On global isomorphisms and a closure property of semigroups

Abstract

Let $S$ be a semigroup (written multiplicatively). Endowed with the operation of setwise multiplication induced by $S$ on its parts, the non-empty subsets of $S$ form themselves a semigroup, denoted by $\mathcal P(S)$. Accordingly, we say that a semigroup $H$ is globally isomorphic to a semigroup $K$ if $\mathcal P(H)$ is isomorphic to $\mathcal P(K)$; and that a class $\mathscr C$ of semigroups is globally closed if a semigroup in $\mathscr C$ can only be globally isomorphic to an isomorphic copy of a semigroup in the same class. We show that the classes of groups, torsion-free monoids, and numerical monoids are each globally closed. The first result extends a 1967 theorem of Shafer, while the last relies non-trivially on the second and on a classical theorem of Kneser from additive number theory.