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On the generalized nilpotent and generalized solvable loops. (English) Zbl 1299.20090

Summary: The paper introduces the notions of commutator-associators of types \((\alpha,\beta,1)\), \((\alpha,\beta,2)\) (respect. \((\mu,1)\), \((\mu,2)\)) of certain weight, related directly with inner mappings of loops. With the help of these notions the authors describe the (transfinite) upper series of loops, lower series of loops, derived central series of loops, of Kurosh-Chernikov classes of loops and characterize the centrally nilpotent, centrally solvable loops, respect. Moufang loops.

MSC:

20N05 Loops, quasigroups
20F14 Derived series, central series, and generalizations for groups
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