Construction of Quasigroups with Invertibility Properties

We consider linear isotopes of commutative groups, i.e., central quasigroups and study the invertibility and orthogonality conditions. It turns out that it is sufficient to study these conditions solely for the unitary isotopes, i.e., for isotopes, which have an idempotent. We establish criteria for the possession of each invertibility property (inverse property, crossed inverse property, and mirroring) for unitary central and matrix quasigroups. In particular, for matrices of the second order, we describe the corresponding matrix quasigroups over the fields of characteristics 2 and 3. The orthogonality criteria are obtained for matrix quasigroups with the indicated invertibility properties.