Structural Properties of Models of N1-Categorical Theories

Abstract

Publisher Summary In the structural theory of categoricity (in uncountable powers), the notions of a strongly minimal set and algebraic closure are introduced and it is shown that the structure of a strongly minimal set with respect to algebraic closure (acl) affects essentially the structure of the model itself. The structure of a strongly minimal set S with respect to the closure operator acl can be essentially characterized by the geometry associated with S. If the geometry associated with S over any non-algebraic element is isomorphic to geometry of a projective space over a division ring then the geometry associated with S is called “locally projective.” If the division ring in the definition is finite, then the main result describes the locally projective geometry as an affine or projective geometry over the division ring. Natural examples of strongly minimal structures with projective geometries are strongly minimal abelian groups and, more generally, modules. Affine spaces over division rings have locally projective geometries, which are not projective. The natural numbers with the successor operation is a typical example of a strongly minimal disintegrated structure. On the other hand such strongly minimal structures, as algebraically closed fields, can hardly be characterized in terms of their geometries.