Chapter 2 Relation, Partial Ordering, Chain, Isomorphism, Cofinality

29

CHAPTER

2

RELATI ON, PARTIAL ORDER1 NG, CHAI N , ISOMORPH I Sp1, COFI WALITY

§

1 - RELATION,

MULTIRELATION,

EXTENSION,

RESTRICTION,

COHERENCE

LEMMA, AXIOM OF DEPENDENT CHOICE

E be a s e t and n an i n t e g e r . I n c h . 1 5 2.3 we d e f i n e d t h e n o t i o n o f

Let

n - t u p l e w i t h values i n

+

denoted

-

and

relation. w i t h e value

R(xl

,...,xn)

. We

E

s e t a s i d e two elements c a l l e d values, which a r e

( f o r i n s t a n c e , t h e s e can be d e f i n e d by 0 and 1). An n-ary E =

, o r based on E , i s a f u n c t i o n R + o r - t o each n - t u p l e x1 ,..., xn

nience, we o f t e n denote t h e n - t u p l e b y i t s i n d i c e s

1

to

which a s s o c i a t e s t h e in n

E

( f o r conve-

instead o f from 0

.

t o n - 1 ) . The s e t E , t h e base o f R , w i l l be denoted I R I The i n t e g e r n w i l l be c a l l e d t h e * o f R . F o r n = 1,2,3, we w i l l say a unary, b i n a r y , ternary r e l a t i o n . n = 0

For on

E

, we

, which - .

adopt t h e c o n v e n t i o n t h a t t h e r e e x i s t two 0 - 9 r e l a t i o n s based

we denote by

(E,t)

and

(E,-)

: the 0-ary r e l a t i o n s w i t h value

t

and value

We adopt t h e c o n v e n t i o n t h a t , f o r each p o s i t i v e

n

, there

e x i s t s a unique n-ary

r e l a t i o n w i t h empty base. However, t h e r e e x i s t two 0 - a r y r e l a t i o n s w i t h empty base:

(O,+)

and

. These

(0,-)

conventions agree w i t h t h e c a l c u l a t i o n o f t h e

number o f n - t u p l e s w i t h v a l u e s t a k e n f r o m a base o f f i n i t ' e c a r d i n a l p

The number o f n-ary r e l a t i o n s based on

p ; i.e.

elements i s " 2 t o t h e power pn

ordinal exponentiation coincides w i t h cardinal exponentiation

np

, for

'I.

p

Here

n and p

finite. Examples o f r e l a t i o n s . The u s u a l o r d e r i n g o f t h e i n t e g e r s i s t h e r e l a t i o n which s a t i s f i e s

R(x1,x2)

=

+

when

a ternary r e l a t i o n t a k i n g t h e value x1.x2 # x3 , where we s h a l l o f t e n use

x1 t

when

.

x2

and

-

when

x1.x2 = x3

x1

> x2 . A

and t h e v a l u e

i s t h e c o m p o s i t i o n l a w o f t h e group. I n s t e a d o f x,y,z .

R

group i s

- when x1,x2,x3

A multirelation with base E i s a f i n i t e sequence R o f r e l a t i o n s R1, ...,Rh ( h i n t e g e r ) , each w i t h base E Each Ri (i= 1, ...,h ) i s c a l l e d a component o f the multirelation R We c a l l t h e arity o f R t h e sequence (nl, ...,nh) o f a r i t i e s o f t h e components R1, Rh We say t h e n t h a t t h e m u l t i r e l a t i o n R i s (nl, ..., n h ) - x . The l e n g t h h o f t h e sequence o f i n d i c e s can be zero: i n t h i s case, t h e m u l t i r e l a t i o n i s reduced t o i t s base E . I n s t e a d o f t h e n o t a t i o n R1, R2,R3 , o f t e n we s h a l l use R,S,T I n t h e case where h = 2 , we w i l l say

.

.

.

...,

.

n

.

THEORY OF RELATIONS

30

a b i r e l a t i o n ; f o r h = 3 a t r i r e l a t i o n , e t c . F i n a l l y , t h e base o f a m u l t i r e l a t i o n R s h a l l be denoted I R I

.

Example. An ordered group i s a (3,2)-ary

b i r e l a t i o n which i s formed o f t h e t e r n a r y

group r e l a t i o n and t h e b i n a r y o r d e r i n g r e l a t i o n . denumeraaccording t o whether i t s base i s f i n i t e , i n f i n i t e , countable, denumerable or continuum-equipotent. The c a r d i n a l o f t h e mu t i re1 a t i o n R i s the c a r d i n a l o f i t s base J R I

A r e l a t i o n o r m u l t i r e l a t i o n w i l l be c a l l e d f i n i t e , i n f i n i t e , countable o r continuum-equipotent,

.

Given two m u l t i r e l a t i o n s

E

w i t h common base

R, S

, we

c a l l t h e concatenation

o f R and S , denoted (R,S) , the sequence o f components o f R f o l l o w e d by the components o f S , i n which case f o r the l a t t e r t h e i n d i c e s a r e increased by the number of terms i n R . 1.1. n-ARY RESTRICTION, n-ARY EXTENSION R be an n-ary r e l a t i o n w i t h base

Let

E

, and

let

F be a subset o f

E

. We

c a l l t h e n - a 3 r e s t r i c t i o n o f R t o F , denoted by R/F , t h e n-ary r e l a t i o n t a k i n g t h e same value f o r each n - t u p l e w i t h values i n F . The n o t i o n o f r e s t r i c t i o n o f a f u n c t i o n i n ch.1 5 1.3, i s more general than t h a t o f n-ary r e s t r i c t i o n : t h e former would a l l o w one t o r e s t r i c t R t o an a r b i t r a r y subset o f t h e s e t 'E o f n-tuples w i t h values i n E , and n o t n e c e s s a r i l y t o a subset o f t h e form 'F w i t h F S E However i n p r a c t i c e , t h e context w i l l make t h e meaning o f t h e ad-

.

j e c t i v e "n-ary" obvious: we t a c i t l y assume t h i s . For t h e a r i t y 0, t h e r e s t r i c t i o n t o F o f t h e 0-ary r e l a t i o n

(F,+) ; s i m i l a r l y w i t h

-

Given a r e l a t i o n R w i t h base E

sion o f Let X

R

R, R '

3

to

E+

be two

and a superset

any r e l a t i o n w i t h base E+

6

n

, we

have

Given a m u l t i r e l a t i o n R = (R1,-..,Rh) we d e f i n e t h e r e s t r i c t i o n o f R t o

.

F

R/X = R ' / X

R, R '

.

of

E

-

of .

, we

E

. If

, then

9-

c a l l an

E

is

R

.

f o r every subset

R = R'

.

w i t h base E and a subset F o f E , by R/F , t o be t h e m u l t i r e l a R w i t h base E and a superset any m u l t i r e l a t i o n w i t h base E+ + where any sequence (R;,. ,Rh)

1, ... ,h)

be two m u l t i r e l a t i o n s o f common a r i t y

I f f o r each subset X base E have R/X = R ' / X , then R = R '

-

E+

w i l l be

(E,+)

, denoted

Given a m u l t i r e l a t i o n t i o n (R1/F, ...,Rh/F) E+ o f E , we c a l l an extension o f R t o Ef whose r e s t r i c t i o n t o E i s R . E q u i v a l e n t l y , + each Ri i s an extension o f Ri t o E+ ( i = Let

.

whose r e s t r i c t i o n t o

n-ary r e l a t i o n s w i t h common base

E with cardinal

F

; t h i s remains v a l i d f o r empty

(nl,

. ...,nh)

E w i t h c a r d i n a l 6 Max(nl,

.

and w i t h common

...,nh) ,

Chapter 2

31

1.2. COMPATIBLE RELATIONS Two r e l a t i o n s ( o r m u l t i r e l a t i o n s ) w i t h t h e same a r i t y a r e s a i d t o be compatible i f f t h e y have t h e same r e s t r i c t i o n t o t h e i n t e r s e c t i o n o f t h e i r bases.

Let

6%

be a s e t o f m u t u a l l y c o m p a t i b l e r e l a t i o n s ( o r m u l t i r e l a t i o n s ) :

&, , based

( 1 ) t h e r e e x i s t s a common e x t e n s i o n o f t h e r e l a t i o n s i n

on t h e u n i o n

o f t h e i r bases; ( 2 ) l e t us denote by

t h e u n i o n o f t h e bases and by

E

n

t h e common a r i t y , o r

t h e maximum o f t h e common a r i t y ( f o r m u l t i r e l a t i o n s ) ; i f each of

E

n-element subset

i s covered by one o f t h e bases, t h e n t h e common e x t e n s i o n i s unique.

1.3. COHERENCE LEMMA Consider a s e t 9 o f s e t s F f o r each o f which we have a f i n i t e non-empty s e t UF o f m u l t i r e l a t i o n s based on F ( a l l o f t h e same a r i t y ) w i t h t h e f o l l o w i n g hypotheses: (1) 3 i s a d i r e c t e d system: i f F, F ' belong t o )3 , t h e n t h e r e e x i s t s an F" i n 3 with F"? F u F' ; ( 2 ) i f F, F ' belong t o 3 and F ' C F , t h e n e v e r y m u l t i r e l a t i o n b e l o n g i n g t o

UF , when r e s t r i c t e d t o F ' , y i e l d s an element o f U F , ; i n t h i s case, t h e r e e x i s t s a m u l t i r e l a t i o n R based on t h e u n i o n o f t h e s e t s F i n 3 , such t h a t f o r each F t h e r e s t r i c t i o n R/F belongs t o UF (uses t h e u l t r a f i l t e r axiom; ZF s u f f i c e s ift h e F a r e f i n i t e and t h e i r u n i o n c o u n t a b l e ) . 0 Denote by E t h e u n i o n o f t h e F i n 3 To each F a s s o c i a t e t h e s e t VF o f e x t e n s i o n s t o E o f m u l t i r e l a t i o n s b e l o n g i n g t o UF . The supersets o f t h e VF c o n s t i t u t e a f i l t e r on t h e s e t o f m u l t i r e l a t i o n s based on E w i t h t h e g i v e n a r i t y . Indeed i f F, F ' belong t o 3 , t h e n t h e r e e x i s t s i n 3 an F " ? FuF' ; hence VFn V F 8 i s a s u p e r s e t o f VF,, . Take an u l t r a f i l t e r e x t e n d i n g t h i s f i l t e r .

.

For each F o f '3 , p a r t i t i o n t h e m u l t i r e l a t i o n s i n VF

i n t o a f i n i t e number

F o f these m u l t i r e l a t i o n s .

o f classes, each c l a s s d e f i n e d by t h e r e s t r i c t i o n t o

One and o n l y one o f t h e s e c l a s s e s i s an element o f o u r u l t r a f i l t e r : denote by

RF

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

RF

belongs t o

UF

. Hence

the

RF

a r e m u t u a l l y c o m p a t i b l e i n t h e sense o f 1.2 above: t h e e x i s t e n c e o f t h e m u l t i relation

R

stated i n our proposition follows. If

a r e f i n i t e subsets o f

E

, then

i s c o u n t a b l e and t h e

E

F

t h e u l t r a f i l t e r becomes s u p e r f l u o u s , so t h a t t h e

axioms o f ZF a r e s u f f i c i e n t . 0

1.4. The coherence lemma i m p l i e s , and hence i s e q u i v a l e n t t o t h e u l t r a f i l t e r axiom. 0

Let

e

be a s e t ,

p(e)

be t h e s e t o f subsets o f

e

, and 'bQ

a f i l t e r on

e

F be a f i n i t e s e t o f subsets o f e which i s c l o s e d w i t h r e s p e c t t o union, i n t e r s e c t i o n and t a k i n g complements ( i n e ) . To each F a s s o c i a t e t h e s e t UF o f unary r e l a t i o n s X w i t h base F which s a t i s f y t h e f o l l o w i n g c o n d i t i o n s : Let

.

