Geodesic
Modification and correction of Medvedev's example of a solvable alternative algebra
Algebra i logika, Tome 61 (2022) no. 5, pp. 619-639.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{AL_2022_61_5_a5,
     author = {I. P. Shestakov},
     title = {Modification and correction of {Medvedev's} example of a solvable alternative algebra},
     journal = {Algebra i logika},
     pages = {619--639},
     publisher = {mathdoc},
     volume = {61},
     number = {5},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2022_61_5_a5/}
}
TY  - JOUR
AU  - I. P. Shestakov
TI  - Modification and correction of Medvedev's example of a solvable alternative algebra
JO  - Algebra i logika
PY  - 2022
SP  - 619
EP  - 639
VL  - 61
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2022_61_5_a5/
LA  - ru
ID  - AL_2022_61_5_a5
ER  - 
%0 Journal Article
%A I. P. Shestakov
%T Modification and correction of Medvedev's example of a solvable alternative algebra
%J Algebra i logika
%D 2022
%P 619-639
%V 61
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2022_61_5_a5/
%G ru
%F AL_2022_61_5_a5
I. P. Shestakov. Modification and correction of Medvedev's example of a solvable alternative algebra. Algebra i logika, Tome 61 (2022) no. 5, pp. 619-639. http://geodesic.mathdoc.fr/item/AL_2022_61_5_a5/

[1] Yu. A. Medvedev, “Primer mnogoobraziya razreshimykh alternativnykh algebr nad polem kharakteristiki 2, ne imeyuschego konechnogo bazisa tozhdestv”, Algebra i logika, 19:3 (1980), 300–313 | MR

[2] V. T. Filippov, “O tsepochkakh mnogoobrazii, porozhdennykh svobodnymi maltsevskimi i alternativnymi algebrami”, Dokl. AN SSSR, 260:5 (1981), 1082–1085 | MR | Zbl

[3] V. T. Filippov, “0 svobodnykh algebrakh Maltseva i alternativnykh algebrakh”, Algebra i logika, 21:1 (1982), 84–107 | MR

[4] V. T. Filippov, “O trivialnykh idealakh svobodnoi alternativnoi algebry”, Algebra i logika, 22:2 (1983), 182–197 | MR | Zbl

[5] V. T. Filippov, “O nilpotentnykh idealakh svobodnoi alternativnoi algebry”, Algebra i logika, 22:3 (1983), 343–354 | MR | Zbl

[6] V. T. Filippov, “O mnogoobraziyakh maltsevskikh i alternativnykh algebr, porozhdennykh algebrami konechnogo ranga”, Gruppy i drugie algebraicheskie sistemy s usloviyami konechnosti, Trudy In-ta matem., 4, Nauka, Novosibirsk, 1984, 139–156

[7] V. T. Filippov, “Trivialnye yadernye idealy svobodnoi alternativnoi algebry”, Algebra i logika, 24:6 (1985), 696–717 | MR | Zbl

[8] V. T. Filippov, “O tsentralnykh idealakh svobodnoi konechno-porozhdennoi alternativnoi algebry”, Algebra i logika, 25:4 (1986), 470–491 | MR | Zbl

[9] V. T. Filippov, “O trivialnykh yadernykh idealakh alternativnykh algebr”, Algebra i logika, 36:1 (1997), 97–115 | MR | Zbl

[10] V. T. Filippov, “O tsentrakh maltsevskikh i alternativnykh algebr”, Algebra i logika, 38:5 (1999), 613–635 | MR | Zbl

[11] I. P. Shestakov, “Superalgebry i kontrprimery”, Sib. matem. zh., 32:6 (1991), 187–196 | MR

[12] A. V. Grishin, “Primery ne konechnoi baziruemosti $T$-prostranstv i $T$-idealov v kharakteristike 2”, Fundament. i prikl. matem., 5:1 (1999), 101–118 | MR | Zbl