Decomposition of Multiplace Functions in Operations Research

Abstract

This extended abstract summarizes the results of a decomposition theory for multiplace functions that generalizes and unifies theories known from a number of areas in Operations Research. The considered decompositions of a multiplace function are representations as terms of functions of fewer variables where variables may be used only once. This restricted “disjoint” functional superposition or “substitution” has been defined independently in switching circuit design, combinatorial optimization over networks and clutters and ordinal and expected utility theory. There, it has led to interesting results on unique “normal form” representations, like additive utility functions. These results have great similarities that are explained by the proposed theory, where the admitted decompositions are characterized set-theoretically: An n-ary operation f on a given set is decomposed into “conditional” functions obtained from f by fixing variables suitably. The following exposition is fairly technical to state results precisely. Proofs are found in [5] and further references in [4][5].