  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.
