Information Algebras and Consequence Operators

We explore a connection between different ways of representing information in computer science. We show that relational databases, modules, algebraic specifications and constraint systems all satisfy the same ten axioms. A commutative semigroup together with a lattice satisfying these axioms is then called an “information algebra”. We show that any compact consequence operator satisfying the interpolation and the deduction property induces an information algebra. Conversely, each finitary information algebra can be obtained from a consequence operator in this way. Finally we show that arbitrary (not necessarily finitary) information algebras can be represented as some kind of abstract relational database called a tuple system.