Characterizations of monotone right continuous functions which generate associative functions

TL;DR

This paper characterizes when a class of functions constructed via monotone right continuous functions and associative functions with neutral elements are themselves associative, based on properties of the range of the monotone functions.

Contribution

It provides necessary and sufficient conditions for the associativity of functions generated through monotone right continuous functions and associative functions with neutral elements.

Findings

Conditions depend only on the properties of the range of the monotone function

Provides a complete characterization of associativity in this class of functions

Extends understanding of the structure of associative functions generated by monotone transformations

Abstract

Associativity of a two-place function $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(T^*(f(x),f(y)))$ where $T^*:[0,1]^2\rightarrow[0,1]$ is an associative function with neutral element in $[0,1]$, $f: [0,1]\rightarrow [0,1]$ is a monotone right continuous function and $f^{(-1)}:[0,1]\rightarrow[0,1]$ is the pseudo-inverse of $f$ depends only on properties of the range of $f$. The necessary and sufficient conditions for the $T$ to be associative are presented by applying the properties of the monotone right continuous function $f$.