Abstract
In this article we presentconcrete results on both the countable and uncountable categoricity of some fragments. Our research on the categoricity of these fragments is conducted within theextended framework of normal Jonsson theory. The fragments are derived through the application of the closure operator of a specified pregeometry, and the involved sets are regular. The resulting models in this context constitute the Kaiser class associated with a studied Jonsson theory. These models exhibitdistinct and interesting structural properties, making them a subject of extensive study. Furthermore, we analyze how variations in the underlying pregeometry influence the classification of fragments. Such an approach makes it possible to identify broader and deeper connections between categoricity and model-theoretic stability, thereby significantly expanding and refining the understanding of their interrelation in the context of contemporary fundamental scientific research.