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
A zygote (, ) is a eukaryote, eukaryotic cell (biology), cell formed by a fertilization event between two gametes. The zygote's genome is a combination of the DNA in each gamete, and contains all of the genetic information of a new individual ...
algebras, and baric algebras (also called weighted algebra). The study of these algebras was started by .
In applications to genetics, these algebras often have a basis corresponding to the genetically different
gamete
A gamete (; , ultimately ) is a haploid cell that fuses with another haploid cell during fertilization in organisms that reproduce sexually. Gametes are an organism's reproductive cells, also referred to as sex cells. In species that produce ...
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 , and .
Baric algebras
Baric algebras (or weighted algebras) were introduced by . A baric algebra over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''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 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
. Every such algebra has idempotents ''e'' of the form
with
. The
Peirce decomposition
In ring theory, a Peirce decomposition is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements.
The Peirce decomposition for associative algebras was introduced by . A similar but more complicated Peirce decomp ...
of ''B'' corresponding to ''e'' is
:
where
and
. Although these subspaces depend on ''e'', their dimensions are invariant and constitute the ''type'' of ''B''. An ''exceptional'' Bernstein algebra is one with
.
Copular algebras
Copular algebras were introduced by
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.
[Tian (2008) p.18] An evolution algebra is necessarily commutative and
flexible
Flexible may refer to:
Science and technology
* Power cord, a flexible electrical cable.
** Flexible cable, an Electrical cable as used on electrical appliances
* Flexible electronics
* Flexible response
* Flexible-fuel vehicle
* Flexible rake re ...
but not necessarily associative or
power-associative In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity.
Definition
An algebra (or more generally a magma) is said to be power-associative if the subalgebra ge ...
.
[Tian (2008) p.20]
Gametic algebras
A ''gametic algebra'' is a finite-dimensional real algebra for which all structure constants lie between 0 and 1.
Genetic algebras
Genetic algebras were introduced by who showed that special train algebras are genetic algebras and genetic algebras are train algebras.
Special train algebras
Special train algebras were introduced by 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.
[
showed that special train algebras are train algebras.
]
Train algebras
Train algebras were introduced by as special cases of baric algebras.
Let be elements of the field ''K'' with . The formal polynomial
:
is a ''train polynomial''. The baric algebra ''B'' with weight ''w'' is a train algebra if
:
for all elements , with defined as principal powers, .
Zygotic algebras
Zygotic algebras were introduced by
References
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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
Population genetics
Non-associative algebras