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 obtain (as a corollary of a theorem of independent interest) that if $H$ is a group and $K$ is a semigroup, then $\mathcal{P}(H)\cong \mathcal{P}(K)$ implies $H\cong K$. The finitary analogue of this statement is considerably more difficult, and we prove it only for $H$ an additive subgroup of the rationals. Most notably, the proof of the second result relies, in a rather circuitous way, on a special case of the Evertse--Schlickewei--Schmidt theorem.

16 pages, no figures