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 Etherington (1939).

In applications to genetics, these algebras often have a basis corresponding to the genetically different gametes, and the structure constant 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.

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) algebra over a field K with a homomorphism w to K satisfying (x²)² = w(x)²x².

Copular algebras

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

Gametic algebras

Copular algebras were introduced by Etherington (1939, section 6)

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. 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.

Zygotic algebras

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

References

• 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, MR0215885 
• Etherington, I. M. H. (1939), "Genetic algebras", Proc. Roy. Soc. Edinburgh 59: 242–258, MR0000597, http://math.usask.ca/~bremner/research/geneticalgebras/etherington/ga.pdf 
• Etherington, I. M. H. (1941), "Special train algebras", The Quarterly Journal of Mathematics. Oxford. Second Series 12: 1–8, doi:10.1093/qmath/os-12.1.1, ISSN 0033-5606, MR0005111 
• Lyubich, Yu.I. (2001), "Bernstein problem in mathematical genetics", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/b/b120160.htm 
• Micali, A. (2001), "Baric algebra", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/b/b110140.htm 
• Micali, A. (2001), "Bernstein algebra", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/b/b110340.htm 
• Reed, Mary Lynn (1997), "Algebraic structure of genetic inheritance", American Mathematical Society. Bulletin. New Series 34 (2): 107–130, doi:10.1090/S0273-0979-97-00712-X, ISSN 0002-9904, MR1414973 
• Schafer, Richard D. (1949), "Structure of genetic algebras", American Journal of Mathematics 71: 121–135, ISSN 0002-9327, JSTOR 2372100, MR0027751 
• Wörz-Busekros, Angelika (1980), Algebras in genetics, Lecture Notes in Biomathematics, 36, Berlin, New York: Springer-Verlag, ISBN 978-0-387-09978-1, MR599179 
• Wörz-Busekros, A. (2001), "Genetic algebra", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/g/g043970.htm