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Peirce Decompositions for Evolution Algebras

  • Irene Paniello
Published/Copyright: December 18, 2023
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ABSTRACT

We address Peirce decompositions for evolution algebras at idempotent elements contained in the associative nucleus of the evolution algebras. If the idempotent elements are natural vectors, the requirement of being nuclear is then proved to be equivalent to the evolution algebra to be baric. A description of baric evolution algebras is also provided.

2020 Mathematics Subject Classification: Primary 17A60; 17C99; 17D92

(Communicated by Anatolij Dvurečenskij)


Funding statement: Partially supported by grant MTM2017-83506-C2-1-P (AEI/FEDER, UE) and PID2021-123461NB-C21, funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”.

REFERENCES

[1] Boudi, N.—Cabrera Casado, Y.—Siles Molina, M.: Natural families in evolution algebras, Publ. Math. 66 (2022), 159–181.10.5565/PUBLMAT6612206Search in Google Scholar

[2] Bustamante, M. D.—Mellon, P.—Velasco, M. V.: Determining when an algebra is an evolution algebra; https://arxiv.org/pdf/2102.04493.Search in Google Scholar

[3] Cabrera Casado, Y.—Cadavid, P.—Reis, T.: Derivations and loops of some evolution algebras, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 117 (2023), Art. No. 119.10.1007/s13398-023-01446-2Search in Google Scholar

[4] Cabrera Casado, Y.—Cardoso Gonçalves, M. I.—Gonçalves, D.—Martín Barquero, D.—Martín González, C.: Chains in Evolution algebras, Linear Algebra Appl. 622 (2021), 104–149.10.1016/j.laa.2021.03.026Search in Google Scholar

[5] Cabrera Casado, Y.—Kanuni, M.—Siles Molina, M.: Basic ideals in evolution algebras, Linear Algebra Appl. 570 (2019), 148–180.10.1016/j.laa.2019.01.010Search in Google Scholar

[6] Cabrera Casado, Y.—Siles Molina, M.—Velasco, M. V.: Evolution algebras of arbitrary dimension and their decompositions, Linear Algebra Appl. 495 (2016) 122–162.10.1016/j.laa.2016.01.007Search in Google Scholar

[7] Cadavid, P.—Rodiño Montoya, M. L.—Rodríguez, P. M.: The connection between evolution algebras, random walks and graphs, J. Algebra Appl. 19(2) (2020), Art. ID 2050023.10.1142/S0219498820500231Search in Google Scholar

[8] Casas, J. M.—Ladra, M.—Omirov, B. A.—Rozikov, U. A.: On evolution algebras, Algebra Colloq. 21(2) (2014), 331–342.10.1142/S1005386714000285Search in Google Scholar

[9] Casas, J. M.—Ladra, M.—Rozikov, U. A.: A chain of evolution algebras, Linear Algebra Appl. 435 (2011), 852–870.10.1016/j.laa.2011.02.012Search in Google Scholar

[10] Ceballos, M.—Falcón, R. M.—Núũez-Valdés, J.—Tenorio, A. F.: A historical perspective of Tian’s evolution algebras, Expo. Math. 40(3) (2022), 819–843.10.1016/j.exmath.2021.11.004Search in Google Scholar

[11] Da Motta Ferreira, J. C.—Micali, A.: Sur les algèbres nucléaires, Indag. Math. 7(3) (1996), 331–342.10.1016/0019-3577(96)83724-4Search in Google Scholar

[12] Elduque, A.—Labra, A.: Evolution algebras and graphs, J. Algebra Appl. 14(7) (2015), Art. ID 1550103.10.1142/S0219498815501030Search in Google Scholar

[13] Etherington, I. H. M.: Genetic algebras, Proc. Roy. Soc. Edinburgh Sect. A 59 (1939), 242–258.10.1017/S0370164600012323Search in Google Scholar

[14] Jacobson, N.: Structure of Rings. Amer. Math. Soc. Colloq. Publ., Vol. 37, Amer. Math. Soc. Providence, R. I., 1956.Search in Google Scholar

[15] Kurdoyberdiyev, A. KH.—Omirov, B. A.—Qaralleh, I.: Few remarks on evolution algebras, J. Algebra Appl. 14(4) (2015), Art. ID 1550053.10.1142/S021949881550053XSearch in Google Scholar

[16] Lyubich, Y. I.: Mathematical Structures in Population Genetics. Biomathematics, Vol. 22, Springer-Verlag, Berlin, 1992.10.1007/978-3-642-76211-6Search in Google Scholar

[17] Mellon, P.—Velasco, M. V.: Analytic aspects of evolution algebras, Banach J. Math. Anal. 13(1) (2019), 113–132.10.1215/17358787-2018-0018Search in Google Scholar

[18] Mukhamedov, F.—Qaralleh, I.: On S-evolution algebras and their envelopping algebras, Mathematics 9 (2021), Art. No. 1195.10.3390/math9111195Search in Google Scholar

[19] Murodov, SH. N.: Classification of two-dimensional real evolution algebras and dynamics of some two dimensional chains of evolution algebras, Uzbek. Math. J. 25(2) (2014), 102–111.10.1142/S0129167X14500128Search in Google Scholar

[20] Ouattara, M.—Savadogo, S.: Power-associative evolution algebras. In: Associative and Non-Associative Algebras and Applications, MAMAA 2018, Springer Proc. Math. Stat., Vol. 311, Springer, 2020, pp. 23–49.10.1007/978-3-030-35256-1_2Search in Google Scholar

[21] Ouattara, M.—Savadogo, S.: Evolution train algebras, Gulf J. Math. 8(1) (2020), 37–51.10.56947/gjom.v8i1.299Search in Google Scholar

[22] Paniello, I.: Markov evolution algebras, Linear Multilinear Algebra 70(19) (2022), 4633–4653.10.1080/03081087.2021.1893636Search in Google Scholar

[23] Reed, M. L.: Algebraic structure of genetic inheritance, Bull. Amer. Math. Soc. 34(2) (1997), 107–130.10.1090/S0273-0979-97-00712-XSearch in Google Scholar

[24] Rich, M.: Rings with idempotents in their nuclei, Trans. Amer. Math. Soc. 208 (1975) 81–90.10.1090/S0002-9947-1975-0371972-9Search in Google Scholar

[25] Schafer, R. D.: An Introduction to Nonassociative Algebras. Pure and Appl. Math., Vol. 22, Academic Press, New York-London, 1966.Search in Google Scholar

[26] Tian, J. P.—Vojtěchovský, P.: Mathematical concepts of evolution algebras in non-Mendelian genetics, Quasigroups Related systems 14 (2006), 111–122.Search in Google Scholar

[27] Tian, J. P.: Evolution Algebras and their Applications, Lecture Notes in Math., Vol. 1921, Springer, Berlin, 2008.10.1007/978-3-540-74284-5Search in Google Scholar

[28] Wörz-Busekros, A.: Algebras in Genetics. Lecture Notes in Biomathematics, Vol. 36, Springer-Verlag, New York, 1980.10.1007/978-3-642-51038-0Search in Google Scholar

Received: 2022-08-18
Accepted: 2023-03-31
Published Online: 2023-12-18

© 2023 Mathematical Institute Slovak Academy of Sciences

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