An alternative generalisation of the concept of duality

ROSE, A. Math. Annalen 147, 318--327 (1962)

An alternative Generalisation of the Concept of Duality By

ALAN ROSE in N o t t i n g h a m

A formula • (P1 . . . . . Pn) of the 2-va lued p ropos i t iona l calculus is sa id to be a dual of a formula ~b(P 1 . . . . , P~,), where P1 . . . . . Pn are syn tac t i ca l var i - ables represen t ing d i s t inc t p ropos i t iona l var iables , if, for all the 2" ass ignments of t r u th -va lue s to the p ropos i t iona l var iables ,

T ( P 1 . . . . , Pn) =T O ( P l . . . . . P n ) .

This def ini t ion m a y be e x t e n d e d to a def ini t ion of a dua l in the m-va lued pro- pos i t ional calculus 1) if the symbol ..... now denotes the nega t ion func to r of LUKASIEWICZ~). W e say t h a t T ( P 1 . . . . . P , ) is the dual of qi(P1 . . . . . P~) if i t is a dua l cons t ruc ted according to a prescr ibed rule (for in te rchanging func tors or the i r a rgumen t s or both) .

Thus, if O(Pl . . . . . Pn), W(P1 . . . . . Pn) t ake the t r u th -va lue s ¢ ( x 1 . . . . . xn), ~p (x 1 . . . . . x~) respec t ive ly when Px . . . . . Pn t ake the t ru th -va lues Xl, . . . , x n respect ive ly , T(P1 . . . . . Pn) is a dua l of ~b(P 1 . . . . . Pn) if and only if 3)

m + 1 - - ~ ) ( x I . . . . ,Xn) = ¢ ( m + 1 -- x 1 . . . . . m + 1 - x,n)

( m = 2 , 3 , . . . ; n = 0 , 1 . . . . ) .

I f m = 2 and the ope ra t ion "+ '" is defined b y the equat ions

1 + ' 1 = 2 , 1 - ~ ' 2 = 2 + ' I = 1, 2 + ' 2 = 2

the above equa t ion is equ iva len t to the equa t ion

~ ( Z 1 . . . . . Xn) + ' 1 = y~(Xl+ ' l . . . . . X ~ + ' l ) ( n = 0, 1 . . . . ) .

Thus we may , as an a l t e rna t ive , general ise the defini t ion of a dua l in the 2-va lued propos i t iona l calculus as follows:

The ]ormula ~ ( P1 . . . . . . P~) o/the m-valued propositional calculus is an m-al o/ type 1 o/the/ormula q~(P1 . . . . . Pn) o/this calculus i/

x + ' y ~- x + y ( m o d m ) ] . ( x , y = l ~ . . . . . r e ; m = 2 , 3 . . . . ) ,

l < x + ' y < m ) and

~ (x x . . . . . xn) + ' 1 = y~(x~+' l . . . . , x . + ' l ) (n = O, 1 , . . . ) .

1) RosE, A.: Math. Ann. 128, 76--78 (1951). ~) LVKAS~WICZ, J., and A. TARSKI: Comptes rendus (Warsaw), 21, 30--50 (1930). *) See footnote 1.

The Concept of Duality 319

This defini t ion is, of course, equ iva len t to defini t ion b y the equa t ion

•(P1 . . . . . Pn) = w ~ ~b( ~ m - l p 1 . . . . . ~ m - l p , )

where " ~ - 1 , , denotes m - 1 consecut ive Pos t a) nega t ion functors . I n the case m = 2 th is equa t ion reduces to t h a t given a t t he beginning of the paper .

W e say t h a t A ( P I . . . . . Pn) is an m-al o / t y p e i of ~b(P 1 . . . . , Pn) if the re exis ts a formula f 2 ( P l . . . . . P~) such t h a t

(i) [2 (P 1 . . . . . Pn) is an m-al of t y p e i - 1 of ~b(P I . . . . . Pn),

(ii) A ( P 1 . . . . . Pn) is an m-al of t y p e 1 of ~2(P 1 . . . . . Pn) (i = 2, 3 . . . . ).

