December  2011, 4(6): 1543-1551. doi: 10.3934/dcdss.2011.4.1543

Conjectures for the existence of an idempotent in $\omega $-polynomial algebras

1. 

Département de Mathématiques et Informatique Appliquées, Université Paul Valéry, Montpellier III, 34199 Montpellier, France, France

Received  March 2009 Revised  October 2009 Published  December 2010

The existence of idempotent elements in baric algebras defined by $\omega$-polynomial identities ($\omega$-PI algebras) is an important problem for the study of genetic algebras. We conjecture here two criteria on the existence of an idempotent. These criteria are based on the existence of 1/2 as double root of a polynomial built from the identity defining a $\omega$-PI algebra. We show that these criteria are true in all the algebras studied until now and for which we have results concerning the existence of idempotent elements.
Citation: Michelle Nourigat, Richard Varro. Conjectures for the existence of an idempotent in $\omega $-polynomial algebras. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1543-1551. doi: 10.3934/dcdss.2011.4.1543
References:
[1]

V. M. Abraham, Linearizing quadratic transformations in genetic algebras, Proc. London Math. Soc., (3) 40 (1980), 346-363. doi: 10.1112/plms/s3-40.2.346.  Google Scholar

[2]

I. M. H. Etherington, Commutative train algebras of ranks 2 and 3, J. London Math. Soc., 15 (1940), 136-149; Corrigendum ibid., 20 (1945) 238.  Google Scholar

[3]

J. C. Gutiérrez Fernández, Principal and plenary train algebras, Comm. Algebra, 28 (2000), 653-667. doi: 10.1080/00927870008826850.  Google Scholar

[4]

A. Labra and A. Suazo, On plenary algebras of rank 4, Comm. Algebra, 35 (2007), 2744-2752. doi: 10.1080/00927870701353589.  Google Scholar

[5]

J. López-Sánchez and E. Rodríguez Santa Maria, On train algebras of rank 4, Comm. Algebra, 24 (1996), 439-445.  Google Scholar

[6]

C. Mallol and A. Suazo, Une classe d'algèbres pondérées de degré 4, (French) [A class of weighted algebras of degree 4], Comm. Algebra, 28 (2000), 2191-2199. doi: 10.1080/00927870008826952.  Google Scholar

[7]

C. Mallol and R. Varro, Les algèbres de mutation, (French) [Mutation algebras], Non-associative algebra and its applications (Oviedo, 1993), 245-250, Math. Appl., 303, Kluwer Acad. Publ., Dordrecht, 1994.  Google Scholar

[8]

C. Mallol and R. Varro, Algèbres de Mutation et Train algèbres, (French) [Mutation algebras and train algebras] East-West J. Math., 4 (2002), 77-85.  Google Scholar

[9]

C. Mallol and R. Varro, Sur la Gamétisation et le Rétrocroisement, (French) [Gametization and backcrossing], Algebras Groups Geom., 22 (2005), 49-60.  Google Scholar

[10]

M. Nourigat, "Étude des $\omega $-PI Algèbres de Degré 4," PhD Thesis, Université de Montpellier II, France, 2008. Google Scholar

[11]

R. Varro, Introduction aux algèbres de Bernstein périodiques (cas Moufang, idempotents, caractéristique 2), (French) [Introduction to periodic Bernstein algebras (Moufang case, idempotents, characteristic 2)], Non-associative algebra and its applications (Oviedo, 1993), 384-388, Math. Appl., 303, Kluwer Acad. Publ., Dordrecht, 1994.  Google Scholar

[12]

S. Walcher, Algebras which satisfy a train equation for the first three plenary powers, Arch. Math. (Basel), 56 (1991), 547-551.  Google Scholar

[13]

A. Wörz-Busekros, "Algebras in Genetics," Lecture Notes in Biomathematics, 36, Springer-Verlag, Berlin-New York, 1980.  Google Scholar

[14]

K. A. Zhevlakov, A. M. Slin'ko and I. P. Shestakov, "Rings that are Nearly Associative,", Pure and Applied Mathematics, 104 ().   Google Scholar

show all references

References:
[1]

V. M. Abraham, Linearizing quadratic transformations in genetic algebras, Proc. London Math. Soc., (3) 40 (1980), 346-363. doi: 10.1112/plms/s3-40.2.346.  Google Scholar

[2]

I. M. H. Etherington, Commutative train algebras of ranks 2 and 3, J. London Math. Soc., 15 (1940), 136-149; Corrigendum ibid., 20 (1945) 238.  Google Scholar

[3]

J. C. Gutiérrez Fernández, Principal and plenary train algebras, Comm. Algebra, 28 (2000), 653-667. doi: 10.1080/00927870008826850.  Google Scholar

[4]

A. Labra and A. Suazo, On plenary algebras of rank 4, Comm. Algebra, 35 (2007), 2744-2752. doi: 10.1080/00927870701353589.  Google Scholar

[5]

J. López-Sánchez and E. Rodríguez Santa Maria, On train algebras of rank 4, Comm. Algebra, 24 (1996), 439-445.  Google Scholar

[6]

C. Mallol and A. Suazo, Une classe d'algèbres pondérées de degré 4, (French) [A class of weighted algebras of degree 4], Comm. Algebra, 28 (2000), 2191-2199. doi: 10.1080/00927870008826952.  Google Scholar

[7]

C. Mallol and R. Varro, Les algèbres de mutation, (French) [Mutation algebras], Non-associative algebra and its applications (Oviedo, 1993), 245-250, Math. Appl., 303, Kluwer Acad. Publ., Dordrecht, 1994.  Google Scholar

[8]

C. Mallol and R. Varro, Algèbres de Mutation et Train algèbres, (French) [Mutation algebras and train algebras] East-West J. Math., 4 (2002), 77-85.  Google Scholar

[9]

C. Mallol and R. Varro, Sur la Gamétisation et le Rétrocroisement, (French) [Gametization and backcrossing], Algebras Groups Geom., 22 (2005), 49-60.  Google Scholar

[10]

M. Nourigat, "Étude des $\omega $-PI Algèbres de Degré 4," PhD Thesis, Université de Montpellier II, France, 2008. Google Scholar

[11]

R. Varro, Introduction aux algèbres de Bernstein périodiques (cas Moufang, idempotents, caractéristique 2), (French) [Introduction to periodic Bernstein algebras (Moufang case, idempotents, characteristic 2)], Non-associative algebra and its applications (Oviedo, 1993), 384-388, Math. Appl., 303, Kluwer Acad. Publ., Dordrecht, 1994.  Google Scholar

[12]

S. Walcher, Algebras which satisfy a train equation for the first three plenary powers, Arch. Math. (Basel), 56 (1991), 547-551.  Google Scholar

[13]

A. Wörz-Busekros, "Algebras in Genetics," Lecture Notes in Biomathematics, 36, Springer-Verlag, Berlin-New York, 1980.  Google Scholar

[14]

K. A. Zhevlakov, A. M. Slin'ko and I. P. Shestakov, "Rings that are Nearly Associative,", Pure and Applied Mathematics, 104 ().   Google Scholar

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