Tolerances as images of congruences in varieties defined by linear   identities
  An identity s=t is linear if each variable occurs at most once in each of the terms s and t. Let T be a tolerance relation of an algebra A in a variety defined by a set of linear identities. We prove that there exist an algebra B in the same variety and a congruence Theta of B such that a homomorphism from B onto A maps Theta onto T. 
Ivan Chajda, G\'abor Cz\'edli, Radomir Halas, and Paolo Lipparini
math.RA