  Information algebras arise from the idea that information comes in pieces
which can be aggregated or combined into new pieces, that information refers to
questions and that from any piece of information, the part relevant to a given
question can be extracted. This leads to a certain type of algebraic
structures, basically semilattices endowed with with additional unary
operations. These operations essentially are (dual) existential quantifiers on
the underlying semilattice. The archetypical instances of such algebras are
semilattices of subsets of some universe, together with the saturation
operators associated with a family of equivalence relations on this universe.
Such algebras will be called {\em set algebras} in our context. Our first
result is a basic representation theorem: Every abstract information algebra is
isomorphic to a set algebra. When it comes to combine pieces of information,
the idea to model the logical connectives {\em and}, {\em or} or {\em not} is
quite natural. Accordingly, we are especially interested in information
algebras where the underlying semilattice is a lattice, typically distributive
or even Boolean. A major part of this paper is therefore devoted to developing
explicitly a full-fledged natural duality theory extending Stone resp.
Priestley duality in a suitable way in order to take into account the
additional operations.
