Spectrum of a separable Dynkin algebra and topology on it

Abstract

We continue a preparation to the construction of generalized axiomatics of probability theory, begined in previous papers of author. Our approach is based on the investigation of set systems more general than usual set algebras and their Boolean versions. We name them Dynkin algebras. The spectrum of separable Dynkin algebra and appropriate Grothendieck topology on it are introduced. Separable Dynkin algebras constitute a natural class of a abstract Dynkin algebras defined by author previously. A structural result was received in this mentioned paper: every separable Dynkin algebra is a union of its maximal Boolean subalgebras. In this paper, we define, basing on this result, the spectrum of a separable Dynkin algebra and an appropriate Grothendieck topology on it. The corresponding constructions are similar to the construction of the simple spectrum of a commutative ring and that of the Zariski topology on it. The analogy here is incomplete: Zariski topology makes the spectrum of a commutative ring to be a usual topological space while Grothendieck topology, which is not a topology in usual sense, transforms the spectrum of a Dynkin algebra into a more abstract object (site or situs via Grothendieck). This is sufficient for our aims.