Independence and simplicity in Jonsson theories with abstract geometry

Abstract

The concepts of forking and independence are examined in the framework of the study of Jonsson theories and the fixed Jonsson spectrum. The axiomatically given property of nonforking satisfies the classical notion of nonforking in the sense of S. Shelah and the approach to this concept by Laskar-Poizat. On this basis, the simplicity of the Jonsson theory is determined and the Jonsson analog of the Kim-Pillay theorem is given. Abstract pregeometry on definable subsets of the Jonsson theory's semantic model is defined. The properties of Morley rank and degree for definable subsets of the semantic model are considered. A criterion of uncountable categoricity for the hereditary Jonsson theory in the language of central types is proved.