On medial Latin quandles and affine modules

TL;DR

This work establishes categorical equivalences between medial quandle varieties and affine-module categories over two rings, enabling a module-theoretic treatment of medial Latin and medial commutative quandles. By mapping quandles to affine data via Alex and mid constructions, it provides explicit descriptions of free objects and yields a structure theorem for finitely generated medial commutative racks, resolving BE7.1 and BE7.3 posed by Bardakov and Elhamdadi. The approach unifies quandle theory with affine module theory, clarifying the role of Alexander and midpoint structures in the medial setting and offering concrete tools for studying quandle rings and knot invariants. The results enhance both the theoretical understanding and practical computation of medial quandles, including decompositions into well-understood building blocks.

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).

Theorems & Definitions (36)

• Lemma 2.1
• Example 2.2: nelsonjedlicka
• Example 2.3
• Remark 2.4
• Lemma 2.5
• Lemma 2.6
• Remark 2.7
• Theorem 3.1
• proof
• Corollary 3.2