Products and Complete Classes

Abstract

The prominent place held by the direct product in group theory, ring theory and in the theories of other classical algebraic objects, depends largely on the fact that the direct product of groups is a group, the direct product of rings is a ring, etc. For the theory of first-order languages, the concept is less convenient since the direct product of systems having a certain first-order property may in general not possess that property. At the end of the 1950’s, so-called ultraproducts of systems were introduced. They in full measure satisfied the requirement that first-order properties be preserved. At the present time the technique of the ultraproduct construction is playing a great role in the solution of rather delicate problems of first-order theory. The basic ideas and results of the theory of ultraproducts are elucidated in this section.