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Algebras arising in theoretical genetics Kwei, John T.P. 1971

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ALGEBRAS ARISING IN THEORETICAL GENETICS by JOHN T. P. KWEI Hons B. Sc. University of B r i t i s h Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of MATHEMATICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1971. In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree tha permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of MfrtiM fries; The University of British Columbia Vancouver 8, Canada Date cZi./AfPU- / 197/ 1 ( i i ) i 1 • • , ABSTRACT i .j i • I: Certain non-associative algebras have important a p p l i -cations i n t h e o r e t i c a l Mendelian Genetics. In t h i s thesis we w i l l give d e f i n i t i o n s to these algebras and study t h e i r propertie Some examples w i l l also be given. TABLE OF CONTENTS Chapter 1 Introduction Chapter 2 M u l t i p l i c a t i o n i n Genetics Chapter 3 Non-Associative Products and Powers Chapter 4 Baric Algebras Chapter 5 Trai n Algebras Chapter 6 Commutative Tra i n Algebras of Ranks 2 and 3 * Chapter 7 Special Train Algebras Chapter 8 ; A S t a b i l i t y Theorem f o r Special Train Algebra Chapter 9 Sequences of powers i n Special T r a i n Algebras Chapter 10 Genetic Algebras " of Symmetrical Inheritance Chapter 11 Some Examples Bibliography (v) ACiKNOWLEDG-EMENTS i ' • I I would l i k e to thank my supervisor Dr. J . V. Whittaker f o r h i s suggestions and encouragement i n wr i t i n g this:_the_tis. My indebtedness i s also due to Dr. D. C. Murdoch f o r reading my thesis and providing valuable suggestions. I am g r a t e f u l to the University of B. C. and the National Research Council f o r t h e i r f i n a n c i a l support. Last, but not le a s t , I wish to thank my wife Serena f o r her help and f o r typing the t h e s i s . CHAPTER 1 INTRODUCTION Certain non-associative algebras have important a p p l i -cations i n t h e o r e t i c a l Mendelian Genetics. The study of such non-associative algebras was in i t i a t e d , i n the 1930's by Dr. I.M. H. Etherington (2)-(6). Such studies were continued i n the years to follow by many other authors such as R.D. Schafer (18) and Geiringer (7). I t i s ca r r i e d further by current researchers i n the f i e l d such as H. Gonshor (9),(10) and P. Holgate (13),(14), to name just a few. The occurrence of the genetic algebras may be described i n general terms as follows. (Etherington, Genetic Algebras 1939 (3) ). The mechanism of chromosome inheritance, i n so f a r as i t determines the pr o b a b i l i t y d i s t r i b u t i o n s of genetic types i n families and f i l i a l generations, and expresses i t s e l f through t h e i r frequency d i s t r i b u t i o n s , may be represented conveniently by algebraic symbols. Such a symbolism i s described, f o r instance by Jennings (16, Chap, i x ) ; many applications are given by Geppert and K o l l e r . The symbolism i s equivalent to the use of a system of rela t e d l i n e a r algebra, i n which m u l t i p l i c a t i o n i s commutative (PQ=QP) but non-associative (PQ R = P QR). A popu-l a t i o n ( i . e . a d i s t r i b u t i o n of genetic types) i s represented by - 2 -a normalized vector i n one or other algebra, according to the point of view from which i t i s s p e c i f i e d . I f P,Q are populations, the f i l i a l generation P x Q ( i . e . the s t a t i s t i c a l population of off s p r i n g r e s u l t i n g from the random mating of individuals of P with i n d i v i d u a l s of Q) i s obtained by multiplying two corresponi-ding representations of P and Q; and from t h i s requirement of the symbolism i t w i l l be obvious why m u l t i p l i c a t i o n must be non-asso-c i a t i v e . I t must be understood that a population may mean a single i n d i v i d u a l , or rather the information which we may have concern-ing him i n the form of a pr o b a b i l i t y d i s t r i b u t i o n . In t h i s t h e s i s , we endeaver to give a d e t a i l e d account of many ,different algebras that ari s e i n t h e o r e t i c a l genetics. D e f i n i t i o n s w i l l be given of baric algebras, t r a i n algebras, s p e c i a l t r a i n algebras and genetic algebras i n that order. Many properties and s t r u c t u r a l theorems w i l l be developed. Some exam-ples are given i n chapter 11. Care was taken to blend the old and the new materials and to ensure the flow of the t h e s i s . Some u s e f u l d e f i n i t i o n s :-gamete : a reproductive c e l l that can unite with another s i m i l a r one to form the c e l l that develops into a new i n d i v i d u a l . zygote : any c e l l formed by the union of two gametes. haterozygote : any c e l l formed by the union of two d i f f e r e n t type of gametes. chromosome : any of the microscopic rod-shaped bodies which carry the genes that convey hereditary c h a r a c t e r i s t i c s . . sex-chromosome : a sex-determining chromosome i n the germ c e l l s of most plants and animals: f e r t i l i z e d eggs containing two X-chromosomes (one from each parent germ c e l l ) develop into females those containing one X- and one X-chromosome (male gem c e l l s carry ei t h e r one or the other) develop into males. autosome : any chromosome other than the sex chromosome. autosomal inheritance : inheritance of c h a r a c t e r i s t i c s through autosomes. progeny : descendants, o f f s p r i n g c o l l e c t i v e l y . genotype : an individual's genetic c o n s t i t i o n ( i n terms of i t s hereditary f a c t o r s ) . phenotype : a type distinguished by v i s i b l e characters rather i - 4 -than by hereditary or genetic t r a i t s ( i . e . r e f e r r i n g to the appearance). genomes : among organisms with chromosomes, each species has a c h a r a c t e r i s t i c set of genes, or genome. autopolyploids : organisms having four s i m i l a r genomes per nucleus. CHAPTER 2 MULTIPLICATION IN GENETICS DEFINITION: (5) The following d e f i n i t i o n i s due to I.M.H. Etherington. The m u l t i p l i c a t i o n of populations - individ u a l s -gametes — means the ca l c u l a t i o n of progeny d i s t r i b u t i