Sequences of Powers in Genetic Algebras

Certain non-associative algebras have important applications intheoretical Mendelian genetics. The following definitions and summary of the theory are taken from papers by Etherington [2]-[7], who initiated their study. A baric algebra is one which admits a non-trivial homomorphism, x-*a)(x), into its coefficient field. co(#) is called the weight of £. In this paper we shall only be concerned with commutative algebras over the real numbers. Clearly, the basis of a baric algebra may be so chosen that one of its elements, say c0, has weight unity, and the remainder, cv ...,cn have weight zero. The set of elements of unit weight is closed with respect to multiplication, while the set of elements of zero weight is an ideal N, the nil ideal. In a non-associative system there is no unambigious k-th power of an element. In genetic algebras the following sequences are important: the principal powers x, defined by