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  "body": "\n## Summary\n\nAn idempotent semiring, also called a dioid, is a [[semiring|D31S3GR51T]] in which $a + a = a$ for every element $a$.\nIn such a semiring the addition defines a natural partial order, with $a \\le b$ when $a + b = b$, and addition becomes the join of this order.\nThe Boolean semiring, the tropical semirings and the semiring of formal languages are examples.\nIdempotent semirings are the algebra of shortest paths, of tropical geometry and of [[Kleene algebras|BK4DDS24XC]].\n\n## Explanations\n\n### L1 human\n\nIn ordinary arithmetic, $1 + 1 = 2$.\nIn an idempotent semiring, adding something to itself changes nothing, like taking the larger of a number and itself.\nA good example uses \"maximum\" as addition and ordinary addition as multiplication: the \"sum\" of $3$ and $3$ is $3$.\nBecause of this, the addition behaves like choosing the best of several options, which is why these semirings fit optimisation problems.\n",
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