Classifying positive equivalence relations

Given two (positive) equivalence relations R,S on the set omega of natural numbers, we say that R is m-reducible to S if there exists a total recursive function h such that for every x, y in omega, we have x R y if hx S hy. We prove that the equivalence relation induced in omega by a positive precomplete numeration is complete with respect to this reducibility (and, moreover, a "uniformity property" holds). This result allows us to state a classification theorem for positive equivalence relations. We show that there exist nonisomorphic positive equivalence relations which are complete with respect to the above reducibility; in particular, we discuss the provable equivalence of a strong enough theory: this relation is complete with respect to reducibility but it does not correspond to a precomplete numeration. From this fact we deduce that an equivalence relation on co can be strongly represented by a formula if it is positive. At last, we interpret the situation from a topological point of view. Among other things, we generalize a result of Visser by showing that the topological space corresponding to a partition in e.i. sets is irreducible and we prove that the set of equivalence classes of true sentences is dense in the Lindenbaum algebra of the theory.