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  "body": "\n## Summary\n\nA quantale is a [[complete lattice|D7A0ZGZT5S]] with an associative multiplication that distributes over arbitrary joins on both sides.\nLattices of ideals of a ring, frames of open sets, and power sets of monoids are quantales.\nQuantales generalise frames and locales to a non-commutative or non-idempotent setting.\nThey appear in noncommutative topology and linear logic.\nThe multiplication of a quantale always has left and right residuals, which links quantales with [[residuated lattices|E337A6WVH3]].\n\n## Explanations\n\n### L1 human\n\nTake the ideals of a commutative ring.\nYou can join any family of ideals by taking the ideal they generate, and you can multiply two ideals.\nMultiplying an ideal by the join of a family gives the join of the products.\nA quantale is any complete lattice with a multiplication with this property.\n",
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