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  "body": "\n## Definition\n\nLet $(S, +, \\cdot, 0, 1)$ be a [[semiring|Q07RA7AMDE]].\nThe semiring $S$ is *idempotent* if $a + a = a$ for all $a \\in S$.\n\n## Remarks\n\nIn an idempotent semiring, the relation $a \\le b \\iff a + b = b$ is a [[partial order|011C29Q1S1]].\nFor this order, $a + b$ is the least upper bound of $a$ and $b$, the element $0$ is the least element, and multiplication is monotone on both sides.\nExample: the Boolean semiring $\\set{0, 1}$ with $a + b = \\max(a, b)$ and $a \\cdot b = \\min(a, b)$ is idempotent, since $\\max(a, a) = a$.\nExample: the subsets of the [[free monoid|27ZPTJP9PP]] $\\Sigma^*$ on an alphabet $\\Sigma$, with union as addition and concatenation as multiplication, form an idempotent semiring, since $L \\cup L = L$.\nNon-example: $\\N$ with the usual operations is a semiring but is not idempotent, since $1 + 1 = 2 \\ne 1$.\nA ring is idempotent only if it is the zero ring, since $a + a = a$ gives $a = 0$.\nIdempotent semirings are also called *dioids*.\n",
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