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Keywords:
quasigroup; parastrophe; parastrophic symmetry; parastrophic orbit; translation; direction
quasigroup; parastrophe; parastrophic symmetry; parastrophic orbit; translation; direction
Summary:
It is proved that every translation in a quasigroup has two independent parameters. One of them is a bijection of the carrier set. The second parameter is called a direction here. Properties of directions in a quasigroup are considered in the first part of the work. In particular, totally symmetric, semisymmetric, commutative, left and right symmetric and also asymmetric quasigroups are characterized within these concepts. The sets of translations of the same direction are under consideration in the second part of the work. Coincidence of these sets defines nine varieties, among them are varieties of $LIP$, $RIP$, $MIP$ and $CIP$ quasigroups. Quasigroups in other five varieties also have some invertibility properties.
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