In mathematical genetics, a genetic algebra is a (possibly non-associative

Non-associative algebra

A non-associative algebra over a field K is a K-vector space A equipped with a K-bilinear map A × A → A. There are left and right multiplication maps...

) 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 .

In applications to genetics, these algebras often have a basis corresponding to the genetically different gamete

Gamete

A gamete is a cell that fuses with another cell during fertilization in organisms that reproduce sexually...

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 (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

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

Copular algebras

Copular algebras were introduced by

Gametic algebras

Copular algebras were introduced by

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. showed that special train algebras are train algebras.

Train algebras

Train algebras were introduced by as special cases of baric algebras.

Zygotic algebras

Zygotic algebras were introduced by