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][1], but I don't have access to this Springer journal so I can't say for sure. 


  [1]: http://www.springerlink.com/content/33167ux0p52622m6/