On power monoids and their automorphisms
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910688579944448 |
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| author | Tringali, Salvatore Yan, Weihao |
| author_facet | Tringali, Salvatore Yan, Weihao |
| contents | Endowed with the binary operation of set addition, the family $\mathcal P_{{\rm fin},0}(\mathbb N)$ of all finite subsets of $\mathbb N$ containing $0$ forms a monoid, with the singleton $\{0\}$ as its neutral element.
We show that the only non-trivial automorphism of $\mathcal P_{{\rm fin},0}(\mathbb N)$ is the involution $X \mapsto \max X - X$. The proof leverages ideas from additive number theory and proceeds through an unconventional induction on what we call the boxing dimension of a finite set of integers, that is, the smallest number of (discrete) intervals whose union is the set itself. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_04439 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On power monoids and their automorphisms Tringali, Salvatore Yan, Weihao Combinatorics Number Theory Rings and Algebras Endowed with the binary operation of set addition, the family $\mathcal P_{{\rm fin},0}(\mathbb N)$ of all finite subsets of $\mathbb N$ containing $0$ forms a monoid, with the singleton $\{0\}$ as its neutral element. We show that the only non-trivial automorphism of $\mathcal P_{{\rm fin},0}(\mathbb N)$ is the involution $X \mapsto \max X - X$. The proof leverages ideas from additive number theory and proceeds through an unconventional induction on what we call the boxing dimension of a finite set of integers, that is, the smallest number of (discrete) intervals whose union is the set itself. |
| title | On power monoids and their automorphisms |
| topic | Combinatorics Number Theory Rings and Algebras |
| url | https://arxiv.org/abs/2312.04439 |