Ultrametric-preserving functions as monoid endomorphisms
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929383486259200 |
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| author | Dovgoshey, Oleksiy |
| author_facet | Dovgoshey, Oleksiy |
| contents | Let $\mathbb{R}^{+}=[0, \infty)$ and let $\mathbf{End}_{\mathbb{R}^+}$ be the set of all endomorphisms of the monoid $(\mathbb{R}^+, \vee)$. The set $\mathbf{End}_{\mathbb{R}^+}$ is a monoid with respect to the operation of the function composition $g \circ f$. It is shown that $g : \mathbb{R}^+ \to \mathbb{R}^+$ is pseudoultrametric-preserving iff $g \in \mathbf{End}_{\mathbb{R}^+}$. In particular, a function $f : \mathbb{R}^+ \to \mathbb{R}^+$ is ultrametrics-preserving iff it is an endomorphism of $(\mathbb{R}^+,\vee)$ with kernel consisting only the zero point. We prove that a given $\mathbf{A} \subseteq \mathbf{End}_{\mathbb{R}^+}$ is a submonoid of $(\mathbf{End}, \circ)$ iff there is a class $\mathbf{X}$ of pseudoultrametric spaces such that $\mathbf{A}$ coincides with the set of all functions which preserve the spaces from $\mathbf{X}$. An explicit construction of such $\mathbf{X}$ is given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07166 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ultrametric-preserving functions as monoid endomorphisms Dovgoshey, Oleksiy General Topology Primary 26A30, Secondary 54E35, 20M20 Let $\mathbb{R}^{+}=[0, \infty)$ and let $\mathbf{End}_{\mathbb{R}^+}$ be the set of all endomorphisms of the monoid $(\mathbb{R}^+, \vee)$. The set $\mathbf{End}_{\mathbb{R}^+}$ is a monoid with respect to the operation of the function composition $g \circ f$. It is shown that $g : \mathbb{R}^+ \to \mathbb{R}^+$ is pseudoultrametric-preserving iff $g \in \mathbf{End}_{\mathbb{R}^+}$. In particular, a function $f : \mathbb{R}^+ \to \mathbb{R}^+$ is ultrametrics-preserving iff it is an endomorphism of $(\mathbb{R}^+,\vee)$ with kernel consisting only the zero point. We prove that a given $\mathbf{A} \subseteq \mathbf{End}_{\mathbb{R}^+}$ is a submonoid of $(\mathbf{End}, \circ)$ iff there is a class $\mathbf{X}$ of pseudoultrametric spaces such that $\mathbf{A}$ coincides with the set of all functions which preserve the spaces from $\mathbf{X}$. An explicit construction of such $\mathbf{X}$ is given. |
| title | Ultrametric-preserving functions as monoid endomorphisms |
| topic | General Topology Primary 26A30, Secondary 54E35, 20M20 |
| url | https://arxiv.org/abs/2406.07166 |