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Algebras of genetic self-incompatibility systems

To every system of inheritance there corresponds a genetic algebra, generally non-associative, that describes its structure. The genetic algebras of a large number of breeding systems have been studied, but they have all been non-selective, and mostly random mating. The object of this paper is to investigate the genetic algebras of a number of systems involving a very strong form of differential fertility, where some pairs of individuals cannot produce viable offspring at all. For these, none of the classical properties of genetic algebra hold. In §2, the phenomenon of pollen incompatibility with m alleles is studied. The case m = 3 occurs in nature in Nicotiana alata For this, but not when m > 3, the genetic algebra is found to be Lie admissible, and some detailed relations consequent on this are obtained. Section 3 is devoted to two systems of style height self-incompatibility, Lythrum salicaria and Oxalis rosea. For these, described essentially by 6- and 26- dimensional genetic algebras respectively, the idempotents are listed, and full and outline descriptions respectively are given of the lattices of subalgebras. In the last section a class of algebras is defined corresponding to a multilocus generalization of the Lythrum mechanism. It is shown that this mechanism always leads to an isoplethic equilibrium.