  We look at equivalence relations on the set of models of a theory -- MERs,
for short -- such that the class of equivalent pairs is itself an elementary
class, in a language appropriate for pairs of models. We provide many examples
of definable MERs, along with the first steps of a classification theory for
them. We characterize the special classes of definable MERs associated with
preservation of formulas, either in classical first order logic or in
continuous logic, and uncover an intrinsic role for the latter. We bring out a
nontrivial relationship with interpretations (imaginary sorts), leading to a
wider hierarchy of classes related to the preservation of reducts. We give
results about the relationship between these classes, both for general theories
and for theories satisfying additional model-theoretic properties, such as
stability.