THEORY OF RELATIONS

32

, i f a E 8 t h e n t h e v a l u e X(a) = + ; i f e-a F , we have o p p o s i t e values X(e-a) # X(a) ;

f o r each

a& F

f o r each

a E

i f a, b E F so a n b e F and X(a) = X(b) = + , t h e n i f a, b E F and a c b and X(a) = + , t h e n X(b) = The s e t

. The

F

i s non-empty f o r each

UF

set o f the

2 then

E

+

X(anb) =

.

+

F

X(a) = -;

;

forms a d i r e c t e d system,

so we can a p p l y t h e coherence lemma. Consequently t h e r e e x i s t s a unary r e l a t i o n

based on T ( e )

whose r e s t r i c t i o n t o each

F

belongs t o

. The

UF

subsets o f

which g i v e t h e value (+) t o t h i s unary r e l a t i o n c o n s t i t u t e an u l t r a f i l t e r on which i s f i n e r t h a n

w.0

e e

1.5. A v a r i a n t o f t h e coherence lemma i s g i v e n by RADO 1949. Consider a s e t o f

f i n i t e mutually d i s j o i n t sets consider a choice f u n c t i o n fI(a)

of

the J

a

, and

a

of

I

. Then

, and

a

fI

there e x i s t s a choice f u n c t i o n

I

f o r each f i n i t e s e t

with

f/I

f o r each f i n i t e s e t

which a s s o c i a t e s t o each o f the

equal t o t h e r e s t r i c t i o n

of

a

,

I an element

whose domain i s t h e s e t o f

f

, there

a

I o f sets

a

e x i s t s a f i n i t e superset

.

f,/I

The preceding RADO's lemma p l u s t h e axiom o f c h o i c e f o r f i n i t e s e t s i s e q u i v a l e n t t o t h e coherence lemma (BENEJAM 1970). 1.6. AXIOM OF DEPENDENT CHOICE Let

E

, such

satisfying

R(x,y)

be a b i n a r y r e l a t i o n w i t h base

R

e x i s t s a t l e a s t one

y

of

E

c h o i c e a s s e r t s t h a t , g i v e n such an ai

of

satisfying

E

R(ai,ai+l)

R =

+

, there

t h a t f o r each =

+ . The

x

of

E

there

axiom o f dependent

e x i s t s an w - s e q u e n c e o f elements

f o r each i n t e g e r

i (MOSTOWSKI 1948).

The axiom o f dependent c h o i c e o b v i o u s l y f o l l o w s f r o m t h e axiom o f c h o i c e . I t i s proved t h a t t h e dependent c h o i c e i s s t r i c t l y weaker t h a n t h e axiom o f

c h o i c e : see f o r i n s t a n c e JECH 1973 p. 122 and f o l l o w i n g . The c o u n t a b l e axiom o f choice, s t a t e d i n ch.1

5

2.5,

f o l l o w s from t h e axiom o f

dependent choice. 0 S t a r t f r o m an

take

R

R(x,y) =

w -sequence o f non-empty m u t u a l l y d i s j o i n t s e t s ai

t o be t h e b i n a r y r e l a t i o n based on t h e u n i o n o f t h e

+

i f f t h e r e e x i s t s an

i with

x

E

ai

ai

and y e ai+l

(ii n t e g e r ) by

, defined

.0

The c o u n t a b l e axiom o f c h o i c e i s s t r i c t l y weaker t h a n t h e axiom o f dependent c h o i c e : see JECH 1973 p. 119 and f o l l o w i n g . F i n a l l y , from t h e axiom o f dependent choice, assumed t o be c o n s i s t e n t , one cannot deduce t h e axiom o f c h o i c e f o r f i n i t e s e t s , s t a t e d i n ch.1

5

2.10.

The p r o o f i s

due t o MOSTOWSKI 1948 w i t h o u t t h e axiom o f f o u n d a t i o n , and t o FEFERMAN 1965 w i t h foundation.

,

Chapter 2

33

1.7. NEGATION, CONJUNCTION, DISJUNCTION Given a r e l a t i o n

, i t s negation

R

i s t h e r e l a t i o n w i t h same base and a r i t y ,

R

1

always t a k i n g t h e o p p o s i t e v a l u e . Given conjunction

R

disjunction

R v S

A

takes t h e v a l u e (+) i f f

S

w i t h t h e same base and a r i t y , t h e

R, S

and

R

t a k e t h e value ( + ) . The

S

t a k e s t h e v a l u e (+) i f f e i t h e r

or

S

, denoted

by

R

t a k e s t h e v a l u e (+).

OF A BINARY RELATION, RETRO-ORDINAL

CONVERSE

Given a b i n a r y r e l a t i o n

R

, the

converse o f

R

,i s

R-

the r e l a t i o n

f o r e v e r y x, y . I n p a r t i c u l a r w i t h t h e same base, such t h a t R-(x,y) = R(y,x) we c o n s i d e r an o r d i n a l o( as t h e b i n a r y r e l a t i o n based on t h e s e t a l r e a d y denot e d by

o(

(the set o f ordinals

QUASI-ORDERING,

P A R T I A L ORDERING,

CHAIN,

WELL-ORDERABLE SET,

WELL-ORDERING,

ORDERING,

) , and t a k i n g t h e value (+) i f f

o(

x E y

or

w i l l be c a l l e d

x ay ) . The converse r e l a t i o n o(

x = y (denoted a l r e a d y b y a retro-ordinal.

2 -

<

WELL-FOUNDED PARTIAL

HAUSDORFF-ZORNAXIOM

A q u a s i - o r d e r i n g ( o r p r e - o r d e r i n g ) i s a b i n a r y r e f l e x i v e and t r a n s i t i v e r e l a t i o n ( t h e s e n o t i o n s assumed t o be known). I f A x 4 y (mod A)

elements i n t h e base, t h e n equal t o y

, means

that

A(x,y)

o r i s g r e a t e r t h a n o r equal t o than

y

,if

o r again y

x 6 y x

. We

+

. We

x

x

, denoted b y x

y

two

y

>, x (mod A) o r

x < y (mod A) A(x,y)

=

or

x

y

follows

s t r i c t l y less

+ and A(y,x)

=

y

if

x

6y

l y (mod A)

or y

6x

. x

t h e base, t h e e q u i v a l e n c e c l a s s of

such

that

A(x,y)

= A(y,x)

=

+

;

; otherwise

An e q u i v a l e n c e r e l a t i o n i s a symmetric q u a s i - o r d e r i n g . Given a element x

-

; a l s o " s m a l l e r " i s synonymous w i t h " l e s s

i s comparable w i t h

i s incomparable w i t h

also write

write

x, y

precedes or i s l e s s t h a n o r

x

; i n o t h e r words

s t r i c t l y greater than

than". We say t h a t x

and y$

=

x

i s a q u a s i - o r d e r i n g and or

(mod A)

i s the s e t o f those y

of

.

A p a r t i a l o r d e r i n g i s an a n t i s y m m e t r i c q u a s i - o r d e r i n g ( n o t i o n assumed t o be

known). We a l r e a d y have t h e example o f i n c l u s i o n .

A

Given a q u a s i - o r d e r i n g

, the

e q u i v a l e n c e r e l a t i o n generated by

A , i s the

r e l a t i o n w i t h t h e same base, t a k i n g t h e v a l u e (+) i f f x < y and y 4 x (mod A ) . F o r each element x , t h e e q u i v a l e n c e c l a i s o f x (mod A) i s t h e c l a s s o f x modulo t h e e q u i v a l e n c e r e l a t i o n generated by

A

.

Take as a new base t h e s e t o f e q u i v a l e n c e c l a s s e s , and w r i t e

.

( e q u i v . c l a s s o f x)

We t h u s o b t a i n a p a r t i a l o r d e r i n g ( e q u i v . c l a s s o f y ) , i f x \ < y (mod A) c a l l e d t h e p a r t i a l o r d e r i n g generated by t h e q u a s i - o r d e r i n g A

4

Let

A

be a p a r t i a l o r d e r i n g ,

n o t i o n of maximum o f minimum, denoted

D (mod A)

Min D

. Recall

D

a subset o f t h e base

IA I

. . We

assume t h a t t h e

, denoted Max D , i s known. S i m i l a r l y f o r t h e t h a t an element i s maximal i n

D (mod A ) ,

i f it

34

THEORY OF RELATIONS

belongs t o D and there i s no element of D which s t r i c t l y follows i t . Analogous notion of a minimal element. The maximum, i f i t e x i s t s , i s maximal, b u t the converse i s f a l se . Similarly f o r the minimum. These notions extend in an obvious manner t o a quasi-ordering. Here there can e x i s t several maximums and several minimums, which are equivalent t o each other in the sense of the qenerated equivalence relation. The reader i s assumed t o know the notion of uoper bound of a s e t D (mod A ) as well as t h a t of lower bound. The supremum of D , denoted by Sup D , i f i t e x i s t s , i s the minimum in the s e t of upper bounds. Hence x > r Sup D i s equivalent t o x greater than or equal t o every element in D . If Sup D belongs t o D , then i t i s the maximum. Analogous definition of the infimum, denoted by Inf D These notions appeared already in ch.1 5 2.1 f o r ordinals, in ch.1 5 4.5 and 4.6 f o r reals.

-

.

INTERVAL, INITIAL AND FINAL INTERVAL The reader i s assumed t o be familiar with the notion of an element z between x and y (mod A) , or z intermediate between x and y , as well as t h a t of an element s t r i c t l y intermediate. An interval of A i s a subset of the base which i s closed with respect t o the notion of intermediate (mod A ) . An i n i t i a l interval o r i n i t i a l segment of A i s a subset closed with respect t o " l e s s than" . A final interval i s a subset closed with respect t o "greater t h a n "

.

