A simple algebraic characterization of nonstandard extensions
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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.References
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Additional Information
- Marco Forti
- Affiliation: Dipartimento di Matematica Applicata “U. Dini”, Universitá di Pisa, Via Buonarroti 1C, 56100 Pisa, Italy
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2910776