o n r e s u l t -ing from t h e i r random mixing - mating - fusion. Defining a population as a pr o b a b i l i t y d i s t r i b u t i o n of genetic types, we may say i n a l l cases that we are multiplying populations, ( that i s , a population may mean a single i n d i v i d u a l , or rather informa-t i o n which we may have concerning him i n the form of a pr o b a b i l i t y d i s t r i b u t i o n ). Unlike m u l t i p l i c a t i o n i n ordinary algebra, the asso-r c i a t i v e law i s not necessarily obeyed i n genetical m u l t i p l i c a t i o n . As an example of non-associativity of genetical m u l t i p l i c a t i o n consider the following two populations (AB)C and A(BC). The progeny (AB)C, a mating between the offspring from A and B and a t h i r d one C i s c l e a r l y d i f f e r e n t from the progeny from A and the hybrid population BC. Genetical m u l t i p l i c a t i o n obeys the d i s t r i b u t i v e law : A( B+C ) = AB + Act *^D=^ B +/3C has genetic i n t e r p r e t a t i o n only i f ^ + / J =l,<<,/*eF. where F i s a f i e l d . In that case i t means that the i n d i v i d u a l D i s composed of 10OX# of B and 100/3$ of c, (or produces 100°$ of B and 100/5$ of C as the case may be. c/f example 1.) - 6 -There are three cases to be considered regarding the commutative law : CASE ( l ) : For autosomal characters, the outcomes of r e c i p r o c a l matings are generally i d e n t i c a l and therefore m u l t i p l i c a t i o n i s commutative, although i n case (3) we s h a l l see that there are exceptions. CASE (2) : For sex-linked characters the commutative law also holds (although one might be tempted to say that i t i s non-commutative). The reason i s that f o r sex-linked characters we can only speak of r e c i p r o c a l matings i n connection with the phenotype c l a s s i f i c a t i o n of. a population. On the other hand, the c a l c u l a t i o n of progeny d i s t r i b u t i o n i s only possible on the basis of the genotype c l a s s i f i c a t i o n . There are only the male genotype (which does not involve Y-chromosome) and the female genotype (which involves Y-chromosome). A r e c i p r o c a l mating between geno-types i s impossible. Therefore AB and BA means the same thing f o r a male genotype A and female genotype B, i . e . the m u l t i p l i c a t i o n i s commutative. CASE (3) • I t i s possible f o r autosomal inheritance to be un-symmetrical i n the sexes, though either crossing-over values or gametic s e l e c t i o n being d i f f e r e n t i n male and female. In such cases we can either treat the male and female genotypes as the same type with the same relevent gene content or as d i s t i n c t types (since they produce d i f f e r e n t s e r i e s of gametes). In the former case, AB and BA are d i f f e r e n t , r e f e r r i n g to r e c i p r o c a l crosses which do not produce s i m i l a r d i s t r i b u t i o n of off s p r i n g , and m u l t i p l i c a t i o n i s non-commutative. In the l a t e r case i t i s as case ( 2 ) . Therefore genetical m u l t i p l i c a t i o n i s commutative and d i s t r i b u t i v e with the exception of c e r t a i n cases where we have the option of using a varied form of the symbolism i n which the m u l t i p l i c a t i o n i s non-commutative. Genetical m u l t i p l i c a t i o n i s however, non-associative. In view of the above remark, we s h a l l assume that the mu l t i p l i c a t i o n s to be encountered i n t h i s paper as both commutative and d i s t r i b u t i v e . - 8 -CHAPTER 3 NON-ASSOCIATIVE PRODUCTS AND POWERS Care must be taken when dealing with non-associative products and powers involving many f a c t o r s . Brackets inserted i n d i f f e r e n t ways would indicate d i f f e r e n t orders of association of the f a c t o r s . To eliminate brackets, we s h a l l use groups of dots to separate factors when necessary, fewness of dots implying precedence i n m u l t i p l i c a t i o n . Thus A :.BC - AD : E means A{ [ (BC) (AD)] E] and represents the pedigree below: We s h a l l mainly be concerned with the two simple types of non-associative powers: the " p r i n c i p a l " and "plenary 1* powers. A, then the successive discrete generations are represented by A A fc. D \ / AD I t i s important to d i s t i n g u i s h between, e.g. (A 2) 1 and A* = A[A (A*)}. I f mating takes place at random i n an i n i t i a l population - 9 -the sequence of plenary powers. .22 X .1 A , A , A , A »* * ' » A , • • • (3 .1 ) i . e . A D 3 = A A = A * A (3 .2 ) While the sequence of p r i n c i p a l powers: A , A 1 ' , A * - , • •.• , A N ,. • • • (3.3) i . e . A * = A N " - A (3.4) refers to a mating system i n which each generation i s mated back to one o r i g i n a l ancestor or ancestral population. There i s ;a t h i r d kind of power which occurs frequently i n genetics and w i l l be discussed here at times. This i s the primary product of the form: x{y n~'} = { [ ( X Y ) Y J J Y . . . . = X Y • Y : Y (3.5) The primary product refe r s to the descendants of a single i n d i -v i d u a l or a subpopulation X mating at random within a population Y . A sequence of primary products i s c a l l e d the operational sequence: X , X Y , X Y • Y , X Y • Y J Y , • - . (3.6) CHAPTER 4 BARIC ALGEBRAS DEFINITION : A baric algebra i s a l i n e a r algebra A, associative or not, that possesses a n o n - t r i v i a l representation onto i t s c o e f f i c i e n t f i e l d F, i . e . W : A > F a i •> W(a) such that W( a+b ) = V. (a) + W(b) y( c * a ) = <*w(a) (4.1) W( ab ) = W(a) . W(b) o ^ F , a , b6A W(a) i s c a l l e d the weight of a, and W i s a weight function of A. I f W(a) i s f 0 , a can be normalized as: * = WTO" (4.2) of unit weight. The set of elements of unit weight i s closed with respect to m u l t i p l i c a t i o n . Elements of zero weight are c a l l e d - 11 -n i l elements. The set N of a l l n i l elements i s an invariant sub-algebra i . e . A N 1 N , the n i l subalgebra. By suitable l i n e a r transformations, the basis of a baric .algebra ( Co , c, , ••• , c„_, ) may be so chosen that one of i t s elements, say c 0 , has weight unity, and the remanider, c ( , , c n_, have weight zero. For, suppose