Enumeration of idempotent-sum subsequences in finite cyclic semigroups and smooth sequences

TL;DR

The paper generalizes the classical zero-sum subsequence problem from finite cyclic groups to finite cyclic semigroups ${C_{k;n}}$ by counting idempotent-sum subsequences via $N(T; {\mathbf e})$. It establishes a tight lower bound $N(T; {\mathbf e}) \ge 2^{|T|-{\lceil k/n \rceil} n+1}-1+\lfloor 1/k \rfloor$ and shows that when this quantity is not large, $T$ must contain a subsequence with a smooth (or signed smooth) structure, with precise dichotomies depending on whether $k>n$ or $k\le n$. A key device is the map $\Psi$ linking ${C_{k;n}}$-subsequences to zero-sum subsequences in $\mathbb{Z}/n\mathbb{Z}$, enabling transfer of structural results (e.g., Theorem B and long-idempotent-sum-free sequences) to the semigroup setting. The work thus extends Erdős–Burgess type results to cyclic semigroups, clarifies the role of smooth structures in inverse zero-sum problems, and connects combinatorial counts to zero-sum invariants like the Erdős–Burgess constant and related smoothness notions.

Abstract

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.

Theorems & Definitions (32)

• Definition 2.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• proof
• Lemma 3.5
• proof
• Lemma 3.6
• proof