On the isomorphism problem for power semigroups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911991113711616 |
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| author | Tringali, Salvatore |
| author_facet | Tringali, Salvatore |
| contents | Let $\mathcal P(S)$ be the semigroup obtained by equipping the family of all non-empty subsets of a (multiplicatively written) semigroup $S$ with the operation of setwise multiplication induced by $S$ itself. We call a subsemigroup $P$ of $\mathcal P(S)$ downward complete if any element of $S$ lies in at least one set $X \in P$ and any non-empty subset of a set in $P$ is still in $P$.
We obtain, for a commutative semigroup $S$, a characterization of the cancellative elements of a downward complete subsemigroup of $\mathcal P(S)$ in terms of the cancellative elements of $S$. Consequently, we show that, if $H$ and $K$ are cancellative semigroups and either of them is commutative, then every isomorphism from a downward complete subsemigroup of $\mathcal P(H)$ to a downward complete subsemigroup of $\mathcal P(K)$ restricts to an isomorphism from $H$ to $K$. This solves a special case of a problem of Tamura and Shafer from the late 1960s and generalizes a recent result by Bienvenu and Geroldinger, where it is assumed, among other conditions, that $H$ and $K$ are numerical monoids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11475 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the isomorphism problem for power semigroups Tringali, Salvatore Rings and Algebras Combinatorics Let $\mathcal P(S)$ be the semigroup obtained by equipping the family of all non-empty subsets of a (multiplicatively written) semigroup $S$ with the operation of setwise multiplication induced by $S$ itself. We call a subsemigroup $P$ of $\mathcal P(S)$ downward complete if any element of $S$ lies in at least one set $X \in P$ and any non-empty subset of a set in $P$ is still in $P$. We obtain, for a commutative semigroup $S$, a characterization of the cancellative elements of a downward complete subsemigroup of $\mathcal P(S)$ in terms of the cancellative elements of $S$. Consequently, we show that, if $H$ and $K$ are cancellative semigroups and either of them is commutative, then every isomorphism from a downward complete subsemigroup of $\mathcal P(H)$ to a downward complete subsemigroup of $\mathcal P(K)$ restricts to an isomorphism from $H$ to $K$. This solves a special case of a problem of Tamura and Shafer from the late 1960s and generalizes a recent result by Bienvenu and Geroldinger, where it is assumed, among other conditions, that $H$ and $K$ are numerical monoids. |
| title | On the isomorphism problem for power semigroups |
| topic | Rings and Algebras Combinatorics |
| url | https://arxiv.org/abs/2402.11475 |