A Structural Fixed-Point Principle in Kunen's Theorem on Quasigroups

TL;DR

This paper reformulates Kunen's theorem on quasigroups with Moufang-type identities using a fixed-point principle, showing such quasigroups are necessarily loops with a unique identity element.

Contribution

It introduces a categorical fixed-point extraction approach to prove that quasigroups satisfying a Moufang-type identity are loops, providing a conceptual reformulation of Kunen's original proof.

Findings

Quasigroups with Moufang-type identity have a unique identity element.

The fixed-point approach simplifies the proof of Kunen's theorem.

The categorical reformulation offers new conceptual insights.

Abstract

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}.