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

Takao Inoué

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}.

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

Abstract

Kunen proved that a quasigroup satisfying a Moufang-type identity () must be a loop. We reformulate the argument in the category as a fixed-point extraction principle. From one canonically obtains an idempotent endomorphism . Its fixed-point object splits off as a retract. The -symmetry forces to coequalize the (regular) translation action, hence factors through the terminal object. Thus , yielding a unique global identity element. This provides a conceptual reformulation of Kunen's original algebraic proof \cite{Kunen}.
Paper Structure (12 sections, 10 theorems, 25 equations)

This paper contains 12 sections, 10 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Quasigroups and the Moufang Identity
  3. The Fixed-Point Extractor $j:G\to G$
  4. Idempotence and the Fixed-Point Object
  5. Translation Action and a Coequalizer Collapse
  6. A Fixed-Point Collapse Lemma in $\mathbf{Set}$
  7. Categorical Collapse of the Fixed Part
  8. Application to Kunen's theorem
  9. Fixed point and identity element
  10. Kunen's Theorem as a Fixed-Point Extraction Principle
  11. Discussion
  12. Conclusion

Key Result

Lemma 3.2

Assume $(G,\cdot)$ satisfies $(N1)$. Then $j(x)=k(x)$ for all $x$. Hence, writing $j$ for this common map, we have

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Lemma 3.2: Two-sided local identity
  • proof
  • Remark 3.3
  • Lemma 4.1: Idempotence of values
  • proof
  • Proposition 4.2: Idempotent endomorphism
  • proof
  • ...and 18 more