Genetic Algebras Satisfying Bernstein's Stationarity Principle

There are several motivating influences behind this paper. Most of the breeding structures studied by algebraists have led to genetic algebras of Schafer's type [11]. It seems a worthwhile next step to seek those systems where although the algebra is not necessarily of Schafer's type it contains an important subalgebra which is, or an ideal with respect to which the difference algebra is of Schafer's type. An example of the former possibility arises in sex linkage [9]. It will be recalled that genetic algebras are commutative but not associative (see [4] for an introductory account) and that among the types of powers of an element x, particular importance attaches to the principal powers x" and the plenary powers x, defined by " ~x, x = x x [ ] , x = x = x.