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  "body": "\n## Summary\n\nMax-plus algebra is linear algebra in which addition is replaced by taking the maximum and multiplication is replaced by ordinary addition.\nThese operations make $\\R \\cup \\set{-\\infty}$ a semiring, the max-plus semiring.\nIt is one form of the tropical semiring.\nMatrices over it describe longest paths in weighted graphs.\nThey also model the timing of systems that wait for several events, such as train schedules.\n\n## Explanations\n\n### L1 human\n\nIn max-plus arithmetic, \"$a$ plus $b$\" means $\\max(a, b)$ and \"$a$ times $b$\" means $a + b$.\nSo $3 \\oplus 5 = 5$ and $3 \\otimes 5 = 8$.\nA matrix product then picks the best route: entry $(i, j)$ of $A \\otimes B$ is $\\max_k (a_{ik} + b_{kj})$.\nIf $a_{ik}$ is the time to go from station $i$ to station $k$, then powers of $A$ give the longest travel times along chains of steps.\n",
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