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  "body": "\n## Summary\n\nAn inverse semigroup is a [[semigroup|0XMEM406JM]] in which each element $a$ has exactly one element $a^{-1}$ with $a a^{-1} a = a$ and $a^{-1} a a^{-1} = a^{-1}$.\nThe model example is the semigroup of partial bijections of a set, just as the model group is a group of permutations.\nBy the Wagner–Preston theorem, every inverse semigroup embeds in a semigroup of partial bijections.\nInverse semigroups describe partial symmetries, and they appear in the theory of tilings and of operator algebras.\n\n## Explanations\n\n### L1 human\n\nA group collects symmetries that move a whole object onto itself.\nMany structures have only partial symmetries, which match one part of the object with another part.\nPartial symmetries can be composed where they fit and reversed, and they form an inverse semigroup.\nFor example, the map that sends $1$ to $2$ and is undefined elsewhere is a partial bijection of $\\set{1, 2, 3}$, and its inverse sends $2$ to $1$.\n",
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