I f A ( P 1 . . . . . P , ) t akes the t r u t h - v a l u e ~(x 1 . . . . . x~) when /)1 . . . . . Pn t ake the t ru th -va lues X l , . . . , x~ respec t ive ly and 1 ~ i ~ m these condi t ions are equ iva len t to the equa t ion

¢(x~ . . . . . x n) + ' i = 2(Xl ÷ ' i . . . . . xn ÷ ' i) (n = O, 1 . . . . )

and to the equa t ion

A(P1 . . . . . P , ) =T ~i~b( ~ - i P 1 . . . . . ~ m - ~ P n ) (n = 0, 1 . . . . ) .

Clear ly every fo rmula is an m-al of t y p e m of itself. W e shall say t h a t W(P1 . . . . . P~) is the m-al o / t ype 1 of ~b(P 1 . . . . . Pn) if i t is an m-al of t y p e 1 of ~5(P 1 . . . . . P~) cons t ruc ted according to a p rescr ibed rule and t h a t A(P1, • . . , P . ) is the m-al o / t y p e i of ~ ( P ~ . . . . . Pn) if

(i) f 2 ( P 1 . . . . . P . ) is the m-al of t y p e i - 1 of qS(p~ . . . . . p~) ,

(ii) A ( P 1 . . . . . Pn) is the m-al of t y p e 1 of f 2 ( P 1 . . . . . P . ) (i = 2, 3 . . . . ).

Clearly the m-al of t y p e m of a fo rmula cannot a lways be t h a t fo rmula unless the set of p r imi t ives is self m-al.

Le t us now consider self m-al sets of pr imi t ives . W e shall t a k e the pa r t i cu l a r case m = 2 of a p rev ious t heo rem of the a u t h o r s) concerning sys t em wi th a single p r imi t ive a n d general ise i t wi th respec t to the concept of m-a l i ty . W e shall t hen show tha t , if m = pa where p is p r ime and a is a pos i t ive integer , a solut ion in which the p r imi t ive has a smal ler n u m b e r of a r g u m e n t s is im- possible. F o r o the r va lues of m some reduc t ion in t he n u m b e r of a rgumen t s is possible and we shall ob t a in a " b e s t poss ib le" resul t . F i n a l l y we shal l general ise the t heo rem of CHUI~CH 6) concerning the self-dual set of p r imi t ives for t h e 2-valued p ropos i t iona l calculus consis t ing of impl i ca t ion and converse non- impl icat ion•

Le t us consider first the m-va lued func tor

4) PosT, E. L.: Am J. Math. 43, 163--185 (1921). 5) RosE, A.: Math. Ann. 126, 144--148 (1953). ~) CHURCh, A. : Portugal. Math. 7, 87--90 (1948).

320 ALA~ ROSE :

of m ÷ 1 arguments whose t ru th- table is determined by the equat ion r

Q, P 1 , . . . , Pm =T ~ [Q,/)1 . . . . . P~, Q]

where "[ . . . . . ]" deno*~es the m-valued condit ioned disjunction 7) functor. I I

Theorem 1. The m-valued propositional calculus with . . . . . as the only primitive/unctor is/unctionally complete. The primitive is sel[ m-al and the m-al o/type 1 o/a/ormula may be obtained by replacing each (proper or improper) sub-

I -I

/ormula o/the ]orm Q, P1 . . . . . Pro, starting/rom the innermost, by I" "1

Q,P , , ,P~ . . . . , P m - 1 •

We show first that , if Qt, P~ . . . . . P*m are the type 1 m-als of Q, P1 . . . . . Pm I t r I

respectively, then the type 1 m-al of Q,/)1 . . . . . P,~ is Qt, P*,~, P~* . . . . . P'm- 1. r I

