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

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 $\mathbf{\text{e}}$ we denote the unique idempotent of the semigroup $\mathcal{S}$. Let $T$ be a sequence over the semigroup $\mathcal{S}$, and let $N(T;\mathbf{\text{e}})$ be the number of distinct subsequences of $T$ with sum being the idempotent $\mathbf{\text{e}}$. We obtain the lower bound for $N(T;\mathbf{\text{e}})$ in terms of the length of $T$, and moreover, prove that $T$ contains subsequences with some smooth-structure in case that $N(T;\mathbf{\text{e}})$ 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.