A simple algebraic characterization of nonstandard extensions

TL;DR

This paper introduces the concept of functional extensions of sets using algebraic properties, linking them to ultrapowers, and provides an algebraic proof of Keisler's characterization of nonstandard models.

Contribution

It offers a new algebraic framework for understanding nonstandard extensions and characterizes those that are limit ultrapowers, simplifying Keisler's theorem proof.

Findings

Functional extensions are characterized algebraically.

Nonstandard extensions correspond to limit ultrapowers.

Provides a purely algebraic proof of Keisler's theorem.

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.