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Loops and semidirect products. (English) Zbl 0974.20049

Let \(B\) be a left loop, and \(H\) a group. The authors construct a semidirect product on the set \(B\times H\) generalizing the well-known construction of L. V. Sabinin [Mat. Zametki 12, 605-616 (1972; Zbl 0258.20066)]. As for the usual semidirect product of groups, there is an internal and an external version of the construction. Both are described in detail.
The authors consider pseudo-automorphisms and the pseudo-\(\text{A}_l\) property and its relation with the internal semidirect product. For the external semidirect product, rather complicated conditions are given which make \(B\times H\) a group.
Also discussed are various identities for loops, such as \(\lambda_x\lambda_y\lambda_y\lambda_x=\lambda_{xy}\lambda_{xy}\). If this identity holds, then the left and automorphic inverse property are equivalent. That seems to be a new result. Many interesting examples are given.

MSC:

20N05 Loops, quasigroups

Citations:

Zbl 0258.20066
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References:

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