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Model companions and k-model completeness for the complete theories of Boolean algebras

Published online by Cambridge University Press:  12 March 2014

J. Mead  and
G. C. Nelson
J. Mead
Affiliation:
University of Iowa, Iowa City, IA 52242
G. C. Nelson
Affiliation:
University of Iowa, Iowa City, IA 52242
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Extract

In studying mathematical logic, Boolean algebras can play an important role in the organization of syntactic properties of first-order theories. In this paper we are concerned with syntactical questions about the complete theories of Boolean algebras, Tarski [12] and Ershov [4]. We give a syntactic characterization of the languages introduced in [4] by Ershov in terms of number of alternations of quantifiers and show that extensions with respect to these natural languages are equivalent to extensions with respect to ∀k(L−1) ⋃ ∃k(L−1) for specified, k. The main motivation for this paper was to examine a generalized notion of model completeness of A. Robinson [14], called k-model completeness, and to find real mathematical structures having these properties in a nontrivial way; this turns out to be realized in the very natural setting of Boolean algebras and our results are very definitive about these notions in that setting. Also, we point out here the connections of these notions with those examined by Chang in [2] and observe that the syntactic properties of the complete theories of Boolean algebras have much in common with those properties enjoyed by ∀n-axiomatizable theories which are categorical for “some” infinite cardinal. Finally, a complete characterization of the model companions of the complete theories of Boolean algebras is given with respect to Ershov's languages [4].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 45 , Issue 1 , March 1980 , pp. 47 - 55
Copyright
Copyright © Association for Symbolic Logic 1980

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References

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