The characterizations of monotone functions which generate associative functions

TL;DR

This paper characterizes the conditions under which a class of two-place functions, constructed from associative functions and monotone functions, are themselves associative, focusing on the properties of the monotone functions involved.

Contribution

It provides necessary and sufficient conditions for the associativity of functions generated by monotone functions and associative functions, based on the properties of the range of the monotone functions.

Findings

Characterization of associative functions generated by monotone functions.

Conditions depend on the range properties of the monotone functions.

Provides a complete criterion for associativity in this class of functions.

Abstract

Associativity of a two-place function $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,\infty]^2\rightarrow[0,\infty]$ is an associative function, $f: [0,1]\rightarrow [0,\infty]$ is a monotone function which satisfies either $f(x)=f(x^{+})$ when $f(x^{+})\in \mbox{Ran}(f)$ or $f(x)\neq f(y)$ for any $y\neq x$ when $f(x^{+})\notin \mbox{Ran}(f)$ for all $x\in[0,1]$ and $f^{(-1)}:[0,\infty]\rightarrow[0,1]$ is a 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 function $f$.