Genetic algebra

In mathematical genetics, a genetic algebra is a (possibly non-associative) algebra used to model inheritance in genetics. Some variations of these algebras are called train algebras, special train algebras, gametic algebras, Bernstein algebras, copular algebras, zygotic algebras, and baric algebras (also called weighted algebra). The study of these algebras was started by Ivor Etherington (1939).

In applications to genetics, these algebras often have a basis corresponding to the genetically different gametes, and the structure constants of the algebra encode the probabilities of producing offspring of various types. The laws of inheritance are then encoded as algebraic properties of the algebra.

For surveys of genetic algebras see Bertrand (1966), Wörz-Busekros (1980) and Reed (1997).

Baric algebras

Baric algebras (or weighted algebras) were introduced by Etherington (1939). A baric algebra over a field K is a possibly non-associative algebra over K together with a homomorphism w, called the weight, from the algebra to K.^[1]

Bernstein algebras

A Bernstein algebra, based on the work of Sergei Natanovich Bernstein (1923) on the Hardy–Weinberg law in genetics, is a (possibly non-associative) baric algebra B over a field K with a weight homomorphism w from B to K satisfying ${\displaystyle (x^{2})^{2}=w(x)^{2}x^{2}}$ . Every such algebra has idempotents e of the form ${\displaystyle e=a^{2}}$  with ${\displaystyle w(a)=1}$ . The Peirce decomposition of B corresponding to e is

${\displaystyle B=Ke\oplus U_{e}\oplus Z_{e}}$ 

where ${\displaystyle U_{e}=\{a\in \ker w:ea=a/2\}}$  and ${\displaystyle Z_{e}=\{a\in \ker w:ea=0\}}$ . Although these subspaces depend on e, their dimensions are invariant and constitute the type of B. An exceptional Bernstein algebra is one with ${\displaystyle U_{e}^{2}=0}$ .^[2]

Copular algebras

Copular algebras were introduced by Etherington (1939, section 8)

Evolution algebras

An evolution algebra over a field is an algebra with a basis on which multiplication is defined by the product of distinct basis terms being zero and the square of each basis element being a linear form in basis elements. A real evolution algebra is one defined over the reals: it is non-negative if the structure constants in the linear form are all non-negative.^[3] An evolution algebra is necessarily commutative and flexible but not necessarily associative or power-associative.^[4]

Gametic algebras

A gametic algebra is a finite-dimensional real algebra for which all structure constants lie between 0 and 1.^[5]

Genetic algebras

Genetic algebras were introduced by Schafer (1949) who showed that special train algebras are genetic algebras and genetic algebras are train algebras.

Special train algebras

Special train algebras were introduced by Etherington (1939, section 4) as special cases of baric algebras.

A special train algebra is a baric algebra in which the kernel N of the weight function is nilpotent and the principal powers of N are ideals.^[1]

Etherington (1941) showed that special train algebras are train algebras.

Train algebras

Train algebras were introduced by Etherington (1939, section 4) as special cases of baric algebras.

Let ${\displaystyle c_{1},\ldots ,c_{n}}$  be elements of the field K with ${\displaystyle 1+c_{1}+\cdots +c_{n}=0}$ . The formal polynomial

${\displaystyle x^{n}+c_{1}w(x)x^{n-1}+\cdots +c_{n}w(x)^{n}}$ 

is a train polynomial. The baric algebra B with weight w is a train algebra if

${\displaystyle a^{n}+c_{1}w(a)a^{n-1}+\cdots +c_{n}w(a)^{n}=0}$ 

for all elements ${\displaystyle a\in B}$ , with ${\displaystyle a^{k}}$  defined as principal powers, ${\displaystyle (a^{k-1})a}$ .^[1][6]

Zygotic algebras

Zygotic algebras were introduced by Etherington (1939, section 7)

References

1. González, S.; Martínez, C. (2001), "About Bernstein algebras", in Granja, Ángel (ed.), Ring theory and algebraic geometry. Proceedings of the 5th international conference on algebra and algebraic geometry, SAGA V, León, Spain, Lect. Notes Pure Appl. Math., vol. 221, New York, NY: Marcel Dekker, pp. 223–239, Zbl 1005.17021
2. Catalan, A. (2000). "E-ideals in Bernstein algebras". In Costa, Roberto (ed.). Nonassociative algebra and its applications. Proceedings of the fourth international conference, São Paulo, Brazil. Lect. Notes Pure Appl. Math. Vol. 211. New York, NY: Marcel Dekker. pp. 35–42. Zbl 0968.17013.
3. Tian (2008) p.18
4. Tian (2008) p.20
5. Cohn, Paul M. (2000). Introduction to Ring Theory. Springer Undergraduate Mathematics Series. Springer-Verlag. p. 56. ISBN 1852332069. ISSN 1615-2085.
6. Catalán S., Abdón (1994). "E-ideals in baric algebras". Mat. Contemp. 6: 7–12. Zbl 0868.17023.

• Bernstein, S. N. (1923), "Principe de stationarité et généralisation de la loi de Mendel", C. R. Acad. Sci. Paris, 177: 581–584.
• Bertrand, Monique (1966), Algèbres non associatives et algèbres génétiques, Mémorial des Sciences Mathématiques, Fasc. 162, Gauthier-Villars Éditeur, Paris, MR 0215885
• Etherington, I. M. H. (1939), (PDF), Proc. R. Soc. Edinburgh, 59: 242–258, MR 0000597, Zbl 0027.29402, archived from the original (PDF) on 2011-07-06
• Etherington, I. M. H. (1941), "Special train algebras", The Quarterly Journal of Mathematics, Second Series, 12: 1–8, doi:10.1093/qmath/os-12.1.1, ISSN 0033-5606, JFM 67.0093.04, MR 0005111, Zbl 0027.29401
• Lyubich, Yu.I. (2001) [1994], "Bernstein problem in mathematical genetics", Encyclopedia of Mathematics, EMS Press
• Micali, A. (2001) [1994], "Baric algebra", Encyclopedia of Mathematics, EMS Press
• Micali, A. (2001) [1994], "Bernstein algebra", Encyclopedia of Mathematics, EMS Press
• Reed, Mary Lynn (1997), "Algebraic structure of genetic inheritance", Bulletin of the American Mathematical Society, New Series, 34 (2): 107–130, doi:10.1090/S0273-0979-97-00712-X, ISSN 0002-9904, MR 1414973, Zbl 0876.17040
• Schafer, Richard D. (1949), "Structure of genetic algebras", American Journal of Mathematics, 71: 121–135, doi:10.2307/2372100, ISSN 0002-9327, JSTOR 2372100, MR 0027751
• Tian, Jianjun Paul (2008), Evolution algebras and their applications, Lecture Notes in Mathematics, vol. 1921, Berlin: Springer-Verlag, ISBN 978-3-540-74283-8, Zbl 1136.17001
• Wörz-Busekros, Angelika (1980), Algebras in genetics, Lecture Notes in Biomathematics, vol. 36, Berlin, New York: Springer-Verlag, ISBN 978-0-387-09978-1, MR 0599179
• Wörz-Busekros, A. (2001) [1994], "Genetic algebra", Encyclopedia of Mathematics, EMS Press

Further reading

• Lyubich, Yu.I. (1983), Mathematical structures in population genetics. (Matematicheskie struktury v populyatsionnoj genetike) (in Russian), Kiev: Naukova Dumka, Zbl 0593.92011