The existence of Buchsteiner and conjugacy-closed quasigroups

We show that a Buchsteiner quasigroup is necessarily a loop but that there exist proper conjugacy-closed quasigroups. This completes the investigation of quasigroups which satisfy identities corresponding to ones for loops obtainable by nuclear amalgamation. Also, we determine in terms of autotopisms the consequence for a quasigroup of Bol–Moufang type of having an inverse property.