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  "body": "\n## Definition\n\nLet $(P, \\le)$ and $(Q, \\le)$ be [[partially ordered sets|011C29Q1S1]].\nA map $f \\colon P \\to Q$ is *residuated* when, for each $y \\in Q$, the preimage $f^{-1}(\\downarrow y)$ is a principal down-set $\\downarrow x$ of $P$, for some $x \\in P$.\nHere $\\downarrow y = \\set{z : z \\le y}$ is the [[principal down-set|TGW5NTV3AA]] of $y$.\n\n## Remarks\n\nA residuated map is [[monotone|0WFF6AZPBP]].\nA 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 $y \\in Q$.\nThen $g(y)$ is the greatest element of $f^{-1}(\\downarrow y)$, and $g$ is called the *residual* of $f$.\nFor an integer $a \u003e 0$, 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 $x \\le \\lfloor y / a \\rfloor$.\nThe map $f \\colon \\R \\to \\R$ with $f(x) = 0$ for $x \u003c 0$ and $f(x) = 1$ for $x \\ge 0$ is monotone but not residuated.\nIndeed $f^{-1}(\\downarrow 0) = (-\\infty, 0)$ has no greatest element, so it is not a principal down-set.\n\n## Explanations\n\n### L1 human\n\nA residuated map is one whose inequalities $f(x) \\le y$ always have a largest solution $x$.\nThe largest solution, as a function of $y$, is the residual.\n",
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