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  "body": "\n## Example\n\nLet $F$ be a [[field|04C5ZBG1WT]] and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.\nThe [[affine space|09XNAPTJQ7]] of $V$, seen as an algebra in the sense of universal algebra, is the set $V$ with these operations:\n\n1. the [[ternary operation|WC47MFR6GW]] $p(x, y, z) = x - y + z$;\n2. for each $r \\in F$, the binary operation $b_r(x, y) = (1 - r)x + ry$.\n\nThere is no constant and no addition of two points.\nEach operation is an [[affine combination|06HYY6DK6X]], since its coefficients sum to $1$.\n\nCheck that every affine combination is a term in these operations.\nTake $x_1, \\dots, x_n \\in V$ and $\\lambda_1, \\dots, \\lambda_n \\in F$ with $\\lambda_1 + \\cdots + \\lambda_n = 1$.\nThen $\\sum_i \\lambda_i x_i = x_1 + \\sum_{i \\ge 2} \\lambda_i (x_i - x_1)$.\nEach point $b_{\\lambda_i}(x_1, x_i) = x_1 + \\lambda_i (x_i - x_1)$ is a term.\nFor points $x_1 + a$ and $x_1 + c$, the term $p(x_1 + a, x_1, x_1 + c)$ equals $x_1 + a + c$.\nApplying $p$ repeatedly therefore gives $x_1 + \\sum_{i \\ge 2} \\lambda_i (x_i - x_1)$.\nSo the subalgebras are exactly the subsets closed under affine combinations, that is, the affine subspaces and the empty set.\n\n## Remarks\n\nEvery translation $x \\mapsto x + v$ is an automorphism of this algebra, because each operation is an affine combination.\nA translation by $v \\ne 0$ does not fix $0$, so it is not a linear map of $V$.\nThis shows that the algebra forgets the origin of $V$.\nThe term $p$ satisfies $p(x, x, y) = y$ and $p(x, y, y) = x$, so it is a [[Mal'cev term|Y203PPTS61]].\n\n## Explanations\n\n### L1 human\n\nA vector space has a special point, the zero vector, but a plane in space has no special point.\nThe affine space keeps only the operations that do not depend on where zero is.\nOne may take weighted averages of points, such as the midpoint, but one may not add two points.\nFor example, in $\\R^2$ the midpoint of $(0, 0)$ and $(2, 4)$ is $b_{1/2}((0,0), (2,4)) = (1, 2)$.\nMoving every point by the same vector keeps all these operations, so such a move is a symmetry of the algebra.\n",
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