Prime types and geometric completeness

Abstract

The geometric form of Hilbert's Nullstellensatz may be understood as a property of "geometric saturation" in algebraically closed fields. We conceptualise this property in the language of first order logic, following previous approaches and borrowing ideas from classical model theory, universal algebra and positive logic. This framework contains a logical equivalent of the algebraic theory of prime and radical ideals, as well as the basics of an "affine algebraic geometry" in quasivarieties. Hilbert's theorem may then be construed as a model-theoretical property, weaker than and equivalent in certain cases to positive model-completeness, and this enables us to geometrically reinterpret model-completeness itself. The three notions coincide in the theories of (pure) fields and we apply our results to group-based algebras, which supply a way of dealing with certain functional field expansions.