2 . 1 . Let A be a p a r t i al ordering. Then every subset of the base I A 1 without a minimal element i s i n f i n i t e . Similarly for a subset without a maximal element. 0 To each element x of the subset 0 under consideration, associate the s e t Dx of elements of D which are less t h a n or equal t o x (mod A ) . None of the Dx i s minimal under inclusion (see ch.1 5 1.1, definition of a f i n i t e s e t ) . 0 2 . 2 . AMALGAMATION LEMMA

Let A , B be two p a r t i al orderings having the same re stric tion t o the intersection of the bases. Then there e x i s t s a partial ordering which i s an extension of b o t h A .-and B , based on the union of the bases. OWrite x G y when x , y G l A l and x s y ( m o d A ) , o r w h e n w e h a v e t h e s a m e condition for B , or when x belongs t o I A l , y belongs t o I B I and there (mod A) and t $ y (mod B ) , e x is t s an element t i n the intersection with x or when we have the same condition when interchanging A and B . Finally write x\y in the other cases. 0

-

st

2.3. CHAIN, ORDERABLE SET A chain, or total orderinq, i s a p ar t i al ordering whose elements are mutually

Chapter 2

35

comparable. For example, we shall denote by Z the chain of the positive and negative integers, and by Q the chain of the rationals. The previous amalgamation lemma 2 . 2 extends t o the case of two chains, the common extension i t s e l f being a chain. However, t h i s lemma does not extend t o t r e e s , defined i n ch.4 5 6. 0 Take a t r e e on a,b,c,d with a,b,c mutually incomparable, d < a , d c b and d I c ; and another t r e e on a,b,c,e with e < b , e < c and e l a . Then e i t h e r d < e < c or e Q d < a : contradiction. 0 W e say t h a t a s e t E i s orderable i f f there e x i s t s a chain based on E . Using only the axioms o f ZF, every f i n i t e s e t i s orderable (induction: see ch.1 5 1.1). ORDERING AXIOM The ordering axiom a s s e r t s t h a t every s e t i s orderable. I t follows from the ultraf i l t e r axiom, or equivalently from the coherence lemma 1.3. 0 Let E be a s e t ; t o each f i n i t e subset F of E , associate the s e t U F of chains based on F . By 1.3 there e x i s t s a relation R based on E every of whose f i n i t e r e s t r i c t i o n i s a chain; thus R i s a chain. 0 The ordering axiom i s s t r i c t l y weaker t h a n the u f t r a f i l t e r axiom (JECH 1973 p.100). The ordering axiom implies the axiom of choice f o r f i n i t e s e t s (see ch.1 5 2.10). 0 Given a s e t of mutually d i s j o i n t f i n i t e s e t s , i t suffices t o take a chain A based on the union: t o each f i n i t e s e t we associate i t s minimum (mod A ) . 0 The axiom of choice f o r f i n i t e s e t s i s s t r i c t l y weaker than the ordering axiom: see LAUCHLI 1964 f o r ZF without foundation, completed f o r ZF by PINCUS 1972. The axiom of choice f o r f i n i t e s e t s does not follow from the axiom of deoendent choice: see 5 1 . 6 . Hence the ordering axiom does n o t follow from dependent choice. 2.4. WELL-FOUNDED PARTIAL ORDERING OR QUASI-ORDERING; WELL-ORDERING

W e say t h a t a partial ordering o r quasi-ordering i s well-founded i f f every non-empty subset of i t s base has a t l e a s t one minimal element. A well-founded chain, o r t o t a l ordering, i s called a well-ordering. Every f i n i t e p a r t i a l ordering i s well-founded. Every r e s t r i c t i o n of a well-founded p a r t i a l ordering i s well-founded. Given a p a r t i a l ordering A , the reader i s assumed t o know the notion of a sequence with values in A which i s incr?asing, decreasing, s t r i c t 1 2 or otherwise.

A s a t i s f i e s the following conditions: (1) there i s no s t r i c t l y decreasing (mod A) W-sequence; ( 2 ) every t o t a l l y ordered r e s t r i c t i o n of A i s well-founded, hence a well-ordering; equivalently every non-empty t o t a l l y ordered r e s t r i c t i o n of A has a minimum. Every well-founded p a r t i a l ordering

THEORY OF RELATIONS

36

Conversely, each of the conditions ( l ) , ( 2 ) implies, hence i s equivalent t o saying t h a t A i s well-founded. This uses the axiom of dependent choice, y e t ZF suffices i f A i s countable, or i f the base I A l i s well-orderable, in the sense below. In the general case, apply dependent choice t o the relation y < x (mod A ) .

2 . 5 . WELL-ORDERABLE SET We say t h a t E i s well-orderable i f f there e x i s t s a well-ordering based on E . For example any f i n i t e or denumerable s e t i s well-orderable. A s e t E i s well-orderable i f f there e x i s t s a choice function on the s e t < non-empty subsets of E 0 Let f be a choice function on non-empty subsets. Let a. = f(E) . Let u be a non-zero ordinal, Du the s e t of a l l a i ( i i u ) Let a u = f(E-DU) , as long as possible, thus reaching a s e t DU = E . 0 WELL-ORDERING AXIOM, TRICHOTOMY AXIOM I t follows t h a t the axiom of choice i s equivalent t o saying t h a t every s e t i s well-orderable. Or again t h a t every cardinal i s an aleph, or t h a t every i n f i n i t e cardinal has the form c . ) ~ ( o( ordinal index: see ch.1 5 6.1 t o 6 . 4 ) . The axiom of choice i s equivalent t o the trichotomy axiom which says t h a t , given any two cardinals a , b , e i t h e r a < b or a = b or a > b . 0 If every cardinal i s an aleph, then trichotomy holds. Conversely, given a s e t a and the Hartogs u of a (see ch.1 5 6.2), i f trichotomy holds then necessar i l y a i s subpotent t o o( , hence a i s well-orderable. 0

-

.

.

2.6. MAXIMAL CHAIN Let A be a partial ordering and C be a t o t a l l y ordered r e s t r i c t i o n of A . The chain C i s said t o be maximal (under inclusion, mod A) i f f every t o t a l l y ordered r e s t r i c t i o n of A extending C i s identical t o C . Let E be a s e t ; denote by X any well-ordering based on a subset of E Write

.

X6 X '

.

i f f X i s an i n i t i a l interval of X ' The well-founded p a r t i a l ordering thus defined on the s e t of X will be called the interval-orderinqon E . (1) A s e t E i s well-orderable i f f there e x i s t s a maximal chain which i s a r e s t r i c t i o n of the interval-ordering on E ( 2 ) Let A be a partial ordering and C be a t o t a l l y ordered r e s t r i c t i o n of A Let U be any chain which i s both a r e s t r i c t i o n of A and an extension of C Every function f which t o each U associates f(U) , a t o t a l l y ordered rest r i c t i o n of A and extension of U , has a fixed point V such t h a t f(V) = V 0 Index by ordinals a sequence of chains U i s t a r t i n g with Uo = C ; s e t Ui+l = f ( U i ) and, f o r i a l i m i t ordinal, l e t Ui be the common extension of U j ( j < i ) t o the union of t h e i r bases. 0

.

.

.

.

37

Chapter 2

2.7. MAXIMAL CHAIN AXIOM, OR HAUSDORFF-ZORN AXIOM T h i s axiom, g o i n g back t o HAUSDORFF 1914, t h e n t a k e n up by KURATOWSKI, MOORE and

A

t h e n ZORN, i s s t a t e d as f o l l o w s . Given a p a r t i a l o r d e r i n g which i s a r e s t r i c t i o n o f

and a c h a i n

C

e x i s t s a c h a i n which i s an e x t e n s i o n o f

C

.

and maximal (mod A) By 2.6.(2)

, there

A

above, t h e axiom o f c h o i c e i m p l i e s t h e maximal c h a i n axiom. By 2.6.(1),

the

By 2.5,

t h e maximal c h a i n axiom i m p l i e s t h a t every s e t i s w e l l - o r d e r a b l e . maximal c h a i n axiom i s t h e n e q u i v a l e n t t o t h e axiom o f c h o i c e . 2.8.

The u l t r a f i l t e r axiom f o l l o w s f r o m t h e axiom o f choice.

0 Consider t h e s e t o f f i l t e r s on a g i v e n s e t , w i t h t h e comparison o r d e r i n g

" f i n e r f i l t e r t h a n " . Take a maximal c h a i n e x t e n d i n g t h e c h a i n reduced t o a g i v e n

. The

filter

u l t r a f i l t e r g i v e n by t h e u n i o n o f t h e f i l t e r s b e l o n g i n g t o t h e

F.0

maximal c h a i n i s f i n e r t h a n

The u l t r a f i l t e r axiom i s s t r i c t l y weaker t h a n t h e axiom o f choice: HALPERN, LEVY 1971 p. 83-134. 2.9. FREE SUBSET, ANTICHAIN, MAXIMAL FREE SUBSET, MAXIMAL ANTICHAIN

,a

A

Given a p a r t i a l o r d e r i n g

subset o f i t s base i s c a l l e d

i t s elements a r e m u t u a l l y incomparable (mod A) f r e e subset

D

i s c a l l e d an a n t i c h a i n (mod A )

r e l a t i o n based on

D

. The

free (mod A)

restriction

. It reduces

A/D

iff

t o such a

t o the i d e n t i t y

.

A f r e e subset, and t h e c o r r e s p o n d i n g a n t i c h a i n , a r e c a l l e d maximal (under i n c l u -

A

s i o n ) i f f t h e r e i s no p r o p e r s u p e r s e t which i s f r e e . Given a p a r t i a l o r d e r i n g and a f r e e subset

D

, there

e x i s t s a maximal f r e e subset i n c l u d i n g

axiom o f choice; ZF s u f f i c e s i f

A

0 (uses

i s c o u n t a b l e ) : a p p l y t h e maximal c h a i n axiom

t o t h e i n c l u s i o n among f r e e subsets.

3 - ISOMORPHISM^

AUTOMORPHISM, H E I G H T OF A WELL-FOUNDED P A R T I A L ORDERING, SUM AND PRODUCT OF CHAINS, HOMOMORPHIC IMAGE §

Let

n

an

be a non-negative i n t e g e r ,

n - a r y r e l a t i o n w i t h base

We say t h a t

f

transforms o r

f

i s an isomorphism o f

(=

+

o r -)

and we s e t

f"(R)

= (El,+)

n-ary r e l a t i o n w i t h base

E' ; l e t

f

takes

into

onto

f o r a l l elements

A relation R' from

R

an

R

x1 or

R'

R

be a b i j e c t i o n f r o m

iff R ' ( f x l

,...,xn R = (E,-)

in

. This

E

, denoted

,...,fx,)

. For

and t h e n

i s s a i d t o be isomorphic w i t h

R onto R '

R'

R

R'

= R(xl

n = 0

E'

onto

E 5

and

E

f"(R)

R'

.

, or that

,..., xn)

, either

f"(R) = (El,-)

R = (E,+)

.

i f f t h e r e e x i s t s an isomorphism

c o n d i t i o n i s r e f l e x i v e , symmetric and t r a n s i t i v e ,

y i e l d i n g an e q u i v a l e n c e r e l a t i o n o n e v e r y s e t o f r e l a t i o n s o f a g i v e n a r i t y .

38

THEORY OF RELATIONS

ISOMORPHISM TYPE, ORDER TYPE Modeled after the definition of "cardinal" in ch.1 0 5.4, we consider the relations isomorphic with R , and among such, those whose base has minimum fundamental rank. These form a set, called the isomorphism type of R (the order type if R is a chain, or total ordering). Thus two relations are isomorphic iff they have the same type. AUTOMORPHISM, EMPTY FUNCTION Given a relation R with base E , a permutation f of E i s called an automorphism of R iff f i s an isomorphism from R onto R . The automorphisms of R form a group of permutations of E . We adopt the convention that the empty function, which is a bijection of the empty set onto itself, is also an automorphism of each relation with empty base. In particular it is an automorphism of the 0-ary relation with empty base and value ( + ) , denoted (O,+) , and also of (0,-). However (O,+) and (0,-) are not isomorphic. These definitions and conventions extend to mu tirelations. Given a multirelation R = (R1 ,..., Rh) with base E and R' = (Ri,. .,Rb) with base E ' , a bijection f from E onto E' transforms R into R' or is an isomorphism from R onto R' , denoted R' = f"(R) , iff for each i = 1, ...,h , the function f is an isomorphism of the component Ri onto the component R; . In other words fo(R1 ,...,Rh) = (fo(R1) ... .,fo(Rh)) . 3.1. (1) Let A be a well-ordering and f be an isomorphism from A onto a restriction of A . Then fx & x (mod A ) for each element x of A . ( 2 ) Given a well-ordering, its unique automorphism is the identity. Given two well-orderings A, B , there exists at most one isomorphism from A onto B . ( 3 ) Given a well-ordering A , no proper initial interval of A is isomorphic with A . In particular two isomorphic ordinals are identical.

