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  "body": "\n## Summary\n\nAn ordered semigroup is a [[semigroup|0XMEM406JM]] with a [[partial order|0QV2JS9RJ3]] that is preserved by multiplication on both sides.\nIt is the most general of the ordered algebraic structures with one associative operation.\nOrdered monoids and [[ordered groups|KRDR04RQ7N]] are special cases.\nOrdered semigroups appear in the theory of formal languages, of semirings and of residuated structures.\n\n## Explanations\n\n### L1 human\n\nThe natural numbers under addition are an ordered semigroup: if $a \\le b$, then $a + c \\le b + c$.\nThe positive integers under multiplication are another: if $a \\le b$, then $ac \\le bc$.\nIn general, an ordered semigroup is any associative operation together with a way to compare elements that the operation respects.\n",
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