Associativity of two-place functions generated by left continuous monotone functions and other properties

TL;DR

This paper characterizes the associativity of a two-place function generated by a left continuous monotone function and an associative function, revealing that associativity depends solely on the properties of the range of the monotone function.

Contribution

It introduces a weak pseudo-inverse for monotone functions and uses it to analyze the associativity and other properties of a class of two-place functions.

Findings

Associativity depends only on the range properties of the monotone function.

The paper establishes conditions for idempotence, limits, and continuity of the two-place function.

It provides a characterization of the function's behavior based on the properties of the generating functions.

Abstract

This article introduces a weak pseudo-inverse of a monotone function, which is applied to characterize the associativity of a two-place function $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=t^{[-1]}(F(t(x),t(y)))$ where $F:[0,\infty]^2\rightarrow[0,\infty]$ is an associative function with neutral element in $[0,\infty]$, $t: [0,1]\rightarrow [0,\infty]$ is a left continuous monotone function and $t^{[-1]}:[0,\infty]\rightarrow[0,1]$ is the weak pseudo-inverse of $t$. It shows that the associativity of the function $T$ depends only on properties of the range of $t$. Moreover, it investigates the idempotence, the limit property, the conditional cancellation law and the continuity of the function $T$, respectively.