-

3.2. HEIGHT IN A WELL-FOUNDED PARTIAL ORDERING

Let A be a well-founded partial ordering. To each element x of IAI , associate as follows an ordinal called the height of x (mod A) and denoted Ht x If x is a minimal element, let Ht x = 0 . Let o( be a non-zero ordinal; assume that each ordinal (o( has been associated to at least one element, but that there still remain elements in the base to which no height (q has been associated. Then associate the height o( to minimal elements among these. Given a well-founded partial ordering A , there i s a unique height associated to each element of the base I A l . Moreover, for each element x of height cr( and every ordinal /3 < d , there exists at least one element < x (mod A) with height /3.

.

Chapter 2 However, g i v e n

>

elements (mod A) 0

Let

x

x


(d

> /s

o f heights and

x (y

(mod A)

, then

, it

i s p o s s i b l e t h a t no element

x

fs. a

< e <.d

and i t s o n l y s t r i c t upper bound i s

If

f5 2 4 , even i f t h e r e e x i s t

of h e i g h t o( and an o r d i n a l

(mod A)

exists with height a

39

<

Ht x

with

elb

, with

d

and

height 3

elc

.

Then

e

has h e i g h t 1

.0

.

Ht y

Two d i s t i n c t elements of t h e same h e i g h t a r e incomparable. But two incomparable elements may have d i f f e r e n t h e i g h t s . 3.3. HEIGHT OF A WELL-FOUNDED PARTIAL ORDERING The h e i g h t s of t h e elements o f a w e l l - f o u n d e d p a r t i a l o r d e r i n g an o r d i n a l , c a l l e d t h e h e i q h t o f H t A = Sup((Ht x ) + l )

for all

A

x (mod A )

The h e i g h t o f an element

and denoted

IAI

X E

.

Ht A

.

A

constitute

E q u i v a l e n t l y we have

i s also the heiqht o f the r e s t r i c t i o n o f

I n p a r t i c u l a r , t h e fundamental r a n k o f a s e t

a

(ch.1

5

5.2) i s t h e h e i g h t o f

t h e w e l l - f o u n d e d p a r t i a l o r d e r i n g based on t h e t r a n s i t i v e c l o s u r e o f d e f i n e d by h

i f f t h e r e e x i s t s a f i n i t e sequence

x
i n t e g e r and

ti

E ti+l

f o r each

3.4. Every w e l l - o r d e r i n g A

i< h

x

into

o f t h e o t h e r , by c h . 1

with

Ht A

t h e isomorphism

.

one i s i s o m o r p h i c t o an

n i t i a l interval

A

2 . 1 ( t r i c h o t o m y ) . Given a w e l l - o r d e r i n g

i s isomorphic w i t h an i n i t i a l i n t e r v a l o f

A

restriction o f

5

, and

.

Ht x

Consequently, g i v e n two w e l l - o r d e r i n g s ,

a

... th = y

to = x,

i s isomorphic t o t h e o r d i n a l

phism t r a n s f o r m s each element

A

.

, each I f two w e l l -

o r d e r i n g s a r e each isomorphic t o a r e s t r i c t i o n o f t h e o t h e r , t h e n t h e y a r e isomorphic. 3.5.

of

(1) L e t

A

A

be a well-founded p a r t i a l o r d e r i n g . Then each r e s t r i c t i o n

i s a well-founded p a r t i a l ordering w i t h

Ht B

6

Ht A

.

.

Given a w e l l - f o u n d e d p a r t i a l o r d e r i n g ordered r e s t r i c t i o n o f

A

A

does n o t r e a c h

B

Indeed f o r each

x o f 1 6 1 we have H t x (mod B ) 4 H t x (mod A) ( 2 ) I f B i s an i n i t i a l i n t e r v a l o f A c o n t a i n i n g x , then H t x (mod H t x (mod A) . Consequence o f 3.2 ( i n d u c t i o n on H t x ) . element

B)

=

, it

can happen t h a t a maximal t o t a l l y H t A . Even t h e supremum o f t h e

.

Ht A i , t a k e a c h a i n isomorphic t o i ; t h e s e c h a i n s a r e assumed t o be m u t u a l l y incomparable. We o b t a i n a well-founded p a r t i a l o r d e r i n g w i t h h e i g h t s o f t h e maximal c h a i n s can be s t r i c t l y l e s s t h a n

0 F o r each i n t e g e r

height

A


t o elements

a ,b u t

i n which e v e r y maximal c h a i n i s f i n i t e .

40

THEORY OF RELATIONS

Another example. Take denumerably many c o p i e s isomorphic t o t h e p r e v i o u s o r d e r i n g Above t h e

jth copy ( j i n t e g e r ) , p u t a c h a i n isomorphic t o

j

. Let

these

d i f f e r e n t o r d e r i n g s w i t h d i s j o i n t bases be m u t u a l l y incomparable. We o b t a i n a

, in

well-founded p a r t i a l o r d e r i n g w i t h h e i g h t w.2

which e v e r y maximal c h a i n

i s finite. 0 3.6. ORDINAL SUM, HOMOMORPHIC IMAGE OF A CHAIN

A, B be two c h a i n s w i t h d i s j o i n t bases. We c a l l t h e o r d i n a l sum, o r s i m p l y t h e sum A+B t h e c h a i n based on t h e u n i o n o f t h e bases, which i s t h e common e x t e n s i o n o f A and B f o r which each element i n I A I precedes each element Let

in

IBI

.

T h i s g e n e r a l i z e s t h e n o t i o n o f sum among o r d i n a l s (ch.1

T h i s sum agrees w i t h isomorphism, i n t h e sense t h a t i f and

6'

to

, then

B

i s isomorphic t o

A'+B'

commutative, even up t o isomorphism:

A

More g e n e r a l l y , l e t ly disjoint intervals

\Ail

t&of

. The

A

chain

, again

Ai

denoted

= rAi

A A

. The

i

.

Take t h e new base t o be t h e s e t

I by p u t t i n g

i s called the

A

i< j

I A J. 1 . T h i s

set o f the intervals

i f f each element

chain

I i s called

i s c a l l e d a decomposi-

Ai

I-=o r t h e sum a l o n g

I

o f the

Ai

:

.

(iE I ) Each homomorphic image o f a c h a i n A notation

A

It i s associative but not

w + 1 and l + w a r e n o t i s o m o r p h i c .

i s s t r i c t l y l e s s t h a n each element o f

a homomorphic image o f

.

3.1).

be a c h a i n . Consider a p a r t i t i o n o f i t s base i n t o mutual-

o f t h e s e i n t e r v a l s , and d e f i n e t h e c h a i n of

A+B

5

i s isomorphic t o

A'

i s isomorphic t o a r e s t r i c t i o n o f

A

(uses axiom o f c h o i c e ) : t a k e one element i n each i n t e r v a l o f t h e decomposition. The converse i s f a l s e . F o r example, t h e c h a i n w o f t h e i n t e g e r s i s a r e s t r i c t i o n o f W+1

, but

i n t e r v a l . Hence

each decomposition o f ~

+ i n1t o i n t e r v a l s has a l a s t

w i s n o t a homomorphic image o f w + l Q

Another counterexample. The c h a i n

.

o f the rationals i s a restriction o f the

c h a i n o f t h e r e a l s , b u t n o t a homomorphic image. Indeed DEDEKIND's theorem

5

4.6) would be v i o l a t e d by a decomposition a l o n g Q o f t h e c h a i n R of t h e r e a l s , and a p a r t i t i o n o f Q , t h u s a p a r t i t i o n o f R i n t o an i n i t i a l i n t e r v a l w i t h o u t a maximum and t h e complementary i n t e r v a l w i t h o u t a minimum. (ch.1

Note t h a t a homomorphic image o f a homomorphic image o f 3.7.

A, B t h e o r d i n a l p r o d u c t

A

.

i s t h e c h a i n based on t h e

A.B

C a r t e s i a n p r o d u c t o f t h e bases, d e f i n e d by p u t t i n g

<

i s an image o f

ORDINAL PRODUCT OF CHAINS

Given two c h a i n s y

A

y ' (mod B )

t o each y

of

t h e ordered p a i r

or

y = y'

and

x

1B1 the chain A (y,x)

Y

6 x'

(mod A)

.

iff (y,x),( (y',x') I n o t h e r words, by a s s o c i a t i n g

o b t a i n e d by r e p l a c i n g each

x

of

A

5

3.2).

, t h e n t a k i n g t h e 8-sum o f t h e A

Y ' T h i s p r o d u c t g e n e r a l i z e s t h e n o t i o n o f p r o d u c t among o r d i n a l s (ch.1

by

41

Chapter 2

A'

T h i s p r o d u c t agrees w i t h isomorphism: i f then

A'.B'

i s isomorphic t o

up t o isomorphism:

~3

and

.2

A.6

.

i s isomorphic t o

A

and

B'

to

B,

I t i s a s s o c i a t i v e b u t n o t commutative, even

a r e n o t isomorphic.

2. W

3.8. CARDINAL SUM, PRODUCT AND EXPONENTIATION REVISITED L e t us r e t u r n t o t h e o p e r a t i o n s between alephs and between c a r d i n a l s (ch.1 § 5.5) w i t h t h e means now a f f o r d e d by t h e n o t i o n s o f isomorphism and w e l l - o r d e r i n g . a

(1)

be an i n f i n i t e aleph; t h e n

( 2 ) k t a, b b = a x b a

a

+a

= a x a = a

.

be two alephs, a t l e a s t one o f which i s i n f i n i t e ; t h e n = Max(a,b) ( f o r t h e p r o d u c t we assume t h a t a, b # 0 ) .

+

0 It s u f f i c e s t o e s t a b l i s h

a x a = a : indeed

+a

a

= a

f o l l o w s by BERNSTEIN-

SCHRODER; by t h e same argument ( 2 ) f o l l o w s from (1). L e t E be t h e f o l l o w i n g t o t a l o r d e r i n g r e l a t i o n between two couples o f o r d i n a l s ( d , f s ) and ( ~ ' , f i ': )e i t h e r M a x ( & , p ) < M a x ( o ( ' , / j ' ) , o r the maximums are equal and fi <& or A = 0 ' anb 4 F o r coup1 es o f o r d i n a l s , t h i s comparison i s a w e l l - o r d e r i n g . Thus t h e r e e x i s t s a c o n d i t i o n ( & , f i , r ) which t o each c o u p l e ( d , ( 3 ) which i s isomorphic w i t h t h e s e t o f associates one and o n l y one o r d i n a l

so('.

couples

) . We have ?f>/ M a x ( d , p ) . f ("i, ~ 0 , ' 6 )a s s o c i a t e s t o each o(

< (6.0)(mod

take /3 = 0 : t h e n This

8

product

increases as o( increases, and

.

o<

o( x

, t h e n Sup gi

3

an o r d i n a l

o(

i s equipotent w i t h the Cartesian

Moreover, g i v e n a s e t o f o r d i n a l s

are associated t o them by

I n particular,

4

xi

and t h e

which

.

i s associated t o

Everything comes down t o p r o v i n g t h a t , f o r o( i n f i n i t e , t h e a s s o c i a t e d 8 i s the smallest = w Denote by F o r o( = w we have

.