the hasis of a baric algebra i s ( bo, b*, ' ••-• , bo - i ) and i t has a weight function W. Without loss of generality assume W( b 0) =f 0. Now, l e t C o = y(b° ) ' C i = b l ~ w(b<-) c ° x:= 1,<* • , n - l then W(c<,) = and • . W ( c l ) = W(b:> - Jtf(bt) W(co) = W(b-J - W(bO = 0 (c„ , c,,, ••• , C r , - , ) i s the required basis. Let the m u l t i p l i c a t i o n table of a l i n e a r algebra A with - 12 -basis ( c 0 , c, , c ^ , ) be: ( i. , : ,k = 0, 1, n-1 (4.3) and l e t the a r b i t r a r y element a 6 A be denoted by a = E o4 c L where o/c€ F (4.4) THEOREM (4-A): A l i n e a r algebra A i s a baric algebra i f and only i f the equation n - i ( i , j ,k = 0, 1, ..., n-1 ) regarded as ordinary simultaneous equations i n F f o r the unknowns XL. t should possess a non-null sol u t i o n XL = WL. Proof: Necessary: Let ML = W( x i ) = i i ( XL regarded as unknowns •< i n F ) (4.5) and since W i s a n o n - t r i v i a l representation there exists a non-n u l l s o l u t i o n X L = w^ . S u f f i c i e n c y : Let ¥( a ) = c/j. wc. (4.6) 3 e A 3 ¥ ( XL ) = WJ. ^ 0 and (4.1) are at once deducible from the d e f i n i t i o n . The W_'s are c a l l e d the basic weights and they form the weight vector of A. Let us denote the kernel of the homomorphism W by R. Then a necessary and s u f f i c i e n t condition that a l i n e a r algebra A be a baric algebra i s that A contains an i d e a l R such that A/R = P (4.7) This i s c l e a r from the d e f i n i t i o n of a baric algebra. Let the general element x G A be denoted by n-l x = X;.CL (4.8) 1 = 0 Let the rank equation or equation of lowest degree connecting the p r i n c i p a l powers x , x 1 , x 3 , • •-• • (4.9) be g( x ) = x r + s, x r _ l + &^x. r~ x + "• + S H I : (4.10) - 14 -THEOREM (4-B): 8«x i n (4.10) i s a homogeneous polynomial of degree m i n the c o e f f i c i e n t s i t of x. : Remsrk: The rank equation i s the same as. the minimal polynomial f o r a l i n e a r transformation or matrix and f o r each x e A, the l i n e a r transformation i n question i s l e f t m u l t i p l i c a t i o n by x. Proof: Suppose A has a unit element. For each x 6 A 3 a one-to-one correspondence ( isomorphism.) h from A into the algebra E of a l l l e f t m u l t i p l i c a t i o n of x, x e A, that i s , h ; A * E (4,11) h( x ) where h( x ) : A * A i s the l e f t m u l t i p l i c a t i o n by x a i ) xa f o r each a e A ° l Ci = I I r : i k c * (4.13) ( t f j , k = 1, 2, - " n ) Then XC-} - £L where y 3 « = • § X ^ 3 k (4.14) (4.15) h( X )( C j ) = T y. k c w i . e . there exists n x n matrix H( x ) such that h( x )•( a ) = H( x ) • [ a ] where *i% y«. H( x ) = (4.16) (4.17) (4.18) Let P be extended to i t s algebraic closure, so that the c h a r a c t e r i s t i c polynomial G( x ) of H( x ) factors compltetely G( x ) = ( x-^, ) - . - ( x - o / r ) (4.19) Then there i s an ordered basis ( c / , c^ ,••• , c R ' ) f o r A i n which H( x ) i s represented by a matrix which i s i n - 16 -Jordan form: H( x ) = el, o 1 V . 0 1 0 • • • • 0 1 </• 0 0 . . 0 .1 o£ 0 0 0 0 1 C^r (4.20) where some of the entries i n the second diagonal may be zero and r v - k C k ( i , i , k = 1, 2, • n) y^' = 1, x ^ ' Ui ~ y ' n x- r-. • (4.20 a) (4.20 b) (4.20 c) (4.21) and the minimal polynomial of H( x ) i s + ( x ) = ( x ) • • • ( x - o^ T ) (4.22) where the , • • • ,o/ rare not necessarily d i s t i n c t . Since the minimal polynomial i s unchanged by l i n e a r transforma-- 17 -t i o n i t i s c l e a r that s m i s homogeneous of degree m i n the co-e f f i c i e n t s X . Suppose A does not have a un i t , and A have a basis [ c, , c^ , . .• , c n J . Consider the algebra A* with u n i t [ c c and basis { c c , c , , « • • , c n } where c c x = c„ , c Dc- = c : c 0 = c-_. (4 .23) Thus c i s a unit of A*. Set X = X* C r *i=l x*= x 0 c 0 + x , (4.24) where x v € F i = G, 1, 2, * * • , n 1 - 1 correspondence ( isomorphism ) h from A into the algebra E of a l l l e f t m u l t i p l i c a t i o n by x , x £ A i . e . h : A* ->E t i o n by x x * i >h( x * ) - (4.25) such that h( x * ) : A ?A i s the l e f t m u l t i p l i c a a 1 > x*a (4.26) Then x* c c = ( x 0 c o + x ) c Q = x* x* c. = ( Xo Co + x ) c^ = ( i = 1, 2, , n ) where n V (4.27) (4.28) h( x * ) ( Cj ) = x 0c 3' + g- y- k c k. (4.29) i.e.there exists n + 1 x n + 1 matrix H( x^ ) H( x* ) = x 0 0 0 x, x 0 +y,, y 2 l X2 y i a . x"o+y« . 0 . y « . . ynx . y « 3 x o "*"ynn (4.50) When we set x Q = 0, the c h a r a c t e r i s t i c polynomial takes the form G( x ) = x ( x - oL, ) . . . ( x - ^r- ) (4.31) and the minimal polynomial^ or equivalently the rank equation) , « f ( X ) = x( X - o4 ) • • • • ( X - <^r ) I! (4.32) where the 0(1*3 may not be d i s t i n c t . This completes the proof. Since g( x ) = x r + s, x r~' +.six.r"* +••••' + s r_, x=0, g( x ) i s of zero weight. Hence the equation i s s a t i s f i e d when we substitute W( x ) f o r x: consequently x - W( x ) I g( x ) i . e . W( x ) i s a root of the rank equation. - 20 -CHAPTER 5 TRAIN ALGEBRAS Consider the rank equation (4.10). In.general the s L w i l l depend on x, hut i f i n so f a r as they depend on x, they depend only on V( x ), then the ba r i c algebra i s c a l l e d a t r a i n  algebra of p r i n c i p a l rank k; ( or rank k ). "' Since s ^ . i s homogeneous of degree m i n the c o e f f i c i e n t s X c of x, i t must i n a t r a i n algebra be a numerical multiple of W( x f. Therefore, the rank equation can be factored: Vy g(x) = x(x - w)(x - X,w)(x - A a w ) • -• = 0 (5.1) i n an extended f i e l d i f necessary. The numbers 1, .A. i , » • • • ° are c a l l e d the p r i n c i p a l t r a i n roots of the algebra. For each element x of unit weight (4.10) becomes g( x ) = x + s, x + • ' • + s r-, x = 0 (5.2) where now the s: 's are constant and (5.1) becomes - 21 -. y(x) = x(x - l ) ( x -A,) » - .• (x - \ j ) - • • = 0 (5 .3) Since (5.2) can be m u l t i p l i e d by x any number of times, i t can be regarded as a l i n e a r recurrence equation with constant c o e f f i c i e n t connecting the p r i n c i p a l powers of the general normalized element x. - 22 -CHAPTER 6 • COMMUTATIVE TRAIN ALGEBRAS OF RANK 2 AND 3 I. Nilproduct and T r i p l e Nilproduct Let X be a commutative baric algebra of order n with basis ( Co » c. » • • • » Cn.