On medial Latin quandles and affine modules
arXiv:2602.08875 · doi:10.4153/S0008439526102306
The paper shows that the categories of Latin and commutative medial quandles are equivalent to categories of affine modules over a Laurent polynomial ring and over the dyadic rationals, respectively, and uses this equivalence to describe free objects and give a structure theorem for finitely generated medial commutative quandles.
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:2601.07057).
Accepted Manuscript version. References added; typos corrected. 13 pages