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On the left-distributive quasigroups. (Russian) Zbl 0592.20066

Main result: A group G with generator set E is the left-associated group of a left-distributive quasigroup if and only if the following two conditions are satisfied: (i) if two of the elements a, b, c satisfy the equation \(ab=bc\) and belong to E then the remaining one is uniquely determined and belongs to E too, (ii) if \(a^ n=c\) is independent of \(a\in E\) for some natural number n then \(c=1\). - Further there are specified special cases (of medial, right-distributive, commutative and symmetric quasigroups and also of TS-quasigroups, respectively). As an interesting fact it results that any group G with generator set E satisfying (i) and (ii) has trivial centre.
Reviewer: V.Havel

MSC:

20N05 Loops, quasigroups
20F05 Generators, relations, and presentations of groups
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