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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.17037 |
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| _version_ | 1866909405325295616 |
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| author | Jardón, Daniel Sánchez, Iván Sanchis, Manuel |
| author_facet | Jardón, Daniel Sánchez, Iván Sanchis, Manuel |
| contents | Given a uniform space $(X, \mathcal{U})$, we denote by $\mathcal{F}(X)$ to the family of all normal upper semicontinuous fuzzy sets $u \colon X \to [0,1]$ with compact support. In this paper, we study transitivity on some uniformities on $\mathcal{F}(X)$: the level-wise uniformity $\mathcal{U}_{\infty}$, the Skorokhod uniformity $\mathcal{U}_{0}$, and the sendograph uniformity $\mathcal{U}_S$. If $f \colon (X, \mathcal{U}) \to (X, \mathcal{U})$ is a continuous function, we mainly characterize when the induced dynamical systems $\widehat{f} \colon (\mathcal{F}(X), \mathcal{U}_{\infty}) \to (\mathcal{F}(X), \mathcal{U}_{\infty})$, $\widehat{f} \colon (\mathcal{F}(X), \mathcal{U}_{0}) \to (\mathcal{F}(X), \mathcal{U}_{0})$ and $\widehat{f} \colon (\mathcal{F}(X), \mathcal{U}_{S}) \to (\mathcal{F}(X), \mathcal{U}_{S})$ are transitive, where $\widehat{f}$ is the Zadeh's extension of $f$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17037 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transitivity of some uniformities on fuzzy sets Jardón, Daniel Sánchez, Iván Sanchis, Manuel General Topology Given a uniform space $(X, \mathcal{U})$, we denote by $\mathcal{F}(X)$ to the family of all normal upper semicontinuous fuzzy sets $u \colon X \to [0,1]$ with compact support. In this paper, we study transitivity on some uniformities on $\mathcal{F}(X)$: the level-wise uniformity $\mathcal{U}_{\infty}$, the Skorokhod uniformity $\mathcal{U}_{0}$, and the sendograph uniformity $\mathcal{U}_S$. If $f \colon (X, \mathcal{U}) \to (X, \mathcal{U})$ is a continuous function, we mainly characterize when the induced dynamical systems $\widehat{f} \colon (\mathcal{F}(X), \mathcal{U}_{\infty}) \to (\mathcal{F}(X), \mathcal{U}_{\infty})$, $\widehat{f} \colon (\mathcal{F}(X), \mathcal{U}_{0}) \to (\mathcal{F}(X), \mathcal{U}_{0})$ and $\widehat{f} \colon (\mathcal{F}(X), \mathcal{U}_{S}) \to (\mathcal{F}(X), \mathcal{U}_{S})$ are transitive, where $\widehat{f}$ is the Zadeh's extension of $f$. |
| title | Transitivity of some uniformities on fuzzy sets |
| topic | General Topology |
| url | https://arxiv.org/abs/2411.17037 |