Stochastic Independence, Algebraic Independence and Abstract Connectedness

Abstract Mutual stochastic independences among σ-algebras and mutual algebraic independences among elements of semimodular lattices are observed to have a very similar behaviour. We suggest abstract independence structures called I -relations describing it. Presented examination of I -relations resembles a theory of abstract connectedness: a dual characterization of I -relations by families of connected sets is found by means of a special Galois connection. Representations of I -relations in the matroid theory sense by σ-algebras and by elements of lattices are discussed.