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  "body": "\n## Definition\n\nThe basic notions of Krohn–Rhodes theory are the following.\nLet $S$ and $T$ be [[semigroups|0XMEM406JM]].\n\n1. $S$ *divides* $T$ if $S$ is the image of a homomorphism from a subsemigroup of $T$ onto $S$.\n2. Let $(X, A)$ and $(Y, B)$ be [[transformation semigroups|7ZB7MRKPQ9]]. Their *wreath product* $(X, A) \\wr (Y, B)$ is the transformation semigroup on $X \\times Y$ consisting of all maps $(x, y) \\mapsto (\\varphi(y)(x), \\, b(y))$, where $b \\in B$ and $\\varphi \\colon Y \\to A$ is any function.\n3. The *flip-flop* is the transformation monoid on $\\set{0, 1}$ consisting of the identity and the two constant maps.\n4. A *Krohn–Rhodes decomposition* of a finite semigroup $S$ is a division of $S$ into the semigroup of an iterated wreath product $(X_1, A_1) \\wr \\cdots \\wr (X_k, A_k)$.\n   Each factor $(X_i, A_i)$ is the flip-flop or a finite [[simple group|0YMYMNQ0QZ]] acting on itself by left multiplication.\n\n## Remarks\n\nThe set in clause 2 is closed under composition: applying $(\\varphi', b')$ and then $(\\varphi, b)$ gives the map of the same form with the function $y \\mapsto \\varphi(b'(y)) \\circ \\varphi'(y)$ and the element $b \\circ b'$.\nIn the wreath product, the coordinate in $Y$ moves on its own, and it controls how the coordinate in $X$ moves; this is a cascade of two machines.\n\nThe Krohn–Rhodes theorem says that every finite semigroup $S$ has a Krohn–Rhodes decomposition in which each simple group divides $S$.\n\nExample: a finite group $G$ with a composition series has a decomposition into its composition factors, with no flip-flops.\nFor $G = \\Z/4\\Z$, two copies of $\\Z/2\\Z$ suffice, as in the carry rule of a two-digit binary counter.\n\nNon-example: the wreath product of two copies of $\\Z/2\\Z$ contains no flip-flop factor, and its semigroup is a finite group of bijections of $X \\times Y$.\nEvery subsemigroup of a finite group is a group, and so is every homomorphic image of a group.\nSo the flip-flop, which is not a group, does not divide it, and every decomposition of a finite semigroup that is not a group needs at least one flip-flop.\n",
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