.

equipotent w i t h o(

i s s t r i c t l y subpotent t o , t h u s t o o( x o ( . T h i s o( does n o t have any o r d i n a l o( ' <' P( which i s e q u i p o t e n t w i t h o( Indeed we

o r d i n a l f o r which o(

.

would have o( Thus q

8

'

e q u i p o t e n t w i t h o(

i s n e c e s s a r i l y an aleph. Denote by o( t h e o r d i n a l a s s o c i a t e d t o ct

Fi 6 4

.

o(

a x 4.

equipotent w i t h

the ordinals

.

<

o(

ri

and by

by By h y p o t h e s i s each is so s t r i c t l y subpotent t o o( , so Ti < 4 ; t h u s By t h e p r e c e d i n g d i s c u s s i o n = Sup x i * % : contradiction. 0

, and

equipotent w i t h o( Sup

' x 6 ', hence

Modulo t h e axiom o f choice, o u r p r o p o s i t i o n extends t o a r b i t r a r y c a r d i n a l s . Note t h a t t h e e q u a l i t y

a x a = a

f o r every i n f i n i t e cardinal

a

i s equiva-

l e n t t o t h e axiom o f c h o i c e (TARSKI 1924; see a l s o RUBIN 1963). 3.9. k t a

be an i n f i n i t e a l e p h and l e t

Indeed a = a x b

~

so

a2 = (axb)2 = a(b2)

2 6 b

>,

6a

ab

.

; then ab = a2

.

Modulo t h e axiom o f choice, t h i s i d e n t i t y extends t o a r b i t r a r y c a r d i n a l s .

THEORY OF RELATIONS

42

3.10. Given an i n f i n i t e c a r d i n a l

a

,let

o(

be the Hartogs aleph f o r sets o f

c a r d i n a l a (ch.1 5 6.2). E i t h e r N = w ; then a i s i n f i n i t e b u t n o t D e d e k i n d - i n f i n i t e (ch.1

5

2 . 6 ) ; then

+

1 i s immediately g r e a t e r than a , i n t h e sense t h a t i t i s t h e c a r d i n a l sum a s t r i c t l y g r e a t e r than a (as a c a r d i n a l ) y e t t h e r e i s no s e t s t r i c t l y intermediat e w i t h respect t o subpotence. O r o( i s a t l e a s t equal t o w 1 ; then the c a r d i n a l sum a g r e a t e r than a i n the preceding sense (see TARSKI 1954).

+

0 Consequence o f t h e f a c t t h a t a r e s t r i c t i o n o f t h e o r d i n a l o(

d

i s immediately

e i t h e r i s isomor-

p h i c w i t h o( o r w i t h an o r d i n a l /3 s t r i c t l y l e s s than o( ; then a + CardfS= a.D Modulo t h e axiom o f choice, every c a r d i n a l i s an aleph. Hence f o r each c a r d i n a l a there e x i s t s a c a r d i n a l b which i s immediately g r e a t e r than a i n t h e stronger f o l l o w i n g sense: every c a r d i n a l > a i s b b = a+ , the successor c a r d i n a l o f a , I n the mentioned paper, i t i s proved Put t h a t the preceding statement i s e q u i v a l e n t t o the axiom o f choice. I n the absence o f t h e axiom o f choice, t h e r e can e x i s t several c a r d i n a l s immedia-

.

t e l y g r e a t e r than a given c a r d i n a l ( i n the weak sense). For example, w i t h t h e continuum hypothesis w i t h o u t t h e axiom o f choice, w 1 and a l s o t h e continuum

w

a r e immediately g r e a t e r than

§

4

-

and p o s s i b l y incomparable: see ch.1

5

6.5.

REINFORCED RELATION, REINFORCED PARTIAL ORDERING

4.1. REINFORCEMENT, WEAKENING

.

L e t A be an n-ary r e l a t i o n w i t h base E An n-ary r e l a t i o n B w i t h t h e same base i s c a l l e d a reinforcement o f A , and A i s a weakening o f B i f f every n - t u p l e i n E having t h e value (+) i n A s t i l l has t h e value (+) i n B . Reinforcement i s a p a r t i a l o r d e r i n g on t h e s e t o f n-ary r e l a t i o n s w i t h base A s t r i c t o r proper reinforcement o f

A

i s a reinforcement d i s t i n c t from

A

E

.

.

S i m i l a r l y we speak o f a s t r i c t o r p r o p e r weakenin&. L e t A be a p a r t i a l o r d e r i n g which i s n o t a chain. We o b t a i n a s t r i c t r e i n f o r c e ment o f A by t a k i n g two elements u, v which are incomparable (mod A) , then d e f i n i n g , f o r a l l elements x, y o f t h e base, x d y i f f e i t h e r x d y (mod A) o r x,< u and v g y (mod A ) Generalize t h i s procedure as f o l l o w s .

.

L e t A be a p a r t i a l o r d e r i n g w i t h base E . L e t D g : E and B be a p a r t i a l order i n g w i t h base D which i s a reinforcement o f t h e r e s t r i c t i o n A/D Then there e x i s t s a p a r t i a l o r d e r i n g C which i s a reinforcement o f A and an

.

extension o f

6

2

E

. Moreover,

minimal among these reinforcements ding i s a reinforcement o f

Co

.

t h e r e e x i s t s a p a r t i a l o r d e r i n g Co C

which i s

, i.e. every C which v e r i f i e s t h e prece-

Chapter 2 0 Let

x, y

be two elements o f

e x i s t two elements

XI,

of

y'

E D

.

.

Co ; t h e

are reinforcements o f

x+ y

iff

x

x 4 x ' (mod A)

with

y ' 6 y (mod A )

C's

Put

43

6 y (mod A ) and

o r i f there

x ' d y ' (mod

B) and

The p a r t i a l o r d e r i n g t h u s o b t a i n e d i s t h e minimal r e i n f o r c e m e n t Co

.0

4.2. REINFORCEMENT AXIOM

-

This axiom a s s e r t s t h a t each p a r t i a l o r d e r i n g has a c h a i n among i t s r e i n f o r c e m e n t s . This f o l l o w s from t h e u l t r a f i l t e r axiom (SZPILRAJN 1930); ZF s u f f i c e s i n t h e case o f a countable p a r t i a l o r d e r i n g .

A

0 Let

we l e t

be a p a r t i a l o r d e r i n g w i t h base

UF

cement o f

E

. F o r each

denote t h e s e t o f a l l c h a i n s w i t h base A/F

. The

f i n i t e subset

F of

E

,

F each o f which i s a r e i n f o r -

a r e non-empty and v e r i f y t h e hypotheses o f t h e coherence

UF

E which s a t i s f i e s C i s a chain, s i n c e i t s f i n i t e r e s t r i c t i o n s a r e chains; i t i s a r e i n f o r c e m e n t o f A 0 I n t h e l i g h t o f t h e p r e c e d i n g 4.1, t h e r e i n f o r c e m e n t axiom can be s t a t e d equival e n t l y : g i v e n a p a r t i a l o r d e r i n g A and a c h a i n C which i s a r e i n f o r c e m e n t o f lemma 1.3. Hence t h e r e e x i s t s a r e l a t i o n C/F E UF

f o r each

F

C

based on

(uses u l t r a f i l t e r axiom). T h i s

.

a restriction o f

A

ment o f

A

,there

e x i s t s a c h a i n which i s s i m u l t a n e o u s l y a r e i n f o r c e -

and an e x t e n s i o n o f

C ,

5 2.3. It s u f f i c e s , g i v e n , t o r e i n f o r c e i n t o a c h a i n t h e p a r t i a l o r d e r i n g which reduces t o t h e

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

E

i d e n t i t y on

E

.

Note t h a t t h e r e i n f o r c e m e n t axiom i s s t r i c t l y weaker t h a n t h e u l t r a f i l t e r axiom: see FELGNER 1974 p. 375, making r e f e r e n c e t o t h e unpublished t h e s i s o f t h e a u t h o r . Also t h e o r d e r i n g axiom i s s t r i c t l y weaker t h a n t h e r e i n f o r c e m e n t axiom: see MATHIAS 1974. 4.3. Each weakened p a r t i a l o r d e r i n g o f a well-founded p a r t i a l o r d e r i n g i s w e l l founded. On t h e o t h e r hand, a r e i n f o r c e m e n t o f a w e l l - f o u n d e d p a r t i a l o r d e r i n g i s n o t n e c e s s a r i l y well-founded: i n t o an w 4.4.

r e i n f o r c e t h e i d e n t i t y w i t h a denumerable base

c h a i n (converse o f w )

.

For each w e l l - f o u n d e d p a r t i a l o r d e r i n g

reinforcement o f

A

A

, there

e x i s t s a well-ordered

(uses axiom o f choice; ZF s u f f i c e s i f A

i s countable o r

has w e l l - o r d e r a b l e base).

i s t r i c t l y l e s s t h a n H t A , t a k e a w e l l - o r d e r i n g Ci based i (axiom o f c h o i c e ) . Then t a k e t h e sum according t o increasing i 0

For each o r d i n a l

on t h e f r e e s e t o f elements w i t h h e i g h t

of the Ci

.

THEORY OF RELATIONS

44 4.5.

If

A, B

Let

be two well-founded p a r t i a l orderings w i t h t h e same base.

. Consequently

H t x (mod A)

5 5 - COFINAL SUBSET,

, then f o r

A

i s a reinforcement o f

B

each

.

Ht B 3 Ht A

x

H t x (mod B) >/

i n t h e base,

Proof by i n d u c t i o n ,

C O - I N I T I A L SUBSET, C O F I N A L I T Y OF A C H A I N

COFINAL. CO-INITIAL SUBSET AND RESTRICTION

5.1.

A be a D a r t i a l orderino. A subset D o f IAI i s s a i d t o be c o f i n a l (mod A ) . and A / D i s s a i d t o be a c o f i n a l r e s t r i c t i o n o f A i f f f o r each x i n I A l Let

there e x i s t s a y

in

D with

. Analogous

y + x (mod A)

definition for a

c o - i n i t i a l subset and a c o - i n i t i a l r e s t r i c t i o n . I f t h e base i s non-empty, then every c o f i n a l o r c o - i n i t i a l subset i s non-empty. Each superset o f a c o f i n a l s e t i s c o f i n a l ; s i m i l a r l y w i t h c o - i n i t i a l . Each c o f i n a l r e s t r i c t i o n K a c o f i n a l re_z_tt--c_t_ion_ i s a qozinal r e s t r i c t i o n ; similarly with co-initial.

A

Let

be a p a r t i a l ordering,

B

. For each

in

E

base

p a i r \x,y)

. Then

then remove t h i s

y

c o f i n a l (mod A)

, and

B

i s a well-ordering,

there e x i s t s an and

x


x

(mod B)

would e x i s t an

x

x

>y

,let x

are never ordered i n

D

i s a well-founded p a r t i a l o r d e r i n g L

.

Indeed f o r each

y

which i s removed,

among those s a t i s f y i n g

x

>y

(mod A)

. i s a well-ordering.

