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 xed Jonsson spectrum. The axiomatically given property of nonforking satises 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 denable subsets of the Jonsson theory's semantic model is dened. The properties of Morley rank and degree for denable 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.