Let us denote the distinct propositional variables occurring in Q, P1 . . . . . Pm by the syntact ical variables R 1 . . . . . R k. Let R 1 . . . . . Rk take the t ruth-values z 1 . . . . , z k respectively and let the corresponding t ruth-values of Q,/)1 . . . . . Pm be y, x 1 . . . . . x,~ respectively. Since

| '1

Q, P~, . . ., Pm = T ~ [Q, P1 . . . . . P,,, Q] 1 1

the corresponding t ru th-value of Q, P1 . . . . . P ~ will be x u + ' l . I f R1, . . . , R k take the t ruth-values z 1 + ' 1 , . . . , z~ + ' 1 respectively then, by their definitions, the formulae Qt, Pa* . . . . . P*m take the t ruth-values y + ' 1, x l + ' 1 . . . . . xm+' 1 respectively. Hence the formula [Qt, P'm, PI* . . . . . P*~-1, Qt] takes the same t ru th-value as the (y + ' l ) - t h of the formulae P,*~, PI* . . . . . P fm-1 , i.e. as the

I I formula Pfv" Thus the formula Qt, P'm, PI* . . . . . P'm- 1 takes the same t ru th-value as ~ P'v, i.e. it takes the t ru th-value (xu+' 1) + ' 1. Thus the result is proved. Hence the primitive is self m-al and the rule for obtaining the type 1 m-al of a formula follows a t once. Clearly the type m m-al of a formula is t h a t formula.

Thus, for example, if m -~ 3, the trials of types 1, 2, 3 of the formula

J l ': 1 ~ i

p , q , r , s , q, r , p , r , q , s are

p , s , q , r , s , q , r , q , p , r ,

l i ~ i '" I ' ' :1 p, r, s, q, r, r, q, p, s, q,

h " : ' 1 I ' ' I

p , q , r , s , q , r , p , r , q , s

respectively. As a fur ther example we note that , in this system, we m a y define t h e primitive negat ion functor of Pos t by

P ----dx P, P , P ,

and t h a t this definition is self-trial.

') See footnote 1.

The Concept of Duality 321

I n order to establish (for all values of m) the funct ional completeness of the system we first note tha t , if we make the definitions

I" I

~ P =dr P , P . . . . . P ,

r

VI(P) =dr P, ~-i p, ~ m - ~ p . . . . . p ,

V i ~ I ( P ) = d r ~ Vi(P) (i = 1, . . . , m - 1)

t hen V/(P) always takes the t ru th-vMue i (i = 1 . . . . , m). Since we can define condi t ioned d is junct ion by

L

[Q, P1 . . . . . Pro, Q] -=-at ~ ~ - 1 Q, P1 . . . . . Pm

we can construct the m ~ formulae

[P, V~I(P) . . . . . V ~ ( P ) , P]

which take the t ru th -va lue xi when P takes the t ru th-vMue i (i = 1, . . . , m; x 1 = 1 . . . . . m ; . . . ; xm = 1 . . . . , m). The remainder of the proof of funct ional completeness is ident ical with t ha t g iven s ) for the system with condi t ioned dis- junc t ion and the m logical constants as primitives.

Theorem 2. I[ m = pa where p is prime and a is a positive integer, then, i/ n ~ m, no sell m-al [unctor of n arguments has a truth-table such that the m-valued propositional calculus with this [unctor as the only primitive is functionally complete.

Let us consider first the case n = m and let us suppose t h a t such a functor ~b( . . . . . ) exists. The type 1 m-al of ~b (Pl . . . . . p~) mus t be of the form q5 (Pil . . . . . Pim) where {i 1 . . . . . i~} is a pe rmu ta t i on of the integers 1, . . . , m.

If this pe rmuta t ion is no t cyclic of order m then there mus t exist integers Jl . . . . . ]k such t h a t

J ~ + " ' + J k = m , / c ~ 2

and the type N m-al of ~b(p 1 . . . . . Pm) is q~(Pl . . . . . p~) where

~¢ = { /1 . . . . . J ~ } -

Since k ~ 2 none of the integers Jl . . . . . Jk is divisible by pa. Hence there exists an integer M such t h a t

l < - M g m - 1 and

N ~ M (modm) .

