Notre Dame Journal of Formal Logic

Boolean Algebras, Tarski Invariants, and Index Sets

Barbara F. Csima, Antonio Montalbán, and Richard A. Shore

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.

Article information

Source
Notre Dame J. Formal Logic, Volume 47, Number 1 (2006), 1-23.

Dates
First available in Project Euclid: 27 March 2006

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1143468308

Digital Object Identifier
doi:10.1305/ndjfl/1143468308

Mathematical Reviews number (MathSciNet)
MR2211179

Zentralblatt MATH identifier
1107.03031

Subjects
Primary: 03D50: Recursive equivalence types of sets and structures, isols

Keywords
Boolean algebras computability index sets Tarski invariants

Citation

Csima, Barbara F.; Montalbán, Antonio; Shore, Richard A. Boolean Algebras, Tarski Invariants, and Index Sets. Notre Dame J. Formal Logic 47 (2006), no. 1, 1--23. doi:10.1305/ndjfl/1143468308. https://projecteuclid.org/euclid.ndjfl/1143468308


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