-< )• Let N denote the n i l i d e a l . N consists of a l l elements of zero weight, the n i l elements. Then N = ( c , , • • • , c . A - I ) the subalgebra generated by c, , c n-, and hence i s of order n - 1. Let x, y & X be of weight j,^ . The nilproduct i s defined as the n i l element x * y = xy - i j y - i f x C6.1) c l e a r l y x y = y x x ' ( y ± z ) = x * y = x z The nilsquare x x = x - X x i s written x DEFINITION: A p-element i s an element of the form i . e . a l i n e a r combination of nilsquares. (6.3) Let P be the set of a l l p-elements. - 23 -THEOREM (6-A) I; . . • ' ' A l l nilproducts are p-elements. Proof: If x =Jx , y =^y , j f 0 , ^  0 , then x • y = tj^{4 m •+ w )"-x - y"]:-'e P (6.4) If ) = 0 or [ = 0, then x or y can be expressed as the difference of two elements of equal non-zero weight; there fore, with (6.2) the theorem follows. Corollary: N? C P C N THEOREM (6-B) (i) P i s an ideal. ( i i ) X/P i s a train algebra with the train equation x( x - 1 ) = 0 Proof: (i) Let x e X and p e P then xp = x ' p + f j p 6 P ( i i ) x( x - 1 ) = x * x e P therefore x( x - 1 ) = 0 £ X/P. This completes the proof. - 24 -j Let \ € F be f i x e d , and l e t x, y, z £ X. be of weights j , ^  , . Their t r i p l e nilproduct i s defined as . *x'y'z=i x(yz)+y(zx)+z(xy)-(l+X)0yz+fzx+^xy!); 4( x-AJ)(y-z)+3(y-Af)(z-x)-ht(zA§)(x'y) (6.5) The t r i p l e nilproduct i s commutative, associative, and d i s t r i -» b u t i v e , i . e . x y z = x z . y = x f y ( z + z ) = x y z = x y z ^ V * ) / ( y ) ' ( y ) = - / r ( x'y'z ) (6.6) The nilcube of X i s denoted "^x* ** and f o r a normalized element >x*-*= x 3 - ( 1 + A ) x 2 + X r = x ( x - 1 ) ( x - -\) = ( x - .X ) x ** (6.7) -DEFINITION: A q-element i s an element of the form xL" " (6,8) i . e . a l i n e a r combination of nilcubes. Let Q„\ be the set of a l l q-eleraents. - 2.5-THBOREM (6-C) A l l t r i p l e nilproducts are q.-elements. Prooft For x, y» z of tinit weight i n X, we have 6 x y z = ( x + y + z ) - \ y + z ) - ( z + x ) - ( x + y ) + X x ' " + * y ***+Az * " (6.9) and together with (6.6) the theorem follows. THEOREM(6-D) QxC P Proof: This follows from (6.7) and the fact that P i s an i d e a l . THEOREM (6-E) '.. • I f A =MC» then 0 x^ + Qjt = P Proof's ( ^ l - X ) x " =-Ax*"-"^x**' implies P C Q > f c but Ct\+ GL&C P by theorem (6-A) therefore - 26 -- QA + CU = ? THEOREM (6-F) ( i ) I f Qx i s an i d e a l of X, then X/Qa i s a t r a i n algebra with t r a i n equation x( x - 1 )( x -X) = 0 ( i i ) I f further X i s a t r a i n algebra, t h e n \ must be one of i t s t r a i n roots. Proof: ( i ) x( x - 1 )( x - > 0 = A x " £ QA therefore x( x - 1 ).( x - A ) = 0 6 X/Q ( i i ) I f X i s a t r a i n algebra, then the rank equation of X i s of the form x( x - j )( x - \j ) f ( x ) = 0, f o r some f ( x ) , i . e . \ i s one of i t s t r a i n roots. I I . ( Commutative ) t r a i n algebra of rank 2 Let X be a t r a i n algebra of rank 2. The t r a i n equation i s x( x - 1 ) = 0 (6.10) But - 27 -therefore • P = N* = 0 (6.11) Hence f o r x, y £ X with weights, x'y = xy - i ^ x + -^y = 0 implies xy = ^ x + \y (6.12) In p a r t i c u l a r , taking the base elements to be a l l of unit weight, say ( c 0 , c, , •• * , c n_, ), then c-tc^ •'= £c : + i c ^ (6.13) Or taking one base element c c of unit weight and c,, • »•• , c^.,, of zero weight, we have c„ a = c „ c . C i . = i d i = 1, •, n - 1 c,.c3. = 0 i , j =1, 2,-- • , n-1 (6.14) THEOREM (6-G) In a commutative t r a i n algebra of rank 2. ( i ) M u l t i p l i c a t i o n i s associative f o r powers. ( i i ) Any sequence of powers forms a t r a i n * with the ^ For d e f i n i t i o n of trains see next chapter. - 2 8 -same t r a i n equation x( x - 1 ) = 0, ( i i i ) The operational sequence forms a t r a i n with the t r a i n equation x {y - l ] (y - = 0. Proof: ( i ) From (6.12) • J x a = ^ x therefore x = ) x f o r a l l n (6.15) ( i i ) Since x 1 = x , a l l powers of x are equal. ( i i i ) xy = i x + 4-y „ or x( y' - i ) = iy . multipluing on both sides by ( y - 1 ), x( y - * )( y - 1 ) = £y( y - 1 ) = 0 (6.16) I I I . ( Commutative ) t r a i n algebra of rank 3 Let the t r a i n equation be v x( x - 1 ) ( x - A ) = o = x 3 - ( 1 + A ) x* +Ax (6.17) The rank equation i s then :( x - y .)( x - A j ) = 0 - 29 -= x 3 - ( 1+A ^x * + A J x (6.18) Take i n (I) as the t r a i n root , then from (6.17) = 0 Therefore Xx-x*x 2 = x( xx2- )+x( x ax )+x J( XX )-("l - f A ) ' ( xx.1+x2x+xx )-fc\( x+x+x2) = 2x4+x***-( 1+A)( 2x 3+x z)+A( 2x+x-) But from (6.17) 2 x 4 - 2( 1 + A ) x 3 + 2Ax z = 0 Hence x*'1 - ( 1 + 2A -Jx2- + 2\x = 0 We have just proved the following theorem. THEOREM (6-H) . A t r a i n algebra with the p r i n c i p a l t r a i n equation (6.17) possesses the plenary t r a i n equation x C x - 1 DC x - 2AD = 0 = x 2 , 1 - ( 1 + 2A )x2+2Ax (6.19) The plenary rank equation i s obtained by multiplying by ^  and i s x x x - ( .1 + 2 A ) j V + ZA^x = 0 (6.20) Substitutingpdx + p y + <fz + £w f o r x i n equation (6.20), andjby <-i + ji + V + f o r some <x/, /3 , Tr , 6 F, simplying and col l e c t -ing terms one has xy ( zw ) + xz ( yw ) + xw ( yz ) - ( \ +A )( yz + zx + xy + xw + yw + zw ) + J:A ( x + y + z + w ) = 0 (6.21) I f we l e t w = "z=y i n (6.21), we obtain 3xy fx - (• * +A )( y2- + 3xy + 2ya- ) : + 1 A ( x + 3y ) = 0 (6.22) Since *xy-y* y=xy(yy)+yCy(xy)J +yD.xy)yj|-(l+A) [yy+y (xy) + (xy )y] +A( xy+y+y) = 0 or (xy)y" +2[(xy) y] y + (1+A) [z(xy)y+y1]-A.(xy+2y) = 0 ( 6 . 2 3 ) Equating equations (6.22) and. (6.23) and eliminating the term ( x y ) y 2 we obtain [(xy)y]y-(l+X) (xy)y+(i+A)xy-i\x - 31 -= i y - iAy or x{y-*]*(y-A) = iy(y-y\) (6.24) Hence multiplying both sides of (6.24) by (y-l) we see that the operational sequence x £ y ^ ' J forms a t r a i n s a t i s f y i n g the equation x { y - 1} {y - * f [ y - ^ j =. o - 32 -CHAPTER 7 • i . • • ' ' SPECIAL TRAIN ALGEBRAS DEFINITION: A s p e c i a l t r a i n algebra i s a commutative l i n e a r algebra f o r which there exists a basis such that x 0 f t o = 1 (7.1) x.jk'= 0 f o r k < j (7.2) xi-jk = 0 f o r i , j > 0 and > k ^  max(i,j) (7.3) and furthermore a l l powers of the i d e a l { a, , a a t •• • , a R | are i d e a l s . Remarks: I t follows from the d e f i n i t i o n that t i ) The map which sends each element into i t s c o e f f i -cient a i s the only n o n - t r i v i a l homomorphisim of the algebra into i t s c o e f f i c i e n t f i e l d . Therefore, a s p e c i a l t r a i n algebra i s a baric algebra. ( i i ) { a, , a, , • •• , a R.