Algebras of genetic self-incompatibility systems. (English) Zbl 0713.17021

Holgate, P.

Proc. R. Soc. Lond., Ser. A 427, No. 1873, 359-369 (1990).

To every system of inheritance corresponds a genetic algebra, generally nonassociative, that describes its structure. Genetic algebras, studied in previous papers, were all non-selective, and mostly random mating. This paper is devoted to some genetic algebras arising from plant populations when matings between individuals of the same, or closely related, genetic types cannot be fertile.
It has two objectives: first, to apply genetic algebraic methods to self- incompatibility; secondly, to exploit the possibilities of genetic algebras to examine a family of self-incompatibility systems based on k loci for \(k\geq 1\), of which the cases of \(k=1,2\), are known in nature. It is shown that for every k, there is an isoplethic equilibrium population.
Pollen elimination algebras, then style height algebras are studied, suggesting the definition and abstract characterization of self- incompatibility. The author defines isoplethic elements, algebras of Lythrum salicaria type, and gives a theorem about these algebras.

Reviewer: M.Bertrand

MSC:

17D92	Genetic algebras
17D25	Lie-admissible algebras
92D10	Genetics and epigenetics

Keywords:

Lie-admissible algebra; genetic algebras; plant populations; self- incompatibility; isoplethic equilibrium population