  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.
