TITLE: Is this "set quotient" known?
QUESTION [2 upvotes]: Let $A,B$ be subsets of a set $X$. Then there is a largest subset $C \subseteq X$ such that $C \cap A \subseteq B$. Explicitly, we have $C = \{x \in X : x \in A \Rightarrow x \in B\} = (X \setminus A) \cup B$. Does $C$ have a name? I would call $C$ the set quotient and denote it by $(B:A)$. Namely, there is an analogy$^1$ to ideal quotients: If $A,B$ are ideals of a commutative ring $R$, then there is a largest ideal $C$ such that $C \cdot A \subseteq B$, namely $C=(B:A) = \{x \in R : x \cdot A \subseteq B\}$.
$^1$ Actually it is more than just analogy. Both quotients are internal homs in monoidal preorders, the one being $(\wp(X),\subseteq,\cap)$ and the other one $(\mathrm{Id}(R),\subseteq,*)$.

REPLY [3 votes]: Let $L$ be a lattice that is cartesian closed as a category. Then $L$ is a Heyting algebra, and the operation you describe is sometimes called the Heyting implication or relative pseudocomplement. If $L$ is moreover a boolean algebra then the Heyting implication $a \to b$ is given by the classical formula $(\lnot a) \vee b$, as you described.
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