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  "body": "\n## Definition\n\nA *residuated lattice* is a structure $(L, \\wedge, \\vee, \\cdot, e, \\backslash, /)$ such that:\n\n1. $(L, \\wedge, \\vee)$ is a [[lattice|WH4SFAACAS]], with order $\\le$;\n2. $(L, \\cdot, e)$ is a [[monoid|0R47MAD57S]];\n3. $\\backslash$ and $/$ are binary operations on $L$ such that for all $x, y, z \\in L$,\n$$x \\cdot y \\le z \\iff y \\le x \\backslash z \\iff x \\le z / y.$$\n\nThe operations $\\backslash$ and $/$ are the *left residual* and the *right residual*.\n\n## Remarks\n\nClause 3 says that $x \\backslash z$ is the largest $y$ with $x \\cdot y \\le z$, and $z / y$ is the largest $x$ with $x \\cdot y \\le z$.\nExample: a [[Heyting algebra|BND6A96S3D]] $H$ with $x \\cdot y = x \\wedge y$, $e = 1$ and $x \\backslash z = z / x = x \\to z$ is a residuated lattice.\nClause 3 is then the defining rule $x \\wedge y \\le z \\iff y \\le x \\to z$ of the Heyting implication.\nExample: the group $\\Z$ with $\\min$, $\\max$, $x \\cdot y = x + y$, $e = 0$ and $x \\backslash z = z / x = z - x$ is a residuated lattice, since $x + y \\le z \\iff y \\le z - x$.\nNon-example: $\\N$ with $\\min$, $\\max$, ordinary multiplication and $e = 1$ satisfies clauses 1 and 2, but has no left residual.\nFor $x = 0$ and $z = 0$, every $y \\in \\N$ satisfies $0 \\cdot y \\le 0$, and $\\N$ has no largest element.\nMany authors also require $L$ to be bounded, or the monoid to be commutative, in which case $x \\backslash z = z / x$ is written $x \\to z$.\nMultiplication in a residuated lattice is monotone and distributes over all existing joins.\n",
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