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  "body": "\n## Definition\n\nLet $(M, \\cdot)$ be a [[magma|9689SFPVE9]].\nIt is *medial* if\n$$(x \\cdot y) \\cdot (u \\cdot v) = (x \\cdot u) \\cdot (y \\cdot v)$$\nfor all $x, y, u, v \\in M$.\n\n## Remarks\n\nExample: $\\Q$ with $x \\circ y = (x + y)/2$.\nBoth $(x \\circ y) \\circ (u \\circ v)$ and $(x \\circ u) \\circ (y \\circ v)$ equal $(x + y + u + v)/4$.\nThis magma is medial but not associative.\n\nExample: every commutative semigroup is medial, since $(xy)(uv) = x(yu)v = x(uy)v = (xu)(yv)$.\n\nNon-example: $\\Z$ with $x \\cdot y = xy + 1$.\nTake $x = y = 1$ and $u = v = 0$.\nThen $1 \\cdot 1 = 2$ and $0 \\cdot 0 = 1$, so the left side is $2 \\cdot 1 = 3$.\nAlso $1 \\cdot 0 = 1$, so the right side is $1 \\cdot 1 = 2 \\ne 3$.\n\nA magma is medial if and only if the map $M \\times M \\to M$, $(x, y) \\mapsto x \\cdot y$, is a homomorphism from the product magma $M \\times M$ to $M$.\n",
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