Universal Computably Enumerable Equivalence Relations

Abstract

We study computably enumerable equivalence relations (ceers), under the reducibility $R \le S$ if there exists a computable function f such that $x\,R\,y$ if and only if $f\left( x \right)\,\,S\,f\left( y \right)$ , for every $x,y$ . We show that the degrees of ceers under the equivalence relation generated by $\le$ form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first-order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer R is universal if and only if $R\prime \le R$ , where $R\prime$ denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are ${\rm{\Sigma }}_3^0$ -complete (the former answering an open question of Gao and Gerdes).

OnΔ20-categoricity of equivalence relations

“…Many results of numbering theory can be translated into results on equivalence relations and vice versa, see the recent paper [1] for more details. Numbering theory has been one of the central topics in the Soviet logic school for over 40 years.…”

“…The theory of effective equivalence relations has grown to a rather broad area; we cite [1,18,9] for recent results on this subject. Many results of this paper can be stated in terms of Δ 0 2 -embeddings between effectively presented equivalence structures.…”

“…A computable algebraic structure A is Δ 0 n -categorical if every two computable presentations of A are 0 (n−1) -isomorphic. 1 In contrast to computable categoricity, obtaining a complete classification of Δ 0 n -categoricity in a given class is typically a difficult task. Already for n = 2 and even for algebraically very well understood classes, 1 Here 0 (n+1) stands for the (n + 1)th iterate of the Halting problem.…”

“…1 In contrast to computable categoricity, obtaining a complete classification of Δ 0 n -categoricity in a given class is typically a difficult task. Already for n = 2 and even for algebraically very well understood classes, 1 Here 0 (n+1) stands for the (n + 1)th iterate of the Halting problem. We note that there are variations of Definition 1.1 such as the notion of relative Δ 0 n -categoricity [3], and also related notions of categoricity spectra [20] and degrees of categoricity [19,10].…”

Weakly precomplete computably enumerable equivalence relations

“…Precomplete ceers have been studied in several papers, of which, particularly relevant to our purposes are . For important examples of precomplete ceers that naturally arise in logic, cf., e.g., .…”