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Complete direct decompositions of quasigroups with an idempotent. (Russian) Zbl 0746.20053
The author proves that if \(Q(\cdot)\) is a quasigroup with the idempotent element \(h\) and \(Q\) has the direct decompositions \(A\times B\) and \(A'\times B\) where \(A\), \(A'\) and \(B\) are normal subquasigroups, \(h\in A\cap A'\cap B\), then there exists an \(\alpha\) \(h\)-central automorphism of \(Q(\cdot)\) with \(id_ B\) and \(\alpha(A)=A'\). The result is extended to direct decompositions of normal subquasigroups \(A_ 1,\dots,A_ n\) and \(A_ 1',\dots,A_ m'\).

MSC:
20N05 Loops, quasigroups
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