Thus if the type M m-al of ~b(p 1 . . . . . p~) is ~ ( P l . . . . , Pro) then

~r/(Pl . . . . . Pro) =T(J)(Pl . . . . , Pro)

and it follows a t once t h a t the type M m-al of a n y formula whatever has the same t ru th - t ab le as t h a t formula. Hence the func tor V 1 ( ), whose type M m-al is VM+ 1( ), is no t definable in te rms of q}( , . . . , ) a nd the system is no t funct ional ly complete.

s) See footnote 1.

3 2 2 A L A ~ R O S E :

If the permuta t ion is cyclic of order m let ~b(P 1 . . . . . Pro) take the t ru th- value ¢ (x 1 . . . . . xm) when P1 . . . . . P,~ take the t ruth-values x 1 . . . . . xm respectively• We m a y suppose, wi thout loss of generality, t ha t the type 1 m-al of ~ ( P l . . . . . pro) is ~b(p~, Pl . . . . , Pro-l). Hence

¢(1 . . . . . m) + ' 1 = ¢(1 . . . . . m ) .

Thus, whenever n = m, we have a contradict ion and the theorem is proved. I f n < m and a functor ~b( . . . . . ) having the required properties

exists let E(P1 . . . . . P~) =T ¢ ( P 1 . . . . . P,~).

The functor ~ ( . . . . . ) is self m-al since, if the type 1 m-al of ~5 (P1 . . . . . P~) is ~b(p~,, P~), the type 1 m-al of ~(P1, P,,,) is ~ ( P ~ , p t

• . . ~ • • . , • . . , Z n ~

P~*+I . . . . . P,*~). The sys tem with E ( . . . . . ) as the only primit ive is funct ional ly complete since we m a y make the definition

qb(p~ . . . . . p , ) =d fE(p1 . . . . . p,, . . . . . p , ) .

Hence if a suitable functor of less than m arguments exists a suitable functor of m arguments exists also. Since it has been shown above tha t this is no t the case the theorem is proved.

Theorem 3. I l m I . . . . . m~ are prime, a 1 . . . . . a~ are positive integers,

J J m= H m > N = S m ?

k = l k = l

and M < N then no sel I m-al /unctor o / M arguments has a truth-table such that the m-valued propositional calculus with this /unctor as the only primitive is ]unc- tionally complete•

I f such a functor exists then there mus t exist integers % , . . . , uz such t h a t l

z ~ , % = M a = l

and the type W m-al of ~b(p 1 . . . . . PM) is ~ b ( p I . . . . . P M ) where

W = { u 1 . . . . . u l } .

Since M < N there exists an integer k (1 ~ k ~< ?') such tha t m~ is not a divisor of any of the integers u 1 . . . . . ut. Hence W is not divisible by m~ k. I t follows at once t h a t W is not divisible by m and it then follows, as in the proof of the previous theorem, t h a t the sys tem is not funct ional ly complete•

Theorem 4. I I N is defined as in the previous theorem and ] ->- 2 then there exists a sel l m-al lunctor o / N arguments whose truth-table is such that the m-valued propositional calculus with this lunctor as the only primitive is lunctionally com- plete.

Let us consider a functor

, • * • ~

of N arguments such t h a t if I "l

P1 . . . . . PA, P1 . . . . . P~v

The Concept of Duality 323

t ake the t r u th -va lue s x I . . . . . Xy, y~(x I . . . . . Xy) respec t ive ly and

v~ = ~ , ~ (k = 1 . . . . . j )

t hen W(Xl . . . . . Xl, x l ÷ ' 1, x 1 . . . . . x I . . . . . xj . . . . . x~, x~÷ ' 1, x~,

. . . . x j ) = i ÷ ' l (x 1= 1 . . . . . m ; . . . ; x ~ = 1 , . . . , m ) where 9)

i ~ wk(modvk) (k = 1 . . . . . ] ) ,

the number of the a r g u m e n t place occupied b y the express ion x ~ ÷ ' l is ] c - - 1

