Ultrafilter Extensions of Models

Abstract

We show that any model $\mathfrak{A}$ can be extended, in a canonical way, to a model $\beta\mathfrak{A}$ consisting of ultrafilters over it. The extension procedure preserves homomorphisms: any homomorphism of $\mathfrak{A}$ into $\mathfrak{B}$ extends to a continuous homomorphism of $\beta\mathfrak{A}$ into $\beta\mathfrak{B}$. Moreover, if a model $\mathfrak{B}$ carries a compact Hausdorff topology which is (in a certain sense) compatible, then any homomorphism of $\mathfrak{A}$ into $\mathfrak{B}$ extends to a continuous homomorphism of $\beta\mathfrak{A}$ into $\mathfrak{B}$. This is also true for embeddings instead of homomorphisms.

Partially supported by an INFTY grant of ESF.