Sequences in Genetic Algebras for Overlapping Generations

When Etherington (2) introduced linear commutative non-associative algebras in connection with problems in theoretical genetics, he pointed out that various sequences of elements in these algebras represented different mating systems. In all such systems it was however assumed that the generations did not overlap, and this restriction has been kept in later work in this field. In this paper we treat sequences which make it possible to find the probability distribution in successive generations in a discrete time model where the generations may be overlapping. We also consider idempotents in genetic algebras and outline how the method used on the overlapping generation sequence may be applied to other sequences. Unless otherwise stated all algebras considered have the complex numbers as the scalar field. Capital letters are used for all algebra elements.