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  "body": "\n## Summary\n\nKrohn–Rhodes theory studies finite [[semigroups|0XMEM406JM]] and finite automata by decomposing them into simple building blocks.\nIts central result is the Krohn–Rhodes theorem.\nIt says that every finite semigroup divides an iterated [[wreath product|W1X984PKJ9]] of finite [[simple groups|0YMYMNQ0QZ]] and copies of the flip-flop, a monoid of resets.\nThe simple groups in such a decomposition can be chosen among the simple groups that divide the semigroup.\nSo every finite automaton can be built in a cascade from reversible parts, which are groups, and memory parts, which are flip-flops.\nThe least number of group levels needed is the complexity of the semigroup, a central object of study in the theory.\n\n## Explanations\n\n### L1 human\n\nA finite automaton is a machine with finitely many states that reads input one symbol at a time.\nKrohn–Rhodes theory says that any such machine can be simulated by a chain of very simple machines, each feeding its state to the next ones.\nThe simple machines are of two kinds: those that permute their states reversibly, like the hands of a clock, and one-bit memories that can be set or reset.\nThis is a kind of prime factorization for machines.\n",
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