A discrete Choquet integral for ordered systems

A model for a Choquet integral for arbitrary finite set systems is pre-sented. The model includes in particular the classical model on the system of all subsets of a finite set. The general model associates canonical non-negative and positively homogeneous superadditive functionals with gen-eralized belief functions relative to an ordered system, which are then ex-tended to arbitrary valuations on the set system. It is shown that the gen-eral Choquet integral can be computed by a simple Monge-type algorithm for so-called intersection systems, which include as a special case weakly union-closed families. Generalizing Lovász ’ classical characterization, we give a characterization of the superadditivity of the Choquet integral relative to a capacity on a union-closed system in terms of an appropriate model of supermodularity of such capacities