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  "body": "\n## Summary\n\nIdempotence is the property of something that gives the same result when it is applied twice as when it is applied once.\nAn element $x$ is idempotent for a [[binary operation|0M418FE7BB]] $\\ast$ when $x \\ast x = x$, and a function $f$ is idempotent when $f \\circ f = f$.\nLattice operations, projections and closure operators are idempotent.\nThe [[idempotent elements of a ring|0N5NJZKKN3]] are the idempotents for its multiplication.\n\n## Explanations\n\n### L1 human\n\nPressing the \"on\" button of a machine that is already on changes nothing; that is idempotence.\nIn numbers, $\\max(x, x) = x$ for every $x$, so the operation $\\max$ is idempotent.\nRounding down is an idempotent function: rounding a number that is already an integer leaves it unchanged, so rounding twice is the same as rounding once.\nAddition of numbers is not idempotent, because $1 + 1 \\ne 1$.\n",
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