Decompositions of functions based on arity gap

TL;DR

This paper classifies functions based on their arity gap, extending previous results to arbitrary sets, and provides unique decompositions especially when the codomain has a group structure, with applications to counting functions.

Contribution

It offers a complete classification of functions by arity gap for arbitrary sets and introduces unique decompositions when the codomain is a group, extending prior finite-function results.

Findings

Complete classification of functions by arity gap

Unique decompositions for functions with group-structured codomain

Counting functions with specific arity gaps for finite sets

Abstract

We study the arity gap of functions of several variables defined on an arbitrary set A and valued in another set B. The arity gap of such a function is the minimum decrease in the number of essential variables when variables are identified. We establish a complete classification of functions according to their arity gap, extending existing results for finite functions. This classification is refined when the codomain B has a group structure, by providing unique decompositions into sums of functions of a prescribed form. As an application of the unique decompositions, in the case of finite sets we count, for each n and p, the number of n-ary functions that depend on all of their variables and have arity gap p.