A Structural Fixed-Point Principle in Kunen's Theorem on Quasigroups
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910028883034112 |
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| author | Inoué, Takao |
| author_facet | Inoué, Takao |
| contents | Kunen proved that a quasigroup satisfying a Moufang-type identity ($N1$) must be a loop. We reformulate the argument in the category $\mathbf{Set}$ as a fixed-point extraction principle. From $N1$ one canonically obtains an idempotent endomorphism $j:G\to G$. Its fixed-point object $\mathrm{Fix}(j)=\mathrm{Eq}(j,\mathrm{id}_G)$ splits off as a retract. The $N1$-symmetry forces $j$ to coequalize the (regular) translation action, hence $j$ factors through the terminal object. Thus $\mathrm{Fix}(j)\cong 1$, yielding a unique global identity element. This provides a conceptual reformulation of Kunen's original algebraic proof \cite{Kunen}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18587 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Structural Fixed-Point Principle in Kunen's Theorem on Quasigroups Inoué, Takao Group Theory Category Theory 20N05 (Primary), 18A20 (Secondary) Kunen proved that a quasigroup satisfying a Moufang-type identity ($N1$) must be a loop. We reformulate the argument in the category $\mathbf{Set}$ as a fixed-point extraction principle. From $N1$ one canonically obtains an idempotent endomorphism $j:G\to G$. Its fixed-point object $\mathrm{Fix}(j)=\mathrm{Eq}(j,\mathrm{id}_G)$ splits off as a retract. The $N1$-symmetry forces $j$ to coequalize the (regular) translation action, hence $j$ factors through the terminal object. Thus $\mathrm{Fix}(j)\cong 1$, yielding a unique global identity element. This provides a conceptual reformulation of Kunen's original algebraic proof \cite{Kunen}. |
| title | A Structural Fixed-Point Principle in Kunen's Theorem on Quasigroups |
| topic | Group Theory Category Theory 20N05 (Primary), 18A20 (Secondary) |
| url | https://arxiv.org/abs/2602.18587 |