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  "body": "\n## Summary\n\nA medial magma is a [[magma|9689SFPVE9]] that satisfies the medial law $(x y)(u v) = (x u)(y v)$: the two middle elements of a product of four may be swapped.\nEvery commutative semigroup is medial, and so is every operation of the form $x \\cdot y = \\varphi(x) + \\psi(y) + c$ on an abelian group with commuting endomorphisms $\\varphi, \\psi$.\nFor quasigroups, the [[Toyoda–Bruck theorem|W0KBNZBCRR]] says that these affine operations are the only medial ones.\nMediality is the binary case of the [[entropic property|9T2XVJNBBY]] of general algebras.\n\n## Explanations\n\n### L1 human\n\nWrite four elements in a square: $x$ and $y$ in the top row, $u$ and $v$ in the bottom row.\nYou can combine each row and then combine the two results, or combine each column and then combine the results.\nA magma is medial when both ways give the same answer.\nFor example, the average $x \\circ y = (x + y)/2$ on the rationals is medial: both ways give the average of all four numbers.\n",
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  "name": "Medial magma",
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