Linear Orders Realized by C.E. Equivalence Relations

Abstract

Let E be a computably enumerable (c.e.) equivalence relation on the set ω of natural numbers. We say that the quotient set ω/E (or equivalently, the relation E) realizes a linearly ordered set L if there exists a c.e. relation respecting E such that the induced structure (ω/E; ) is isomorphic to L. Thus, one can consider the class of all linearly ordered sets that are realized by ω/E; formally, K(E) = {L | the order-type L is realized by E}. In this paper we study the relationship between computability-theoretic properties of E and algebraic properties of linearly ordered sets realized by E. One can also define the following pre-order lo on the class of all c.e. equivalence relations: E 1 lo E 2 if every linear order realized by E 1 is also realized by E 2 . Following the tradition of computability theory, the lo-degrees are the classes of equivalence relations induced by the pre-order lo . We study the partially ordered set of lo-degrees. For instance, we construct various chains and antichains and show the existence of a maximal element among the lo-degrees.

Quantifier Free Definability on Infinite Algebras

“…As we mentioned, Malcev [20] and Rabin [23] initiated this line of research, and their work lead to the modern development of the theory of computable structures [12]. Recently, the third approach has been suggested that introduces effectiveness into the study of infinite algebraic structures [2] [3] [8]. In this approach the primary objects are domains of the type ω/E, where E is an equivalence relation.…”

“…For instance, such domains do not realise infinite complete graphs. In [8], it is proved that there are domains that realise linear orders of the type ω + 1 + ω only, where ω is the order of negative integers. These all show that some domains allow us to tame algebraic structures.…”

The theory of ceers computes true arithmetic

“…Additional interest for computable reducibility on ceers comes from the study of c.e. presentations of structures, as is shown for instance in [12,16]. It is also worth noticing that ceers have been investigated in computability theory also not in connection with computable reducibility: for instance, Carroll [7] studied the lattice of ceers under inclusion; Nies [26] studied ceers modulo finite differences: they both showed that the first order theory of the resulting structures is computably isomorphic to Th 1 pNq.…”

Word problems and ceers

“…Given a ceer E (having possibly some interesting computational property) it is natural to ask which algebras can be positively presented having E as their equality relation (cf., e.g., [19, 22]). Surprisingly, much less is known about the reverse problem, namely, given a class of structures , which ceers are “realized” by members of ?…”