Residuated mapping

Remarks

A residuated map is monotone. A map $f$ is residuated if and only if there is a map $g \colon Q \to P$ with $f(x) \le y \iff x \le g(y)$ for all $x \in P$ and $y \in Q$. Then $g(y)$ is the greatest element of $f^{-1}(\downarrow y)$, and $g$ is called the residual of $f$. For an integer $a > 0$, the map $x \mapsto a x$ on $\Z$ is residuated with residual $y \mapsto \lfloor y / a \rfloor$, since $a x \le y$ if and only if $x \le \lfloor y / a \rfloor$. The map $f \colon \R \to \R$ with $f(x) = 0$ for $x < 0$ and $f(x) = 1$ for $x \ge 0$ is monotone but not residuated. Indeed $f^{-1}(\downarrow 0) = (-\infty, 0)$ has no greatest element, so it is not a principal down-set.