Linear orders realized by C.e. Equivalence relations

Journal of Symbolic Logic 81 (2):463-482 (2016)

Abstract

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.

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Citations of this work

The Theory of Ceers Computes True Arithmetic.Uri Andrews, Noah Schweber & Andrea Sorbi - 2020 - Annals of Pure and Applied Logic 171 (8):102811.
Initial Segments of the Degrees of Ceers.Uri Andrews & Andrea Sorbi - 2022 - Journal of Symbolic Logic 87 (3):1260-1282.
Word Problems and Ceers.Valentino Delle Rose, Luca San Mauro & Andrea Sorbi - 2020 - Mathematical Logic Quarterly 66 (3):341-354.

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