Genetic algebra

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]

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 were introduced by Etherington (1939, section 8)

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]

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

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 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 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 were introduced by Etherington (1939, section 7)

• 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), "Genetic algebras" (PDF), Proc. Roy. 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. Oxford. 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", American Mathematical Society. Bulletin. 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, 1921, Berlin: Springer-Verlag, ISBN 978-3-540-74283-8, Zbl 1136.17001
• Wörz-Busekros, Angelika (1980), Algebras in genetics, Lecture Notes in Biomathematics, 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

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