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

TL;DR

This paper extends the classical problem of counting zero-sum subsequences from finite cyclic groups to finite cyclic semigroups, providing bounds and structural insights into subsequences with specific sum properties.

Contribution

It generalizes Gao's theorem to semigroups, establishing lower bounds for the number of subsequences summing to the idempotent and identifying smooth-structure subsequences.

Findings

Established lower bounds for the number of subsequences summing to the idempotent.

Proved existence of smooth-structure subsequences when the count is not large.

Generalized zero-sum enumeration results from groups to semigroups.

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. Erd\H{o}s. 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.