  The basic idea behind information algebras is that information comes in
pieces, each referring to a certain question, that these pieces can be combined
or aggregated and that the part relating to a given question can be extracted.
This algebraic structure can be given different forms. Questions were
originally represented by subsets of variables. Pieces of information were then
represented by valuations associated with the domains of variables. This leads
to an algebraic structure called valuation algebras. The basic axiomatics of
this algebraic structure was in essence proposed by Shenoy and Shafer. Here a
much more general view of systems of questions is proposed and pieces of
information are related to the elements of this system of questions. This leads
to a new and extended system of axioms for information algebras. Classical
valuation algebras are essentially a special case of this new system. A full
discussion of the algebraic theory of this new information algebras is given,
including local computation, duality between labeled and domain-free versions
of the algebras, order of information, finiteness of information and
approximation, compact and continuous information algebras. Finally a rather
complete discussion of uncertain information, based on random maps into
information algebras is presented. This is shown to represent a generalisation
of classical Dempster-Shafer theory.
