On the compounding of higher order monotonic pseudo-Boolean functions

TL;DR

This paper provides a simplified proof demonstrating that the property of being a higher order monotonic pseudo-Boolean function is preserved under compounding, extending previous results for submodular and k-alternating functions.

Contribution

It offers a more accessible proof of the preservation of higher order monotonicity under compounding, generalizing earlier complex formulas.

Findings

Higher order monotonic functions are preserved under compounding.

Simplified proof improves understanding of function properties.

Extends previous results on submodular and k-alternating functions.

Abstract

Compounding submodular monotone (i.e. 2-alternating) set functions on a finite set preserves this property, as shown in 2010. A natural generalization to k-alternating functions was presented in 2018, however hardly readable because of page long formulas. We give an easier proof of a more general result, exploiting known properties of higher order monotonic functions.