On monoids of metric preserving 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_ | 1866911846316900352 |
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| author | Bilet, Viktoriia Dovgoshey, Oleksiy |
| author_facet | Bilet, Viktoriia Dovgoshey, Oleksiy |
| contents | Let $\mathbf{X}$ be a class of metric spaces and let $\mathbf{P}_{\mathbf{X}}$ be the set of all $f:[0, \infty)\to [0, \infty)$ preserving $\mathbf{X},$ $(Y, f\circρ)\in\mathbf{X}$ whenever $(Y, ρ)\in\mathbf{X}.$ For arbitrary subset $\mathbf{A}$ of the set of all metric preserving functions we show that the equality $\mathbf{P}_{\mathbf{X}}=\mathbf{A}$ has a solution iff $\mathbf{A}$ is a monoid with respect to the operation of function composition. In particular, for the set $\mathbf{SI}$ of all amenable subadditive increasing functions there is a class $\mathbf{X}$ of metric spaces such that $\mathbf{P}_{\mathbf{X}}=\mathbf{SI}$ holds, which gives a positive answer to the question of paper [1]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13280 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On monoids of metric preserving functions Bilet, Viktoriia Dovgoshey, Oleksiy General Topology Primary 26A30, Secondary 54E35, 20M20 Let $\mathbf{X}$ be a class of metric spaces and let $\mathbf{P}_{\mathbf{X}}$ be the set of all $f:[0, \infty)\to [0, \infty)$ preserving $\mathbf{X},$ $(Y, f\circρ)\in\mathbf{X}$ whenever $(Y, ρ)\in\mathbf{X}.$ For arbitrary subset $\mathbf{A}$ of the set of all metric preserving functions we show that the equality $\mathbf{P}_{\mathbf{X}}=\mathbf{A}$ has a solution iff $\mathbf{A}$ is a monoid with respect to the operation of function composition. In particular, for the set $\mathbf{SI}$ of all amenable subadditive increasing functions there is a class $\mathbf{X}$ of metric spaces such that $\mathbf{P}_{\mathbf{X}}=\mathbf{SI}$ holds, which gives a positive answer to the question of paper [1]. |
| title | On monoids of metric preserving functions |
| topic | General Topology Primary 26A30, Secondary 54E35, 20M20 |
| url | https://arxiv.org/abs/2404.13280 |