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  "body": "\n## Definition\n\nLet $X$ be a [[set|0SJ3RX9TXA]].\nLet $T_X$ be the set of all [[maps|0XHDTE5YMZ]] $X \\to X$, with the product $fg = f \\circ g$ given by [[composition|0THBK9V825]].\nA *transformation semigroup* on $X$ is a nonempty subset $S \\subseteq T_X$ such that $f \\circ g \\in S$ for all $f, g \\in S$.\nIt is often written as the pair $(X, S)$.\n\n## Remarks\n\nComposition of maps is associative, so a transformation semigroup is a [[semigroup|0XMEM406JM]].\nThe whole set $T_X$ is the full transformation [[monoid|0N6R902FG1]]; for $X$ with $n$ elements it has $n^n$ elements.\nIts invertible elements form the [[symmetric group|0515A6D2Y1]] on $X$.\nEvery semigroup $S$ is isomorphic to a transformation semigroup on the set $S^1$, by sending $a$ to the map $x \\mapsto ax$.\n\nExample: the constant maps $c_a \\colon x \\mapsto a$, for $a \\in X$.\nFor all $a, b$, $c_a \\circ c_b = c_a$, so they form a transformation semigroup.\n\nNon-example: on $X = \\set{1, 2}$, let $s$ swap $1$ and $2$, and let $c_1$ be the constant map to $1$.\nThe set $\\set{\\id, s, c_1}$ is not closed under composition, because $s \\circ c_1 = c_2$ is not in it.\n",
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