| = N, the n i l - i d e a l , i s n i l -potent. - 33 -In such algebras, there are many other sequences whose properties resemble those of the sequence of p r i n c i p a l power, i.e, (1) sequences of elements derived from a general element, which s a t i s f y l i n e a r recurrence equations. (2) c o e f f i c i e n t s of the equations depend only on the weight of the element. (3) c o e f f i c i e n t s become constants f o r each element of unit weight. Such sequences w i l l be c a l l e d t r a i n s . For example: the sequence of plenary powers: x, x a , x 2 , i , x a , (7.4) forms a t r a i n i n a s p e c i a l t r a i n algebra and so does the opera-t i o n a l sequence of primary products: x ,. xy , xy.y,:-: xy.y.y , ... (7.5) We w i l l denote the r t h element of a t r a i n as x , and regard i t as a symbolic r t h power of x, and we s h a l l use t h i s symbolism throughout t h i s paper i n algebraic operations, l e t the normalized recurrence equation, or t r a i n equation be f LxJ = x + t, x + • • • + ts - i x = 0 (7.6) and. symbolically f C x J = x C x - l ] [x--«-0---[x-^s-J = 0 , (7.7) - 34 -The square brackets indicate that a f t e r expansion, powers of x are to be interpreted as symbolic powers. Assuming no repeated factors present, s i s the rank of the t r a i n , and the number 1, jilt » • * * are the t r a i n roots. By the t r a i n property we s h a l l mean that the deter-mination of the nth term ( or generation i n genetics ) depends ultimately on a l i n e a r recurrance equation with constant c o e f f i -c i e n t s . The s i g n i f i c a n c e of s p e c i a l t r a i n algebra i s seen i n the a p p l i c a t i o n to genetics. I t w i l l be seen that a l l the funda-mental symmetrical genetic algebras are s p e c i a l t r a i n algebras. Various trains have genetical s i g n i f i c a n c e ; the x'r' represent successive discrete generations of an evolving population, and the t r a i n equation i s the recurrence equation which connects them. For example: plenary powers(7.4) r e f e r to populations with random mating, p r i n c i p a l powers (.3.3) r e f e r to a mating system i n which each generation i s mated back to one o r i g i n a l ancestor or ancestral population, and the primary products r e f e r to the descendants of a single i n d i v i d u a l or subpopulation x mating at random within a population y. Other mating system are described by other sequences, and i n various well-known cases these have the t r a i n property. ; I t usually happens that the t r a i n .roots are r e a l , d i s -! ; • i t i n c t and between zero and one. Therefore i t may be shown that xtr^'—-> equilibrium as r »°° ; the rate i s ultimately. that of a geometrical progression with common r a t i o equal to the largest t r a i n root excluding 1, but i t may be some generations . ( depending oh the number of t r a i n roots ) before the rate of approach i s manifest. Tr a i n roots may be described as the eigen values of the « operation of symbolic m u l t i p l i c a t i o n by x or i n genetic language, t r a i n root may be described as the operation of passing from one generation to the next. - • ' . THEOREM (7-A) A s p e c i a l t r a i n algebra i s a t r a i n algebra. Proof: Let B be a s p e c i a l t r a i n algebra, then there exists a . basis { a„ , a • , • • • , a n} A • such that a ta^ = ZI x-L-k a K (7.8) X O O c = 1 (7.9) x ( j k = 0 f o r k < j (7.10) x t-k = 0 f o r i , j > 0 and • ' • ' k ^  mnx(i,j) (7.11) Let x £ B - 36 -. n x = wc 0 +21 o4.cc (7.12) where w = W( x ) i s i t s weight Then using (7.8), (7.9), (7.10), (7.11) X C 0 = X C 0 + • • • X C L = x A ' c C L + •• . (7.13) which implies •- 0 = ( w - x )c0 + ' • • 0 = ( wJ\»- x )c, + • • • 0 = ( w x ) c z + - » etc. (7.14) The c h a r a c t e r i s t i c equation of the algebra i s obtained by equation to zero x times the determinant of t h i s set of equa-tions which i s x( X-W )( X-Xw ) • • • ( X-AKW ) = 0 (7.15) Powers of x i n the expanded form of t h i s equation are to be interpreted as p r i n c i p a l powers. I f the rank equation of B i s g( x ) = 0, then g( x ) must be a factor of the left-hand side of the c h a r a c t e r i s t i c equation (7.15). - 37 -i.e. X ( X - W ) ( X-Jl,\i )( X-JlL\r ) •••• ( X - J r W ) = 0 where ( - A . • • • ,Jtr) C ( X, • • \ , X i O B has thus the e s s e n t i a l ( defining ) property of a t r a i n algebra. This completes the proof. - 38 -• 1 • . CHAPTER 8 M - • • A STABILITY THEOREM FOR SPECIAL TRAIN ALGEBRA • The X c ^ j ( abbreviated A j ) that appeared i n (7.1), (7.2) and (7.3) w i l l be c a l l e d the t r a i n roots of the algebra. C l e a r l y A« = 1. THEOREM (8-A) Every s p e c i a l t r a i n algebra which has no t r a i n r o o t \ s a t i s f y i n g 2X= 1 has a unique non-zero idempotent. Proof: Construct x i inductively such that ( jL x L a ) = Z. x«taL + Z_ y t a L (8.1) Since X e o c = 1, we have x 0 = 1 f o r a non-zero idempotent. Suppose x c , x, , ••• x m has been chosen. Then ( S x L a L + z a ^ , ) = ( ,jsL x.jaj. f + 2z&y~+t /f^ xL&i + z ^ a ^ , x ta ca- V A+\+ zxa2w,+< 1.= o = .21 x- a;: + -21, y L a L + 2zx a M-v, ^ 4, a^-+ , + Q b y (6.2) and (6.3)] - 39 -= x t a L + L 2 z ^ w , + f ( x,, x i t ••• f )] where y^-n = f ( x», x 2, - , x M ) i s a function of X , ' , X w v Since 2 A«4> 4 1, the equation i n Z 2 } w + 1 Z,+ f ( x, , x 2 , • • • , x w ) = Z (8.2) has a unique s o l u t i o n which i s taken as x M , . This proves the existence and uniqueness simultaneously. In the following the algebra i s taken over the r e a l s . We may then define x -• a ^ ^ 2L x ^ a w ^ ^ x w ^ T> x ^ THEOREM (8-B) The sequence of plenary powers of an element of weight 1 i n a s p e c i a l t r a i n algebra whose t r a i n roots other than X° = 1 a l l have absolute value less than \ tends to an idempotent. Proof: Suppose ai. = . 