Skip to main content
×
Home
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 36
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Bazhenov, N. A. 2015. 2-Computably Enumerable Degrees of Categoricity for Boolean Algebras with Distinguished Automorphisms. Journal of Mathematical Sciences, Vol. 211, Issue. 6, p. 738.


    Bazhenov, N. A. 2015. The Branching Theorem and Computable Categoricity in the Ershov Hierarchy. Algebra and Logic, Vol. 54, Issue. 2, p. 91.


    Bazhenov, N. A. 2014. Δ 2 0 -Categoricity of Boolean Algebras. Journal of Mathematical Sciences, Vol. 203, Issue. 4, p. 444.


    Bazhenov, N. A. 2014. Hyperarithmetical Categoricity of Boolean Algebras of Type B $$ \mathfrak{B} $$ (ω α × η). Journal of Mathematical Sciences, Vol. 202, Issue. 1, p. 40.


    Bazhenov, N. A. and Tukhbatullina, R. R. 2013. Computable categoricity of the Boolean algebra $ \mathfrak{B}\left( \omega \right) $ with a distinguished automorphism. Algebra and Logic, Vol. 52, Issue. 2, p. 89.


    Bazhenov*, N. A. 2013. Degrees of categoricity for superatomic Boolean algebras. Algebra and Logic, Vol. 52, Issue. 3, p. 179.


    Alaev, P.E. 2012. Computably categorical Boolean algebras enriched by ideals and atoms. Annals of Pure and Applied Logic, Vol. 163, Issue. 5, p. 485.


    Khoussainov, Bakhadyr and Kowalski, Tomasz 2012. Computable Isomorphisms of Boolean Algebras with Operators. Studia Logica, Vol. 100, Issue. 3, p. 481.


    Alaev, P. E. 2010. Autostable atomic-ideal enrichments of computable Boolean algebras. Doklady Mathematics, Vol. 82, Issue. 1, p. 528.


    Calvert, Wesley Cenzer, Douglas Harizanov, Valentina S. and Morozov, Andrei 2009. Effective categoricity of Abelian <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>p</mml:mi></mml:math>-groups. Annals of Pure and Applied Logic, Vol. 159, Issue. 1-2, p. 187.


    Cenzer, Douglas Harizanov, Valentina Marker, David and Wood, Carol 2009. Preface. Archive for Mathematical Logic, Vol. 48, Issue. 1, p. 1.


    Chubb, Jennifer Frolov, Andrey and Harizanov, Valentina 2009. Degree spectra of the successor relation of computable linear orderings. Archive for Mathematical Logic, Vol. 48, Issue. 1, p. 7.


    Calvert, Wesley Cenzer, Douglas Harizanov, Valentina and Morozov, Andrei 2006. Effective categoricity of equivalence structures. Annals of Pure and Applied Logic, Vol. 141, Issue. 1-2, p. 61.


    Semukhin, P. M. 2005. The Degree Spectra of Definable Relations on Boolean Algebras. Siberian Mathematical Journal, Vol. 46, Issue. 4, p. 740.


    Harizanov, Valentina S. 2003. Turing degrees of hypersimple relations on computable structures. Annals of Pure and Applied Logic, Vol. 121, Issue. 2-3, p. 209.


    Harizanov, Valentina S. 2002. Computability-Theoretic Complexity of Countable Structures. Bulletin of Symbolic Logic, Vol. 8, Issue. 04, p. 457.


    Hirschfeldt, Denis R. 2002. Degree spectra of relations on structures of finite computable dimension. Annals of Pure and Applied Logic, Vol. 115, Issue. 1-3, p. 233.


    Downey, R.G. 1998. Handbook of Recursive Mathematics - Volume 2: Recursive Algebra, Analysis and Combinatorics.


    Ershov, Yu.L. and Goncharov, S.S. 1998. Handbook of Recursive Mathematics - Volume 1: Recursive Model Theory.


    Goncharov, S.S. 1998. Handbook of Recursive Mathematics - Volume 1: Recursive Model Theory.


    ×

Recursive isomorphism types of recursive Boolean algebras

  • J. B. Remmel (a1)
  • DOI: http://dx.doi.org/10.2307/2273757
  • Published online: 01 March 2014
Abstract

A Boolean algebra is recursive if B is a recursive subset of the natural numbers N and the operations ∧ (meet), ∨ (join), and ¬ (complement) are partial recursive. Given two Boolean algebras and , we write if is isomorphic to and if is recursively isomorphic to , that is, if there is a partial recursive function f: B1B2 which is an isomorphism from to . will denote the set of atoms of and () will denote the ideal generated by the atoms of .

One of the main questions which motivated this paper is “To what extent does the classical isomorphism type of a recursive Boolean algebra restrict the possible recursion theoretic properties of ?” For example, it is easy to see that must be co-r.e. (i.e., N is an r.e. set), but can be immune, not immune, cohesive, etc? It follows from a result of Goncharov [4] that there exist classical isomorphism types which contain recursive Boolean algebras but do not contain any recursive Boolean algebras such that is recursive. Thus the classical isomorphism can restrict the possible Turing degrees of , but what is the extent of this restriction? Another main question is “What is the recursion theoretic relationship between and () in a recursive Boolean algebra?” In our attempt to answer these questions, we were led to a wide variety of recursive isomorphism types which are contained in the classical isomorphism type of any recursive Boolean algebra with an infinite set of atoms.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[4]S. S. Goncharov , Constructivizable superatomic Boolean algebras, Algebra i Logika, vol. 12(1973), pp. 3140.

[6]S. S. Goncharov and A. T. Nurtazin , Constructive models of complete solvable theories, Algebra i Logika, vol. 12 (1973), pp. 125142.

[9]D. A. Martin , Classes of recursively enumerable sets and degrees of unsolvability, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 295310.

[10]J. B. Remmel , Realizing partial orderings by classes of co-simple sets, Pacific Journal of Mathematics, vol. 76(1978), pp. 169184.

[14]G. E. Sacks , The recursively enumerable degrees are dense, Annals of Mathematics, vol. 80(1964), pp. 300312.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×