The arity gap of order-preserving functions and extensions of pseudo-Boolean functions

TL;DR

This paper classifies order-preserving functions based on their arity gap, focusing on aggregation functions and Lovász extensions of pseudo-Boolean functions, providing explicit classifications for these classes.

Contribution

It provides the first explicit classification of the arity gap for Lovász extensions and order-preserving functions between partially ordered sets.

Findings

Classified Lovász extensions of pseudo-Boolean functions by arity gap

Established explicit classification for order-preserving functions between posets

Enhanced understanding of aggregation functions and their properties

Abstract

The aim of this paper is to classify order-preserving functions according to their arity gap. Noteworthy examples of order-preserving functions are so-called aggregation functions. We first explicitly classify the Lov\'asz extensions of pseudo-Boolean functions according to their arity gap. Then we consider the class of order-preserving functions between partially ordered sets, and establish a similar explicit classification for this function class.