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  "body": "\n## Definition\n\nLet $S$ be a [[semigroup|0XMEM406JM]].\nAn element $a \\in S$ is *regular* if there is $x \\in S$ with $axa = a$.\nThe semigroup $S$ is *regular* if every element of $S$ is regular.\n\n## Remarks\n\nIf $axa = a$, then $y = xax$ satisfies $aya = a$ and $yay = y$; such a $y$ is called an inverse of $a$.\nSo $S$ is regular if and only if every element has at least one inverse in this sense.\nIf $axa = a$, then $ax$ and $xa$ are [[idempotents|0N5NJZKKN3]].\nA semigroup is regular if and only if every $\\mathcal{R}$-class, or equivalently every $\\mathcal{L}$-class, of [[Green's relations|S7WJW2YES2]] contains an idempotent.\n\nExample: the monoid $T_X$ of all maps $X \\to X$ under composition, $(fg)(x) = f(g(x))$.\nGiven $f$, choose $g$ such that $g(y)$ is a preimage of $y$ under $f$ for each $y$ in the image of $f$, and $g(y)$ is arbitrary otherwise.\nThen $f(g(f(x))) = f(x)$ for all $x$, so $fgf = f$.\n\nNon-example: the positive integers under addition.\nFor $a = 1$, the equation $1 + x + 1 = 1$ has no positive solution $x$, so $1$ is not regular.\n",
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