~ , ~ + 2 7 v~ (k = ] . . . . . j ) l ~ l

and the numbers of the a r g u m e n t places occupied b y the express ion x k are k - - 1 k - - 1 k--i k

l + X v ~ . . . . . w ~ - l + X v ~ , ~ + ] + Z ~v, . . . . . Z : v ~ ( k = l , . . , i ) l = 1 / = 1 / = 1 / = 1

Clearly, for the m j + l ass ignments of va lues to x 1 . . . . . x y so far considered, if the i n t e g e r s / q . . . . . k~v, i l . . . . . i y are defined by the equat ions

then

1 ~ g - - ~ v t g vka+l, I = 1

k~ (:¢ = 1 . . . . . N ) , 1 <= i s - Z, v~ <= vk~+ 1,

/ = 1

1 ÷ i ~ :¢ (modvk~+ 1)

~0(xi + ' 1 . . . . . x i ~ + ' l ) = ~ (x I . . . . . xy) ÷ ' 1 . (A)

I t will no t be neccessary to s t ipu la te the exac t va lues of ~ ( x I . . . . . Xy) in the remain ing m y - m ~" + 1 cases. All t h a t is necessary is to s t ipu la te t h a t equa t ion (A) remains va l id in these cases. This l a t t e r r equ i remen t canno t be self-contradic- t o ry 1°) since, if so, the re m u s t exis t t r u th -va lues x 1 . . . . . x y and an in teger i (1 -< i -< m - 1) such t h a t the cor responding as s ignment x l ÷ ' i . . . . . X y ÷ ' i

I" | to t he var iab les of the t y p e i m-al of Pl . . . . . p y is ident ica l , as regards a r g u m e n t places, wi th the or iginal ass ignment . Thus, if t he ope ra t ion " ÷ .... is defined wi th respect to the in teger v 1 in a m a n n e r analogous to t h a t in which the opera- t ion " ÷ ' " is defined wi th respec t to the in teger m, then

x : ~ ÷ ' i = x:+, , i (o~ = 1 . . . . . vl) .

Hence x ~ , ÷ ' V l i = x~+ , , v l~= x~(:¢ = 1 . . . . . vl)

and v i i is d ivis ible b y m. I t follows a t once t h a t i is d ivis ible b y %. Hence, if " + ..... is defined wi th respec t to the in teger v~ in a m a n n e r s imilar to t h a t used in previous defini t ions,

xvl + (1 + ' " 0 = x~,~ + l ~= x,J, + l ÷ ' i

~) Since vl . . . . . vj are relatively prime, i is uniquely determined. See, for example, KL~E~, S. C. : Introduction to recta-mathematics (Amsterdam 1952), p. 240.

10) Cf. the cyclic case in the proof of Theorem 2.

324 AbAN ROSE:

and the t ru th-values of the variable occurring in the @1 + (1 + ' " i))-th a rgument places in the two cases are unequal. Thus we should have a contradict ion a n d t h e consistency is proved.

We then consider the functor

such t h a t if

takes the t ruth-value ¢ ( x l , . . . , X=v) then

¢ (x . . . . . x ) = x + ' l (x----1 . . . . . m)

and, in all o ther cases,

" [I I . . . . = T . . . . . . . . . . . . . . .

, ,] D~(Pz. , . . . . P.~) . . . . . - Q m ( P w P ~ + . . . . . . . P s ) , t ) i . . . . . P~. , (B)

where D~(P., , . . . . Pz~) occupies the (y + 1) th a rgument place of the condi- t ioned disjunction,

k - - 1 k

l = 1 l = l k - - 1

y- -=zk- - ~ v t ( m o d v ~ ) ( k = 1 . . . . . j) l = 1

and Di (ql . . . . . qi) is the m-al of type i - 1 of the formula n)

q l ) ( q , , D ( q 3 . • • D ( q j - 1 ) q , ) • • .) (i = 1 . . . . . m) .

We show first t h a t the type 1 m-al of ))1 . . . . . P v is

We first note t h a t

¢ ( x + ' l . . . . . x + ' l ) = x + ' l + ' l = ¢ ( x . . . . . x ) + ' l ( x = 1 . . . . . m ) .

