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 relevant variables of the problem under consideration. In this paper we investigate the decomposability of functions which take values into an Abelian Group. Examples of such functions include: probability distributions, energy functions, value functions, fitness functions, and relations. For such problems we define a notion of conditional independence between subsets of the problem's variables. We prove a decomposition theorem that relates independences between subsets of the problem's variables with a factorization property of the respective function. As particular cases of this theorem we retrieve the Hammersley-Clifford theorem for probability distributions; an Additive Decomposition theorem for energy functions, value functions, fitness functions; and a relational algebra decomposition theorem.