Construction and properties of (r,s,t)-inverse quasigroups. I Original Research Article

In an earlier paper, we discussed the construction and possible application to cryptography of m-inverse quasigroups, a generalization of CI-quasigroups introduced originally by Karklin′š and Karklin′ in 1976 (but then only for the special case of loops). In this paper, we generalize this concept further to that of an (r,s,t)-inverse quasigroup and show that CI-quasigroups, WIP-quasigroups and m-inverse quasigroups may all be regarded as special examples of this new structure. We are thus able to put some of the properties which the aforementioned types of quasigroup and loop have in common in a more general setting. In Part I of the paper, we provide constructions for, and concrete examples of, (r,s,t)-inverse quasigroups and introduce a direct product construction.