Pushpa Publishing House

Journal Menu

Content

Volume 64 (2025)

	Volume 64, Issue 5 (In Progress) Pg 451 - 545 (September 2025)
	Volume 64, Issue 4 Pg 329 - 450 (July 2025)
	Volume 64, Issue 3 Pg 221 - 327 (May 2025)
	Volume 64, Issue 2 Pg 117 - 219 (March 2025)
	Volume 64, Issue 1 Pg 1 - 115 (January 2025)

Volume 63 (2024)

Volume 62 (2023)

Volume 61 (2023)

Volume 60 (2023)

Volume 53 (2022)

Volume 52 (2021)

Volume 51 (2021)

Volume 50 (2021)

Volume 49 (2021)

Volume 48 (2020)

Volume 47 (2020)

Volume 46 (2020)

Volume 45 (2020)

Volume 44 (2019)

Volume 43 (2019)

Volume 42 (2019)

Volume 41 (2019)

Volume 40 (2018)

Volume 39 (2017)

Volume 38 (2016)

Volume 37 (2015)

Volume 36 (2015)

Volume 35 (2014)

Volume 34 (2014)

Volume 33 (2014)

Volume 32 (2014)

Volume 31 (2013)

Volume 30 (2013)

Volume 29 (2013)

Volume 28 (2013)

Volume 27 (2012)

Volume 26 (2012)

Volume 25 (2012)

Volume 24 (2012)

Volume 23 (2011)

Volume 22 (2011)

Volume 21 (2011)

Volume 20 (2011)

Volume 19 (2010)

Volume 18 (2010)

Volume 17 (2010)

Volume 16 (2010)

Volume 15 (2009)

Volume 14 (2009)

Volume 13 (2009)

Volume 12 (2008)

Volume 11 (2008)

Volume 10 (2008)

Volume 9 (2007)

Volume 8 (2007)

Volume 7 (2007)

Volume 6 (2006)

Volume 5 (2005)

Volume 4 (2004)

Volume 3 (2003)

Volume 2 (2002)

Volume 1 (2001)

Important: All future articles and volumes will be published only on our new website: www.pphmjopenaccess.com. Authors are requested to submit their papers through the new website only. Visit now: www.pphmjopenaccess.com

JP Journal of Algebra, Number Theory and Applications

JP Journal of Algebra, Number Theory and Applications
Volume 64, Issue 5 , Pages 451 - 488 (September 2025)
http://dx.doi.org/10.17654/0972555525024

EVOLUTION ALGEBRAS SATISFYING DEGREE FOUR IDENTITIES NOT IMPLIED BY COMMUTATIVITY AND WITH NO UNIT ELEMENT

Savadogo Souleymane, Konkobo Amidou and Ouattara Moussa

Abstract:

The aim of this paper is to study evolution algebras satisfying identity not implied by commutativity and with no unit element. We prove that the class of power-associative evolution algebras is contained in the given class, and that the latter also admits one and only one finite-dimensional non-nil indecomposable evolution algebra without idempotent. We then present a weighting criterion and a classification, up to isomorphism, in dimension up to three. Finally, we conclude with a derivation study of the aforementioned algebras.

Keywords and phrases:

degree four identity, evolution algebra, baric algebra, derivation

Received: January 27, 2025; Revised: April 3, 2025; Accepted: May 18, 2025; Published: June 30, 2025

How to cite this article: Savadogo Souleymane, Konkobo Amidou and Ouattara Moussa, Evolution algebras satisfying degree four identities not implied by commutativity and with no unit element, JP Journal of Algebra, Number Theory and Applications 64(5) (2025), 451-488. https://doi.org/10.17654/0972555525024

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

References:

[1]A. A. Albert, A theory of power-associative commutative algebras, Trans. Amer. Math. Soc. 69 (1950), 503-527. https://doi.org/10.2307/1990496.

[2]N. Boudi, Y. C. Casado and M. S. Molina, Natural families in evolution algebras, Publ. Mat. Barc. 66(1) (2022), 159-181.

https://doi.org/10.5565/PUBLMAT6612206.

[3]L. Carini, I. R. Hentzel and G. M. Piacentini-Cattaneo, Degree four identities not implied by commutativity, Comm. Algebra 16(2) (1988), 339-356.

https://doi.org/10.1080/00927878808823575.

[4]C. Y. Casado, M. S. Molina and M. V. Velasco, Evolution algebras of arbitrary dimension and their decompositions, Linear Algebra Appl. 495 (2016), 122-162. https://doi.org/10.1016/j.laa.2016.01.007.

[5]J. M. Casas, M. Ladra, B. A. Omirov and U. A. Rozikov, On evolution algebras, Algebra Colloq. 21(2) (2014), 331-342.

https://doi.org/10.1142/S1005386714000285.

[6]M. Ceballos, R. M. Falcón, J. Núñez-Valdés and Á. F. Tenorio, A historical perspective of Tian’s evolution algebras, Expo. Math. 40(3) (2022), 819-843. https://doi.org/10.1016/j.exmath.2021.11.0.

[7]A. Elduque and A. Labra, Evolution algebras and graphs, J. Algebra Appl. 14(7) (2015), 1550103. https://doi.org/10.1142/S0219498815501030.

[8]A. Labra and C. Rojas-Bruna, On trace forms on a class of commutative algebras satisfying an identity of degree four, Algebra Colloq. 17 (2010), 875-880. https://doi.org/10.1142/S1005386710000817.

[9]M. Ouattara and S. Savadogo, Derivations in power-associative evolution algebras, JP J. Algebra Number Theory Appl. 43(1) (2019), 23-52.

https://doi.org/10.17654/NT043010023.

[10]M. Ouattara and S. Savadogo, Evolution train algebras, Gulf J. Math. 8(1) (2020), 37-51. https://gjom.org/index.php/gjom/article/view/299/280.

[11]M. Ouattara and S. Savadogo, Power-associative evolution algebras, Associative and Non-associative Algebras and Applications, M. Siles Molina, M. El Kaou, L. ben Yakoub, M. Benslimane, eds., MAMAA 2018, Springer Proceedings in Mathematics and Statistics, Springer, Cham, Vol. 311, 2020, pp. 23-49. https://doi.org/10.1007/978-3-030-35256-1_2.

[12]J. M. Osborn, Commutative non-associative algebras and identities of degree four, Canad. J. Math. 20 (1968), 769-794. https://doi.org/10.4153/CJM-1968-077-4.

[13]R. D. Schafer, An introduction to nonassociative algebras, Pure Appl. Math., Academic Press, Vol. 22, 1966, p. 166.

https://doi.org/10.1017/s0008439500028952.

[14]J. P. Tian and P. Vojtěchovský, Mathematical concepts of evolution algebras in non-Mendelian genetics, Quasigroups Related Systems 14(1) (2006), 111-122.

[15]J. P. Tian, Evolution algebras and their applications, Lect. Notes Math., 1921, 2008. https://doi.org/10.1007/978-3-540-74284-5.

[16]A. Wörz-Busekros, Algebras in genetics, Lect. Notes Biomath., Vol. 36, 1980. https://doi.org/10.1007/978-3-642-51038-0_2.

Number of Downloads: 97 | Number of Views: 221