A. D. Keedwell and V. A. Shcherbacov
Quasigroups with an inverse property and generalized parastrophic identities

We study quasigroups (and loops) which have an inverse property. We show that each such quasigroup satisfies a generalized parastrophic identity and that, when investigating properties related to the nuclei, quasigroups which possess any type of inverse property can all be treated in the same way. By means of our approach using autostrophies, we obtain results concerning isomorphisms between or equality of these nuclei. Also, we find conditions for a groupoid which satisfies a generalized parastrophic identity to be a quasigroup. Some of these results generalize our results on (r,s,t)-inverse quasigroups.