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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some properties of $ \forall\exists$ models in the isols


Author: T. G. McLaughlin
Journal: Proc. Amer. Math. Soc. 97 (1986), 495-502
MSC: Primary 03D50; Secondary 03C62, 03H15
MathSciNet review: 840636
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Abstract: It is a consequence of theorems proved by Nerode [10] and Hirschfeld [7] that every countable model of $ \forall \exists $ arithmetic is isomorphic to a subsemiring of a one-generator semiring of isols. We characterize, in terms of the generators of "Nerode semirings", the contents of arbitrary semirings $ {\mathbf{R}}$ of isols that are models of $ \forall \exists $ arithmetic, and we show that all such $ {\mathbf{R}}$ are in fact models of the $ \omega $-true $ \forall \exists $ sentences of isol theory. We solve one of the chief problems left open in [8], and in $ \S3$ we provide an example of the applied virtues of $ \forall \exists $-correct subsemirings of the isols.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0840636-X
PII: S 0002-9939(1986)0840636-X
Keywords: Isol, Model of $ \forall \exists $ arithmetic, Nerode semiring
Article copyright: © Copyright 1986 American Mathematical Society