Cartesian product of combinatorially rich sets -- algebraic, elementary and dynamical approaches
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917809272913920 |
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| author | Debnath, Pintu |
| author_facet | Debnath, Pintu |
| contents | Using the methods of topological dynamics, H. Furstenberg introduced the notion of a central set and proved the famous Central Sets Theorem. D. De, N. Hindman, and D. Strauss introduced $C$-set, satisfying the strong central set theorem. Using the algebraic structure of the Stone-Čech compactification of a discrete semigroup, N. Hindman and D. Strauss proved that the Cartesian product of two $C$-sets is a $C$-set. S. Goswami has proved the same result using the elementary characterization of $C$-sets. In this article, we will prove that the product of two $C$-sets is a $C$-set, using the dynamical characterization of $C$-sets. Recently, S. Goswami has proved that the Cartesian product of two $CR$-sets is a $CR$-set, which was a question posed by N. Hindman, H. Hosseini, D. Strauss, and M. Tootkaboni in [Semigroup Forum 107 (2023)]. Here we also prove that the Cartesian product of two essential $CR$-sets is an essential $CR$-set. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_12693 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cartesian product of combinatorially rich sets -- algebraic, elementary and dynamical approaches Debnath, Pintu Combinatorics 05D10, 22A15, 54D35 Using the methods of topological dynamics, H. Furstenberg introduced the notion of a central set and proved the famous Central Sets Theorem. D. De, N. Hindman, and D. Strauss introduced $C$-set, satisfying the strong central set theorem. Using the algebraic structure of the Stone-Čech compactification of a discrete semigroup, N. Hindman and D. Strauss proved that the Cartesian product of two $C$-sets is a $C$-set. S. Goswami has proved the same result using the elementary characterization of $C$-sets. In this article, we will prove that the product of two $C$-sets is a $C$-set, using the dynamical characterization of $C$-sets. Recently, S. Goswami has proved that the Cartesian product of two $CR$-sets is a $CR$-set, which was a question posed by N. Hindman, H. Hosseini, D. Strauss, and M. Tootkaboni in [Semigroup Forum 107 (2023)]. Here we also prove that the Cartesian product of two essential $CR$-sets is an essential $CR$-set. |
| title | Cartesian product of combinatorially rich sets -- algebraic, elementary and dynamical approaches |
| topic | Combinatorics 05D10, 22A15, 54D35 |
| url | https://arxiv.org/abs/2307.12693 |