I n the remaining m N - m cases, if -Qv(P1 . . . . . Pj) takes the t ru th-value to N(x I . . . . . xj) (y = 1 . . . . . m).

~ b ( X i ~ + ' l , . . . , X i N + ' I ) = O ) y ( X z , + ' l . . . . . X Z / + ' I )

where Y = ~ ( x h + ' l , • • . , xt , ,+' 1)

and, since Z k = i~ where ~ ~ Z k + 1 (modvk), k - - 1

Y - ~ Z k + I - - ~ v t ( m o d v k ) ( k = l , . . . , j ) . l = l

But, if the operat ion " - - " ' is the inverse of the operat ion " + " ' ,

~ r ( x z , + ' l . . . . . X z j + ' l ) = ¢ O y - , l ( x z . . . . . . Xzj) + ' 1 ,

Y - - ' 1 = ~(xx . . . . . x~.)

n) The m-al of type 0 of a formula is considered to be that formula.

The Concept of Duality 325

and

Hence

k--1 Y--'I------Ze-- ~ v ~ ( m o d v k ) .

/=1

P i

Q

~Q

In order to justify this definition we note tha t

m-~ vk(modvk) (k = 1 . . . . . ])

and tha t the formula

~or(xz,+' 1 . . . . . x z j + ' 1) = ¢(Xl . . . . . xx) + ' 1 .

Thus the functor . . . . . is self m-al. Finally we shall show tha t the m-valued propositional calculus with this

funetor as the only primitive is functionally complete. To this end we shall define the funetors ~ and v of Post. I t is clear tha t we may make the definition

m R =df ]D, • . . , p . We then make the definition

QI(P1 . . . . . PJ'-1, Q) =dr

P l . . . . . P1, ~ P l . . . . , P J - 1 . . . . . e j - i , ~ P ~ - I , Q . . . . , Q, ~ Q ,

the numbers of the argument places occupied by the subformulae Pi , ~ P i

(i ---- 1 . . . . . ] - 1), Q, ~ Q being as s tated below.

Sub-formula Argument place number(s)

i--1 i--1 ] l+~--~lV~= . . . . . vi-- 1+~ ~=lva

( i = 1 . . . . . j - l ) 2~ v~

c~=l 1--1

1 + ~ v . . . . . . N - - 1 4=1

N .

i

P1 . . . . . /)1, ~ P x . . . . . PJ-~ . . . . . PJ-~, ~ P~-I , Q . . . . , Q, ~ Q '

therefore takes the t ruth-value 1. Use of equation (B) then completes the justification. We may then define implication and disjunction by

V(P) =drQ1 (P . . . . . P ) ,

P ~ Q =dr Q~(V(P) . . . . . V(P), P, Q),

PvQ=~(PDQ) DQ

and the functional completeness of the system follows a t once. Finally we consider the generalisation of CHURCH'S theorem concerning

implication and non-implication. Let us denote by

Q,(p, q) Math. Ann. 147 22

326 ALAN ROSE:

the m-al of type i of ZO(p, q) (i = 0 . . . . . m - 1) where, if p, q, ZO(p, q) take the t ru th-values x, y, eo (x, y) respectively

~o(1, x ) = x ( x = l . . . . . m ) ,

o)(2, x ) = 1 ( x = 1 . . . . . m - - l ) ,

(o(2, m) = 2 ,

~o(x, y) = 1 ( x = 3 . . . . . m ; y = l . . . . . m) .

Theorem 5. The ]unctors ZOo ( , ) . . . . . ~(~m--1 ( , ) / o r m a complete set o/ independent p r imi t i ve s /or the m-valued proposit ional calculus. The set o / p r i m i - tives is sell m-al and the m-al o/ type 1 o/ a /ormula may be obtained by replacing each /unctor ZOi( , ) by ZOi+I( , ) (i = 0 . . . . . m - 2) and each /unctor

ZOrn-l( , ) by zOo( , ). The proofs of the last two parts of the theorem are trivial. We shall now

establish the functional completeness of the system by a method similar to t h a t used in a previous paper 12) of the author. We make the definitions

V~(P) =d~ZOi_I(P, P) (i = 1 . . . . . m) ,

g m - l ( P ) = d f ~ Q m - i ( V l ( P ) , P ) ,

W i ( P ) =dfzoi_I(P , V m _ l ( P ) ) , (i : ] . . . . . m)

J i ( P ) = d f J m _ l ( W i ( P ) ) . (i = 1 . . . . . m - 2, m) .

