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For monoids (which are associative) the Krohn-Rhodes theorem gives a powerful decomposition result: every finite monoid is a quotient of a submonoid of an alternating wreath product of finite groups and monoids. Google suggests that Joel Wanderwerf may have generalized this theorem to arbitrary algebras in a 1996 article for the Semigroup Forum journal, but I don't have access to this Springer journal so I can't say for sure.