Power semigroups and two rigidity theorems for groups

Abstract: Let $\mathcal P(H)$ be the semigroup obtained by endowing the family of all non-empty subsets of a semigroup $H$ with the setwise operation naturally induced by $H$ on its power set, and denote by $\mathcal P_\text{fin}(H)$ the subsemigroup of $\mathcal P(H)$ consisting of all non-empty finite subsets of $H$. We call $\mathcal P(H)$ and $\mathcal P_\text{fin}(H)$ the large power semigroup and the finitary power semigroup of $H$, respectively.

We show that if $H$ is a group and $K$ is an arbitrary semigroup, then for $\mathcal P(H)$ to be isomorphic to $\mathcal P(K)$ it is necessary (and sufficient) that $H$ is isomorphic to $K$ (and hence $K$ is itself a group). The finitary analogue of the same statement appears to be considerably more difficult, and we establish it only when $H$ is an additive subgroup of the rationals. The proof of this second result relies, in a circuitous way, on a special case of the Evertse--Schlickewei--Schmidt theorem. The talk is based on joint work with Shuolin Liu.

group theorynumber theory

Audience: researchers in the topic

( paper )

Combinatorial and additive number theory seminar (CANT 2026)