is a set, $S$ is a subset thereof, and $R$ is a equivalence relation on $X$, then:

$$S/R = \{y \in X : \exists x \in S(xRy)\}$$

So basically, what I'm denoting $R \diamond S$ can be denoted $S/R$, as long as we don't worry too much about the distinction between partitions and equivalence relations.