On automorphism groups of power semigroups over numerical semigroups or over numerical monoids

A numerical semigroup $S$ is a cofinite subsemigroup of $ \mathbb{N}$, where $\mathbb{N}$ is the additive monoid of non-negative integers. Denote by $\mathcal{P}_{\rm fin} (S)$ the semigroup consisting of all non-empty finite subsets of $S$ endowed with the operation of setwise addition defined by $$X+Y=\{x+y:x\in X, y\in Y\}, \qquad\text{for all } X, Y \in \mathcal P_\text{fin}(S).$$ We call $\mathcal{P}_{\rm fin} (S)$ the finitary power semigroup of $S$. When $0\in S$ (and hence $S$ is a numerical monoid), the family $\mathcal P_{\text{fin},0}(S)$ of all finite subsets of $S$ containing $0$ is a submonoind of $\mathcal P_\text{fin}(S)$; we call $\mathcal{P}_{{\rm fin}, 0}(S)$ the reduced finitary power monoid of $S$ with the singleton $\{0\}$ as zero-element. For a non-empty finite subset $X$ of $\mathbb{N}$, we denote by $ \min X$ and $\max X $ the minimum and the maximum in $X$. Tringali and Yan have recently proved in [J.\ Combin.\ Theory Ser.\ A 209 (2025)] that the only non-trivial automorphism of $\mathcal{P}_{{\rm fin},0}(\mathbb{N})$ is the involution $X \mapsto \max X - X$. By applying Tringali-Yan's result, we in this article determined the automorphism group of the finitary power semigroup $\mathcal{P}_{\rm fin}(S)$ of an arbitrary numerical semigroup $S$. More precisely, if $S$ is the set of all integers larger than or equal to a fixed $k \in \mathbb N$, then the only non-trivial automorphism of $\mathcal{P}_{\rm fin}(S)$ is the involution $X \mapsto \max X - X+ \min X$; otherwise, $\mathcal{P}_{\rm fin}(S)$ has only the identity automorphism.