In this paper, we first give some general results specifying the necessary conditions that must be satisfied by an almost Bernstein algebra. In the second step, we proceed to the classification of almost Bernstein algebras of dimension four. Finally, we describe the derivations of an almost Bernstein algebra.

Peirce decomposition, type of algebra, Bernstein algebra, almost Bernstein algebra, train algebra, idempotent, derivation.

Received: March 4, 2022; Accepted: April 27, 2022; Published: July 1, 2022

How to cite this article: Paul Beremwidougou and André Conseibo, Classification and derivations of four-dimensional almost Bernstein algebras, JP Journal of Algebra, Number Theory and Applications 56 (2022), 1-25. http://dx.doi.org/10.17654/0972555522022

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] R. D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, New York, 1966.
[2] P. Holgate, Genetic algebras satisfying Bernstein’s stationarity principle, J. Lond. Math. Soc., II Ser. 9 (1975), 613-623.
[3] M. T. Alcalde, C. Burgueño, A. Labra and A. Micali, Sur les algèbres de Bernstein (On Bernstein algebras), Proc. Lond. Math. Soc. (3) 58(1) (1989), 51 68.
[4] S. Bernstein, Solution of a mathematical problem connected with the theory of heredity, Ann. Math. Stat. 13 (1942), 53-61.
[5] S. Walcher, Bernstein algebras which are Jordan algebras, Arch. Math. 50(3) (1988), 218-222.
[6] J. López-Sánchez and S. M. E. Rodríguez, On train algebras of rank 4, Comm. Algebra 24(14) (1996), 4439-4445.
[7] J. Bayara, A. Conseibo, M. Ouattara and A. Micali, Train algebras of degree 2 and exponent 3, Discrete Contin. Dyn. Syst. Ser. S 4(6) (2011), 1371-1386.
[8] P. Beremwidougou and A. Conseibo, Algebras satisfying identity of almost Bernstein algebras, Far East J. Math. Sci. (FJMS) 131(2) (2021), 131-152.
[9] A. Wörz-Busekros, Algebras in genetics, Lecture Notes in Biomathematics, 36, Springer-Verlag, Berlin, New York, 1980.