Characterizations of monotone right continuous functions which generate associative functions
Yun-Mao Zhang, Xue-ping Wang
TL;DR
This work characterizes when the binary operation $T(x,y)=f^{(-1)}(T^{*}(f(x),f(y)))$, built from an associative $T^{*}$ with neutral element and a monotone right-continuous generator $f$, is associative on $[0,1]$. By representing $M=\mathrm{Ran}(f)$ via a unique interval-set decomposition and introducing an induced operation $x\otimes y=G_{M}(T^{*}(x,y))$ on $M$, the authors prove that $T$ is associative if and only if $\otimes$ is associative, which equivalently requires $\mathfrak{T}(M)\cap M=\emptyset$ when $f(1)=1$ or $I(M)\cap (M\setminus\{f(1)\})=\emptyset$ when $f(1)<1$. The results yield concrete criteria for the associativity of $T$ and provide conditions under which $T$ becomes a t-norm or a t-conorm, linking the range structure of $f$ to classical fuzzy-logic operators. The study extends and unifies prior additive-generator approaches by leveraging right-continuous monotone generators and range-based decompositions.
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 $f$.