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A simple algebraic characterization of nonstandard extensions


Author: Marco Forti
Journal: Proc. Amer. Math. Soc. 140 (2012), 2903-2912
MSC (2010): Primary 03H05, 03C07, 03C20; Secondary 26E35
DOI: https://doi.org/10.1090/S0002-9939-2011-11104-3
Published electronically: November 28, 2011
MathSciNet review: 2910776
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Abstract: We introduce the notion of functional extension of a set $ X$, by means of two natural algebraic properties of the operator ``$ *$'' on unary functions. We study the connections with ultrapowers of structures with universe $ X$, and we give a simple characterization of those functional extensions that correspond to limit ultrapower extensions. In particular we obtain a purely algebraic proof of Keisler's characterization of nonstandard (= complete elementary) extensions.


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Additional Information

Marco Forti
Affiliation: Dipartimento di Matematica Applicata “U. Dini”, Universitá di Pisa, Via Buonarroti 1C, 56100 Pisa, Italy

DOI: https://doi.org/10.1090/S0002-9939-2011-11104-3
Received by editor(s): December 24, 2010
Received by editor(s) in revised form: January 31, 2011, and March 3, 2011
Published electronically: November 28, 2011
Additional Notes: This work was partially supported by MIUR Grants PRIN 2007, 2009, Italy.
Communicated by: Julia Knight
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.