Barycentrically associative and preassociative functions

TL;DR

This paper explores barycentric associativity and preassociativity in functions with indefinite arities, generalizing classical mean characterizations and expanding understanding of these properties in mathematical functions.

Contribution

It introduces the concept of barycentric preassociativity and generalizes Kolmogoroff-Nagumo's characterization to this broader class of functions.

Findings

Defined barycentric preassociativity as a generalization of associativity.

Provided a characterization of barycentrically preassociative functions.

Extended classical mean function results to new function classes.

Abstract

We investigate the barycentric associativity property for functions with indefinite arities and discuss the more general property of barycentric preassociativity, a generalization of barycentric associativity which does not involve any composition of functions. We also provide a generalization of Kolmogoroff-Nagumo's characterization of the quasi-arithmetic mean functions to barycentrically preassociative functions.