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Matric representation of Boolean algebras

Item Type text; Thesis-Reproduction (electronic)

Authors Mobley, Charles Lee, 1920-

Publisher The University of Arizona.

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MA T R I C RFPRESENTAT I ON OF ROOLEAN ALOERRAS

by

C h a r l e s L . Mob I e y

A T h e s i s

s u b m i t t e d t o t h e f a c u l t y o f t h e

D e p a r t m e n t o f M a t h e m a t i c s

i n pa r t i a I , f u I I f i I I me n t o f

t h e r e q u i r e m e n t s f o r t h e d e g r e e o f

M a s t e r o f S c i e n c e

In t h e G r a d u a t e Col l e g e

U n i v e r s i t y o f A r i z o n a

1947

A o o r o v ed / i , / f t ? D i r e c t o r o f T h e s i s

£1111\ f f ^ 7

TABLE OF CONTENTS

I n t r o d u c t i o n 8 e ® ® ® ® ® ® ® ® ® ® ® e- ® ®

C h a p t e r '

I . D e f i n i t i o n s i n t h e M a t r i c S y s t e m

i I « P r o o f t h a t a n y B o o l e a n A l g e b r a Can Be

R e p r e s e n t e d i n t h e M a t r i c S y s t e m „ c

I I I . ® D e d u c t i o n o f A l l P o s s i b l e B i n a r y Ope r a t o r s

IV® E x a m i n a t i o n o f t h e S h e f f e r ,5' t r o k e F u n c t i o n

V® On t h e E x i s t e n c e o f G r o u p s i n B o o l e a ny > *

AI 0 0 b r 3 s o o• © © 0 © ■ © 0 0 o o o o

VI® On t h e E x i s t e n c e o f Ri mgs i n B o o l e a n

A l g e b r a s ® ® ® ® ® ® ® ® ® ® ® ® ® ©

VI I® Sum m ary o f R e s u l t s a n d I m p o r t a n c e o f t h e

M a t r i c R e p r e s e n t a t i o n «

B i b I i o g r a p h y © Q 0 O 0 O C O 0 0-0 O O 0 0 0 0 d 0 0

. 5

. 8

. 16

, 20

o 26

„ 31

® 42

18

INTRODUCTION

Bool ean A l g e b r a , wh i c h i s t he A l g e b r a of L o g i c , had i t s

b e g i n n i n g s e a r l y in c i v i l i z a t i o n because i t i s b a s i c i n i t s

a p p l i c a t i o n t o r e a s o n i n g . However , Lewi s and L a n g f o r d say

t he t o I I owi n g :

w L e i b n i t z may be s a i d t o be t he f i r s t s e r i o u s s t u d e n t o f s y m b o l i c l o g i c , t hough he r e c o g n i z e d the ? A r s Magna ! o f Raymond Lul j_ and c e r t a i n o t h e r s t u d i e s as p r e c e d i n g him i n t he f i e I d . w '

"The f o u n d a t i o n s f r om wh i ch s y mo b I i c l o g i c has had c o n t i n u o u s d e v e l o p me n t were l a i d in G r e a t B r i t a i n between 1825 and 1850, The s i g n a l I y i m p o r t a n t c o n t r i b u t i o n was t h a t o f t he m a t h e m a t i c i a n George Bo o l e ; but t h i s was p r e c e d e d by , and i n p a r t r e s u l t e d f r o m , a renewed i n t e r e s t in l o g i c t he main i n s t i g a t o r s o f ct w h i c b were S i r ' W i l l i a m H a m i l t o n and Au g us t us De Morgan, ' ^

Thus we f i n d t he d e r i v a t i o n of t he name Boo l ean A l g e b r a ,

B o o l e ’ s c o n t r i b u t i o n was t h a t o f c o m p l e t i n g a m a t h e m a t i c a l

r e p r e s e n t a t i o n o f l o g i c , ^ bu t much work has been done s i n c e

t o p e r f e c t and e x t e n d t h e knowl edge of B o o l e ’ s t i m e , A few

o f t he o u t s t a n d i n g w o r k e r s in t he f i e l d were C h a r l e s S, P i e r c e

and E, Sch r°6der who each d i d much t o p^r i z c i B o o l e ’ s sys t em

f r o m abou t 1870 t o 1890, E, V, H u n t i n g t o n who i s n o t ed f o r

some o f t he f i r s t g e n e r a l l y used p o s t u l a t e s f o r Boo l ean

-'cC. I . Lewi s and C;» H L a n g f o r d , Symbo l i c Log f"c;s t fThe C e n t u r y Company, 1932.) , p , 5. .

v * * l b i d . , p. 7. •f fGzorge B o o l e , An I n v e s t i g a t i o n o f t h e Laws o f T h o u g h t ,

( L o n d o n , 1854; r e p r i n t e d , Open C o u r t P u b l i s h i n g Company,I 940) , .

- 2 -

A l g e b r a s , ' ^ Hv M» S h e f f e r who p o s t u l a t e d Boo l ean A l g e b r a s

in t e rms o t a c l a s s and one b i n a r y o p e r a t i o n , ' ^ B« A,

B e r n s t e i n ho showed t h e e x i s t e n c e , o f g r oups and a b e l i a n

g r oups in Boo l ean A l g e b r a s and l i n k e d i t more c l o s e l y to

t he co nc ep t o f ' 'Mode r n A l g e b r a " , ^ and Mo H« St one who f i r s t

p u b l i s h e d work on t he e x i s t e n c e o f r i n g s i n Boo l ean A l g e b r a s

Of c o u r s e , t h e r e were many o t h e r c o n t r i b u t o r s in t h e f i e l d ,

so many, i n f a c t , t h a t i f wou l d be a s u b j e c t i n i t s e l f to

s t u d y them and t h e i r c o n t r i b u t i o n s . However , t he men men

t i o n e d . a b o v e were t he ones who d i d i n i t i a l wor k on the t o p i c s

c o v e r e d i n t h i s p a p e r ,

- Th e p u r p o s e of t h i s paper i s t o d e f i n e ! ; j u s t i f y , and e x

p l a i n t he use o f a m a t r i c r e p r e s e n t a t i o n of f i n i t e Bool ean

A l g e b r a s , The work i n v o l v e d is not b e l i e v e d t o be o r i g i n a l ,

bu t i t i s n e i t h e r b e l i e v e d by t he a u t h o r t o have been p r e v i o u s -

’"A* E. V c Hunt i n . t on , '’ Sets o f P o s t u l a t e s f o r t h e A l g e b r a o f Log i c s' , T r a n s a c t i o n s o f t h e Ame r i c a n M a t h e m a t i c a l S o c i e t y , V, ( 1 9 0 4 ) , p p „ 2 3 8 - 3 0 9 , A l s o , ftA New Set o f I nd e p end e n t P o s t u l a t e s f o r t h e A l g e b r a of L o g i c , w i t h S p e c i a l R e f e r e n c e t o Wh i t e h e a d and R u s s e l l ' s " P r i n c i p i a M a t h e m a t i c a l , I b i d , , XXXV, ( 1 9 3 3 ) , p p , 2 7 4 - 3 0 4 , See C o r r e c t i o n s t h e r e t o , I b i d , , pp, 5 5 7 - 558, , , 97 I ,

''‘'"“'H , M, S h e t f e r , ?IA Set o f F i ve P o s t u l a t e s f o r Boolean A l g e b r a s ” , I b i d , , X I V , ( 1 9 1 3 ) , p , 48 2 ,

t B, A, B e r n s t e i n , ’’ O p e r a t i o n s w i t h Respec t t o wh i ch t he El ement s o f a Boo l ean A l g e b r a f orm" a Gr oup” , I b i d , , "XXVI ,( 1924) , pp , 17 1 - 175 , See ’’ E r r a t a ” , I b i d , , XXVI I , (1925.) p, 600, A l s o , ’’ P o s t u l a t e s f o r Boo l ean A l g e b r a I n v o l v i n g t he O p e r a t t o n o f Comp I e & t D i s j u n c t i o n , ” Anna l s o f M a t h e m a t i c s ,XXXVLL, No, 2 , ( Apr M, 1936) „

##H , M, S t o n e , " S u b s u mp t i o n of t he T h e o r y o f Boo l ean A l g e b r a s Under t he Theor y o f R i n g s , ” P r o c e e d i n g s o f t h e N a t i o n a l Academy o f S c i e n c e , XXI , ( 1 9 3 5 ) , pp , 10 3 - 105,

I y p r e s e n t e d in a u n i f i e d man n e r » As t o where c r e d i t f o r the

f i r s t m a t r ] c r e p r e s e n t a t i o n b e l o n g s , t he f a c t s are u n d e t e r

mi ned in t h e a u t h o r ' s mi nd . 8. A. B e r n s t e i n ^ h a s used t a b u l a r

sys t ems t o show p r o p e r t i e s o f Boo l ean A l g e b r a s , t h e s e t a b u l a r

sys t ems b e i n g somet i mes used in a sense p a r a l l e l t o t h e m e t r i c

r e p r e s e n t a t i o n d i s c u s s e d i n t h i s paper . " ' 1" A l s o , p o s s i b l y w i t h

ou t r e a l i z i n g i t , L . I . Di nes has a c t u a l l y used a m e t r i c r e

p r e s e n t a t i o n o f a Boo l ean A l g e b r a in h i s paper on t he e x i s t s

ence p r o o f o f She f f e r '-s p o s t u l a t e s . ^ " C r e d i t f o r g u i d a n c e in

t he work o f t h i s t h e s i s goes t o Dr . Dona l d L, Webb, D e p a r t

ment o f M a t h e m a t i c s , U n i v e r s i t y o f A r i z o n a .

