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 non-associative 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 0.4-01.5 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(AB) C ≠ A (BC)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
