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Boolean algebras, Tarski invariants, and index sets. (English) Zbl 1107.03031
Summary: Tarski defined a way of assigning to each Boolean algebra, \(B\), an invariant \(\text{inv}(B)\in\text{In}\), where In is a set of triples from \(\mathbb{N}\), 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 \(x\in\text{In}\) we define a complexity class \(\Gamma_x\) that could be either \(\Sigma_n\), \(\Pi_n\), \(\Sigma_n \wedge\Pi_n\), or \(\Pi_{\omega+1}\), depending on \(x\), and we prove that the set of indices for computable Boolean algebras with invariant \(x\) is complete for the class \(\Gamma_x\). 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.

MSC:
03C57 Computable structure theory, computable model theory
03D45 Theory of numerations, effectively presented structures
03D50 Recursive equivalence types of sets and structures, isols
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