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Keywords:
quasigroup; parastrophe; parastrophic symmetry; parastrophic orbit; translation; direction; matrix quasigroup
quasigroup; parastrophe; parastrophic symmetry; parastrophic orbit; translation; direction; matrix quasigroup
Summary:
In each quasigroup $Q$ there are defined six types of translations: the left, right and middle translations and their inverses. Two translations may coincide as permutations of $Q$, and yet be different when considered upon the web of the quasigroup. We shall call each of the translation types a direction and will associate it with one of the elements $\iota, l, r, s, ls $ and $rs$, i.e., the elements of a symmetric group $S_3$. Properties of the directions are considered in part 1 of "Classification of quasigroups according to directions of translations I" by F. M. Sokhatsky and A. V. Lutsenko. Let ${^{\sigma}\mathcal{M}}$ denote the set of all translations of a direction $\sigma\in S_{3}$. The conditions ${^{\sigma}\mathcal{M}}={^{\kappa}\mathcal{M}}$, where $\sigma,\kappa\in S_{3}$ and $\sigma\ne\kappa$, define nine quasigroup varieties. Four of them are well known: $LIP$, $RIP$, $MIP$ and $CIP$. The remaining five quasigroup varieties are relatively new because they are left and right inverses of $ CIP$ variety and generalization of commutative, left and right symmetric quasigroups.
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