Boolean Algebras, Tarski Invariants, and Index Sets

Abstract
Tarski defined a way of assigning to each Boolean algebra, B, an invariant inv(B) ∈ In, where In is a set of triples from ℕ, such that two Boolean algebras have the same invariant if and only if they are elementarily equivalent. Moreover, given the invariant of a Boolean algebra, there is a computable procedure that decides its elementary theory. If we restrict our attention to dense Boolean algebras, these invariants determine the algebra up to isomorphism. In this paper we analyze the complexity of the question "Does B have invariant x?" For each x ∈ In we define a complexity class Γx that could be either Σⁿ, Πⁿ, Σⁿ ∧ Πⁿ, or Πω+1 depending on x, and we prove that the set of indices for computable Boolean algebras with invariant x is complete for the class Γx. Analogs of many of these results for computably enumerable Boolean algebras were proven in earlier works by Selivanov. In a more recent work, he showed that similar methods can be used to obtain the results for computable ones. Our methods are quite different and give new results as well. As the algebras we construct to witness hardness are all dense, we establish new similar results for the complexity of various isomorphism problems for dense Boolean algebras
Keywords Boolean algebras   computability   index sets   Tarski invariants
Categories (categorize this paper)
Options
 Save to my reading list
Follow the author(s)
My bibliography
Export citation
Find it on Scholar
Edit this record
Mark as duplicate
Revision history Request removal from index
 
Download options
PhilPapers Archive


Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 12,101
External links
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library
References found in this work BETA

No references found.

Citations of this work BETA
Ekaterina B. Fokina (2009). Index Sets for Some Classes of Structures. Annals of Pure and Applied Logic 157 (2):139-147.
Similar books and articles
Saharon Shelah (2000). Applications of PCF Theory. Journal of Symbolic Logic 65 (4):1624-1674.
Analytics

Monthly downloads

Added to index

2010-08-24

Total downloads

2 ( #366,722 of 1,102,103 )

Recent downloads (6 months)

1 ( #306,622 of 1,102,103 )

How can I increase my downloads?

My notes
Sign in to use this feature


Discussion
Start a new thread
Order:
There  are no threads in this forum
Nothing in this forum yet.