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Classification of quasigroups according to directions of translations II

Fedir Sokhatsky;  Alla Lutsenko — 2021

Commentationes Mathematicae Universitatis Carolinae

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 ι , l , r , s , l s and r s , i.e., the elements of a symmetric group S 3 . Properties of the directions are considered in part 1 of “Classification of quasigroups according...

Classification of quasigroups according to directions of translations I

Fedir Sokhatsky;  Alla Lutsenko — 2020

Commentationes Mathematicae Universitatis Carolinae

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...

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