A characterization of central extensions in the variety of quandles

TL;DR

This paper characterizes central extensions within the variety of quandles, focusing on their algebraic description in relation to abelian symmetric quandles and the associated adjunction.

Contribution

It provides a novel algebraic description of central extensions in the variety of quandles, especially in relation to abelian symmetric quandles and the Mal'tsev property.

Findings

Characterization of central extensions in quandles

Description of abelian symmetric quandles as an abelian subvariety

Insight into the adjunction between quandles and abelian symmetric quandles

Abstract

The category of symmetric quandles is a Mal'tsev variety whose subvariety of abelian symmetric quandles is the category of abelian algebras. We give an algebraic description of the quandle extensions that are central for the adjunction between the variety of quandles and its subvariety of abelian symmetric quandles.