Independence, Decomposability and Functions which Take Values into an Abelian Group

Decomposition is an important property that we exploit in order to render problems more tractable. The decomposability of a problem implies the existence of some “independences” between the variables that are in play in the problem under consideration. In this paper we investigate the decomposability of functions which take values into an Abelian Group. For the problems which involve these types of functions we define a notion of conditional independence between subsets of the problem’s variables. We then prove a decomposition theorem that relates independences and a factorisation property of functions taking values into an Abelian Group. As particular cases of this theorem we retrieve the Hammersly-Clifford theorem for probability distributions, an Additive Decomposition theorem for functions such as value functions or fitnesses and a relational algebra decomposition theorem.