Inverses, Conditionals and Compositional Operators in Separative
  Valuation Algebra
  Compositional models were introduce by Jirousek and Shenoy in the general
framework of valuation-based systems. They based their theory on an axiomatic
system of valuations involving not only the operations of combination and
marginalisation, but also of removal. They claimed that this systems covers
besides the classical case of discrete probability distributions, also the
cases of Gaussian densities and belief functions, and many other systems.
  Whereas their results on the compositional operator are correct, the
axiomatic basis is not sufficient to cover the examples claimed above. We
propose here a different axiomatic system of valuation algebras, which permits
a rigorous mathematical theory of compositional operators in valuation-based
systems and covers all the examples mentioned above. It extends the classical
theory of inverses in semigroup theory and places thereby the present theory
into its proper mathematical frame. Also this theory sheds light on the
different structures of valuation-based systems, like regular algebras
(represented by probability potentials), canncellative algebras (Gaussian
potentials) and general separative algebras (density functions).