Appr oach t o the p r o b l e m was made f i r s t by a c o n c e n t r a t e d

s t u d y o f Boo l ean AI geb r a in i t s usua l n o t a t i o n f r o m -’ Sy mb o l i c

L o g i c ” by C, I . Lewi s and C« H . L a n g f o r d . ^ O t h e r books and

pape r s were used , c h i e f l y ttA Sur vey of Modern A l g e b r a ” by

B. B i r k h o f f and S. M a c L a n e ^ - a n d E. V. H u n t i n g t o n ’ s papers

on p o s t u l a t e s e t s f o r Bool ean A l g e b r a s .

Nex t t he ma t r i e r e p r e s e n t a t i o n is v e r i f i e d i n t he l i g h t

o f H u n t i n g t o n ' s w o r k . I t i s them used t o deduce t he e n t i r e t y

■*"6. A, B e r n s t e i n , " C o mp l e t e Se t s of R e p r e s e n t a t i o n o f Two- E l emen t A l g e b r a s 11, Bui l e t i n o f t h e Amer i can Ma t h e ma t i c s So.c ? e t y , XXX, 1924, p. 26 . A l s o , t!On U n i t - Z e r o Boo l ean Rep r e s e n t a t i o n s o f O p e r a t i o n s and R e l a t i o n s , " I b i d . , XXXVI I I , 11932) , pp. 7 0 7 - 7 1 2 .

***L» L. D i n e s , " Co mp l e t e E x i s t e n t i a I T h e o r y o f S h e f f e r ' s P o s t u l a t e s f o r Boo l ean A l g e b r a s " , I b i d . , XXI , ( 1 9 1 4 ) ,pp. 183- 188 . '

#7?'Lewis and L a n g f o r d , S y m b o l i c L o g i c ."SoFg . B i r k h o f f and S. MacLane, A Sur vey o f Modern A l g e b r a .

( The M a c m i l l a n Company, . 1944) .

c=» Zj, «*»

o f b i n a r y o p e r a t o r s p o s s i b l e in any Boo l ean A l g e b r a * A f t e r

w a r d s , S h e f f e r ^ s S t r o k e F u n c t i o n i s exami ned f r o m t he ms t r i e

v i e w p o i n t * F i n a l l y , t he rea t r i 'c r e p r e s e n t a t i o n i s used t o

d e t e r m i n e a l l o f t he o p e r a t i o n s w i t h r e s p e c t t o w h i c h Bool ean

A l g e b r a s f o r m R i ngs * The work done on t h e s e t o p i c s f o r m the

body o f t h e t h e s i s wh i c h i s c o n t a i n e d i n Ch a p t e r s I t h r o u g h V I *

Cha p t e r V I I i s a summary o f r e s u l t s and d i s c u s s i o n o f t h e f a c t

t h a t t he m a t r i c met hod i s a f i n i t e met hod o f e x a m i n a t i o n o f

a f i n i t e Boo l ean A l g e b r a w i t h r e s p e c t t o any p o s t u l a t e set

end w i l l f u r t h e r m o r e g i v e a l l and o n l y a l l t he o p e r a t o r s wh i ch

s a t i s f y any s e t o f c o n d i t i o n s *

CHAPTER I

D e f i n i t i o n s i n t h e Ma t r i e S y s t e m

I . 1 Assume a m a t r i x o f one r ow a n d m c o l u m n s , such

t h a t i n g e n e r a l t h e t e r ms a r e o f t h e f o r m:

( ^ j , & rp t 9 3 » > a m) c 0 , I $ rn — I , 2 , 3 , 4 , , *

( b | , f b g , • • • , ) bj r 0 , I

s .

( n | , n^ , p3 , ' • • , nm) n .= 0 , I

1.2 I t can be r e a d i l y seen t h a t s i n c e a j , b • , or any

e l e me n t o f any m a t r i x can assume o n l y t he v a l u e s 0 or I ,

t h e r e w i l l be two p o s s i b i l i t i e s f o r a | , two p o s s i b i l i t i e s f o r

a g , e t c . T h e r e f o r e t h e r e w i l l be 2m p o s s i b l e d i s t i n c t m a t r i c e

Th i s i s a d i r e c t a p p l i c a t i o n o f t he s o - c a l l e d Fundamenta l

P r i n c i p l e o f C h o i c e :

I f one t h i n g can be done i n any one o f a ways : and i f a f t e r i t has been done i n some one o f t hese a ways , a second t h i n g can be done in any one o f ^ b ways ; t he two can be done t o g e t h e r , in t h a t o r d e r in a b w a y s • *!r

I t i s o f g r e a t i mp o r t a n c e in t h e r e p r e s e n t a t i o n of f i n i t e

Boo l ean A l g e b r a s t h a t t h e r e are e x a c t l y 2m d i s t i n c t m a t r i c e s ,

1.21 Let us c a l l t he c l a s s o f a l l t hese m a t r i c e s K.

1.3 We w i l l assume the f o l l o w i n g p r o p e r t i e s o f t he s i g n a l

' " L . M. Reagan, E. R. C t t , and D. T. S i g l e y , Co I l ege A l g e b r a , ( F a r r a r & R i n e h a r t , I n c . , 1940) , p. 82.

- 5 -

- 6 -

A. I f a i s i n K , f h e n a r a . ( R e f l e x i v e l aw h o l d s ) .

B. I f a - b , t h e n b - a . ( S y m m e t r i c l aw h o l d s ) .

C. I f a r b , a n d b - c , t h e n a = c . ( T r a n s i t i v e l aw

h o l d s ) .

1 . 3 1 F u r t h e r m o r e , t wo m a t r i c e s w i l l be s a i d t o be e q u a l i f

and o n l y i f a l l o f t h e e l e m e n t s o f o n e m a t r i x a r e e q u a l r e s

p e c t i v e l y ' t o t h e e l e m e n t s i n t h e c o r r e s o o n d i n g c o l u mn s o f t h e

o t h e r m a t r i x .

T h i s d e f i n i t i o n l e a d s t o a m e t h o d o f p r o o f c o n s i s t i n g o f

e x h a u s t i n g t h e p o s s i b i l i t i e s o f t h e r e l a t i o n s h i p o f e l e m e n t s

i n c o r r e s p o n d i n g c o l u mn s i n g e n e r a l a n d t hus s h o w i n g t he r e

l a t i o n s h i p o f t h e m a t r i c e s i n q u e s t i o n .

1 . 4 L e t + a n d X be b i n a r y o p e r a t o r s a n d 1 be a u n a r y

o p e r a t o r h a v i n g o r o o e r t i e s as i n t h e t a b l e b e l o w :

a i b i a , + b. a j X b . V

1 1 1 1 01 0 1 0 00 1 1 0 i0 0 0 0 i

F u r t h e r m o r e , 1e t :

1 . 5 1 (a a a n , " , a ) 4* ( b , bn .r 2 ' 3 m 1 2 m

( a I 4 b r a 2 4 b2 ’ a 3 4 b 3 ’ , a m4 b m]

We w i l l c a l l t h e r e s u l t o f t h e o p e r a t o r + t h e u n i o n o f

t h e t wo m a t r i c e s ( o r e l e m e n t s ) u n d e r g o i n g t h e o p e r a t i o n .

A I s o , I e t :

, a i

- ( a | X b | , a 2 X b g , a ^ X b ^ , * » 3 m X b m)

. 5 2 (a , a , a , ' , a ) X ( b , b , b , • • • , b )I z d m I A d m

m

- 7 -

W? w i l l c a l l t he r e s u l t o f t he o p e r a t o r X t he i n t e r

s e c t i o n o f t he two m a t r i c e s ( o r e l e me n t s } u n d e r g o i n g t he

ope r a t i o n .

And, l e t :

1.53 (a , a , a , • • • , a ) ' = (a' a' a' • • • , a ' }1 2 3 m 1 2 3 m

and c a l l t he r e s u l t o f t h i s u n a r y o p e r a t i o n T t he compl ement

of the m a t r i x ( o r e l e me n t ) u n d e r g o i n g the o p e r a t i o n .

