On medial Latin quandles and affine modules

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).