On global isomorphisms and a closure property of semigroups
Lingxi Li, Salvatore Tringali
TL;DR
The paper investigates when the large power semigroup $\mathcal{P}(S)$ of a semigroup $S$ determines $S$ up to isomorphism via global isomorphisms, addressing the global-closure of classes of semigroups. It establishes global-closure for three key classes: groups, torsion-free monoids, and numerical monoids, using a blend of semigroup theory, unit-structure arguments, and additive combinatorics, notably Kneser’s theorem. A central theme is that global isomorphisms preserve essential structural features (e.g., unit groups for monoids, torsion-freeness), which in turn forces the target to lie in the same class, often yielding an actual isomorphism. The results connect factorization- and additive-theoretic perspectives, extending Shafer’s classical group result and reinforcing the role of density and small-doubling structure in semigroup contexts, with concrete implications for numerical monoids and related classes.
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.