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The basic idea behind information algebras is that information comes inpieces, each referring to a certain question, that these pieces can be combinedor aggregated and that the part relating to a given question can be extracted.This algebraic structure can be given different forms. Questions wereoriginally represented by subsets of variables. Pieces of information were thenrepresented by valuations associated with the domains of variables. This leadsto an algebraic structure called valuation algebras. The basic axiomatics ofthis algebraic structure was in essence proposed by Shenoy and Shafer. Here amuch more general view of systems of questions is proposed and pieces ofinformation are related to the elements of this system of questions. This leadsto a new and extended system of axioms for information algebras. Classicalvaluation algebras are essentially a special case of this new system. A fulldiscussion of the algebraic theory of this new information algebras is given,including local computation, duality between labeled and domain-free versionsof the algebras, order of information, finiteness of information andapproximation, compact and continuous information algebras. Finally a rathercomplete discussion of uncertain information, based on random maps intoinformation algebras is presented. This is shown to represent a generalisationof classical Dempster-Shafer theory.

Algebras of Information. A New and Extended Axiomatic Foundation