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Compositional models were introduce by Jirousek and Shenoy in the generalframework of valuation-based systems. They based their theory on an axiomaticsystem of valuations involving not only the operations of combination andmarginalisation, but also of removal. They claimed that this systems coversbesides the classical case of discrete probability distributions, also thecases of Gaussian densities and belief functions, and many other systems.  Whereas their results on the compositional operator are correct, theaxiomatic basis is not sufficient to cover the examples claimed above. Wepropose here a different axiomatic system of valuation algebras, which permitsa rigorous mathematical theory of compositional operators in valuation-basedsystems and covers all the examples mentioned above. It extends the classicaltheory of inverses in semigroup theory and places thereby the present theoryinto its proper mathematical frame. Also this theory sheds light on thedifferent structures of valuation-based systems, like regular algebras(represented by probability potentials), canncellative algebras (Gaussianpotentials) and general separative algebras (density functions).

Inverses, Conditionals and Compositional Operators in Separative Valuation Algebra