Journal of Symbolic Logic

Classifying Positive Equivalence Relations

Claudio Bernardi and Andrea Sorbi

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Given two (positive) equivalence relations $\sim_1, \sim_2$ on the set $\omega$ of natural numbers, we say that $\sim_1$ is $m$-reducible to $\sim_2$ if there exists a total recursive function $h$ such that for every $x, y \in \omega$, we have $x \sim_1 y \operatorname{iff} hx \sim_2 hy$. 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 (Theorem 2). 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 $\omega$ can be strongly represented by a formula (see Definition 8) iff 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.

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J. Symbolic Logic, Volume 48, Issue 3 (1983), 529-538.

First available in Project Euclid: 6 July 2007

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Bernardi, Claudio; Sorbi, Andrea. Classifying Positive Equivalence Relations. J. Symbolic Logic 48 (1983), no. 3, 529--538.

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