  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.
