A simple algebraic characterization of nonstandard extensions

Forti, Marco

doi:10.1090/S0002-9939-2011-11104-3

A simple algebraic characterization of nonstandard extensions
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Proc. Amer. Math. Soc. 140 (2012), 2903-2912 Request permission

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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