21 x a . W'e want to show x f c l > x ^ f o r x defined as i n theorem (8-A). Cl e a r l y x o L = 1 f o r a l l i ( weight = 1 ) - 40 -suppose x^- -> x^ f o r m = 1, • , k. • consider X k - r | = 2 y l k + , x ^ , ,-u + f( x t c , ' *• x>,c ) (8.3) which i s derived i n the same way as i n theorem (8-A) and where f( xy'L , • • • , X » « . L ) i s a quadratic function of xxl , • • • ,x^c and depends on the m u l t i p l i c a t i o n table of the algebra. Let XK+I , i'+t = a i + , x k+\ » L = a L 2 \ k + t = A \X| + i f ( x % i , • • • , Xvvi. ) = b c • we have a = AtU + bi. (8.5) and b i. — — ^ b = f ( xc» '• • • , x ^ ) Let a be such that a =Aa + b (8.6) write . ac = a + u c (8*7) bj. = b + y c. (8.8) we have u l +, = X u c + y< where IA|<1 - 41 -and yi > 0, and i f u; =>• 0, then a c r> a and we are done. Since = X u , + X y, + • ' ' + y i i f we define Z A = A yy + A yx + ' ' ' + y^ then Z K — > 0 also -X u, 7 o implies > UL *0 This completes the proof. - 42 -CHAPTER 9 SEQUENCES OF POWERS IN SPECIAL TRAIN ALGEBRAS Let A denote the s p e c i a l t r a i n algebra over a space of dimension n + 1 whose canonical basis i a { c. , c, f , c«] i n which a t y p i c a l element of unit weight i s x = c c + u,c, + • • • + = ( 1 , U , , U n ) (9.1) .We s h a l l assume that the algebra contains an idempotent element and i t can be taken as c 0 , i n which case A....k = 0 , k > 0 . . L e t AK-1 be a subspace of A.w whose basis elements are { Ce , c v r • • • , Crv-jJ . Then A, has basis elements { C c » C- iJ with m u l t i p l i c a t i o n table C A . « i , C , 0 (9.2) - 43 -and A,has basis elements [ c e , c, , c a j with m u l t i p l i c a t i o n table c, C a c. A*,, C,-'Xo.a C x AoizCi c, •P 0 Let E * be the operator on which transform x into x2"* For example v , Ej.x = x 1 = Co + 2 Ao,, u, c, + ( 2 A.iiU , + Aiis. u,2* + 2Xoa3u3. ) c a ¥e think of E»\ as operating on the c o e f f i c i e n t u: ( which i s regarded as a va r i a b l e ). For example, Eo. 1 = 1 Ej.u, = 2 Ao,,u, E z U i = 2AcuU, + AUAU , " " + 2 AcnUi (9.3) THEORE, (9-A) The plane of unit weight i n An may be mapped into a variety^Vn l y i n g i n a space B*v of minimal dim B* > n + 1 by a t i n biology, a v a r i e t y i s a group having c h a r a c t e r i s t i c s of i t s own species; subdivision of a species, subspecies. Here, a v a r i e t y means a subset consisting of elements of the form (1, v, , ••• , T « ) . 44 -function R R«( 1, u, , • - • , ux ) = ( 1, v, , ,, v^) where v i = u, . u a ... u * (9.4) i n such a way that a l i n e a r transformation of iBV with matrix E can be found having the following properties: ( i ) E \ ( R*X ) = R*( EAX ) ( i i ) E * i s lower triangular ( i i i ) The c h a r a c t e r i s t i c value of Eyv such that ( Erv - ^ t l ) (R«vx ) has no component i n vt given-by (9.4.) i s ( 2 Aoi «• ) ( 2 A « i ) . . . ( 2 A » M ) Informally, EnUn. involves u^-i . Hence i f B H has been found corresponding to A^~, , and a mapping R given by (9 .4 ) , Bn w i l l need to contain a dimension corresponding to each d i s t i n c t product of powers given by multiplying pairs of expressions on the r i g h t of (9.4) including squares and one corresponding to i u . Proof: For the case A; Since ( 1 , u, ) • '( 1, u, ) = ( 1, 2Ao„u, ) we have Ri = I the i d e n t i t y mapping - 45 -and E \ 0 2\ou / (9.5) For the case A ; Since ( 1, u » » )( 1, U ! , U a ) = ( 1 , 2 A m u, » 2A 0(a Ui + A» 2 l i U i + we have 1, ,u a ) = ( 1, v ( f- • x t = ( 1, VLS induced transformation E a i s G 0 0 0 2 A.CMI 0 0 E a = 0 0 4 A o,» 0 V o 2 A*»o- 2 A«aa (9.6) The theorem i s thus seen to be true f o r A , and A a . Suppose that i t i s true f o r An- i . Let a general element of Bn- i be ( v.. , v, : v>vJ ) (9.7) • • •, m . and the elements of E A - i denoted by e ^ i , j = 0, The required space Bn and matrix E rvwi l l be constructed i n two - 46 -stages and l e t the intermediate construction be B and E. For B we take a space of dimension m + 1 = -H m' + 1 ) ( m' + 2 ) + 1 Define: I f : Arv > B ( 1, u, , , ••• , u„ ) ' > ( b„ , • • • , b>»v-, , b w ) (9.8) as follows: The f i r s t m coordinates be , b , , • • • , bw - i are the products v tv^, 'L'2= 0, , m and j ^ i- , ordered so that Vr-vs precedes v K v e i f either r ^ k or r = k, S^JL. Then bv^u^. Define the matrix Hf as follows;: f o r the f i r s t m rows and columns of E we take the kroneeker square > and delete the row and column corresponding to v tv^ f o r which • The ( m + 1 )th row and column of E are defined by . = 0 , j'•= 0, • • • ,m - 1 (9.9) e^^= 2 AO.A (9.10) e>*k = 2ALi * i f the kth column of E ex-v pressed i n terms of (9.4) corresponds to u;,u^  , i + j . = AlLw i f the kth column of E corresponds to u i a = 0 otherwise (9.11) t Jacobson, Abstract Algebra. Vol, 2, pages 211-213. E s a t i s f i e s ( i i ) and ( i i i ) of theorem (9-A), which i s clear from the construction. I t i s possible that V^VA = v fv s f o r some k, 1, r,s» That i s ( u , • . . u * ) ( u , • • • u k ) = ( U, • " U f )( U, — • Urv ) Therefore the corresponding c h a r a c t e r i s t i c values would also be equal. For each occurrence of t h i s type, add the column of E corresponding to vr vs to that corresponding to v K v e , and delete the row and column corresponding to vr v s . Also delete the co-ordinate of B corresponding to v r v$ . Further, since the l a s t row of E has non-zero entries only f o r coordinate corresponding to , the elements of t h i s row are not affected by the r e -duction procedure except f o r r e l a b e l l i n g of the column number. The r e s u l t s of the reduction procedure are the required space BA and matrix and the associated map R^. To prove ( i ) i s true f o r AA , B^, E ^ and E*, and R* so constructed, consider x e AA , x = ( 1, u» , • • • , u ^ - i , u f t ) and l e t P be the projection of x into i t s f i r s t ( n - 1 ) coordinates, i . e . Px = P ( 1, u , , • • • , U K-( , U»v) = ( 1, u , , • • •• , u*-t , 0 ) (9.12) - 48 -Then E*x = E A ( Px + u A c ^ ) = ( Px ) + f 2 X Z Xi-T«U(.uc+ i-« . J (9.13) From the d e f i n i t i o n of R*v, one sees t h a t the f i r s t term of (9.13) shows t h a t the a p p r o p r i a t e t r a n s f o r m of the f i r s t m c o o r