On medial Latin quandles and affine modules

TL;DR

This paper establishes an equivalence between categories of medial Latin quandles and affine modules over specific rings, providing structural insights and solving open problems in the theory of quandles and racks.

Contribution

It introduces a categorical equivalence linking medial Latin quandles to affine modules over Laurent polynomial rings, and characterizes finitely generated medial commutative quandles.

Findings

Equivalence between medial Latin quandles and affine modules over Laurent polynomial rings

Structure theorem for finitely generated medial commutative quandles

Characterization of racks with duals that are commutative

Abstract

In this note, we show that the category of Latin (resp. commutative) medial quandles is equivalent to the category of affine modules over a certain Laurent polynomial ring (resp. the dyadic rationals). As applications, we describe free objects in these categories and obtain a structure theorem for finitely generated medial commutative quandles. We also characterize racks whose duals are commutative. Collectively, this solves two open problems of Bardakov and Elhamdadi (arXiv:2601.07057v2).