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arXiv · 2409.02941

The characterizations of monotone functions which generate associative functions

Chen Meng·Yun-Mao Zhang·Xue-ping Wang
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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 $f$.

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Chen Meng, Yun-Mao Zhang, Xue-ping Wang. 2024-08-24. The characterizations of monotone functions which generate associative functions. https://doi.org/10.1016/j.fss.2024.109201

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