B

B/X : then

be the minimum o f

d i c t i n g the minimality o f

,

(mod B)

D : i f i t were removed, then there (mod A) and x ' < x < y (mod B) , c o n t r a d i c t i n g the

, we do n o t have

D

>x

(POUZET 1979, unpublished).

For otherwise there would e x i s t a y belong t o

and

belongs t o

x

Now consider t h e case t h a t D

A/D

then A

o f l e a s t h e i g h t (mod B)

. This

minimality o f the height o f of

x (mod A)

.

B

i s c o f i n a l (mod A)

D

x'>

<

y

D obtained a f t e r a l l suc~-removals i s

and modulo

which i s a c o f i n a l r e s t r i c t i o n o f 0 F i r s t we see t h a t

, if

E

the set

any two d i s t i n c t elements o f

opposite senses modulo A Moreover i f

a well-founded p a r t i a l o r d e r i n g w i t h t h e same

x

y

>

in

For each non-empty subset

.

.

X

i s a minimal element (mod A/X)

with y

X

x (mod 8)

(mod B/X)

x

<

Hence y

.

.

x (mod A) Since x, y < x (mod B) , contra-

B i s a well-ordering, i f we i (mod B) , then D i s t h e s e t o f

From an i n t u i t i v e p o i n t o f view, note t h a t when denote by elements = the

Then

bi

bi c

the element w i t h h e i g h t

defined as f o l l o w s among t h e o f least height

c2 = bi(*)

elements that the

.&

co

= the

and

cu = b .

1(u)

bi

4 c1 (u

i(1)

O(

)

.

L e t , ,c

= bo

. Then

# 0 (mod B) , among t h e elements

o f l e a s t height (mod A)

<

b

.

i(2)

>

i(1)

(mod B)

I n general f o r each o r d i n a l

are defined. Then

c o( - bi(%)

c1 = bi(l)

$

co (mod A)

, among t h e o( , assume = the

bi

of

.

Chapter 2

45

l e a s t height i ( % ) 7 i ( u ) f o r a l l u
. ..

9.

founded). Take a w e l l - o r d e r i n g o f

+

IA1

.

and use t h e preceding p r o p o s i t i o n .

5.2. STRATIFIED PARTIAL ORDERING A p a r t i a l o r d e r i n g A i s s a i d t o b e . s t r a t i f i e d i f f t h e union o f t h e r e l a t i o n s o f incomparability (mod A) and i d e n t i t y forms a t r a n s i t i v e r e l a t i o n , hence an equivalence r e l a t i o n . Consequently x 4 y and y I z i m p l i e s t h a t x < z ; s i m i l a r l y with > i n the place o f < . The equivalence classes o f i n c o m p a r a b i l i t y - i d e n t i t y form a t o t a l ordering, by p u t t i n g "class o f x " < "class o f y I' i f f x < y (mod A) . This t o t a l order i n g o f t h e equivalence classes w i l l be c a l l e d t h e p r i n c i p a l t o t a l o r d e r i n g of t h e s t r a t i f i e d p a r t i a l ordering A Every t o t a l l y ordered r e s t r i c t i o n o f A which i s maximal under i n c l u s i o n i s isomorphic w i t h t h e p r i n c i p a l t o t a l ordering.

.

5.3. L e t A be a well-founded p a r t i a l ordering. Then t h e r e e x i s t s a c o f i n a l r e s t r i c t i o n C o f A s a t i s f y i n g H t C 6 Card H t A (POUZET 1979, unpublished). 0 To each o r d i n a l i < H t A associate the c l a s s Bi o f elements o f h e i g h t i (mod A) Order t h e s e t o f t h e Bi by a w e l l - o r d e r i n g isomorphic w i t h i t s cardinal, which i s Card H t A Denote by B the well-founded s t r a t i f i e d p a r t i a l ordering w i t h base IA I , defined by p u t t i n g x < y (mod B) i f f "class o f x I' "class o f y I' according t o t h e preceding well-ordering. Then H t B = Card H t A By 5.1, t h e r e e x i s t s a c o f i n a l r e s t r i c t i o n C o f A such t h a t any two elements o f I C I are never ordered i n opposite senses by A and B

-

.

<

.

.

.

Hence i f x, y belong t o I C l and x < y (mod A) , then x ( y (mod B) , as incomparability (mod 8) i s impossible since x and y belong t o two d i s t i n c t classes Bi Thus B / K I i s a reinforcement o f C = A/)CI . By 4.5, we have

.

Ht C

4

Ht(B/ICI)

6

H t B = Card H t A

.

5.4. COFINALITY, CO-INITIALITY Let A be a p a r t i a l ordering. If, among t h e c o f i n a l sets (mod A) , there e x i s t s one o f l e a s t c a r d i n a l , then t h i s c a r d i n a l i s c a l l e d t h e c o f i n a l i t y o f A , denoted by Cof A Analogous d e f i n i t i o n o f c o - i n i t i a l i t y . With the axiom o f choice, every c a r d i n a l i s an aleph, hence t h e c o f i n a l i t y and coi n i t i a l i t y e x i s t f o r each p a r t i a l ordering. With o n l y t h e axioms o f ZF, these

.

46

THEORY OF RELATIONS

o n l y e x i s t i n p a r t i c u l a r cases, f o r example when t h e base i s w e l l - o r d e r a b l e . T h e i r s t u d y i s v e r y d i f f e r e n t i n t h e case o f a t o t a l o r d e r i n g , t h e c l a s s i c a l case c o n s i d e r e d i n t h e p r e s e n t and n e x t s e c t i o n s , f r o m t h e general case o f a p a r t i a l

5

o r d e r i n g , such a case i n t r o d u c e d and s t u d i e d by POUZET: see

be a t o t a l o r d e r i n g w i t h w e l l - o r d e r a b l e base. Then t h e r e e x i s t s a c o f i n a l

A

Let

7 below.

subset

o f t h e base, w i t h

U

A/U

a w e l l - o r d e r i n g isomorphic t o

Cof A ; same

result with co-initiality. 0 Take a c o f i n a l subset

Totally order and

A

by

D

A/D

D

w i t h l e a s t c a r d i n a l , hence o f c a r d i n a l equal t o

a c c o r d i n g t o i t s c a r d i n a l . Then a p p l y 5.1,

, and

C o r o l l a r y . L e t o( exists a

D ) , which i s

be an o r d i n a l ;

~~t

be

4 . To

Cof

Then we have

N i ( i E I ) = (Sup d i )

o r d i n a l sum o f t h e

Mi

Obviously

a

cofinal

subset

of

i of

each element

f o r which t h e r e

4 ).

o (

-

I

with

w / I

we asso-

zcxi

= o(, where


designates the : see 5 3.6.

i

along the order o f increasing

>, (Sup

oCi

D

c( ( e q u i v a l e n t l y :

di , t h e i n i t i a l i n t e r v a l o f d formed by t h e elements

ciate the ordinal

0

I

indecomposable ordinal.

an

u

i s the least ordinal

u-sequence o f successor o r d i n a l s whose u n i o n i s

o f l e a s t o r d e r t y p e , hence equal t o

.

Cof A by

.0

Cof A

u-sequence of successor o r d i n a l s whose u n i o n i s

a s t r i c t l y increasing 5.5

Cofd

E

D isomorphic t o t h e c a r d i n a l o f D

by a w e l l - o r d e r i n g o f

B

( i . e . t h e l e a s t o r d i n a l based on

replacing

o( . It remains t o e l i m i n a t e t h e case o f 2 Hi > o( and l e t u be t h e l e a s t e l e z o( i(i< u ) >/ o( . E i t h e r u has a predecessor v

Ni)

=

s t r i c t i n e q u a l i t y . Suppose t h a t

I

ment o f (u = v+l)

f o r which

,

<

i n which case o( i s equal t o

< d V hence

which i s


o( i(i v ) p l u s a non-zero o r d i n a l I n t h i s case o( i s n o t indecomposable: con-

tradiction. Or

u

i s a l i m i t o r d i n a l , hence

o f the disjoint intervals i s cofinal f o r greater than

d

di(i<

Ni(i<

. Hence

u

>, Cof

u)

u ) = o(

.

Then t h e s e t o f minimums

y i e l d s an o r d i n a l isomorphic t o

, and

o(

so t h e o r d i n a l o f

d/I

u

, and

is strictly

C o f M : contradiction. 0

N o t i c e t h a t i t i s necessary t o assume t h a t o( i s indecomposable: t a k e t h e counterexample o( = W . 2

with

i i n t e g e r and di = 6~+ i

t o t a k e OC / I

isomorphic t o 2 t h e counterexample LX= Cr) and

o(

=

LJ.2

+

~

(

i W- )

, and

Cof d with

i

for J ,(

<

d . 2

6

.

It i s a l s o necessary

not only o f cardinal and o( i < (2.2

.

=

Cof o( : t a k e

c3 + i f o r f i n i t e i

,

41

Chapter 2

§

6 - REGULAR

AND SINGULAR

ACCESSIBILITY

ALEPH,

6.1. REGULAR AND SINGULAR ALEPH An a l e p h ( i . e . t h e c a r d i n a l of a w e l l - o r d e r a b l e s e t ) i s s a i d t o be r e g u l a r i f f , considered as an o r d i n a l , e v e r y c o f i n a l subset i s e q u i p o t e n t w i t h i t . I n o t h e r words, i t s c o f i n a l i t y i s equal t o i t . An a l e p h i s s a i d t o be s i n g u l a r i n t h e o p p o s i t e case where i t s c o f i n a l i t y i s s t r i c t l y s m a l l e r . For example

3

1 and w a r e r e g u l a r . Each i n t e g e r

s i n g u l a r . The c a r d i n a l ww o r Modulo t h e c o u n t a b l e z i ' l m

2

has c o f i n a l i t y 1, hence i s

has c o f i n a l i t y r i j

o f choice,

, hence

i s singular.

i s r e g u l a r . However t h e i n e q u a l i t y

U1

w 1 > W i s o b t a i n e d i n c h . 1 5 6.3 u s i n g o n l y ZF. 0 Associate t o each c o u n t a b l e subset D o f ul t h e c o u n t a b l e u n i o n o f those

.

D

countable o r d i n a l s which a r e elements o f

5 2.5, a1 . 0

By c h . 1

countable o r d i n a l , hence cannot be t h e e n t i r e s e t L e t A be a c h a i n w i t h w e l l - o r d e r a b l e base; t h e n

Cof A

t h i s union i s a

i s a r e g u l a r aleph.

Consequence o f t h e f a c t t h a t a c o f i n a l r e s t r i c t i o n o f a c o f i n a l r e s t r i c t i o n i s i t s e l f c o f i n a l ( 5 . 1 above). 6.2. Every successor a l e p h i s - y e g - u x (uses axiom o f c h o i c e ) . 0 Our a l e p h i s o f t h e f o r m

c o f i n a l i t y . Take a see c o r o l l a r y 5.4. Suppose

u

ONtl

where

o(

i s an o r d i n a l . L e t

u be i t s

u-sequence o f successor o r d i n a l s whose u n i o n i s

c3d+1 : From some p o i n t on, t h e s e o r d i n a l s a r e e q u i p o t e n t w i t h wq

, hence

s t r i c t l y less than

Card u

6 ad . Then

.

the union

o f the ordinals i n our

u-sequence has c a r d i n a l a t most equal t o fdH x Ld4 hence a t most equal t o fAd (axiom o f c h o i c e g i v i n g a b i j e c t i o n o f each o r d i n a l

6.3. L e t

a

be an i n f i n i t e a l e p h .

sufficient that there exists a set s t r i c t l y subpotent w i t h 0

If a

a

, whose

For

.0

u = Wdtl

Contradiction proving that onto do(). a

t o be s i n g u l a r , i t i s necessary and

u , s t r i c t l y subpotent w i t h a , o f elements union y i e l d s

a

(uses axiom o f c h o i c e ) .

i s s i n g u l a r , t h e n o u r c o n c l u s i o n i s obvious. Conversely i f

regular, then l e t

u be a s e t o f subsets o f a whose u n i o n i s a

o f t h e subsets i s c o f i n a l , hence o f c a r d i n a l

a

and t h e s e t o f l e a s t upper bounds i s c o f i n a l i n

. Or a

a

.

is

E i t h e r one

each subset i s bounded above,

, hence

o f cardinal

a

.

