The characterizations of monotone functions which generate associative functions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908623502835712 |
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| author | Meng, Chen Zhang, Yun-Mao Wang, Xue-ping |
| author_facet | Meng, Chen Zhang, Yun-Mao Wang, Xue-ping |
| contents | 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 $f$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02941 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The characterizations of monotone functions which generate associative functions Meng, Chen Zhang, Yun-Mao Wang, Xue-ping General Mathematics 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 $f$. |
| title | The characterizations of monotone functions which generate associative functions |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2409.02941 |