CHAPTER I I

P r o o f T h a f Any B o o l e a n A l g e b r a Can Be R e p r e s e n t e d I n t h e M a t r l c S y s t e m

2 . 0 To p r o c e e d w i t h an e x a m i n a t i o n of t h i s p r o b l e m , i t

i s M r s t n e c e s s a r y t o show t h a t any f i n i t e Boo l ean A l g e b r a can

be e x p r e s s e d in a m e t r i c r e p r e s e n t a t i o n . Th i s i s a c c o m p l i s h e d

by snowi ng two t h i n g s , name l y ; t h a t t he a l g e b r a as d e f i n e d

above does s a t i s f y t he p o s t u l a t e s f o r a Bool ean A l g e b r a , and

t h a t a Boo l ean A l g e b r a c o n s i s t i n g o f 2 m terms i s e x p r e s s i b l e

i n e s s e n t i a l l y o n l y one way.

Our d e f i n i t i o n h a v i n g been made i n t erms of t he o p e r a t i o n s

+ and X, and t h e n e g a t i v e ( M , we f i n d i t c o n v e n i e n t t o s e l e c t

f r om the many i n d e p e n d e n t se t s o f p o s t u l a t e s f o r Bool ean

A l g e b r a s wh i ch ar e a v a i l a b l e t o d a y the " f i r s t s e t " g i v e n by

E. V. H u n t i n g t o n . These p o s t u l a t e s are q u o t ed here ( e x c e p t

f o r a s l i g h t a l t e r a t i o n o f t he symbol s u s e d ' ' f o r t h e i n f o r m a

t i o n of t he r e a d e r :

l a . a + b i s in t he c l a s s whenever a and b are in t hec l a s s .

l b . a X b i s i n t he c l a s s whenever a and b are in t hec I a s s .

I I a . There i s an e l emen t 0 such t h a t a 4- 0 - a f o r e v e r ye l e me n t a .

M b . There i s an e l emen t I such t h a t a X I - a f o r e v e r ye l emen t a .

‘"‘ E . V. H u n t i n g t o n , " S e t s of P o s t u l a t e s f o r t h e A l g e b r a o f L o g i c " , T r a n s . Ar ne r . M a t h . So c . , V , ( 1 9 0 4 ) , p . 2 9 2 .

‘' ' " ' ' ' Hu n t i n g t o n ’ s (D, <9, V , A , a n d a a r e h e r e + , X, I , 0 , a nd a 1 r e s p e c t i v e l y .

- 8 —

- 9 -

I l i a . a + b = b + a whenever a, b, a + b , and a X bar e In t he c l a s s .

I I I b . a X b r b X a whenever a , b , a X b , and b X aa r e i n t he c l a s s .

IVa. a 4 ( b X c ) = (a 4- b ) X ( a + c ) whenever a, b, c ,a + b, a + c , b X c , a 4- ( b X c ) , and (a + b) X(a + c) a r e in t he c l a s s .

i Vb. a X ( b + c ) = (a X b ) + (a X c ) whenever a, b, c ,a X b , a X c , b + c , a X ( b + c i , and (a X b) +( a X c) are in t he c l a s s .

V. I t t he e l e me n t s 0 and I i n p o s t u l a t e s I I a and M b e x i s t and ar e u n i q u e , t hen t o r e v e r y e l e me n t at h e r e i s an e l emen t a * such t h a t a + a 1 = I anda X a 1 - 0 .

V I . There are a t l e a s t two e l e m e n t s , x and y , i n t he c l a s s such t h a t x £ V•

We w i l l now p r o c e e d t o s t a t e each o t t hese p o s t u l a t e s as

a t heo r em and p r o v e them by use ot t he t o r e g o i n g de t i n i t i on s .

2.11 Theorem: I t two m a t r i c e s a r e i n K , t hen t h e i r u n i o n

is in K. P r o o t : Let (a ,a , , a ) and ( b , b . " , b ) be1 2 m 1 2 m

m a t r i c e s in K. The u n i o n o t t h e s e m a t r i c e s i s :

(d | + b | , ag + b g , ’ ** , am + bm) ( 1 . 5 1 )

B u t , e x a m i n i n g in g e n e r a l t he e l e m e n t s , we t i n d

a i 4- b; = 0 , I ( 1 . 4 )

Hence t he u n i o n is i n K. ( I . I and 1.21)

2 . 1 2 Theorem: I t two m a t r i c e s a r e in K , t hen t h e i r

i n t e r s e c t i o n is in K• P r o o t : Let (a , a , " , a ) andI 2 m

( b , , b _ , * * • , b ) be m a t r i c e s in K. The i n t e r s e c t i o n ot t heseI 2 m

m a t r i c e s i s : ( a | X b j , a^ X b ^ , • • • , am )

Exa mi n i n g i n g e n e r a l the e l e me n t s we t i n d ; a . X b. = 0 , I ( 1 . 4 )

Hence t he i n t e r s e c t i o n i s in K . ( I . I and 1.21)

— 10 —

2.91 Theorem: I f ( a | , a ^ , • • • , a^ ) be a m a t r i x in K ,

t hen the un i o n o f (a , a , * , a ) and (0 , 0 , , 0)I 2 m

1 s ( a , a , • • • , a ) .I 2 m

P r o o f : ( a , a , , a ) + ( 0 , 0 , • • • , 0 }» I 2 m

= ( a | + 0 , a + 0 , , a + 0) ( 1 . 5 ! )

- ( a , a , • • • , a ) ( 1 . 4 )I 2 m

2 . 2 2 Theorem: I f ( a j , a ^ , • • • , a ) be a m a t r i x in K,

t hen t he i n t e r s e c t i o n of ( a j , a^ > * * * , a ^ ) and ( I , I , * * * , I )

i s (a , a , • • • , a ) .I <£ m

' P r o o f : (a , a , • • • , a ) X ( 1 , 1, , I )I °2 m

- (a , X I , a X I , . , a X I ) ( I . 52)I 2 m

= ( a , a , * * * , a ) ( 1 . 4 )I 2 m

2.31 Theorem: I f ( a | , a ^ , * " , a^) and ( b j , b ^ , * * * , b^ )

be m a t r i c e s in K, t hen ( a . a , " , a ) + (b , b , , b )1 2 m 1 2 m

2 l b | ’ b2 ’ * ” ’ bm' + l a | ’ a2 ’ ' am'

P r o o f : (a , a ^ , ,a ) + ( b , , b „ , ' , b )1 2 m 1 2 m

= (a + b , a0 + b . , • # * .a + b ) ( 1 . 5 1 )1 1 2 2 m m

and ( b | , b ^ , " ' , b ^ ) + ( a | , a , • • • , a ^ )

- ( b . + a, , b0 + a0 , • • • , b + a ) ( 1 . 5 1 )i \ z c. m mE x h a u s t i n g a l l p o s s i b i l i t i e s , we f i n d t h a t :

T h i s p r o v e s t h e t h e o r e m : ( 1 . 3 1 )

2 . 3 2 T h e o r e m : I t ( a ( , * " , a ^ ) and ( b j , b9 , • • • , b

be m a t r i c e s i n K , t h e n ( a f , a ^ , " , a ^ ) X ( b j , b ^ , • • • , b ^ )

r ( b | , b2> • • • , b _ ) X ( a , , a 0 , • • • , a m)m m

P r o o f : ( a j , a ^ , • • • , a ^ ) X ( b j , b ^ , • • • , b ^ )

Z ( a | X b | , a <2 X b ^ , , a m X b m) ( 1 . 5 2 )

a n d ( b , , b 2 , • • • , b m) X ( a , , a ^ , • • • , a ^ )

* ( b j X a | , b^ X a ^ , • b m X a m> " • 5 2 >

E x h a u s t i n g a l l p o s s i b i l i t i e s we ha v e t h e f o l l o w i n g t a b l e

a i b; a | X b ; b j X a j

1 1 1 01 0 0 00 1 0 00 0 0 0 ( 1 . 4 )

In v i e w o f t h e d e f i n i t i o n ( 1 . 3 1 ) t h i s p r o v e s t h e t h e o r e m

2 . 4 1 T h e o r e m : I f ( a , a . • • • , a ) , ( b , b , • • • , b )1 2 m I 2 m

P r o o f :

( 4 , , a2> • • • - b m' X , e l * C2 ’ • ” ’ Cm' ]• am’ + [ , b | ’ b2 ’ "

= ( « , , a 2 , • • • , a m) + [ ( b, X c , , X c 2 , • • • , b m X c ^ l ] ( 1 . 5 2 )

- [ a , + l b , X c , ) . a , + ( b2 X c2 ) , • • • ,*m 4- ( b ^ X c ^ ) ] ( 1 . 5 1)

a n d [ ( a , , a2 , , a ^ ) 4- ( b , , b g . • • • , b ^ ) J

X [ l a , . a 2 , • • • . a j + ( c | f e 2> • • • , c j ]

