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In mathematical genetics, a '''genetic algebra''' is a (possibly [[non-associative algebra|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 {{harvs|txt|authorlink=Ivor Malcolm Haddon Etherington|last=Etherington|year=1939}}.

In applications to genetics, these algebras often have a basis corresponding to the genetically different [[gamete]]s, 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 {{harvtxt|Bertrand|1966}}, {{harvtxt|Wörz-Busekros|1980}} and {{harvtxt|Reed|1997}}.

==Baric algebras==

Baric algebras (or weighted algebras) were introduced by {{harvtxt|Etherington|1939}}. A baric algebra over a [[field (mathematics)|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 {{harvs|txt|first=Sergei Natanovich |last=Bernstein|authorlink=Sergei Natanovich Bernstein|year=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>(x^2)^2 = w(x)^2 x^2</math>. Every such algebra has idempotents ''e'' of the form <math>e = a^2</math> with <math>w(a)=1</math>. The [[Peirce decomposition]] of ''B'' corresponding to ''e'' is
:<math> B = Ke \oplus U_e \oplus Z_e </math>
where <math>U_e = \{ a \in \ker w : ea = a/2 \}</math> and <math>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>U_e^2 = 0</math>.<ref>{{cite book | last=Catalan | first=A. | chapter=E-ideals in Bernstein algebras | zbl=0968.17013 | editor-last=Costa | editor-first=Roberto | title=Nonassociative algebra and its applications. Proceedings of the fourth international conference, São Paulo, Brazil. | location=New York, NY | publisher=Marcel Dekker | series=Lect. Notes Pure Appl. Math. | volume=211| pages=35-42 | year=2000 }}</ref>

==Copular algebras==

Copular algebras were introduced by {{harvtxt|Etherington|1939|loc=section 8}}

==Gametic algebras==

Copular algebras were introduced by {{harvtxt|Etherington|1939|loc=section 6}}

==Genetic algebras==

Genetic algebras were introduced by {{harvtxt|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 {{harvtxt|Etherington|1939|loc=section 4}} as special cases of baric algebras. {{harvtxt|Etherington|1941}} showed that special train algebras are train algebras.

==Train algebras==

Train algebras were introduced by {{harvtxt|Etherington|1939|loc=section 4}} as special cases of baric algebras.

Let <math>c_1, \ldots, c_n</math> be elements of the field ''K'' with <math>1 + c_1 + \cdots + c_n = 0</math>. The formal polynomial
:<math>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>a^n + c_1 w(a)a^{n-1} + \cdots + c_n w(a)^n = 0 </math>
for all elements <math>a \in B</math>, with <math>a^k</math> defined as <math>(a^{k-1})a</math>.<ref>{{cite journal | last=Catalán S. | first=Abdón | title=''E''-ideals in baric algebras | zbl=0868.17023 | title=Mat. Contemp. | volume=6 | pages=7-12 | year=1994 }}</ref>

==Zygotic algebras==

Zygotic algebras were introduced by {{harvtxt|Etherington|1939|loc=section 7}}

==References==
{{reflist}}
*{{citation|first=S. N. |last=Bernstein|title=Principe de stationarité et généralisation de la loi de Mendel|journal=C. R. Acad. Sci. Paris|volume= 177 |year=1923|pages= 581–584}}.
*{{Citation | last1=Bertrand | first1=Monique | title=Algèbres non associatives et algèbres génétiques | publisher=Gauthier-Villars Éditeur, Paris | series=Mémorial des Sciences Mathématiques, Fasc. 162 | mr=0215885 | year=1966}}
*{{Citation | last1=Etherington | first1=I. M. H. | title=Genetic algebras | mr=0000597 | zbl=0027.29402 | year=1939 | journal=Proc. Roy. Soc. Edinburgh | volume=59 | pages=242–258 | url=http://math.usask.ca/~bremner/research/geneticalgebras/etherington/ga.pdf}}
*{{Citation | last1=Etherington | first1=I. M. H. | title=Special train algebras | doi=10.1093/qmath/os-12.1.1 | mr=0005111 | zbl=0027.29401 | jfm=67.0093.04 | year=1941 | journal=The Quarterly Journal of Mathematics. Oxford. Second Series | issn=0033-5606 | volume=12 | pages=1–8}}
*{{eom|id=Bernstein_problem_in_mathematical_genetics&oldid=16709 |title=Bernstein problem in mathematical genetics |first=Yu.I. |last=Lyubich}}
*{{eom|id=Baric_algebra&oldid=16628|first=A.|last=Micali|title=Baric algebra}}
*{{eom|id=Bernstein_algebra&oldid=11704|first=A.|last=Micali|title=Bernstein algebra}}
*{{Citation | last1=Reed | first1=Mary Lynn | title=Algebraic structure of genetic inheritance | doi=10.1090/S0273-0979-97-00712-X | mr=1414973 | year=1997 | journal=American Mathematical Society. Bulletin. New Series | issn=0002-9904 | volume=34 | issue=2 | pages=107–130 | zbl=0876.17040 }}
*{{Citation | last1=Schafer | first1=Richard D. | title=Structure of genetic algebras | jstor=2372100 | mr=0027751 | year=1949 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=71 | pages=121–135}}
*{{Citation | last1=Wörz-Busekros | first1=Angelika | title=Algebras in genetics | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Biomathematics | isbn=978-0-387-09978-1 | mr=599179 | year=1980 | volume=36}}
*{{eom|id=g/g043970|first=A.|last= Wörz-Busekros}}

==Further reading==
* {{citation | last=Lyubich | first=Yu.I. | title=Mathematical structures in population genetics. (Matematicheskie struktury v populyatsionnoj genetike) | language=Russian | zbl=0593.92011 | location=Kiev | publisher=Naukova Dumka | year=1983 }}

[[Category:Population genetics]]
[[Category:Non-associative algebras]]

File:Binary Boolean Expression Tree.JPG - Revision history

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