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  "body": "\n## Definition\n\nLet $\\R_{\\max} = \\R \\cup \\set{-\\infty}$.\nFor $a, b \\in \\R_{\\max}$, define\n$$a \\oplus b = \\max(a, b), \\qquad a \\otimes b = a + b,$$\nwith the conventions $\\max(a, -\\infty) = a$ and $a + (-\\infty) = -\\infty$.\nThe *max-plus semiring* is $(\\R_{\\max}, \\oplus, \\otimes)$.\nThe *max-plus algebra* is the algebra of vectors and [[matrices|0VD9ZSWWJG]] with entries in $\\R_{\\max}$, with the operations\n$$(A \\oplus B)_{ij} = a_{ij} \\oplus b_{ij}, \\qquad (A \\otimes B)_{ij} = \\bigoplus_{k} a_{ik} \\otimes b_{kj} = \\max_k (a_{ik} + b_{kj}).$$\n\n## Remarks\n\n$(\\R_{\\max}, \\oplus, \\otimes)$ is a commutative semiring with zero element $-\\infty$ and unit element $0$.\nIt is idempotent: $a \\oplus a = a$ for all $a$.\nExample: $2 \\oplus 7 = 7$, $2 \\otimes 7 = 9$, and $2 \\otimes (3 \\oplus 7) = 9 = (2 \\otimes 3) \\oplus (2 \\otimes 7)$, which checks distributivity in one case.\nNon-example: $\\R_{\\max}$ is not a ring, since $5$ has no additive inverse: $\\max(5, b) = -\\infty$ has no solution $b$.\nLet $A$ be an $n \\times n$ matrix, with $a_{ij}$ the weight of an arc from $i$ to $j$.\nThen $(A^{\\otimes k})_{ij}$ is the largest weight of a path with $k$ arcs from $i$ to $j$.\nThe min-plus semiring $(\\R \\cup \\set{+\\infty}, \\min, +)$ is isomorphic to it by $x \\mapsto -x$.\n",
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