. - [ a , + b , . a 2 + b 2 , • " . a m + b j ]

x [ 3 | + c , , a 2 + c 2 , * • ' , a m + c j ] ( 1 . 5 1 )

= [ j a j + b | ) X ( a j + c | ) » ( a ^

X ( a 0 4- c n ) , * e * , ( a + b ) X ( a + c )12 2 m m m m J

( I . 5 2 )

E x h a u s t i n g a l l p o s s i b i l i t i e s t o r t he g e n e r a l t e r ms

we f i n d :

( 1 . 4 )

2 . 4 2 T h e o r e m: I f ( a ^ , a , • • • , a ^ ) , ( b ^ , b ^ , • • • , b^ )

- I 1 —

P r o o f :, r o o t ; i-( a , , a , • • • , a m. X [ l b , , b2 , • • • , b m) + ( c , , c 2 , • • • , e j ]

= (a , a , a ) X ( b + c , b + c , ' * • , b + c ) ( 1 . 5 1 )1 2 m I 1 2 2 m m

= [», X ' b , + c | > - a 9 X , b2 + c 2 ’ ’ ’ a m X , b m + " • 521

a n d f l a , . a o , " , a ) X ( b , , bn, " " , b )1L I 2 m 1 2 m J

4 [ a , . a 2 . , a m) X I c , . c , . • • • . c ^ , ]

c ( a X b , a X b ^ , , a X b )1 1 2 2 m m

+ l a , X c , . a 2 X e 2 , • • • , a m X c j

= H a | X b | ) 4 ( a , X c , ) , ( a , X b^ l

• , ( a X b ) + ( a X c )1m m m m J+ { a 2 X c 2 ] >

E x h a u s t i n g a l l p o s s i b i l i t i e s f o r t h e g e n e r a l t e r ms

we f i n d :

d i b i c i a . X ( b . + c . ) ( a . X b . ) 4- ( a . X c . )

11

11

I0

11

11

1 0 1 1 11 0 0 0 00 1 1 0 00 1 0 0 00 0 1 0 00 0 0 0 0 (( 1 . 4 )

T h i s p r o v e s t h e t h e o r e m . ( 1 . 3 1 )

2 . 5 1 T h e o r e m: T h e r e e x i s t s a m a t r i x , name I v ( I , I , " * , l )

w h i c h is t h e u n i o n o f a n y m a t r i x i n K a n d t h e c o m p l e m e n t o f

t h a t m a t r i x . T h i s m a t r i x ( I , I , * ‘ * , 1 ) we w i l l c a l l t he

u n i v e r s e .

— 14 —

P r o o f : Assume t h e m a t r i x (a , a , * , a ) i n K .I 2 m

' V V ’ V + ' V V ’ V

- ( a . a , • • • , a ) + (a , a ’ , , ! ) ( 1 . 5 3 )1 2 m I 2 m

= ( a I + <3|» 8^ + a p , * * * , a m 4- a m) ( 1 . 5 1 )

5 ( 1 , I , • • • , I ) ( 1 . 4 )

9 . 5 2 T h e o r e m: T h e r e e x i s t s a m a t r i x , n a m e l y ( 0 , 0 , " " , 0 ) ,

w h i c h I s f h e i n t e r s e c t i o n o f any m a t r i x i n K and t h e c o m p l e

ment o f t h a t m a t r i x . We w i l l c a l l t h e m a t r i x ( 0 , 0 , * " * , 0 )

z e r o .

P r o o f : Assume t h e m a t r i x ( a ^ , a ^ , * * * , a ^ ) i n K .

( a j , d<2 * * * * * 3 ^ ) X ( a | , d t y , * * * , a ^ ) '

- ( a , a , • • • , a ) X ( a * a ’ • • • , a ' ) ( | . 5 3 )1 2 m 1 2 m

= ( a . X a ! , a 0 X a ' • * • , a X a ' ) ( I . 5 2 )i I z z m m

= ( 0 , 0 , • • • , 0 ) ( 1 . 4 )

2 . 6 1 T h e o r e m: T h e r e a r e a t l e a s t t wo m a t r i c e s i n K .

P r o o f : T h e r e a r e e x a c t l y 2 m m a t r i c e s i n K w h e r e m - 1 , 2 , 3 , . . .

( 1 . 2 1 )

The l a s t n umb e r of m a t r i c e s w i l l o c c u r when m = I , g i v i n g a

t o t a l o f ( 2 ) ^ = 2 m a t r i c e s i n K ,

2 . 7 Thus we h a v e shown t h a t our m a t r i c e s s a t i s f y t h e

o o s t u l a t e s o f a f i n i t e B o o l e a n A l g e b r a , s i n c e e a c h o f t h e s e

p o s t u l a t e s c a n be d e r i v e d f r o m t h e d e f i n i t i o n s o f t he m a t r i c

s y s t e m .

2 . 8 The f o l l o w i n g t h e o r e m has been s t a t e d and p r o v e d by

- 15 -

E . V. H u n t i n g t o n :

Any c l a s s t h e number o f wh o s e e l e m e n t s i s a p o we r o t 2 , say 2 m, c^n be made i n t o a l o g i c a l f i e l d by o r o p e r l y d e f i n i n g + , X , a n d a n d t h i s i n e s s e n t i a l l y o n l y o n e w a y . ( No t a - . t i on m o d i f i e d as b e f o r e , )

2 . 9 In v i e w o f t h e p r e c e e d i n g t h e o r e m we c a n now s t a t e

t h a t any f i n i t e B o o l e a n A l g e b r a can be e x o r e s s e d i n t h e m a t r i c

r e p r e s e n t a t i o n as d e f i n e d a b o v e .

*'s-@ is H u n t i n g t o n f s r e l a t i o n o f i n c l u s i o n b u t is no t u s e d i n t h i s p a p e r .

*'5"55'E . V . H u n t i n g t o n , " S e t s o f P o s t u l a t e s f o r t h e A l g e b r a of L o g i c " , T r a n s . Ar ne r . M a t h . S o c . , V , ( 1 9 0 4 ) , p . 3 0 9 .

CHAPTER I I I

D e d u c t i o n o t A l l P o s s i b l e B i n a r y O p e r a t o r s

3 . 0 The wo r k t o f o l l o w i n t h e n e x t f ew p a r a g r a p h s has

been p r e v i o u s l y done i n a l m o s t t h e same ma n n e r by B . A. B e r n

s t e i n . ' H o w e v e r , B e r n s t e i n has n o t , t o t h e k n o w l e d g e o f t h e

a u t h o r , made use o f t h e i n f o r m a t i o n i n t h e f a s h i o n o f t he

r e m a i n d e r o f t h e p a p e r .

3 . 1 F i r s t , i t is t o be n o t i c e d t h a t i n c o m b i n a t i o n o f

t wo e l e m e n t s , e a c h of w h i c h can h a v e e i t h e r o f t wo v a l u e s ,

t h e r e w i l l be t h e f o l l o w i n g p o s s i b i l i t i e s u n d e r a g e n e r a l

b i n a r y o p e r a t o r 0 :

a b a 0 b

1 1 0 , 11 0 0 , 10 1 0 , 10 0 0 , 1

T h i s r e p r e s e n t s f o u r i n d e p e n d e n t c h o i c e s o f e i t h e r o f

t wo v a l u e s g i v i n g , by t h e F u n d a m e n t a l P r i n c i p l e o f C h o i c e ,

a t o t a l o f 16 p o s s i b l e d i f f e r e n t c o m b i n a t i o n s . Thus t h e r e

a r e 16 d i f f e r e n t b i n a r y o p e r a t o r s i n any B o o l e a n A l g e b r a

( a l t h o u g h some o f t h e s e w i l l be s e e n t o be t r i v i a l ) .

3 . 9 We w i l l now p r o c e e d t o b u i l d up a c o m o l e t e t a b l e o f

'^B. A. B e r n s t e i n , " C o m p l e t e S e t s of R e p r e s e n t a t i o n o f T w o - E l e m e n t A l g e b r a s " , B u l l e t i n o f t h e A me r . M a t h . S o c . , XXX, ( 1 0 9 4 ) , p . 2 6 .

■"”« 'R e a g a n , O t t , a n d S i g l e y , C o l l e g e A l g e b r a . ( F a r r a r & R i n e h a r t , I n c . , 1 9 4 0 ) , p . 8 2 .

- 16 -

- 1 7 -

t h e s e o p e r a t o r s ( S e e T a b l e I ) .

3 . 2 1 F i r s t , we h a v e g i v e n i n 1 . 4 t he b i n a r y o p e r a t o r s

+ a nd X w h i c h c a n b e e n t e r e d i mmed i a t e I v .

