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 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.[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 [math]\displaystyle{ (x^2)^2 = w(x)^2 x^2 }[/math]. Every such algebra has idempotents e of the form [math]\displaystyle{ e = a^2 }[/math] with [math]\displaystyle{ w(a)=1 }[/math]. The Peirce decomposition of B corresponding to e is
- [math]\displaystyle{ B = Ke \oplus U_e \oplus Z_e }[/math]
where [math]\displaystyle{ U_e = \{ a \in \ker w : ea = a/2 \} }[/math] and [math]\displaystyle{ Z_e = \{ a \in \ker w : ea = 0 \} }[/math]. Although these subspaces depend on e, their dimensions are invariant and constitute the type of B. An exceptional Bernstein algebra is one with [math]\displaystyle{ U_e^2 = 0 }[/math].[2]
Copular algebras
Copular algebras were introduced by (Etherington 1939)
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) 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) as special cases of baric algebras.
Let [math]\displaystyle{ c_1, \ldots, c_n }[/math] be elements of the field K with [math]\displaystyle{ 1 + c_1 + \cdots + c_n = 0 }[/math]. The formal polynomial
- [math]\displaystyle{ x^n + c_1 w(x)x^{n-1} + \cdots + c_n w(x)^n }[/math]
is a train polynomial. The baric algebra B with weight w is a train algebra if
- [math]\displaystyle{ a^n + c_1 w(a)a^{n-1} + \cdots + c_n w(a)^n = 0 }[/math]
for all elements [math]\displaystyle{ a \in B }[/math], with [math]\displaystyle{ a^k }[/math] defined as principal powers, [math]\displaystyle{ (a^{k-1})a }[/math].[1][6]
Zygotic algebras
Zygotic algebras were introduced by (Etherington 1939)
References
- ↑ 1.0 1.1 1.2 González, S.; Martínez, C. (2001), "About Bernstein algebras", in Granja, Ángel, 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., 221, New York, NY: Marcel Dekker, pp. 223–239
- ↑ Catalan, A. (2000). "E-ideals in Bernstein algebras". in Costa, Roberto. Nonassociative algebra and its applications. Proceedings of the fourth international conference, São Paulo, Brazil.. Lect. Notes Pure Appl. Math.. 211. New York, NY: Marcel Dekker. pp. 35–42.
- ↑ Tian (2008) p.18
- ↑ Tian (2008) p.20
- ↑ Cohn, Paul M. (2000). Introduction to Ring Theory. Springer Undergraduate Mathematics Series. Springer-Verlag. p. 56. ISBN 1852332069.
- ↑ Catalán S., Abdón (1994). "E-ideals in baric algebras". Mat. Contemp 6: 7–12.
- 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
- Etherington, I. M. H. (1939), "Genetic algebras", Proc. R. Soc. Edinburgh 59: 242–258, archived from the original on 2011-07-06, https://web.archive.org/web/20110706211658/http://math.usask.ca/~bremner/research/geneticalgebras/etherington/ga.pdf
- 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
- Hazewinkel, Michiel, ed. (2001), "Bernstein problem in mathematical genetics", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
- Hazewinkel, Michiel, ed. (2001), "Baric algebra", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
- Hazewinkel, Michiel, ed. (2001), "Bernstein algebra", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
- 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
- Schafer, Richard D. (1949), "Structure of genetic algebras", American Journal of Mathematics 71: 121–135, doi:10.2307/2372100, ISSN 0002-9327
- Tian, Jianjun Paul (2008), Evolution algebras and their applications, Lecture Notes in Mathematics, 1921, Berlin: Springer-Verlag, ISBN 978-3-540-74283-8
- Wörz-Busekros, Angelika (1980), Algebras in genetics, Lecture Notes in Biomathematics, 36, Berlin, New York: Springer-Verlag, ISBN 978-0-387-09978-1
- Hazewinkel, Michiel, ed. (2001), "Genetic algebra", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
Further reading
- Lyubich, Yu.I. (1983) (in Russian), Mathematical structures in population genetics. (Matematicheskie struktury v populyatsionnoj genetike), Kiev: Naukova Dumka
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