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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.17418 |
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| _version_ | 1866909904425451520 |
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| author | Hom, Ujjal Kumar Singha, Manoranjan |
| author_facet | Hom, Ujjal Kumar Singha, Manoranjan |
| contents | Characterizations of ultrafilters belong to the smallest ideal of Stone-Čech compactification of a discrete semigroup are exhibited using syndetic sets, strongly central sets and very strongly central sets respectively. These lead to represent piecewise syndetic sets of a semigroup in terms of the sets that contain a broken $\mathcal{A}$ set, where $\mathcal{A}\in\{$ syndetic, quasi-central, central, strongly central, very strongly central$\}$. Also, a characterization of broken IP$^{n}$ sets using ultrafilters, and the equivalence between the sets that contain a broken IP set and sets that contain a broken IP$^{n}$ are established, $n\in \mathbb{N}$. Without assuming the countability of a semigroup, it is shown that piecewise syndetic sets i.e., sets that contain a broken syndetic set (broken IP set) force uniform recurrence (recurrence respectively) and vice versa. In addition, all the said results are established near idempotent of a semitopological semigroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17418 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exhibition of piecewise syndetic and broken IP sets near idempotent Hom, Ujjal Kumar Singha, Manoranjan General Topology 54D80, 22A15, 54D35, 37B20, 37B05 Characterizations of ultrafilters belong to the smallest ideal of Stone-Čech compactification of a discrete semigroup are exhibited using syndetic sets, strongly central sets and very strongly central sets respectively. These lead to represent piecewise syndetic sets of a semigroup in terms of the sets that contain a broken $\mathcal{A}$ set, where $\mathcal{A}\in\{$ syndetic, quasi-central, central, strongly central, very strongly central$\}$. Also, a characterization of broken IP$^{n}$ sets using ultrafilters, and the equivalence between the sets that contain a broken IP set and sets that contain a broken IP$^{n}$ are established, $n\in \mathbb{N}$. Without assuming the countability of a semigroup, it is shown that piecewise syndetic sets i.e., sets that contain a broken syndetic set (broken IP set) force uniform recurrence (recurrence respectively) and vice versa. In addition, all the said results are established near idempotent of a semitopological semigroup. |
| title | Exhibition of piecewise syndetic and broken IP sets near idempotent |
| topic | General Topology 54D80, 22A15, 54D35, 37B20, 37B05 |
| url | https://arxiv.org/abs/2501.17418 |