3 . 2 9 N e x t , we e v a l u a t e a * X b , “ a X b 1 , and a f X b 1 . I f

i s t o be n o t i c e d t h a t each o f t he co l umns under t h e s e o o e r a t o r s

c o n t a i n s one I and t h r e e C ’ s, each I b e i n g in a d i f f e r e n t

p o s i t i o n . From 1 . 4 we i m m e d i a t e l y see t h a t we can f o r m a l l o f

t he n e c e s s a r y co l umns by c o m b i n i n g any t wo , t h r e e , or f o u r o f

t hese o p e r a t o r s a X b, a* X b, a X b T, a 1 X b 1 under t he

o p e r a t o r 4-. However , e v a l u a t i o n o f a 1 4 b, a + b ! , and

a 1 + b 1, a 1, b 1, a, and b en ab l e us t o w r i t e our o p e r a t o r s

more c o m p a c t l y , and a l s o shows i m m e d i a t e l y t he e q u i v a l e n c e o f

c e r t a i n f orms of our b i n a r y o p e r a t o r s . T h i s , in e f f e c t , p r o v e s

many o f t he t heorems o r d i n a r i l y deduced in oa pe r s p r e s e n t i n g

p o s t u l a t e s f o r Boo l ean A l g e b r a s . We now have r e m a i n i n g the

o p e r a t o r ( a 1 X b) 4- (a X b M wh i ch we w i l l de n o t e by a V b

and t h e o p e r a t o r ( a X b ) 4- ( a T X b M w h i c h we w i l l d e n o t e by

a @ b • The o p e r a t o r s g i v i n g t h e u n i v e r s e and t h e z e r o f i l l

t h e r e m a i n i n g c o l u m n s .

3 . 9 4 The s o - c a l l e d " c o m p l e t e e x p a n s i o n o f I " g i ven as

I ' ( a X b) 4- ( a * X b) 4- ( a X b M 4- ( a * X b M

i s r e a d i l y seen f r o m t h e t a b l e . I n f a c t , t h e s y m m e t r i e s a nd

'“'H e nee f o r t h t h e s u b s c r i p t s a r e o m i t t e d , b u t i t i s t o be r e me mb e r e d t h a t we a r e s p e a k i n g o f a g e n e r a l s e t o f e l e m e n t s i n c o r r e s p o n d i n g r ows o f t h e m a t r i c e s i n q u e s t i o n ,

- Hr Lewi s a n d L a n g f o r d , S y m b o l i c L o g i c , ( T h e C e n t u r y Co mp a n y , ( 1 9 3 2 ) , p . 3 7 .

— I 8 —

d u a l i t i e s a p p a r e n t i n t h i s t a b u l a r r e p r e s e n t a t i o n o t t h e o p e r a

t o r s seem t o be s u g g e s t i v e o t many o f t h e t r u t h s ot B o o l e a n

A l g e b r a s .

3 . 2 3 I t is now s e e n t h a t t h e r e a r e o n l y t e n t r u e b i n a r y

o p e r a t o r s p o s s i b l e i n any B o o l e a n A l g e b r a . The o p e r a t o r s

w h i c h can be e x p r e s s e d as a , b f , b , b 1 , z e r o a n d t h e u n i v e r s e

d e p e n d on o n l y one e l e m e n t a n d do n o t v a r y a c c o r d i n g t o t h e

v a l u e o t a n y s e c o n d e l e m e n t . T h u s , t h e r e i s no f u r t h e r need

t o e x a mi n e t he m as b i n a r y o p e r a t o r s .

- 1 9 -

TABLE IA l l P o s s i b l e B i n a r y O p e r a t o r s

a b a + b a + b» a ’ 4. b a ’ + b ' 1

1 1 1 1 1 0 11 0 1 1 0 1 10 1 1 0 1 1 10 0 0 1 1 1 1

a b a X b a X b» a ' X b a» X b» 0

1 1 1 0 0 0 01 0 0 1 0 0 00 1 0 0 1 0 00 0 0 0 0 1 0

a b(a X b) + ( a X b» ) - a X ( b + b f )

- a

(a » X b) + ( a X b) — ( a * 4* a ) X b

= b

11

10

11

10

0 1 0 10 0 0 0

a b( a * X b) 4- ( a ' X b M

= a » X ( b 4- b* )= a T

(a X b M 4- ( a * X b M = (a 4- a M X b '

= b T

1 1 0 01 0 0 10 1 1 00 0 1 1

CHAPTER I V

E x a m i n a t i o n o f t h e S b e t t e r S t r o k e F u n c t i o n

4 . 0 H. M. S b e t t e r has shown t h a t a c o m p l e t e s e t o f

i n d e p e n d e n t p o s t u l a t e s f o r a n y B o o l e a n A l g e b r a can be s e t up

u s i n g a s i n g l e b i n a r y o p e r a t i o n w h i c h he c a l l s t h e " s t r o k e

f u n c t i o n " . H i s p o s t u l a t e s a r e q u o t e d b e l o w f o r t h e b e n e f i t

o f t h e r e a d e r : ’" y

We a s s u me :I . A c I a s s K.

I I . A b i n a r y K - r u l e o f c o m b i n a t i o n | ,I I I . The f o l l o w i n g p r o p e r t i e s o f K . and ( :

1. T h e r e a r e a t l e a s t t wo d i s t i n c t K - e l e m e n t s .2 . W h e n e v e r a and b a r e K - e l e m e n t s , a | b i s a

K - e I ernen t .De f . a ’ ~ a |a

3 . W h e n e v e r a and t h e i n d i c a t e d c o m b i n a t i o n s o f a a r e K - e l e m e n t s ,

( a M 1 = a4 . W h e n e v e r a , b , and t h e i n d i c a t e d b o m b i n a t i o n s

o f a and b a r e K - e l e m e n t s ,a | ( b | b ’ ) = a 1

5 . W h e n e v e r a , b , c a n d t h e i n d i c a t e d c o m b i n a t i o n o f a , b , a n d c a r e K - e l e m e n t s ,

( a | ( b |c 1 ) t r ( b ’ | a ) | ( c T | a )

4 . 1 To b e g i n t h e s t u d y o f t h i s s t r o k e f u n c t i o n we know

a t l e a s t t h a t , s i n c e i t i s a b i n a r y o p e r a t o r , i t i s one or

mor e o f t h e o p e r a t o r s l i s t e d i n T a b l e I . We mu s t i s o l a t e i t

by a s t u d y o f t he p r o p e r t i e s r e q u i r e d by t h e p o s t u l a t e s l i s t e d

a b o v e .

* H . M. S h e f f e r , "A Se t o f F i v e P o s t u l a t e s f o r B o o l e a n A l g e b r a s " , T r a n s . Ar ne r . M a t h . S o c . , X I V , ( 1 9 1 3 ) p . 4 8 2 .

- 20 -

- 21 -

4. 11 The d e f i n i t i o n under P o s t u l a t e 2 r e q u i r e s a * = a |a •

(At t h i s s t age we do not have any r eason t o b e l i e v e a 1 i s t he

compl ement o f a s i n c e i t i s me r e l y a symbol emp l oyed by

S h e f f e r . ) Assume f o r a r gument t h a t a r | and a 1 =■ I , We wou l d

t hen have t h a t I j I = I , Th i s wou l d mean t h a t i n P o s t u l a t e s 4

and 5 i f a - I , b - I , c = I , we wou l d never get any 0 at any

oI a c e i n a n y m a t r i x . P o s t u l a t e 4 w o u l d r e a d I | ( I | I ) ~ I

and P o s t u l a t e 5 w o u l d r e a d ( I j ( I | I ) ) f - ( I | I ) | ( I | I )

wh e n c e I - I i n b o t h c a s e s . T h i s means t h a t t h e a l g e b r a has

b r o k e n down s i n c e o ne mu s t be a b l e t o g e t a 0 a t a n y p l a c e i n

a m a t r i x i f t h e a l g e b r a i s t o s u c c e e d . O t h e r w i s e t h e r e i s no

means o f o b t a i n i n g t h e c o m p l e m e n t o f a n y e l e m e n t . The r e m a i n

i n g p o s s i b i l i t y i s t h a t l J l - 0 .

4 . 1 2 S i m i l a r r e a s o n i n g , s u b s t i t u t i n g 0 f or I i n t he above

d i s c u s s i o n , r e q u i r e s t h a t o | o - I . Th i s a s s u r e s t h a t a 1 i s

t he compl ement o f a.

s ume t h e f o l l o w i n g

We s e e a t o n c e t h a t t h i s me a n s t h a t a | b i s t h e o p e r a t o r

a T i n T a b l e I . As has b e e n p o i n t e d o u t , t h i s i s n o t a t r u e

b i n a r y o p e r a t o r s i n c e i t d o e s n o t d e p e n d on b u t o n e v a r i a b l e .

T h u s we c a n c a s t o u t t h i s p o s s i b l e t a b l e . I f o n e c a r e s t o

t a k e t h e t r o u b l e he w i l l f i n d t h a t t h e t a b l e d o e s n o t s a t i s f y

P o s t u l a t e 5 . H o w e v e r , o n e i n e o n s i s t e n c v i s e n o u g h t o t h r o w

out any t a b l e , and t he c o mp u t a t i o n t o r P o s t u l a t e 5 wo u l d be

u n n e c e s s a r y wo r k .

