Computably enumerable equivalence relations via primitive recursive reductions

Abstract

The complexity classification of computably enumerable equivalence relations (or ceers, for short) has received much attention in the recent literature. A measure of complexity is typically provided by an appropriate notion of a reduction. Given binary relations $R$ and $S$ on natural numbers, a total function $f$ is a reduction from $R$ to $S$ if for arbitrary $x$ and $y$, the conditions $x~R~y$ and $f(x)~S~f(y)$ are always equivalent. If the function $f$ can be chosen primitive recursive, then we say that $R$ is primitively recursively reducible to $S$, denoted by $R \leq _{pr} S$. We investigate the degree structure $(\textbf {Ceers},\leq _{pr})$ of $\leq _{pr}$-degrees of ceers. We examine when pairs of incomparable degrees have an infimum and a supremum. In particular, we show that $(\textbf {Ceers},\leq _{pr})$ is neither an upper semilattice nor a lower semilattice. We also study first-order definable subclasses of $(\textbf {Ceers},\leq _{pr})$. In particular, we prove that the set of equivalences that have only finitely many classes is definable in $(\textbf {Ceers},\leq _{pr})$. Finally, we show that the structure of $\leq _{pr}$-degrees of computably enumerable preorders has a hereditarily undecidable theory.