Replace each upper bound by one o f t h e c o r r e s p o n d i n g subsets (axiom o f c h o i c e ) : the s e t

u has a t l e a s t c a r d i n a l

a

.0

6.4. The p r e c e d i n g p r o p o s i t i o n suggests t h e f o l l o w i n g g e n e r a l i z a t i o n . A c a r d i n a l a

( n o t n e c e s s a r i l y an a l e p h ) i s s a i d t o be s i n g u l a r i f f i t i s t h e u n i o n o f a s e t

THEORY OF RELATIONS

48

a , whose elements a r e s t r i c t l y subpotent w i t h a ; i t i s s a i d t o be r e g u l a r o t h e r w i s e . I n t h e presence o f t h e axiom o f choice, e v e r y c a r d i s t r i c t l y subpotent w i t h

n a l i s an aleph, and we have t h e c l a s s i c a l d e f i n i t i o n i n 6.1. I n t h e absence o f t h e axiom o f choice, we do n o t know whether t h i s g e n e r a l i z e d d e f i n i t i o n o f r e g u l a r and singular cardinal yields interesting results. With t h e axiom o f c h o i c e and t h e continuum h y p o t h e s i s , we know t h a t t h e c a r d i n a l o f t h e continuum equals

CJ

and so i s r e g u l a r . With o n l y t h e axiom o f c h o i c e , t h e r e

e x i s t models where t h e continuum i s a r e g u l a r aleph, and o t h e r s where t h e continuum i s a s i n g u l a r aleph. I t can have any i n f i n i t e c o f i n a l i t y , except i t cannot equal

. This

ww

w

: f o r example

r e s t r i c t i o n on t h e c o f i n a l i t y r e s u l t s f r o m t h e f a c t

t h a t , f o r any p a r t i t i o n o f t h e continuum i n t o a c o u n t a b l e number o f subsets, t h e r e i s a t l e a s t one which i s e q u i p o t e n t w i t h t h e continuum: see c h . 1

6.5.

(1) Let

a

b (14 b

be a r e g u l a r aleph; f o r e v e r y

< a)

5

4.3.

we have

ba = a

(TARSKI 1938; uses g e n e r a l i z e d continuum h y p o t h e s i s ; ZF s u f f i c e s f o r

a = W ;

a = cJ1 , ZF p l u s c h o i c e p l u s continuum h y p o t h e s i s ) . ( 2 ) L e t a be a l i m i t aleph; f o r e v e r y c, d c a we have prop. 9 ; g e n e r a l i z e d continuum h y p o t h e s i s i s used).

(Ibid.

for

0 (1) The s t a t e m e n t i s t r u e f o r

a = W . Suppose t h a t

a = ‘2

s o r aleph, hence o f t h e f o r m


i s an i n f i n i t e succes-

( g e n e r a l i z e d continuum h y p o t h e s i s ) , and b ba = b(C2) = ( b x c ) 2 = ‘2 = a s i n c e b x c

1 b $ c . Then we have by 3.8. Now suppose t h a t a i s a l i m i t a l e p h which i s s t i l l r e g u l a r and

satisfies = c

a

‘d

a .There e x i s t s an i n c r e a s i n g a-sequence o f successor

s t r i c t l y greater than ai

<

a (iruns t h r o u g h a)

ai7

b

, hence b(ai)

cardinals we have

r e g u l a r , no values i n i n each

ai

= ai

b-sequence i s c o f i n a l i n a

a = Sup ai

with

a

. Hence

the set o f a l l

i varies, o f the sets o f b ba = Sup( (ai)) = Sup ai = a 0

i s t h e union, as

.

. From a

c e r t a i n p o i n t on,

by t h e p r e v i o u s d i s c u s s i o n . Since

is

b-sequences w i t h values

.

Hence

a

b-sequences w i t h

( 2 ) L e t u be a successor a l e p h s a t i s f y i n g c < u < a and d < u < a u i s r e g u l a r , hence ‘d \< JI‘ = u by (1); hence ‘d < a . 0

0

. Then

Statement (1) does n o t h o l d f o r s i n g u l a r alephs. Let

a = OU and

b = U. Decompose

and a s s o c i a t e t o each W i t h e union o f the

cc, i+l

. Hence we

a

i n t o t h e u n i o n o f t h e W i (ii n t e g e r )

. By

i t s successor

K O N I G ‘ S theorem ( c h . 1

5

i s s t r i c t l y subpotent t o t h e Cartesian product o f the

Oi

have

<

w (

W

)

.

Even i n ZF (KONIG‘s theorem u s i n g t h e axiom of c h o i c e ) , we have t h e f o l l o w i n g counterexample. L e t

a.

= (d , and f o r each i n t e g e r

i l e t us d e f i n e :

1.8)

49

Chapter 2 = ( 2 t o t h e power ai) , and then a = Sup ai . Then we have a2 5 @ a , ai+l hence a G t a : t o get the f i r s t i n e q u a l i t y , associate t o each subset u o f

<

ui = a,.n

the sequence o f the i n t e r s e c t i o n s Let

6.6.

a

be a r e g u l a r l i m i t aleph; then t h e aleph rank o f

i n o t h e r words 0

a = W

( b < a)

a

.

wa = a

Assume t h e contrary, t h a t

so

u , and n o t i c e t h a t ui 6 ai+l

. Then

a

< ma , and

either

b

a

.

itself;

be t h e aleph rank o f

i s a successor o r d i n a l , so ( i < b) i s c o f i n a l i n a

b

wi

r t h e sequence o f the sor aleph. O

let

& a

a

a

,

i s a succes-

, hence

a

is

not r e g u l a r . 0 On t h e o t h e r hand, we can have t h e l i m i t aleph o f t h e sequence

ba= a

where

a(0) =

u ,a(1)

a

i s a s i n g u l a r c a r d i n a l . Take = Wu

,...,

i : t h e aleph obtained has c o f i n a l i t y w

f o r each i n t e g e r

a(i+l)

, hence

= W a(i) i s singular.

6.7. ACCESSIBLE CARDINAL, A X I O M OF ACCESSIBILITY

An i n f i n i t e cardinal a set with

, strictly

b

, which

a

subpotent w i t h

a

i s s a i d t o be accessible i f

subpotent w i t h

a

a = CJ

, or

i f there e x i s t s

and whose elements a r e s t r i c t l y subpotent

u b = a ; o r f i n a l l y i f there exists a set c s t r i c t l y (c) A cardinal i s said a , and where a i s subpotent w i t h

satisfies

.

t o be inaccessible otherwise. The axiom o f a c c e s s i b i l i t y asserts t h a t every s e t has accessible c a r d i n a l i t y .

ZF i s consistent, t h e theory

Under the assumption t h a t

ZF plus the axiom o f acces-

s i b i l i t y i s consistent: see SHEPHERDSON 1952. Every accessible l i m i t aleph d i f f e r e n t from W i s s i n g u l a r (uses generalized continuum hypothesis). Consequently i n view o f 6.2, this 0 Let

a

i s regular i f f

, hence of

f o r an accessible aleph

a # k~,

i s a successor aleph.

be our aleph, where

c3,

smaller c a r d i n a l i s an

+c

a

Ui ( i

4 i s a non-zero l i m i t o r d i n a l . Every s t r i c t l y

<

c a r d i n a l i t y lrJi

&)

. There

i s no s e t s t r i c t l y subpotent w i t h

, w i t h wd

subpotent w i t h

?(wi)

: indeed

(generalized continuum hypothesis) and i+l< d . F i n a l l y as ?(mi) = o u r aleph i s assumed t o be accessible and d i s t i n c t from .LI , t h e r e e x i s t s a s e t strictly LJ4

subpotent w i t h a d

, and

whose union i s

1J,

, whose

elements are as w e l l s t r i c t l y subpotent w i t h

: our aleph i s thus s i n g u l a r (see 6 . 3 ) . 0

An aleph i s s a i d t o be weakly i n a c c e s s i b l e i f i t i s d i f f e r e n t from 0 , r e g u l a r and a l i m i t aleph. The preceding p r o p o s i t i o n i s e q u i v a l e n t t o saying t h a t , w i t h t h e axiom o f choice and t h e generalized continuum hypothesis, every weakly inaccess i b l e aleph i s inaccessible. Using o n l y t h e axiom o f choice, every inaccessible aleph i s weakly inaccessible.

THEORY OF RELATIONS

50

Indeed, i t i s d i f f e r e n t from w , r e g u l a r , and t h e r e i s no s t r i c t l y smaller c f o r which i t i s subpotent w i t h T ( c ) : hence i t i s not a successor aleph. Problem. Assume t h a t ZF p l u s choice p l u s t h e existence o f a weakly i n a c c e s s i b l e aleph l e s s than the continuum i s consistent. Then does t h i s theory remain c o n s i s t e n t i f we r e q u i r e t h a t t h e continuum i t s e l f be weakly inaccessible. C l a s s i c a l l y we have an a f f i r m a t i v e answer i f we assume the consistency o f ZF p l u s choice p l u s t h e existence o f a weakly i n a c c e s s i b l e aleph g r e a t e r than t h e continuum. 6.8. Some unpublished r e s u l t s of BLASS, i n a l e t t e r t o HODGES, 1982.

Consider t h e f o l l o w i n g statements: (1) every countable union o f countable sets i s countable (see ch.1 5 2 . 5 ) ; ( 2 ) i f a s e t a and i t s elements have c a r d i n a l s cdl , then t h e union o f a has c a r d i n a l $ u 1 ; (3) given a f u n c t i o n f , if Rng f = d l , then U1 i s subpotent t o Dom f

.

Then i t i s proved t h a t n e i t h e r o f (1) and (2) i m p l i e s the other, and t h a t (2) i s equivalent t o (3) p l u s t h e r e g u l a r i t y o f w1 .

§

7 - COFINALITYOF

A PARTIAL ORDERING, COFINAL HEIGHT

The c o f i n a l i t y of a p a r t i a l o r d e r i n g A , denoted Cof A and introduced i n 5.4, was s t u d i e d above o n l y i n t h e case o f a t o t a l ordering. I f A i s n o n - t o t a l l y ordered, then Cof A can A by taking, and p u t t i n g them together Given a p a r t i a l o r d e r i n g

0 Construct

c o f i n a l subset Consequently

U

of

Cof(A/D)

D

be a s i n g u l a r aleph. f o r each i n t e g e r i , a c h a i n of order type wi mutually incomparably: then Cof A = W, 0 A and a c o f i n a l s e t D (mod A) , t h e r e e x i s t s a

.

with

= Cof

A

Card U = Cof A

,

.

f o r each c o f i n a l s e t

D

(uses axiom o f choice;

ZF s u f f i c i e n t i f A i s countable o r has well-orderable base). Take a s e t V c o f i n a l (mod A ) and w i t h l e a s t c a r d i n a l , hence Card V = Cof A , Then replace each element x o f V by an element i n D which i s g r e a t e r

0

than

x

.0

.

