In each quasigroup there are defined six types of translations: the left, right and middle translations and their inverses. Two translations may coincide as permutations of , 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 and , i.e., the elements of a symmetric group . Properties of the directions are considered in part 1 of “Classification of quasigroups according...
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 article considers a problem from Trokhimenko paper [13] concerning the study of abstract properties of commutations of operations and their connection with the Menger and Mann superpositions. Namely, abstract characterizations of some classes of operation algebras, whose signature consists of arbitrary families of commutations of operations, Menger and Mann superpositions and their various connections are found. Some unsolved problems are given at the end of the article.
We are investigating quasigroup functional equation classification up to parastrophic equivalence [Sokhatsky F.M.: On classification of functional equations on quasigroups, Ukrainian Math. J. 56 (2004), no. 4, 1259–1266 (in Ukrainian)]. If functional equations are parastrophically equivalent, then their functional variables can be renamed in such a way that the obtained equations are equivalent, i.e., their solution sets are equal. There exist five classes of generalized distributive-like quasigroup...
We study invertibility of operations that are composition of two operations of arbitrary arities. We find the criterion for quasigroups and specifications for -quasigroups. For this purpose we introduce notions of perpendicularity of operations and hypercubes. They differ from the previously introduced notions of orthogonality of operations and hypercubes [Belyavskaya G., Mullen G.L.: Orthogonal hypercubes and -ary operations, Quasigroups Related Systems 13 (2005), no. 1, 73–86]. We establish...
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