4 . 1 4 Nex t assume t h a t t he f o l l o w i n g h o l d s :

a b a | b

1 1 01 0 10 1 00 0 1

Ex a mi n a t i o n o f Tab l e I shows t h a t a j b i s b f i n t h i s case

and aga i n t he p o s s i b i l i t y o f t h i s t a b l e must be c a s t out f o r

t he same r eason as i n p a r a g r a p h 4 . 1 3 .

4 . 1 5 Now assume t h a t t he f o l l o w i n g t a b l e h o l d s :

a b a|b

1 1 01 0 00 1 00 0 1

Th i s we f i n d i s t he o p e r a t o r l i s t e d as a 1 X b f i n Tabl e I

We now t e s t P o s t u l a t e 4 i n t a b l e f or m:

a b b | b ! a | ( b | b M a 1

1 1 0 0 01 0 0 0 00 1 0 1 10 0 0 1 1

P o s t u l a t e 4 i s t hus s a t i s f i e d s i nc e a | ( b | b M - a 1 f o r a l l

a and b .

Nex t we check P o s t u l a t e 5 by t he f o l l o w i n g t a b l e s :

- 23 -

00 0 0

Thus P o s t u l a t e 5 i s s a t i s f i e d s i n c e ( a | ( b | c ) ) ! =

( b * |a ) | ( c * | a ) t o r a l l v a l u e s o f a , b , and c . Thus t h i s

o p e r a t o r s a t i s f i e s a l l o f S h e t f e r 1s p o s t u l a t e s .

r e ma i n s t o as s ume:

Fr om T a b l e I we see t h a t t h i s is t he o p e r a t o r a 1 + b * .

T h i s g i v e s t he f o l l o w i n g t a b l e f o r P o s t u l a t e 4:

Thus t h e r e s u l t s a r e c o n s i s t e n t s i n c e a ( b b *) c a ' f o r

a i l a and b .

- 24 -

We p r o c e e d t o c h e c k P o s t u l a t e 5 by t h e t a b l e s :

a b c b 1 | a c » | a ( b 1 | a ) | ( c ' I a )

1 1 1 1 1 01 1 0 1 0 11 0 1 0 1 11 0 0 0 0 10 1 1 1 1 00 1 0 1 1 00 0 1 1 1 00 0 0 1 1 0

T h u s t h i s o p e r a t o r a l s o s a t i s f i e s a l l o f S h e f f e r ’ s

p o s t u l a t e s .

4 . 1 7 As a m a t t e r o f f a c t , t h e t w o o p e r a t o r s f o u n d t o

s a t i s f y t h e p o s t u l a t e s a r e d u a l s . T h i s i s n o t i c e a b l e i n t h e

s t a n d a r d n o t a t i o n as w e l l as i n t h e s y m m e t r i c p r o p e r t i e s

w h i c h a p p e a r i n T a b l e I . I f t h e s t r o k e f u n c t i o n is t o be

made u n i q u e i t r e q u i r e s t h a t t h e z e r o o r t h e u n i v e r s e be

d e f i n e d i n t e r m s o f t h a t o p e r a t o r . ' *

4 . 1 8 I n h i s p a p e r S h a t t e r s t a t e s t h e same r e s u l t s as

d e t e r m i n e d a b o v e . T h e w o r k d o n e i n t h i s c h a p t e r r e a l I y s hows

o n l y t wo t h i n g s . F i r s t , i t s h o ws t h e m e t h o d o f e x a m i n a t i o n

o f p o s t u l a t e s i n t h e m a t r i c r e p r e s e n t a t i o n , a n d s e c o n d , i t

x-Sh e f f e r g i v e s ( a | a 1 ) ~ u w h e r e u i s o u r I . I b i d . , p . 4 8 5 .

- 25 -

shows some of t h e p r o p e r t i e s o f t he s t r o k e f u n c t i o n in t he

m a f r i c r e p r e s e n t a t i o n *

CHAPTER V

On t he E x i s t e n c e ot Gr oups i n Boo l ean A l g e b r a s

5 . 0 In t h i s s e c t i o n we w i l l p r o c e e d t o d e t e r m i n e a l l

o t t he o p e r a t o r s under wh i ch a f i n i t e Bool ean A l g e b r a f or ms

a Gr oup and a l s o a l l of t he o p e r a t i o n s under wh i c h a f i n i t e

Boo l ean A l g e b r a f or ms an A b e l i a n Gr oup . The e x i s t e n c e of

t hese o p e r a t o r s and t he c o n d i t i o n s wh i c h d e t e r m i n e t hese

o p e r a t o r s have been shown by B. A. B e r n s t e i n . T h i s p a p e r ,

howev e r , p r e s e n t s an e n t i r e l y d i f f e r e n t a pp r oac h t o t h i s

d e t e r m i n a t i p n .

5.1 To be&i n we s e l e c t f r o m ,f Mo d e r n e A l g e b r a " by B, L.

van der Wa e r d e n a set o f p o s t u l a t e s f o r a Gr oup G. L i b e r a l

t r a n s l a t i o n of t hese p o s t u l a t e s i s as f o l l o w s :

A non empt y c o l l e c t i o n G of e l e me n t s o f any t vpe i s c a l l e da Gr oup when t he f o l l o w i n g c o n d i t i o n s are f u l f i l l e d :

11) Ther e i s g i v e n a r u l e of c o m b i n a t i o n 0 wh i c h a s s i g n st o each e l e me n t p a i r a, b, o f G a t h i r d e l ement o f t he samec o l l e c t i o n G ( u s u a l l y c a l l e d t he P r o d u c t ) . (The Pr o d uc t can depend on t he sequence of f a c t o r s and does not mean a 0 b = b 0 a )

(9) A s s o c i a t i v e Law: For each t h r e e e l e me n t s a, b , andc o f G t he f o l l o w i n g h o l d s :

( a O b ) O c = a O ( b O c )(3) Ther e e x i s t s a t l e a s t one l e f t u n i t y e l emen t e of G

w i t h t he p r o p e r t y :

-»*B. A. B e r n s t e i n , " O p e r a t i o n s w i t h Re s p e c t t o wh i c h t heE l e me n t s o f a B o o l e a n A l g e b r a For m a Or o u o " , T r a n s . Arner . Ma t h .S o c . , XXVI , ( 1 9 2 4 ) , pp . 171- 1 7 5 .

•k-x-R. L. van de r Wa e r d e n , Moder ne A l g e b r a , ( V e r l a g vonJ u l i u s S p r i n g e r , B e r l i n , 19 3 0 ) , p . 15 .

e 0 a = a f o r a l l a o f G.(4) For each a of G t h e r e e x i s t s at l e a s t one l e f t i n

ve r se e l emen t a™ of G w i t h t he o r o p e r t y :a “ 0 a = e

- 26 -

A Gr oud Is c a l l e d A b e l i a n I f , i n a d d i t i o n t o t he above, a l ways a 0 b = b 0 a ( Commut a t i v e Law. )

5 . 2 We nex t p r o c e e d t o e l i m i n a t e the b i n a r y o p e r a t o r s

wh i ch do not c o n f o r m t o t hese Gr oup p o s t u l a t e s and r e t a i n

t hose t h a t do. Th i s i s done by f i r s t c h e c k i n g c o mp l i a n c e

w i t h P o s t u l a t e s 3 and 4 by mean o f t a b l e s of a l l p o s s i b l e

v a l u e s and, i f t hey be s a t i s f i e d , c h e c k i n g P o s t u l a t e 2 i n t he

same manner . A l l o f ou r b i n a r y o p e r a t o r s s a t i s f y P o s t u l a t e I .

In any case work on any o p e r a t o r can s t o p a t t he p o i n t at

wh i ch t he f i r s t i n e o n s i s t a n c y i s met .

5. 201 Ex a mi n i n g a + b :

e T a - a , Mak i ng e = 0

a-1 4- a = 0 I n c o n s i s t e n t by ( 1 . 4 ) s i nc e i t

does not h o l d f o r any v a l u e of a.

5 . 202 Ex ami n i ng a + b 1 : I f a = I , i ' t I s r e q u i r e d t h a t

e + a ’ = a e = l , a n d i f a « 0 , i t is r e q u i r e d

t h a t e = 0. T h e r e f o r e i n c o n

s i s t e n t s i nce no u n i q u e s o l u

t i o n o f e e x i s t s f o r a l l a.

