Submodularity, supermodularity, and higher-order monotonicities of pseudo-boolean functions

Classes of set functions defined by the positivity or negativity of the higher-order derivatives of their pseudo-Boolean polynomial representations generalize those of monotone, supermodular, and submodular functions. In this paper, these classes are characterized by functional inequalities and are shown to be closed both under algebraic closure conditions and a local closure criterion. It is shown that for every m : 1, in addition to the class of all set functions, there are only three other classes satisfying these algebraic and local closure conditions: those having positive, respectively negative, mth-order derivatives, and those having a polynomial representation of degree less than m.