The lattice of r.e. equivalence relations has not been carefully examined even though r.e. equivalence relations have proved useful in logic. A maximal r.e. equivalence relation has the expected lattice theoretic definition. It is proved that, in every pair of r.e. nonrecursive Turing degrees, there exist maximal r.e. equivalence relations which intersect trivially. This is, so far, unique among r.e. submodel lattices.