5 . 203 Ex a mi n i n g a f 4- b :

e 1 4- a = a , e » I

( a " 1 ) ’ 4- a - I a" 1 - a

Thus we must t e s t P o s t u l a t e 2 as f o l l o w s :

— 2 8 —

a b a ’ + b c ( a f 4- b) t + c b 1 + c a ’ 4- ( b T 4 c )

11

11

11

10

10

10

10

1 0 0 1 1 1 11 0 0 0 1 1 10 1 1 1 1 1 10 1 1 0 0 . 0 10 0 1 1 1 10 0 1 0 0 1

5 . 2 0 4 E x a m i n i n g a f + b 1 :

» 1 4- a 1 = a I f a = I , e r 0

and i f a = . 0 , 2 = 1

T h e r e f o r e i n c o n s i s t e n t .

5 . 205 Ex ami n i ng a X b :

e X a - a , e - I

a "" * X a z I , I n c o n s i s t e n t by 1.4

5 . 2 0 6 Ex ami n i ng a X b 1:

e X a f r a , I f a z 0 , e = I

or i f a = I , e = 0


5 . 207 Ex a mi n i n g a 1 X b :

e ! X a z a , e ~ 0

( a ” 1 ) ’ X a = 0 aa»

b ' X c

H e n c e i t i s i n c o n s i s t e n t w i t h P o s t u l a t e 7 .

— 29 —

5 . 2 0 8 Ex a mi n i n g a ’ X b ’ :

e f X a ’ - a I f a = I , e = 0

and i f a = 0, e = I


5 . 2 0 9 Ex ami n i ng a V b :

e 7 a = a , e = 0 ( T a b l e I )

( a " * ) 7 a = 0 , a - I = a

T h e r e f o r e we must check P o s t u l a t e 2:

a b a 7 b c ( a7b) 7 c b 7 c a 7 ( b7c)

1 1 0 1 1 0 11 1 0 0 0 1 01 0 1 1 0 1 01 0 1 0 1 0 10 1 1 1 0 0 00 1 1 0 1 1 10 0 0 1 1 1 10 0 0 0 0 0 0

Thus t he b i n a r y o p e r a t o r 7 s a t i s f i e s t he p o s t u l a t

g r oup and must be check ed f o r !■ h e c o n d i t i o n o f an Abel

Gr oup:

a b a 7 '"ib b 7 a

1 1 0 01 0 1 10 1 1 10 0 0 0

Hence t he o p e r a t o r 7 s a t i . s f i e s a l l p o s t u l a t e s f o r

Abe 1 i a n Gr oup.

5 . 210 Ex a mi n i n g a @ b:

e @ a ■= a , e = 1 ( Tab l e 1 )

a” 1 @ a = 1 , a"-1 , a

f o r a

Thus we must check c o mp l i a n c e w i t h P o s t u l a t e 2.

- 30 -

a b a @ b c ( a©b ). @ c b @ c a @ ( b @c )

11

11

11

10

10

10

10

1 0 C 1 0 0 01 0 0 0 1 1 10 1 0 1 0 1 00 1 0 0 1 0 1 •0 0 1 1 1 0 10 0 1 0 0 1 0

Thus P o s t u l a t e 2 i s a l s o s a t i s f i e d and @ i s a b i n a r y

o p e r a t o r s a t i s f y i n g t he p o s t u l a t e or a Gr oup. Nex t we check

i t a g a i n s t t he c o n d i t i o n s f o r an A b e l i a n Gr oup.

a b a @ b b @ a

1 1 0 01 0 1 10 1 1 10 0 0 0

Thus we f i n d @ i s a b i n a r y o p e r a t o r s a t i s f y i n g the p o s t u

l a t e f o r an A b e l i a n Gr oup .

5 . 3 As has been o r e v i o u s l y me n t i o n e d , t he above t en

o p e r a t o r s a r e t he o n l y " t r u e " b i n a r y o p e r a t o r s i n a f i n i t e

Boo l ean A l g e b r a . Thus we have t wo and o n l y t wo b i n a r y o p e r a

t o r s wh i ch s a t i s f y t he p o s t u l a t e s f o r a g r o u p . I t so happens

t h a t bo t h of t hese a l s o s a t i s f y t he c o n d i t i o n f o r an A b e l i a n

Gr oup.

5 . 4 B. A. B e r n s t e i n has g i v e n , i n t er ms o f Bool ean

A l g e b r a , t he c o n d i t i o n s wh i c h must be i mposed upon an o p e r a

t o r t o make i t s a t i s f y t he p o s t u l a t e s f o r a g r o u p . He men

t i o n s t he o p e r a t o r s whi ch v . e have l o c a t e d above bu t does not

s t a t e t h a t o t h e r s do not e x i s t .

CHAPTER VI

On t he E x i s t e n c e of Ri ngs i n Boo l ean A l g e b r a s

6, I The f a c t t h a t Boo l ean A l g e b r a s f o r m Ri ngs under c e r

t a i n s e t s o f o p e r a t o r s has been shown by M. H. St one . The

l o c a t i o n o f t he se t s of o p e r a t o r s wh i c h c o n f o r m t o t he p o s t u

l a t e s f o r a r i n g i s a much e a s i e r t a s k a f t e r l o c a t i o n of a l l

p o s s i b l e A b e l i a n Cr oup o p e r a t o r s i s c o mp l e t e d . For t h i s r e a

son some c r e d i t f o r t he e x i s t e n c e o f knowl edge o f t n e p r e s en c e

o f r i n g s ' i n Boo l ean A l g e b r a s is due B. A. B e r n s t e i n who d i s

c o v e r e d t he p r e s e n c e of t he o p e r a t o r s under w h i c h Bool ean

A l g e b r a s f o r med A b e l i a n Gr oups .

6 . 2 F i r s t , we w i l l l i s t a set o f p o s t u l a t e s f o r a Ri ng R

f r om B. L. van der Wa e r d e n ’ s M Mo d e r n e A l g e b r a " # * ( L i b e r a l

t r a n s l a t i o n ) :

Under a " s y s t e m w i t h d o ub l e c o m p o s i t i o n " one comprehends a set o f e l e me n t s a, b , , i n wh i c h f o r each two e l emen t sa, b i s u n i q u e l y d e f i n e d a " Sum" a 4 b and a " P r o d u c t " a X b , wh i c h b e l o n g t o t he same s e t .

A sys t em w i t h doub l e c o m p o s i t i o n i s c a l l e d a Ri ng when t he f o l l o w i n g c o n d i t i o n s are f u l f i l l e d f o r a l l e l e me n t s o f t he sy s tern.

I . Laws of Add i t i o n :(a) A s s o c i a t i v e Law: a 4 (b 4 c ) - (a 4 b ) 4 c(b) Commut a t i v e Law; a 4 b = b 4 a( c) S o l u b i l i t y o f t he Eq u a t i o n a 4 X = b f o r a l l

a and b

* M. H. S t o n e , " Su b s u mp t i o n o f t he Theor y of Boo l ean A l g e b r a under t he Theor y o f R i n g s " , P r o c e e d i n g s o f the N a t i o n a l Academy o f S c i e n c e , XXI , ( 1 9 3 5 ) , p p . 10 3 - 105.

* * B . L. van der Waer den , Moderne A l g e b r a , ( V o r l a g von J u l i u s S p r i n g e r , B e r l i n , 1930) , p. 37.

- 3 1 -

- 32 -

I I . Law o f M u l t i p l i c a t i o n :(a) A s s o c i a t i v e Law: a X ( b X c ) = ( a X b ) X c

I I I . D i s t r i b u t i v e Laws:(a) a X (b + c) c ( a X b ) 4- (a X c )(b) (b + c) X a = (b X a) 4 (c X a)

A l s o , i f t he Commut a t i v e Law h o l d f o r M u l t i p l i c a t i o n , t h a t i s , I I . ( b ) a X b = b X at hen t he Ri ng i s c a l l e d Co mmu t a t i v e .

6 . 3 The e q u i v a l e n c e of t he l aw of s o l u b i l i t y of e q u a t i o n s

and t he p o s t u l a t e 3 and 4 f o r a Gr oup i s shown on t he same

page w i t h t he above . Th i s means t h a t f i r s t one must an o p e r a

t o r w i t h r e s p e c t t o wh i ch t he set f o r ms an Abel i an Gr oup, and

t hen t h e r e must be a n o t h e r o p e r a t o r wh i ch obeys p o s t u l a t e s

I I and I I I . Thus , we have a l r e a d y e s t a b l i s h e d a l l p o s s i b l e

o p e r a t o r s under wh i c h a Bool ean A l g e b r a f orms an A b e l i a n

Gr oup , and i t r emai ns o n l y t o f i n d w h i c h , i f any , o p e r a t o r s

s a t i s f y p o s t u l a t e s I I and I I I i n c o n j u n c t i o n w i t h our o p e r a

t o r s 7 and (§:. Th i s we do i n t a b u l a r f o r m i n t he new few pages •

O p e r a t o r s are t e s t e d i n o r d e r as t h e y appear i n Tab l e I , and

work on any o p e r a t o r ceases as soon as the f i r s t i n c o n s i s t e n c y

i s met . P o s t u l a t e I I I i s exami ned f i r s t , t hen i f ne c es s a r y

p o s t u l a t e I I i s exami ned .

