Title:

Researches in nonassociative algebra

I have frequently been asked by biologists for mathematical help in connection with their problems. I was working on one such problem when an algebraist, observing my work without knowing what it was about, remarked that I was apparently using hypercomplex numbers. I was considering a certain type of inheritance specified by formulae which could be regarded as forming the multiplication table of a non associative linear algebra; and my calculations could be regarded as manipulations of hypercomplex numbers in this algebra, or in another algebra derived from it by a process which I later called "duplication; I then realised that there are many such "genetic algebras ", representing different types of inheritance. They are in all cases nonassociative as regards multiplication, though they can always be taken to be commutative. I found that a large class of genetic algebras (viz. those for "symmetrical inheritance" as defined in Paper VI, p. 2) possessed certain distinctive properties which seemed worthy of investigation for their own sake, and also for the sake of possible exploitation in genetics. Part Three, the main part of this thesis, consists of four papers in which this investigation is given  or rather is begun, for there are a good many problems left untackled. Part One consists of four papers (one written in collaboration with Dr A. Erdélyi) on some purely combinatory problems of non  associative algebra, suggested by the notations which I employed for products and powers in the genetic algebras. The combinatory per t 0.401.5 rt. theory is continued in theAconcluding postscipt which follows Paper X. Part Two shows how genetic algebras arise and are manipulate The multiplication table of a genetic algebra, the multiplication of hypercomplex numbers, and the above mentioned process of duplication, are simply a translation into symbols of the relevant essentials in the processes; of inheritance; and the symbolism as a whole is a convenient shorthand for reckoning with combinations and statistical distributions of genetic types, enabling one to dispense with some of the verbal arguments and the chessboard diagrams commonly used for the same purpose. In paper VI the treatment is made as general as possible with the object of showirg the relationship between different genetic algebras and something of their structure; and the concepts to be discussed in Part Three are here defined. In Paper V, which was published later but mostly written earlier than VI, the explanation is given in very much simpler mathematical language (for it was intended to be read by geneticists), and with more attention to practical applications. It can be explained very simply why multiplication in the genetic algebras is non associative, that is to say This statement is interpreted: "If the offspring of A and B mates with C, the probability distribution of genetic types in the progeny will not be the same as if A mates with the offspring of B and C." My symbolism was not essentially new: the novelty lay in is interpretationlin terms of hypercomplex numbers. In fact it could be said that genetic algebras had been used by geneticists in a primitive way for quite a long time without having been recognised. explicitly. Their explicit recognition is I believe more than a mere change of notation. Apart from greater brevity achieved in some applications, general theorems on linear algebras can be applied; transformations can be used which are quite meaningless genetically but which lead to genetically significant conclusions; and the use of an index notation and summation convention reduces the symbolism to manageable proportions when, with inheritance involving many genes, it threatens to become too heavy to handle. Biological considerations were thus the root of these researches, and I intend to return to the genetical applications later; for I believe that genetic algebras may throw light on some deeper problems of genetics. I cannot at present give solid justification for this belief in the sense of having successfully tackled problems otherwise unsolved, and I therefore wish that this thesis may not be judged as a finished achievement in biological investigation; but may be judged primarily as a contribution to algebra, suggested by biological problems, and perhaps having possibilities of application beyond the simple ones so far demonstrated.
