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 constants of the algebra encode the probabilities of producing offspring of various types.
For surveys of genetic algebras see Bertrand (1966), Wörz-Busekros (1980) and Reed (1997).
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
Every such algebra has idempotents e of the form
Although these subspaces depend on e, their dimensions are invariant and constitute the type of B.
An exceptional Bernstein algebra is one with
[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.
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.
defined as principal powers,
[1][6] Zygotic algebras were introduced by Etherington (1939, section 7)