Enumeration of idempotent-sum subsequences in finite cyclic semigroups and smooth sequences
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| Format: | Preprint |
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2025
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| _version_ | 1866910924639567872 |
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| author | Wang, Guoqing Zhao, Yang Yi, Xingliang |
| author_facet | Wang, Guoqing Zhao, Yang Yi, Xingliang |
| contents | The enumeration of zero-sum subsequences of a given sequence over finite cyclic groups is one classical topic, which starts from one question of P. Erdős. In this paper, we consider this problem in a more general setting -- finite cyclic semigroups. Let $\mathcal{S}$ be a finite cyclic semigroup. By $\textbf{e}$ we denote the unique idempotent of the semigroup $\mathcal{S}$. Let $T$ be a sequence over the semigroup $\mathcal{S}$, and let $N(T; \textbf{e})$ be the number of distinct subsequences of $T$ with sum being the idempotent $\textbf{e}$. We obtain the lower bound for $N(T; \textbf{e})$ in terms of the length of $T$, and moreover, prove that $T$ contains subsequences with some smooth-structure in case that $N(T; \textbf{e})$ is not large. Our result generalizes the theorem obtained by W. Gao [Discrete Math., 1994] on the enumeration of zero-sum subsequences over finite cyclic groups to the setting of semigroups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_00486 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enumeration of idempotent-sum subsequences in finite cyclic semigroups and smooth sequences Wang, Guoqing Zhao, Yang Yi, Xingliang Combinatorics Number Theory The enumeration of zero-sum subsequences of a given sequence over finite cyclic groups is one classical topic, which starts from one question of P. Erdős. In this paper, we consider this problem in a more general setting -- finite cyclic semigroups. Let $\mathcal{S}$ be a finite cyclic semigroup. By $\textbf{e}$ we denote the unique idempotent of the semigroup $\mathcal{S}$. Let $T$ be a sequence over the semigroup $\mathcal{S}$, and let $N(T; \textbf{e})$ be the number of distinct subsequences of $T$ with sum being the idempotent $\textbf{e}$. We obtain the lower bound for $N(T; \textbf{e})$ in terms of the length of $T$, and moreover, prove that $T$ contains subsequences with some smooth-structure in case that $N(T; \textbf{e})$ is not large. Our result generalizes the theorem obtained by W. Gao [Discrete Math., 1994] on the enumeration of zero-sum subsequences over finite cyclic groups to the setting of semigroups. |
| title | Enumeration of idempotent-sum subsequences in finite cyclic semigroups and smooth sequences |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2505.00486 |