d i n a t e s o f R,vX i s the Kroneeker square of E * - i reduced t o all o w f o r i d e n t i t i e s among the v i . v ^ » and the remainder of (9 .13) shows t h a t (9.10) and (9.11) g i v e s the r e q u i r e d l a s t row of E*.. To prove t h a t B»v so c o n s t r u c t e d i s minimal, c o n s i d e r 2 En.Un = A n - l , t\-\ , rv U tv- \ + ' ' • (9.14) By h y p o t h e s i s , E r t_ t u A - i generates a l l the d i s t i n c t products o f powers of the u : i n v o l v e d i n the r i g h t - h a n d s i d e of (9.4). Hence a f t e r the r e d u c t i o n s of the above, no f u r t h e r r e d u c t i o n o f B i \ i s p o s s i b l e . T h i s completes the p r o o f . F o r p a r t i c u l a r s e t s of X^k, spaces of s m a l l e r dimen-s i o n than the B* j u s t c o n s t r u c t e d may s a t i s f y the c o n d i t i o n s of the theorem. THEOREM (9-B) Pl e n a r y powers i n A* form a t r a i n . The pl e n a r y t r a i n r o o t s o f An. are i n c l u d e d i n the f o l l o w i n g s e t : the product taken I - 49 -in pairs of those plenary train roots of the AA-I , including 2Ae*n and squares of those plenary train root of k*—\ . Proof: Prom theorem (9-A), Erv( Rx ) = R( E n X ) Hence, i f f( EA ) i s a polynomial operator which annihilates B , •A. f( E w ) w i l l annihilate AA. Hence the minimal polynomial of EA contains as a factor a polynomial which corresponds to the plenary rank equation of AA. In view of i t s lower triangular form, i t can be seen that the proper values of EA, and hence the plenary train roots of AA, are included in the set stated in the theorem. - 50 -CHAPTER 10 GENETIC ALGEBRAS " OF SYMMETRICAL INHERITANCE * Let U be a commutative non-associative algebra of order n over a f i e l d F. 1st Rx f o r x e U be the hononorphism defined as r i g h t m u l t i p l i c a t i o n by x on U, that i s , R> : U > U a j ax a £ 13. (10.1) Let m be a subset of Homp( TJ,U ) DEFINITION: The enveloping algebra of M i s the smallest sub-algebra of Homp( U,U ) containing the homomorphisms i n M. ¥e s h a l l denote i t by env ( M ). A l t e r n a t i v e l y , env ( M ) i s the algebra of a l l polynomials i n the transformations ( homomor-phisms ) i n M with c o e f f i c i e n t s i n F. DEFINITION: The enveloping algebra of the set which consists of the i d e n t i t y I i n Homp( U,U ) together with the r i g h t m u l t i p l i -cations of U, i s the transformation algebra T( U ) of U. Cl e a r l y any T i n T( U ) may be written i n the form T =o(I + f ( Rx, , Rx>. , ' • • ) ( 10.2) where ©t! 6- F and f i s a polynomial. - 51 -I f U i s a baric algebra, then T( U ) i s also a baric algebra. For i f U has weight function W, a weight function Z f o r T( U ) i s defined by Z( T ) = W( Tu ) (10.3) f o r u i s an a r b i t r a r y element i n U of unit weight. By (10.2) we have Z(T) f ( W( X )), W(xJ, ) (10.4) and Z n o n t r i v i a l , since Z( I ) = 1. Let ' T =*(l + f ( Rx, , Rx*, ' ' * ) (10.5) « / 6 F , x c t U , T 6 T ( U ) . The c h a r a c t e r i s t i c function or determinant [ . \ l - T | of T i n (10.5) has c o e f f i c i e n t s which are polynomials i n <^  and the co-ordinates of the x l , polynomials which depend on the function f» DEFINITION: A commutative baric algebra U over F with weight function W i s c a l l e d a genetic algebra i n case the c o e f f i c i e n t s of the c h a r a c t e r i s t i c function of T i n (10.5), insofar as they depend on the xi, depend only on the weight W( x ). THEOREM: (10-A) A genetic algebra U over F i s a t r a i n algebra. - 52 -Proof: Let T = R* i n (10.5) and write w( x ) =^ . Then by the definition of a genetic algebra, the characteristic function 4> ( X ) = IAI - Bxl has the form J tv-1 -VS. yC + • " • • + rv y r : 6 P (10.6) Now (10.6) factors in an algebraic closure R of F as <£( A ) = (X - X.^  )( A - A > j ) ( V ^ j ) Xi- e R. (10.7) If >T + f, X"' + ••• + tr-,A (10.8) is the rank function of U, where ^  a homogeneous polynomial of degree j in the coordinates of x then (10.8) divides A4>(A). The 'Xi in (10.7) may then be ordered so that (10.8) equals A(X-X.j )(X-Xrn) ) from which i t follows that *5 = ( - l i y 2 x u ---Avj j = 1, 2,--- , r - l (10.9) The rank equation i s xr + p,P(x)xr~' +-;-+pM [w(x)j x = 0 - 53 -with i n F. Therefore U i s a t r a i n algebra. THEOREM (10-B) A s p e c i a l t r a i n algebra U over F i s a genetic algebra, Proof: I t was shown in.theorem (7 - A ) that 3- s c a l a r s \ = 1,A*, ••• , A»y. such that the matrix of R x has the form R> = JX- * 0 0 » . . . . /JS (10.10) Then the c h a r a c t e r i s t i c function of R has the form x( x - j )( x - Xj ) (. x - Xj ) = 0 with Ail as i n (10.10) (10.11) ( -1 r t = Z XL, ' ' ' \ i - t (t = 1, 2,/- ;n) (10.12) Hence - 54 -From (10.10) we obtain f(Rx,,R,u, ) = 0 0 where JK = W ( x k ) Then T i n (10.5) has c h a r a c t e r i s t i c equation (10.13) where then |Xl - T| = | ( A - . e ^ ) l - f( R^RJU, • ... )| = [( x - oL) ->Q [( A - cL) -rQ~> [ (A-<0-#J = o ; ' . . . jaA*. ) (10.14) - TT = (x-^r-t^,( A - ^ r v - t ^ * (10.15) where ( -1 f^Jls i s a l i n e a r combination of )^ (, which i s i n turn a l i n e a r combination o f ^ i . with c o e f f i c i e n t s which are symmetric function of the A<^  Hence the c o e f f i c i e n t s of the c h a r a c t e r i s t i c function are independent of xc, that i s U i s a genetic algebra. This completes the proof. - 55 -We have thus f a r established the following r e l a t i o n : a s p e c i a l t r a i n algebra i s a genetic algebra and a genetic alge-bra i s a t r a i n algebra and a l l are baric algebras. THEOREM (10-C) Structure of genetic algebra. Let R be the kernel of the weight function W of a genetic algebra U over F. Then R i s the r a d i c a l rod U of U, and i s n i l p o t e n t . Proof: Let T i n T( U ) have the form (10 .5 ) , and write w(xi )=ji. Then the c h a r a c t e r i s t i c function jxi - T| = x + t.xr' + • ••+vk- (10.16) of T has c o e f f i c i e n t s Sft= 0 f o r any T i n T( U ) whcih may be written i n the form (10.5) with w( x t ) = 0 • i = l , 2 , ' - - (10.17) For by d e f i n i t i p n of a genetic algebra,*^ depends only on w( x c ) and T = 0 i s such a function with o (= X L =0. In t