Ultraproducts which are not saturated

Abstract

In this paper we continue our study, begun in [5], of the connection between ultraproducts and saturated structures. IfDis an ultrafilter over a setI, andis a structure (i.e., a model for a first order predicate logicℒ), the ultrapower ofmoduloDis denoted byD-prod. The ultrapower is important because it is a method of constructing structures which are elementarily equivalent to a given structure(see Frayne-Morel-Scott [3]). Our ultimate aim is to find out what kinds of structure are ultrapowers of. We made a beginning in [5] by proving that, assuming the generalized continuum hypothesis (GCH), for each cardinalαthere is an ultrafilterDover a set of powerαsuch that for all structures,D-prodisα⁺-saturated.