Clearly Vi (P) always takes the t ruth-vMue i (i = 1 . . . . . m). I f P, Q, J m - 1 ( P ) , ZOi (P, Q) take the t ruth-values x, y, ]m-1 (X), (1) i (X, y) respectively then

j m _ l ( m - - 1) = (9m_1(1 , m - - 1)

= O~o(2 , m ) - - ' 1

= 2 - - ' 1 = 1 ,

?m-l(x) = o)m-l(1, X)

= ~%(2, x + ' 1 ) - - ' 1

= 1 - - ' l = m . ( x : ~ m - - 1)

I f P, W~(P) , J t ( P ) take the t ruth-values x, wi(x) , ]i(x) respectively (i = 1, . . . . m - - 2, m)

wi( i ) = O i _ l ( i , m - 1)

---- ~0(1, m - - ' i ) + ' i - - ' 1

- - - - m - - ' i + ' i - - ' l = m - - 1 ,

w i ( x ) ---- w i _ l ( x , m -- 1)

= Wo(X-- ' i + ' 1 , m - - ' i ) + ' i - - ' l

= i ( x = l . . . . . i - - 1, i + 1 . . . . . m ; i = l . . . . . m - - 2 a n d

x = 2 . . . . . i - - 1, i '+ l . . . . , m ; i = m ) ,

1,) Ros~, A.: Ma~h. Ann. 148, 448--462 (1961).

The Concept of Duality 327

w, . (1 ) = ~o, ._1(1, m - 1)

= eo 0 (2, m) - - ' 1

= 2 - - ' 1 = 1 . Hence

)i (i) = ira-1 (wi (i))

= im_l (m-- 1 ) = 1 ,

]i (X) = i m -- 1 (Wi (X))

= jm_l(~) (where (0~-- i) (~ - - 1) = 0)

= r e ( i = 1 . . . . . m - - 2 , m ; x = 1 . . . . . i - - l , i ÷ l . . . . . m).

We then make the definition

P & Q =d~g,~(~2( P, gm(4(Q)) ) ) •

Thus if P, Q, P & Q take the t ru th -va lues x, y, k (x, y) respect ively

k(1, 1) = 1, k(1, m) = k(m, 1) = k(m, m) = m.

We are now in a posit ion to define la) the functors S1j ( , ). We m a k e the definitions

Sjj(P, Q) =dr S2j_I(P, ~2j_ l (P , Q)), / Slj(P, Q) =dfSj j (~2(P, ~(P, Vj(P))), Q), (i = 2 . . . . . m ) .

The funct ional completeness of the sys tem now follows as in 14) the previous paper , except, tha t , in the case k = 0, the required formula is now

f 2 I P 1 , t 2 ( P 2 . . . . . t 2 ( P . _ . V~(P) ) ) . . . .

Final ly we establish the independence of the primit ives. I f

5~i= {1 . . . . . i - - 1, i ÷ 1 . . . . . m}

and P, Q, ~2j (P, Q) t ake the t ru th-va lues x, y, o~j(x, y) respect ively it follows t h a t

if x, y ~ ~i then

~Oo(X, y ) , . . . , o~j_ ~(x, y), % @ , y) . . . . . com_l(x, y) ~ d~i (]" = 1 . . . . . m) .

Since, if k # ?', k E d~ • and ~ _ l(k, k) = ] ~ d~j (] = 1 . . . . . m) the independence of the pr imit ives follows 15) a t once.

(Received November 24, 1961)

13) See footnote 12. 14) See footnote 12. 15) See, for example, the paper referred to in footnote 5.

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