2

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 $(\mathrm{Id}(R),\subseteq,*)$.

Asaf Karagila
  • 393,674
Martin Brandenburg
  • 163,620
  • 3
    It has many names: Heyting implication, relative pseudocomplement... – Zhen Lin Oct 31 '13 at 09:25
  • $(B:A)$ isn't good notation, because it omits the dependence on $X$. The notation $X!\setminus!! A\cup B$ has the advantage that everyone can understand it immediately (add parentheses if you will). – John Bentin Oct 31 '13 at 12:17
  • @John: Since I regard $A,B$ as objects of the partial order of subsets of $X$, this $X$ implicitly belongs to the datum. If $X \subseteq X'$, then (for a category theorist) $A$ isn't really a subset of $X'$, but rather there is a functor $\wp(X) \to \wp(X')$ and $A$ shouldn't be confused with its image (of course set theory is full of such confusions). – Martin Brandenburg Nov 01 '13 at 10:32
  • @John: Besides, if $A,B$ are ideals of a ring $R$, then $(B:A)$ is a univerally accepted notation for the ideal quotient. Note that also here $R$ belongs implicitly to the datum. If $R \subseteq R'$ is a ring extension with $R'A \subseteq A$ (likewise $B$), then of course $A,B$ can be "seen" as ideals of $R'$ and in the partial order of ideals of $R'$ we can form $(B:A)$. – Martin Brandenburg Nov 01 '13 at 10:38
  • @John: This confusing abuse of notation can be avoided if we really stick to the general rule (which I try to promote here on math.SE (see for example http://math.stackexchange.com/questions/389328) and other places) that different categories should be seen as disjoint, and that forgetful functors shouldn't be seen as identities. This clarifies and organizes a lot of mathematics! Objects of mathematics shouldn't "swim" between different categories, but rather each object lies in a distinguished category (and will never leave it). – Martin Brandenburg Nov 01 '13 at 10:38

1 Answers1

3

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.

Zhen Lin
  • 90,111