Krohn–Rhodes decomposition of finite monoids into aperiodic and simple-group factors
A Lean 4 and Mathlib formalization of the Krohn–Rhodes prime decomposition theorem for finite monoids, in a form with aperiodic factors: every finite monoid M divides an iterated wreath product of finite monoids, each of which is aperiodic or a simple group, and every simple group among the factors divides M. The proof follows the local-divisor proof of Diekert, Kufleitner and Steinberg. Unlike the classical statement, the aperiodic factors are not reduced to copies of the three-element flip-flop monoid U₂; for an aperiodic M the statement holds with M as its only factor, and in general it…
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