Groups, triality, and hyperquasigroups

Hyperquasigroups were recently introduced to provide a more symmetrical approach to quasigroups, and a far-reaching implementation of triality (S3-action). In the current paper, various connections between hyperquasigroups and groups are examined, on the basis of established connections between quasigroups and groups. A new graph-theoretical characterization of hyperquasigroups is exhibited. Torsors are recognized as hyperquasigroups, and group representations are shown to be equivalent to linear hyperquasigroups. The concept of orthant structure is introduced, as a tool for recovering classical information from a hyperquasigroup.