Information and Set Algebras: Interpretation and Uniqueness of Conditional Independence

A new seemingly weak axiomatic formulation of information algebras is given. It is shown how such information algebras can be embedded into set (information) algebras. In set algebras there is a natural relation of conditional independence between partitions. Via the embedding of information algebras this relation carries over to information algebras. The new axiomatic formulation is thereby shown to be equivalent to the one given in arXiv:1701.02658. In this way the abstract concept of conditional independence in information algebras gets a concrete interpretation in terms of set theoretical relations.