Modification and correction of Medvedev's example of a solvable alternative algebra

I. P. Shestakov<sup>ab

a</sup> Universidade de São Paulo
^b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Abstract: Yu. A. Medvedev [Algebra and Logic, 19, No. 3, 191—201 (1980)] constructed an example of alternative algebra that he used to prove that a certain variety of alternative algebras possess the non-Specht property over a field of characteristic 2. Though his result concerned the characteristic 2 case, the example was claimed to be alternative over an arbitrary field, and it was later used by V. T. Filippov in a series of papers. Unfortunately, Medvedev's example is in fact not alternative in any characteristic. Therefore, whether the variety considered by Medvedev has the non-Specht property is still not clear. Moreover, the results of Filippov's papers, in which Medvedev's example was used, also become questionable. We construct new examples and employ them to prove that the results of Filippov remain true.

Keywords: alternative algebra, superalgebra, Specht variety.

UDC: 512.554.5

Received: 30.06.2022
Revised: 09.08.2023

DOI: 10.33048/alglog.2022.61.506

 English version: Algebra and Logic, 2022, 61:5, 420–436