Question: For a family of sets $\mathscr{A} \subseteq 2^E$, is there a term for $\{ B: B=A^c = E\setminus A, A \in \mathscr{A} \}$?
I.e. is there a term for the family of all complements of sets in another family?
(If it matters, we can assume that the ground set $E$ is finite.)
Ideally a pointer to a reference using the suggested term would be appreciated, but isn't necessary.
Attempt: If $E$ is the ground set of a matroid, and $\mathscr{A}$ is the family of bases of that matroid, then the aforementioned family of sets, $\{B : B = E \setminus A = A^c, \text{ for some }A \in \mathscr{A} \}$ is exactly the family of bases of the dual matroid.
So perhaps one could, in general, call such families the dual family of sets?
Another thing I considered was to just call this family of sets $\mathscr{A}^c$, but then I realized that this is ambiguous and would lead to a lot of confusion. Specifically, one can also define $$opp(\mathscr{A}):= \{B \subseteq E: B \not\in \mathscr{A} \} $$ (I have seen this definition before in the literature, why I'm not also asking if it has a name too).
But then, w.r.t. $2^E$, $opp(\mathscr{A})$ is just the set complement of $\mathscr{A}$. So if one said "the complement of $\mathscr{A}$", suddenly one would have the ambiguity of whether (i) the family of sets which are complements of the sets in $\mathscr{A}$ (w.r.t. the ground set $E$), or (ii) the complement of $\mathscr{A}$ w.r.t. $2^E$, i.e. $opp(\mathscr{A})$, is meant.
It makes sense to call it, as you suggest, the dual family of sets, although the term dual is overloaded in mathematics. Indeed, if you can consider $\mathscr{A}$ as a partially ordered set (with respect to inclusion), then your set is exactly the dual partially ordered set.