Non-finitely based varieties of right alternative metabelian algebras
Alexey Kuz'min

TL;DR
This paper constructs a specific non-finitely based variety of right alternative metabelian algebras over certain fields, expanding understanding of their algebraic structure and Spechtian properties.
Contribution
It introduces a new non-finitely based variety generated by Grassmann algebra of rank 2, which can also be generated by a five-dimensional superalgebra, advancing the theory of these algebras.
Findings
Constructed a non-finitely based variety of rank 2 Grassmann algebra
Demonstrated the variety can be generated by a five-dimensional superalgebra
Extended the understanding of Spechtian properties in right alternative metabelian algebras
Abstract
Since 1976, it is known from the paper by V. P. Belkin that the variety of right alternative metabelian (solvable of index 2) algebras over an arbitrary field is not Spechtian (contains non-finitely based subvarieties). In 2005, S. V. Pchelintsev proved that the variety generated by the Grassmann -algebra of finite rank over a field , for , is Spechtian iff . We construct a non-finitely based variety generated by the Grassmann -algebra of rank of certain finitely based subvariety over a field , for , such that can also be generated by the Grassmann envelope of a five-dimensional superalgebra with one-dimensional even part.
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