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  "body": "\n## Summary\n\nThe free monoid on a set $A$ is the [[monoid|0N6R902FG1]] $A^*$ of all finite words over the alphabet $A$.\nIts operation is concatenation, and its identity is the empty word.\nIt is free in the sense that any map from $A$ to a monoid extends in exactly one way to a monoid homomorphism from $A^*$.\nFree monoids are the setting of formal languages and automata, and they are the monoid analogue of [[free groups|0KTW26C80J]].\n\n## Explanations\n\n### L1 human\n\nTake an alphabet, such as the letters $a$ and $b$.\nThe free monoid consists of all strings you can write with them, including the empty string: $a$, $ba$, $abba$, and so on.\nTo multiply two strings, write one after the other, so $ab$ times $ba$ is $abba$.\nNo two different strings are ever identified, which is what \"free\" means.\n",
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