Certain Observations on Ideals Associated With Weighted Density Using Modulus Functions
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arXiv
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| Format: | Preprint |
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2025
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| author | Das, Pratulananda Das, Subhankar |
| author_facet | Das, Pratulananda Das, Subhankar |
| contents | In this article our main object of investigation is the simple modular density ideals $\mathcal{Z}_g(f)$ introduced in [Bose et al., Indag. math., 2018] where $g$ is a weight function, more precisely, $g\in G$, $G=\{g:ω\to [0,\infty):\frac{k}{g(k)}\not\to 0 \text{ and }\:\: g(k)\to \infty \text{ as }\:\:k\to \infty \}$ and $f$ is an unbounded modulus function. We mainly investigate certain properties of these ideals in line of [Kwela et al, J. math. Anal. Appl., 2019]. For an unbounded modulus function $f$ it is shown that there are $1$ or $\ck$ many functions $g\in G$ generating the same ideal $\mathcal{Z}_g(f)$. We then obtain certain interactive results involving the sequence of submeasures $\{ϕ_k\}_{k\in ω}$ generating the ideal $\mathcal{Z}_g(f)$ and the functions $g,f$. Finally, we present some observations on $\mathcal{Z}_g(f)$ ideals related to the notion of increasing-invariance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02682 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Certain Observations on Ideals Associated With Weighted Density Using Modulus Functions Das, Pratulananda Das, Subhankar General Topology 03E15, 11B05 In this article our main object of investigation is the simple modular density ideals $\mathcal{Z}_g(f)$ introduced in [Bose et al., Indag. math., 2018] where $g$ is a weight function, more precisely, $g\in G$, $G=\{g:ω\to [0,\infty):\frac{k}{g(k)}\not\to 0 \text{ and }\:\: g(k)\to \infty \text{ as }\:\:k\to \infty \}$ and $f$ is an unbounded modulus function. We mainly investigate certain properties of these ideals in line of [Kwela et al, J. math. Anal. Appl., 2019]. For an unbounded modulus function $f$ it is shown that there are $1$ or $\ck$ many functions $g\in G$ generating the same ideal $\mathcal{Z}_g(f)$. We then obtain certain interactive results involving the sequence of submeasures $\{ϕ_k\}_{k\in ω}$ generating the ideal $\mathcal{Z}_g(f)$ and the functions $g,f$. Finally, we present some observations on $\mathcal{Z}_g(f)$ ideals related to the notion of increasing-invariance. |
| title | Certain Observations on Ideals Associated With Weighted Density Using Modulus Functions |
| topic | General Topology 03E15, 11B05 |
| url | https://arxiv.org/abs/2505.02682 |