The characterizations of monotone functions which generate associative functions
Chen Meng, Yun-Mao Zhang, Xue-ping Wang
TL;DR
The paper characterizes when the binary function $T(x,y)=f^{(-1)}(F(f(x),f(y)))$, with $F$ associative and $f$ a monotone generator, is itself associative. The central idea is to transfer the problem to an induced operation $\otimes$ on $\mathrm{Ran}(f)$ via $x\otimes y=G_M(F(x,y))$, showing that $T$ is associative if and only if $\otimes$ is, and then providing two main criteria (one under cancellativity of $F$ and a range containment condition, and another under a containment $F(C,M)\cup F(M,C)\subseteq M\setminus C$) that are necessary and sufficient for associativity. The work formalizes a canonical decomposition of $\mathrm{Ran}(f)$ into a countable family of intervals and a countable set, with a map $G_M$ linking $M$ to $F$ via $x\otimes y$, and introduces $F$-conditions and $I_k$-intervals to establish when associativity holds. By connecting to additive generators and t-norm theory, the results generalize prior findings (e.g., Viceník, Zhang & Wang, and Yao et al.) to a broader class of monotone generators, including non-strictly monotone and non-strictly monotone right-continuous cases, and even non-increasing generators. The conclusions offer practical criteria for constructing associative $T$ from a given $f$ and $F$, with implications for generalized t-norms and related aggregation operators.
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 $f$.