Fokina, Ekaterina, Khoussainov, Bakhadyr, Semukhin, Pavel and Turetsky, Daniel
(2016)
LINEAR ORDERS REALIZED BY CE EQUIVALENCE RELATIONS.
JOURNAL OF SYMBOLIC LOGIC, 81 (2).
pp. 463-482.
Text
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Abstract
<jats:title>Abstract</jats:title><jats:p>Let<jats:italic>E</jats:italic>be a computably enumerable (c.e.) equivalence relation on the set<jats:italic>ω</jats:italic>of natural numbers. We say that the quotient set<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481215000110_inline1" /><jats:tex-math>$\omega /E$</jats:tex-math></jats:alternatives></jats:inline-formula>(or equivalently, the relation<jats:italic>E</jats:italic>)<jats:italic>realizes</jats:italic>a linearly ordered set<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481215000110_inline2" /><jats:tex-math>${\cal L}$</jats:tex-math></jats:alternatives></jats:inline-formula>if there exists a c.e. relation ⊴ respecting<jats:italic>E</jats:italic>such that the induced structure (<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481215000110_inline4" /><jats:tex-math>$\omega /E$</jats:tex-math></jats:alternatives></jats:inline-formula>; ⊴) is isomorphic to<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481215000110_inline5" /><jats:tex-math>${\cal L}$</jats:tex-math></jats:alternatives></jats:inline-formula>. Thus, one can consider the class of all linearly ordered sets that are realized by<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481215000110_inline6" /><jats:tex-math>$\omega /E$</jats:tex-math></jats:alternatives></jats:inline-formula>; formally,<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481215000110_inline7" /><jats:tex-math>${\cal K}\left( E \right) = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$</jats:tex-math></jats:alternatives></jats:inline-formula>. In this paper we study the relationship between computability-theoretic properties of<jats:italic>E</jats:italic>and algebraic properties of linearly ordered sets realized by<jats:italic>E</jats:italic>. One can also define the following pre-order<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481215000110_inline8" /><jats:tex-math>$ \le _{lo} $</jats:tex-math></jats:alternatives></jats:inline-formula>on the class of all c.e. equivalence relations:<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481215000110_inline9" /><jats:tex-math>$E_1 \le _{lo} E_2 $</jats:tex-math></jats:alternatives></jats:inline-formula>if every linear order realized by<jats:italic>E</jats:italic><jats:sub>1</jats:sub>is also realized by<jats:italic>E</jats:italic><jats:sub>2</jats:sub>. Following the tradition of computability theory, the<jats:italic>lo</jats:italic>-degrees are the classes of equivalence relations induced by the pre-order<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481215000110_inline10" /><jats:tex-math>$ \le _{lo} $</jats:tex-math></jats:alternatives></jats:inline-formula>. We study the partially ordered set of<jats:italic>lo</jats:italic>-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among the<jats:italic>lo</jats:italic>-degrees.</jats:p>
Item Type: | Article |
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Uncontrolled Keywords: | linear orders, equivalence relations, computability |
Depositing User: | Symplectic Admin |
Date Deposited: | 02 Nov 2016 11:19 |
Last Modified: | 19 Jan 2023 07:37 |
DOI: | 10.1017/jsl.2015.11 |
Related URLs: | |
URI: | https://livrepository.liverpool.ac.uk/id/eprint/3001054 |