The cone of nondecreasing set functions

Abstract

A real valued set function, defined on a finite set E = {l,...,n}, is said to be non-decreasing if r(I) ≤ r(J), ∀I ⊆ J ⊆ E. Let 2 E be the set of all subsets of E and n 0 = 2|E| Every real valued function can be represented by a vector x = (x J ) J∈2 E , with x J = r(J). On the other hand, every x ∈ R n ° generates a real valued function r: 2 E → R. Depending on the concrete situation we use the notation $${F_{m}}\left( E \right) = \left\{ {r:{2^{E}} \to R;\quad I \subseteq J \to r\left( I \right) \leqslant r\left( J \right);\quad \forall I \subseteq J \subseteq E} \right\} $$ or $$R_m^{{n_0}} = \{ x \in {R^{{n_0}}}:{x_I} \leqslant {x_J},\forall I \subseteq J \subseteq E\}$$ It is easy to see that F m,o (E) = {r ∈ F m (E): r(0) = 0} is a convex cone. In [2] we find a more general definition of a cone. If c ∈ R then F m,c (E) = {r ∈ F m (E): r(0) = c} is a cone with the vertex r(J) = c,∀J ⊆ E. Further we investigate the cone Fm,0o(E), because F m,c (E) differs from F m,o (E) only by a translation. We show that F m,o (E) is generated by all nondecreasing Boolean set functions.