Free Akivis algebras, primitive elements, and hyperalgebras. (English) Zbl 0993.17002

The notions of free Akivis algebra, its enveloping algebra, and its primitive element are given. It is proved that subalgebras of a free Akivis algebra are also free and finitely generated subalgebras are finitely residual. Decidability of the word problem for the variety of Akivis algebras is also proved. It is proved that a set of multilinear operations can be constructed for every algebra \(B\) such that \(B\) becomes a hyperalgebra called \(G(B)\). The following problem is stated: is it true that any hyperalgebra can be isomorphically embedded into a hyperalgebra \(G(B)\) for a suitable algebra \(B\)?


17A50 Free nonassociative algebras
20N05 Loops, quasigroups
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