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  "body": "\n## Summary\n\nA regular semigroup is a [[semigroup|0XMEM406JM]] in which each element $a$ has a partner $x$ with $axa = a$.\nGroups, the full [[transformation semigroups|7ZB7MRKPQ9]], and the semigroups of all $n \\times n$ matrices over a field under multiplication are regular.\nRegular semigroups are the class in which [[Green's relations|S7WJW2YES2]] give the most detailed structure theory.\nThe [[inverse semigroups|3M724K3B50]] are the regular semigroups in which each $a$ has exactly one $x$ with $axa = a$ and $xax = x$.\n\n## Explanations\n\n### L1 human\n\nIn a group, every element can be undone by its inverse.\nIn a regular semigroup, every element can at least be undone partly: some $x$ brings $a$ back to itself in the product $axa$.\nFor a function $f$ that is not a bijection, one can take $x$ to map each value of $f$ back to one of its preimages.\nThen applying $f$, then $x$, then $f$ again gives the same result as $f$.\n",
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