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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
##### Keywords:
direct decompositions; normal subquasigroups; automorphism
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