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  "body": "\n## Definition\n\nA monoid is a triple $(M, \\ast, e)$, where $(M, \\ast)$ is a [[semigroup|0XMEM406JM]] and $e \\in M$ is an [[identity element|07WYQPB8E3]] for $\\ast$.\nThus $(a \\ast b) \\ast c = a \\ast (b \\ast c)$ and $e \\ast a = a \\ast e = a$ for all $a, b, c \\in M$.\n\n## Remarks\n\nThe natural numbers $\\mathbb{N} = \\{0, 1, 2, \\ldots\\}$ with addition form a monoid with identity $0$.\nThe integers with multiplication form a monoid with identity $1$.\nThe identity element of a monoid is unique, thus some authors write a monoid as the pair $(M, \\ast)$.\nEach [[group|G5S1M9]] is a monoid in which each element has an [[inverse|0ZNZH0EXFX]].\n\n## Explanations\n\n### L1 human\n\nA monoid is a set with an associative operation and an element that does nothing.\nThe strings of letters with concatenation form a monoid, and the empty string is the identity.\nIn a monoid, an element need not have an inverse: you cannot undo the addition of a letter.\n",
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