Assume we have a partition $\mathcal{F}$ of a set $E$ and $X\subset E$ is there a canonical name for
$$\mathcal{G} = \{X\cap A|A\in \mathcal{F}\} $$
?
I'd say induced partition but it is a bit of a mouthful, or trace, borrowing from filters/filtrations, but I'd like to know if it already has another name.
Bonus points : if $X\not\subset E$, how would you call
$$\mathcal{G}' = \{X\cap A|A\in \mathcal{F}\} \cup \{\{x\}|x\in X\smallsetminus E\}$$
?
If anyone wonders where this comes from, the notion appears in some clustering quality metrics for NLP