Projectivity and isomorphisms of strictly simple algebras
  We describe a sufficient condition for the localization functor to be a categorical equivalence. Using this result we explain how to simplify the test for projectivity. This leads to a description of the strictly simple algebras which are projective in the variety they generate. A byproduct of our efforts is the result that if A and B are strictly simple and generate the same variety, then A=B or else both are strongly abelian. 
Keith A. Kearnes (University of Arkansas), {\AA}gnes Szendrei (Bolyai   Insttitute)