Extreme Supermodular Set Functions over Ve Variables

The class of supermodular functions on the power set of a non-empty nite set N forms a cone. It can be viewed as the direct sum of a linear subspace and of a cone of standardized supermodular functions which has nitely many extreme rays. Every extreme ray can be described by a standardized integer-valued set function. We analyse the situation in the case when N has ve elements (= variables). A computer program was used to obtain a catalogue of all classes of permutably equivalent extreme standardized supermodular functions on the power set of N. We consider several alternative ways of representation of these equivalence classes and use various characteristics to describe them. Moreover, two relevant hypotheses valid in case of four variables are disproved in case of ve variables.