Subquandles of affine quandles

TL;DR

This paper characterizes quasi-affine quandles, a class of algebraic structures, using group-theoretic and algebraic conditions, and provides algorithms for their recognition and enumeration.

Contribution

It offers a comprehensive characterization of quasi-affine quandles through multiple algebraic perspectives and develops algorithms for their identification and enumeration.

Findings

Characterization of quasi-affine quandles via group properties

Development of algorithms for recognizing affine and quasi-affine quandles

Enumeration of small quasi-affine quandles

Abstract

A quandle will be called quasi-affine, if it embeds into an affine quandle. Our main result is a characterization of quasi-affine quandles, by group-theoretic properties of their displacement group, by a universal algebraic condition coming from the commutator theory, and by an explicit construction over abelian groups. As a consequence, we obtain efficient algorithms for recognizing affine and quasi-affine quandles, and we enumerate small quasi-affine quandles. We also prove that the "abelian implies quasi-affine" problem of universal algebra has affirmative answer for the class of quandles.