Skip to main content
×
Home
    • Aa
    • Aa

Recursive isomorphism types of recursive Boolean algebras

  • J. B. Remmel (a1)
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? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 2 *
Loading metrics...

Abstract views

Total abstract views: 59 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 21st September 2017. This data will be updated every 24 hours.