The automorphism group of the power semigroup P(H) of any numerical semigroup H is trivial.
Tringali,Power monoids and their arithmetic: a survey, preprint (https://arxiv.org/abs/2602.15754)
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abstract
The non-empty finite subsets of a multiplicatively written monoid form a monoid under setwise multiplication. The same holds for finite subsets containing the identity element. Partly due to their unusual arithmetic properties, these structures, generically known as power monoids, have attracted increasing attention in recent years, stimulating new perspectives in the study of factorizations in non-cancellative or non-commutative settings. We survey these developments and briefly review some related aspects.
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Rigidity theorems establish that P(H) ≅ P(K) implies H ≅ K for group H and semigroup K, with the finite-subset version holding only for additive subgroups of the rationals via a diophantine theorem.
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On the automorphisms of the power semigroups of a numerical semigroup
The automorphism group of the power semigroup P(H) of any numerical semigroup H is trivial.