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  "body": "\n## Summary\n\nA residuated lattice is a [[lattice|WH4SFAACAS]] with a [[monoid|0N6R902FG1]] multiplication that has left and right division operations, called residuals.\nThe residuals are characterised by an adjunction with the multiplication and the order.\n[[Heyting algebras|NG943XNQWH]], [[Boolean algebras|0XADJKM699]], lattice-ordered groups and the lattices of ideals of a ring are examples.\nResiduated lattices are the algebraic semantics of substructural logics, where the multiplication is a conjunction and the residuals are implications.\n\n## Explanations\n\n### L1 human\n\nIn the positive real numbers, $xy \\le z$ holds exactly when $y \\le z/x$.\nSo division is the best answer to the question \"how large can $y$ be so that $x$ times $y$ stays below $z$?\".\nA residuated lattice asks for such a best answer in an abstract ordered setting, from the left and from the right.\nIn logic, \"$x$ and $y$ imply $z$\" holds exactly when \"$y$ implies ($x$ implies $z$)\", and this is the same rule.\n",
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