We study the degree structure of the ω-c.e., n-c.e., and Π10 equivalence relations under the computable many-one reducibility. In particular, we investigate for each of these classes of degrees the most basic questions about the structure of the partial order. We prove the existence of the greatest element for the ω-c.e. and n-computably enumerable equivalence relations. We provide computable enumerations of the degrees of ω-c.e., n-c.e., and Π10 equivalence relations. We prove that for all the degree classes considered, upward density (...) holds and downward density fails. (shrink)

LetEbe a computably enumerable equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized (...) byE. One can also define the following pre-order$ \le _{lo} $on the class of all c.e. equivalence relations:$E_1 \le _{lo} E_2 $if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order$ \le _{lo} $. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees. (shrink)

This book describes a program of research in computable structure theory. The goal is to find definability conditions corresponding to bounds on complexity which persist under isomorphism. The results apply to familiar kinds of structures (groups, fields, vector spaces, linear orderings Boolean algebras, Abelian p-groups, models of arithmetic). There are many interesting results already, but there are also many natural questions still to be answered. The book is self-contained in that it includes necessary background material from recursion theory (ordinal notations, (...) the hyperarithmetical hierarchy) and model theory (infinitary formulas, consistency properties). (shrink)

We introduce a reducibility preordering between classes of countable structures, each class containing only structures of a given similarity type (which is allowed to vary from class to class). Though we sometimes work in a slightly larger context, we are principally concerned with the case where each class is an invariant Borel class (i.e. the class of all models, with underlying set $= \omega$, of an $L_{\omega_1\omega}$ sentence; from this point of view, the reducibility can be thought of as a (...) (rather weak) sort of $L_{\omega_1\omega}$-interpretability notion). We prove a number of general results about this notion, but our main thrust is to situate various mathematically natural classes with respect to the preordering, most notably classes of algebraic structures such as groups and fields. (shrink)

We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF-reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all ${\mathrm{\Sigma }}_{1}^{1}$ equivalence relations on hyperarithmetical subsets of ω.

We investigate dependence of recursively enumerable graphs on the equality relation given by a specific r.e. equivalence relation on ω. In particular we compare r.e. equivalence relations in terms of graphs they permit to represent. This defines partially ordered sets that depend on classes of graphs under consideration. We investigate some algebraic properties of these partially ordered sets. For instance, we show that some of these partial ordered sets possess atoms, minimal and maximal elements. We also fully describe the isomorphism (...) types of some of these partial orders. (shrink)

Given two (positive) equivalence relations ∼ 1 , ∼ 2 on the set ω of natural numbers, we say that ∼ 1 is m-reducible to ∼ 2 if there exists a total recursive function h such that for every x, y ∈ ω, we have $x \sim_1 y \operatorname{iff} hx \sim_2 hy$ . We prove that the equivalence relation induced in ω 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 ω 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. (shrink)

We study computably enumerable equivalence relations (ceers) on N and unravel a rich structural theory for a strong notion of reducibility among ceers.