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 and 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.

[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.

[3]

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

[4]

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

[5]

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

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[12]

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

[13]

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

[14]

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

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.

[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.

[3]

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

[4]

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

[5]

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

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[12]

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

[13]

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

[14]

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

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