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

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

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

Michelle Nourigat ^1, and Richard Varro ^1,

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

Abstract
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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.

Keywords: Bernstein algebra, mutation algebra, $\omega$-PI algebra, backcross algebra., train algebra, plenary train algebra, Baric algebra, idempotent element, $\omega$-polynomial identity.

Mathematics Subject Classification: Primary: 17D92; Secondary: 17A3.

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, Academic Press, New York-London, 1982.

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, Academic Press, New York-London, 1982.

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