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  "body": "\n## Definition\n\nLet $S$ be a set with two binary operations $+$ and $\\cdot$, and two elements $0$ and $1$ of $S$.\nThe tuple $(S, +, \\cdot, 0, 1)$ is a *semiring* if the following hold.\n\n1. $(S, +, 0)$ is a commutative [[monoid|0N6R902FG1]].\n2. $(S, \\cdot, 1)$ is a monoid.\n3. Multiplication distributes over addition on both sides: $a \\cdot (b + c) = a \\cdot b + a \\cdot c$ and $(a + b) \\cdot c = a \\cdot c + b \\cdot c$ for all $a, b, c \\in S$.\n4. Zero is absorbing: $0 \\cdot a = a \\cdot 0 = 0$ for all $a \\in S$.\n\nThe semiring is *commutative* if its multiplication is commutative.\n\n## Remarks\n\nThe natural numbers $\\N = \\set{0, 1, 2, \\dots}$ with the usual addition and multiplication form a commutative semiring.\nThe four clauses are the familiar rules of arithmetic on $\\N$.\nThe tropical semiring is $\\R \\cup \\set{\\infty}$ with $\\min$ as addition, ordinary $+$ as multiplication, $\\infty$ as zero and $0$ as one.\nFor instance, $a + \\min(b, c) = \\min(a + b, a + c)$ gives distributivity, and $a + \\infty = \\infty$ gives the absorbing clause.\nThe positive integers $\\set{1, 2, 3, \\dots}$ with the usual operations are not a semiring: there is no additive identity, so clause 1 fails.\nA [[ring|0B1261B5ED]] is exactly a semiring in which $(S, +)$ is a group; clause 4 then follows from the others.\nSome authors omit $0$ or $1$ from the definition, and some do not require clause 4.\n",
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