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 ${\mathcal{P}}_{\mathrm{f}\mathrm{i}\mathrm{n}}(S)$ the semigroup consisting of all non-empty finite subsets of $S$ endowed with the operation of setwise addition defined by $$X+Y=\{x+y:x\in X,y\in Y\},\text{for all }X,Y\in {\mathcal{P}}_{\text{fin}}(S)\mathrm{.}$$ We call ${\mathcal{P}}_{\mathrm{f}\mathrm{i}\mathrm{n}}(S)$ 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 ${\mathcal{P}}_{\mathrm{f}\mathrm{i}\mathrm{n},0}(S)$ 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 ${\mathcal{P}}_{\mathrm{f}\mathrm{i}\mathrm{n},0}(\mathbb{N})$ 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 ${\mathcal{P}}_{\mathrm{f}\mathrm{i}\mathrm{n}}(S)$ 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 ${\mathcal{P}}_{\mathrm{f}\mathrm{i}\mathrm{n}}(S)$ is the involution $X\mapsto \max X-X+\min X$; otherwise, ${\mathcal{P}}_{\mathrm{f}\mathrm{i}\mathrm{n}}(S)$ has only the identity automorphism.