Algebraic and topological closure conditions for classes of pseudo-Boolean functions

Abstract

We examine classes of real-valued functions of 0–1 variables closed under algebraic operations as well as topological convergence, and having a certain local characteristic (requiring that any function not in the class should have a k -variable minor not belonging to this class). It is shown that for k = 2 , the only 4 maximal classes with these properties are those of submodular, supermodular, monotone increasing and monotone decreasing functions. All the 13 locally defined closed classes are determined and shown to be intersections of the 4 maximal ones. All maximal classes for k ≥ 3 are determined and characterized by the sign of higher order derivatives of the functions in the class.