h i s case X i - T I = X = 0 f o r any T ^ T( U ) which has the form (10.15) and s a t i s f i e s (10.17), that i s T* = 0, T i s nil p o t e n t . Now l e t T be i n the enveloping algebra R of the nilsubalgebra R. Then (10.17) i s s a t i s f i e d , T i s nilpotent. Since env ( R ) i s an associative algebra consisting of nilpotent elements, env ( R . ) i s ni l p o t e n t . Thus R i s a nilpotent i d e a l of U, and i s contained i n the r a d i c a l rad U of U. On the other hand: U/R which implies rad U C R, or rad U = R. - 57 -CHAPTER 11 SOME EXAMPLES EXAMPLE 1: An example of a t r a i n algebra, Mendelian Goemetic Algebra ( D, R ) 1 Consider a single autosomal gene difference ( D, R ) and the corresponding genotypes A = DD , B = DR , C = RR (11.l) According to the Mendelian p r i n c i p l e s : DD = D , DR = D^ + 4,-R , R2" = R (11.2) These give the series of gametes produced by each type of zygote For example the second of equations ( 11.2 ) means that a detero zygote produces D and R gametes i n equal numbers, ( that i s 50 percent D and 50 percent R ). A population P can be described by the frequencies of the gametes which i t produces: Gametic representation P = s D + t R (11.3) with the nomalizing condition: 58 -s > 0 , t > 0 , s + t = l 8 , t Reals (11 .4) The algebra of the symbols D, R defined by the mul-t i p l i c a t i o n table (11.2) w i l l be c a l l e d the gametic algebra f o r  single mendelian inheritance, and referred to as Ga.. I f x e G A , then x = sB + tR some s, t £• Reals (11.5) x Grx i s interpreted as a population only i f the c o e f f i c i e n t s s a t i s f y (11.4). The p r i n c i p a l rank equation i s x z - ( s + t ) x = 0 (11.6) A population P i s represented by an element of unit weight i n the algebra G . The r a t i o s:t gives the r e l a t i v e frequencies of the gametic types which i t produces. In the case (11.6) becomes the t r a i n equation P* = P or P ( P - 1 ) = 0 (11.7) and 1 i s the unique p r i n c i p a l t r a i n root of the algebra G2.. - 59 -EXAMPLE 2: An example of a s p e c i a l t r a i n algebra. The gametic algebra G3 with m u l t i p l i c a t i o n table x z = x , y a = xz = -£x + -fy + -^z . ( 11.8, 11.9 ) zx = z , yz = £y + \ z ( 11.10, 11.11 ) xy = £x + -s-y (11.12) r e f e r s to the inheritance of a single autosomal gene difference ( x, y, z ) i n autotetraploids ( c/f Haldane, 1930 chapter 11 ) the case m=2, with x, y, z written f o r x* = x, xa_ , a * . Let Then Since a 0 = x , a, = x - y , a a = x - 2 y + z (11.13) a 0 a e = x = a 0 (11.14) a*a, = i x - i y = a, (11.15) a 0 a A = £ x - £ y +-£-z =-^aa (11.16) a, a, =-^aJ (11.17) a,a z = 0 (11,18) a^a 2 =»0 (11.19) a i"* = "i" a 2 » a i aa. = a a a i = ^ a l l powers of the i d e a l ( a,, ) are i d e a l s , Therefore, G3 i s - 60 -a s p e c i a l t r a i n algebra. It has the p r i n c i p a l t r a i n equation P ( P - 1 )( P - -p) = 0 (11.20) and plenary t r a i n equations P [ P - 1 ] [ P --^-] = 0 (11.21) where P = a e + sa, + t a z s r t 6 Reals (11.22) EXAMPLE 3t An example of a genetic algebra. R e c a l l that the gametic algebra G 2 of simple mendelian inheritance i s a t r a i n algebra. I t s m u l t i p l i c a t i o n table i s DX = D , DR = iD + 4-R , R* = R (11.23) and ( D, R ) i s i t s basis. . Now l e t u = i D. + £R (11.24) z = £D - i R (11.25) Then ( u, z ) i s a new basis of Gr* such that MX = U , U 2 = , Z 2 = 0 (11.26, 11.27, 11.28) Writing x = ^ u + ^ z , we have weight function w : x ^ W ( x ) =^ . - 61 -For the nil-subalgebra R = ( z. ), we have R A = 0, and Gj? i s a s p e c i a l t r a i n algebra. The transformation algebra T( G a ) has order 3 over F, and any element T of T( G x ) may be written i n the form T = J 1 + J R ^ + ^R £ =p(l + Rx (11.29) The c h a r a c t e r i s t i c function of T i n (11.29) i s I - T A 0 0 X r x - ( ^ + i ) o = [ A- [A- U+j/2)J Therefore the c o e f f i c i e n t s are independent of x, and are dependent only on i t s weight W( x ) =^. Thus G^ i s a genetic algebra. - 62 -BIBLIOGRAPHY Dickson, L. E, 1914 ( reprinted 1930 ), Linear Algebra. Cambridge Tract, no. 16. Etherington, I. M. H., n On Non-associative Combinations ", Proceeding Royal Society Edinburgh. Vol. 59, 1939, p.153-162. Etherington, I. M. H. * Genetic Algebras ", Proceeding Royal  Society Edingurgh. Vol. 59,1939, p.242-258. Etherington, I. M. H. " Non-associative Algebra and the Symbolism of Genetics *, Proceeding Royal Society Edinburgh B, Vol.61, 1941, p.24-42. Etherington, I, M. H., n Commutative Train Algebras of Rank 2 and 3 n , Journal of London Mathematics Society, Vol.15, 1940, p. 136-148. Vol.20, 1945, p.238. . Etherington, I. M. H., rt Special Train Algebras ", Quarterly  Journal of Mathematics, Oxford, Vol.12, 1941, p.1-8. Geiringer, H., " On Some Mathematical Problems Arising in the Development of Mendelian Genetics ".Journal of American  Statiscal Association. Vol.44, 1949, p.526-547. Geppert, H., and Koller, S., 1938, Erbmathematik. Leipzip, Gonshor, H., " Special Train Algebras Arising in Genetics ", Proceeding Edinburgh Mathematical Society (2), Vol.12,I960, p.41-53. Gonshor, H., " Special Train Algebras Arising in Genetics I ", - 65 -Prodeeding Edinburgh Mathematical Society (2). Vol.14, 1965, p.333-338. Halgane, J. B. S., n Theoretical Genetics of Autopolyploids w , Journal of Genetics. Vol. XXII, 1930, p.359-372. Hoffman, K. and Kunze, R. Linear Algebra. Prentice Hall Inc. Engle-wood C l i f f s , N. J. 1965. Holgate, P., * Genetic Algebras Associated with Polyploidy u , Proceeding Edinburgh Mathematical Society (2), Vol.15, 1966, p.1-9. Holgate, P., " Sequences of Powers in Genetic Algebras Journal  of London Mathematical Society. Vol.42. 1967, p.489-496. Jacobson, N..Lectures in Abstract Algebras. Vol.2 - Linear Algebra. D. Van Nostrand Co. Inc. Princeton, N. J. 1951-1964. Jennings, H. S., Genetics, London, 1935. Knoop, K;, Theory and Application of Infinite Series, Blackie, 1928. . Schafer, R. D., " Structure of Genetic Algebras n , American  Journal of Mathematics. Vol.71. 1949, p.121-135. SRB, A..M., Owen, R. D., Edgar, R. S., General Genetics. Second Edition 1965, W.. H. Freemen and Co. San Francisco and London. 

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