6 . 301 Ex ami n i ng a V b , a + b :


- 33 -

6 . 3 0 2 Exami ni ng a 7 b , a 4- b f :


6 . 3 0 3 Ex a mi n i n g a V b, a f + b:


6 . 3 0 4 Ex a mi n i n n a 7 b, a 1 4- b f :


6 . 3 0 5 Ex ami n i ng a 7 b , a X b :

- 34 -

A l s o :

A n d :

a b c a X (b X c) (a X b) X c

11

11

10

10

1c

1 0 1 c 01 0 0 0 00 1 1 0 00 1 0 0 00 0 1 0 00 0 0 0 0

T h e r e f o r e t h i s set o f o p e r a t o r s s a t i s f i e s t he p o s t u l a t e s

f o r a Ri ng and we check t he p o s t u l a t e f o r a Commut a t i v e R i n g :

a b a X b b X a

11

10

10

10

0 1 0 00 0 0 0

Thus , under t he o p e r a t o r s a V b , a X b , a f i n i t e Bool ean

A l g e b r a is a Commut a t i v e R i ng .

- 35 -

6 . 3 0 6 Exami n i ng a 7 b , a X t


X b :6 . 3 0 7 Ex ami n i ng a 7 b , a »

Pu t :

T h e r e f o r e i n c o n s i s t e n t ;

- 36 -

6 . 3 0 8 Exami ni ng a V b , a ! X b 1:


6 . 3 0 9 Ex ami n i ng a V b, a @ b:


6 . 3 1 0 Ex a mi n i n g : a@ b, a + b :

A I so:

- 37 -

A n d :

late?

Ri ng

can

a b c ( b @ c ) 4* a ( b 4- a ) @ ( c 4- a )

11

11

10

11

11

1 0 1 1 11 0 0 1 10 1 1 10 1 0 0 00 0 1 0 00 0 0 1 1

a b c a 4- ( b 4- c ) (a 4- b ) 4 c

11

11

10

11

11

1 0 1 1 11 0 0 1 10 1 1 1 10 1 0 1 10 0 1 1 10 0 0 0 0

T h e r e f o r e t h i s s e t o f o p e r a t o r s c o m p l i e s

and we check t he pos

a b a 4 b b 4 a

11

1C

11

11

0 1 1 10 0 0 0

Thus , under t he o p e r a t o r s a @ b , a 4- b , any f i n i t e Boo I

A l a e b r a is a Commut a t i ve Ri ng ,

— 3 8 -

6 . 311 Exami ni ng a @ b, a + b 1 :


6 . 3 1 2 Ex ami n i ng a @ c , a ’ 4- b :

Bu t :


- 39 -

6 . 3 1 3 Exami n i ng a @ b , a 1 4- b T:

T h e r e f o r e i n con s is t e n t .

6 . 3 1 4 Ex ami n i ng a @ b , a X b :


6 . 3 1 5 Ex a mi n i n g a b , a X K ’ :


184196

- 40 -

6 , 3 1 6 Exami n i ng a @ b , a 1 X b:


6 . 3 1 7 Ex ami n i ng a @ b , X b» :


6 . 3 1 8 Ex ami n i ng a @ b , a 7 b:


— 41 —

T h e r e f o r e t h e r e are two and o n l y t wo set s o f o p e r a t o r s

under wh i c h any f i n i t e Boo l ean A l g e b r a i s a R i n g . They ar e

a 7 b, a X b, and a @ b, a + b and t h e y s a t i s f y a l l t he p o s t u

l a t e s f o r a Commut a t i v e R i n a .

CHAPTER VI I

1 Summary o f Re s u l t s and I mp o r t a n c e- o f t he M a t r i c R e p r e s e n t a t i o n

In t he p r o c e e d i n g c h a p t e r s i t has been shown t h a t t he

m a t r i c r e p r e s e n t a t i o n can be used f o r any f i n i t e Boo l ean

A l g e b r a * . F u r t h e r m o r e , i t has been shown t h a t t h e r e e x i s t s

a f i n i t e met hod o f e x a mi n i n g a l l p o s s i b l e b i n a r y o p e r a t o r s

w i t h r e s p e c t t o any g i v e n ' se t o f c o n d i t i o n s * In f a c t , t he

same met hod o f e x h a u s t i n g a l l p o s s i b i l i t i e s , when a p p l i e d

t o an m- a r y o p e r a t o r w i t h r e s p e c t t o any g i v e n se t o f c o n d i

t i o n s , must g i v e t he d e s i r e d s o l u t i o n a f t e r a f i n i t e number

o f o p e r a t i o n s . The s o l u t i o n i s r e l a t i v e l y s i mp l e i f t he

number o f t a b l e s r e q u i r e d is not e x c e s s i v e , b u t i t wou l d be

come v e r y l a b o r i o u s i f a g r e a t number o f t r i a l s wer e r e q u i r e d ,

S h e t f e r ' s s t r o k e f u n c t i o n was exami ned , and f i n i t e B o o l

ean A l g e b r a s wer e shown t o $ orm a Gr oup under t he o p e r a t o r s

7 and @ and t o #orm a Ri ng under t he o p e r a t i o n s 7 , X and @, f »

In each t e s t c o n d u c t e d in t h i s paper r e s u l t s c o n c u r r e d

w i t h t hose o b t a i n e d p r e v i o u s l y by o t h e r me t hods . In a d d i t i o n

t h i s met hod showed a I I p o s s i b l e s o l u t i ons wher eas some o t h e r

appr oaches show o n l y some of t he s o l u t i o n s .

The i mp o r t a n c e o f showi ng t he p r o p e r t i e s o f Boo l ean

A l g e b r a s under c e r t a i n o p e r a t i o n s i s , o f c o u r s e , t o e n ab l e one

' t o assume i m m e - d l a t e l y . a l l o f t he many o t h e r t heo r ems t h a t have

' - 42 -

- 43 -

been deduced i n modern a l g e b r a on t he b a s i s o f any a l g e b r a

h a v i n g t he g i v e n p r o p e r t i e s . For e x a mp l e , once i t is known

t h a t a Bool ean A l g e b r a f orms, a Gr oup under c e r t a i n o p e r a t i o n s ,

t hen i t i s known t h a t un der f hose ope r a t i o n s i t w i l l have a l l

t he p r o p e r t i e s o f a Gr oup .

To t h i s end, t h i s met hod may a t some t i me f a c i l i t a t e

new r e s e a r c h i n t he showi ng o f new p r o p e r t i e s o f f i n i t e Bool ean

A l g e b r a s .

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Bo Ao B e r n s t e i n , " O p e r a t i o n s w i t - Respec t t o whi ch Boo l ean A l p e b r a s Form a Gr oup" , T r a n s . Arne r . Mat h . S o c . . XXV l .( 1924) , pp, 17 1- 175 , See " E r r a t a ” , I b i d . , XXVI I ,( 1 9 2 5 ) , p. 600.

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E. V. H u n t i n g t o n , " S e t s o f P o s t u l a t e s f o r t he A l g e b r a of Log i c T r a n s . Arne r . Mat h . S o c . . V, ( 1 9 04 ) , pp. 2 3 8 - 3 0 9 .

____________________ , "A New Set o f I ndependen t P o s t u l a t e s f o r t heA l g e b r a o f L o g i c w i t h Sp e c i a l Re f e r en c e t o Wh i t e h e a d and R u s s e l l ’ s P r i n c i p i a Ma t h e me t i c a , I b i d . , XXXV, I 933, pp. 2 7 4 - 3 0 4 . See C o r r e c t i o n s t h e r e t o , I b i d . , pp. 557 - 558, 971.

C. I < Lewi s and C. H. L a ng t o r d , Sv mb o l i e Log i c , The Ce n t u r y Company, 19 32 .

L. M. Reagan, E. R. O t t , and D. T . ' S i q l e v . C o l l e g e A l g e b r a . F a r r a r & R i n e h a r t , I n c . , 19 4 0 .

H, M. S h e f f e r , " A Set o f F i ve P o s t u l a t e s f o r , B o o l ean A l g e b r a sUnder the. Theor y of R i n g s " , P r o c e e d i n g s o f t he N a t i o n a l Academy of S c i e n c e , X X I , ( I 9 3 5 ) , pp. I 0 3 - I 0 5 .

M» Ho S to n e , " Su b s u mp t i o n of t he Theor y o f Bool ean A l g e b r a s Under t he Th e o r y o f R i n g s M, P r o c e e d i n g s o f t he N a t i o n a l Academy o f S c i e n c e , XXI , ( 1 9 3 5 ) , p p 0 10 3 - 105»

Bo Lo van de r WaerdCn, Moderne A l g e b r a * V e r l a g von J u l i u s S p r i n g e r , B e r l i n ^ 1930*

i , / f t - the university of arizona campus...

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