7.1. L e t A be a p a r t i a l o r d e r i n g and u = Cof A There e x i s t s a c o f i n a l subset U o f c a r d i n a l u , t h e r e s t r i c t i o n A/U being a well-founded p a r t i a l

.

o r d e r i n g w i t h h e i g h t Ht(A/U) 6 u A refinement o f c o r o l l a r y 5.1, due t o PDUZET 1979, unpublished. Uses t h e axiom o f choice; ZF s u f f i c e s i f A I s countable o r has we1 1-orderable base. 0

Let

D be a c o f i n a l subset w i t h l e a s t c a r d i n a l u

= Cof A

. Order

D by

Chapter 2

51

bi

(i< u)

ai

i t s c a r d i n a l i t y , which we assume t o be an aleph, and c a l l quence (axiom o f c h o i c e ) . E x t r a c t a sequence

t h i s se-

by removing, f o r each

i

, the

.

a such t h a t j > i and a . < a i (mod A) The sequence o f t h e bi has l e n g t h j J a t most equal t o u . By 5.1, t h e i n e q u a l i t y i < i ' i m p l i e s bi < o r I bin

.

(mod A)

Moreover, t h e s e t

U

o f values

b

i s cofinal i n

A

and

is a

A/U

well-founded p a r t i a l o r d e r i n g . For each s u b s c r i p t (mod A/U)

is

& i

i< u

, show

. First

that there exists a l e a s t

i with

bi

,

i

by i n d u c t i o n on

o f a l l , a l l the

bi

o f height

t h a t the height o f

2

are

>

bk

o f height

7.2. For e v e r y w e l l - f o u n d e d p a r t i a l o r d e r i n g A (uses axiom o f choice; ZF s u f f i c e s i f

A

, we

. Then

i (mod A/U)

i , w i t h b k < bi (mod A) ; and hence which c o n t r a d i c t s t h e m i n i m a l i t y o f i . F i n a l l y t h e h e i g h t o f exists a

have

k

bi

. Assume

o r \ b o (mod A)


A/U

Cof H t A

there

by 5.1,

is

,<

6 u .

0

Cof A

i s countable o r w i t h well-orderable

base).

D o f l e a s t c a r d i n a l i t y , so Card D = Cof A . The h e i g h t s D f o r m a s e t H , which i s c o f i n a l i n t h e o r d i n a l H t A . By t h e axiom o f c h o i c e , H i s subpotent w i t h D , hence we have Cof H t A ,< Card H 4 Card D = Cof A . 0

0

Take a c o f i n a l s e t

(mod A) o f t h e elements o f

If

A

i s isomorphic w i t h

A

On t h e o t h e r hand, t a k e t h e w e l l - f o u n d e d p a r t i a l o r d e r i n g

A

h

Cof A

i s a well-ordering,

then

Ht A

, hence

Cof H t A =

.

nent c3

,a

component

formed o f a compo-

w+1 , t h e elements o f one c h a i n p u t incomparable w i t h

those o f t h e o t h e r . Then

Cof A = W

,

H t A = W+1 so

Cof H t A

=

1

.

7.3. COFINAL HEIGHT Given a p a r t i a l o r d e r i n g

, then

of

A

of

A , and denoted

A

,

i f there e x i s t s a well-founded c o f i n a l r e s t r i c t i o n

t h e l e a s t h e i g h t o f such r e s t r i c t i o n s i s c a l l e d t h e c o f i n a l h e i g h t Cofh A

f o r every p a r t i a l ordering

.

With t h e axiom o f choice, t h e c o f i n a l h e i g h t e x i s t s

A : see c o r o l l a r y 5.1. Using o n l y t h e axioms o f ZF,

the c o f i n a l height e x i s t s a t l e a s t f o r a l l well-founded p a r t i a l orderings. (1) F o r e v e r y p a r t i a l o r d e r i n g

A ,.we have

Cofh A

6

.

Cof A

quence o f 7 . 1 and uses t h e axiom o f choice; ZF s u f f i c e s i f

A

T h i s i s a consei s countable o r

has w e l l - o r d e r a b l e base. S t r i c t i n e q u a l i t y can happen: f o r example when i t s base

E , t h e n Cofh A

= 1 and

A

Cof A = Card E

i s reduced t o t h e i d e n t i t y on

.

( 2 ) I,f A i s a t o t a l o r d e r i n g , t h e n Cofh A = Cof A . Consequence o f (1) and o f t h e f a c t t h a t t h e c a r d i n a l o f a w e l l - o r d e r e d r e s t r i c t i o n o f A i s a t most equal t o i t s h e i g h t ; same c o n d i t i o n s as i n (1).

52

(3) If

THEORY OF RELATIONS

A

i s a well-founded p a r t i a l ordering, then

Cofh A

4

Card H t A : another

form of 5.3. 7.4. Given a p a r t i a l ordering A , t h e c o f i n a l height i s t h e l e a s t height of wellfounded cofinal r e s t-r.i c t i o n s of A ,whose cardinal i s equal t o Cof A (uses axiom of choice; ZF s u f f i c e s i f A i s countable o r has well-orderable base). S t a r t w i t h a cofinal subset D such t h a t A/D i s a well-founded p a r t i a l ordering of l e a s t height H t ( A / D ) = Cofh A Take a subset D ' of D which is cofinal and of l e a s t cardinal Cof A (see beginning of present 5 7, using t h e axiom of c h o i c e ) . Then Ht(A/D') $ H t ( A / D ) = Cofh A by 3.5. 0

.

-

7.5. I f B i s a cofinal r e s t r i c t i o n of t h e p a r t i a l ordering A , then Cofh B i Cofh A (assuming t h a t these c o f i n a l h e i g h t s e x i s t ) . Indeed each cofinal r e s t r i c t i o n of B i s a cofinal r e s t r i c t i o n of A The following example, due t o POUZET 1979, shows t h a t s t r i c t i n e q u a l i t y i s p o s s i b l e , contrary t o t h e s i t u a t i o n f o r c o f i n a l i t y (beginning of present 5 7 ) . . For each ordered p a i r ( i , j ) 0 Take a s base t h e Cartesian product w x with i i n t e g e r , j countable o r d i n a l , p u t ( i , j ) 6 ( i ' , j ) f o r every i n t e g e r i For each even i n t e g e r i , p u t ( i , j ) s (i,j') for all j . For each odd i n t e g e r i , we l e t t h e ordered p a i r s w i t h f i r s t term i be mutually incomparable. Then we complete by t r a n s i t i v i t y . The p a r t i a l ordering A thus obtained has a cofinal s u b s e t of a l l ordered p a i r s w i t h odd f i r s t term: thus we have Cofh A = a.The r e s t r i c t i o n t o ordered p a i r s with even f i r s t term has cofinal height w1 . 0

.

i'a

.

j'a

For each p a r t i a l ordering A , t h e cofinal height of A i s a cardinal (uses axiom of choice; ZF s u f f i c e s i f A i s countable, o r has well-orderable base, o r i s well-founded; POUZET 1979, unpublished). 0 Take a well-founded cofinal r e s t r i c t i o n B of A w i t h l e a s t h e i g h t , so H t B = Cofh A : see c o r o l l a r y 5.1. By 7.5 we have Cofh A $ Cofh B . By 7 . 3 . ( 3 ) we have Cofh B 6 Card H t B = Card Cofh A . 0 7.6.

Let A be a well-founded p a r t i a l ordering: any of t h e possible comparisons " s t r i c t l y l e s s than", " s t r i c t l y g r e a t e r " , "equal" can be obtained f o r the cardin a l s Cofh A and Cof H t A . (1) S t a r t from t h e chain of i n t e g e r s . To each i n t e g e r i a s s o c i a t e an element i ' . P u t i ' 7 i ; p u t i ' incomparable with i n t e g e r s b i t 1 ; f i n a l l y f o r a l l i n t e g e r s i and j # i , p u t i ' incomparable with j ' . Denote by A t h i s p a r t i a l ordering: we have H t A = Cof H t A = c3 and Cofh A = 1

7.7.

.

(1') In t h e above example 7.5 we have

Cofh A = W and

Cof H t A

= W1

.

Chapter 2

( 2 ) For each ordinal q

, we

have

Cofh o(

=

53

Cof o( = Cof H t o(

.

( 3 ) Construct A by s t a r t i n g with the ordinal c3,; f o r each integer i , add by a bifurcation another chain w i a f t e r the i n i t i a l interval c d i . Then we have H t A = awso Cof H t A = W and Cofh A = w w . W e can also have Cof H t A 4 Cofh A where Cofh A i s a regular aleph. See below

in ch.7

§

5

3.10 where Cof A = Cofh A

8 - SET

= W1

and H t A =

OF SUBSETS, SET OF INJECTIVE

ON THE REALS,

INTERMEDIACY,

dl.w

FUNCTIONS,

CONSECUTIVITY,

CYCLE,

thus

INCREASING

Cof H t A = W . FUNCTIONS

CYCLIC RELATION

8.1. Let E be an i n f i n i t e s e t , I a s e t of subsets Ei ( i 6 I ) of E where I i s subpotent with E and each Ei i s equipotent with E . (1) There e x i s t two d i s j o i n t subsets C , D E such t h a t f o r each i & t intersections CnEi and DnEi are non-empty. ( 2 ) More strongly, there e x i s t d i s j o i n t C , D such t h a t CnEi and DnEi are equipotent with E (uses axiom of choice; ZF suffices when E i s countable; see BERNSTEIN 1908 or KURATOWSKI 1966 p. 514. 0 (1) Well-order E according t o i t s cardinal a , and I according t o i t s cardi nal b ( l e s s t h a n or equal t o a ) . P u t the f i r s t element xo of Eo into C , the second yo into 0 . I n general f o r each ordinal i < b , p u t the f i r s t xi of Ei which i s d i s t i n c t from a l l x j and y j ( j < i ) into C , and the second yi d i s t i n c t from a l l x and y into D . 0 j j 0 ( 2 ) Replace the s e t I by a sequence of the terms Ei with repetitions, as follows. If b = a so t h a t I i s equipotent with E , then take the f i r s t term Eo , then Eo followed by E l , and in general f o r each i , take the sequence of

of

.

E j ( 0 6 j ,c i ) I f b < a , then keep the above sequence of the Ei , b u t keep repeating so as t o obtain a sequence with length a = Card E In e i t h e r case, repeat the preceding proof: f o r each s e t Ei , the intersections CnEi and DnEi have cardinal a 0 Consequently, i f E i s denumerable and 3 i s a non-trivial u l t r a f i l t e r on E , then there i s no countable s e t of subsets Ei f E which generates g ,in the sense t h a t the elements of are obtained by taking supersets of f i n i t e intersections of the E i . 0 Denote by Ei the Ei from the statement and also t h e i r f i n i t e intersections. Then the s e t s C a n d D , which are d i s j o i n t , would b o t h belong t o the ultraf i l t e r : contradiction. 0 More generally, l e t E be i n f i n i t e and 3 be an u l t r a f i l t e r on E whose Then elements are equipotent with E cannot be generated by a s e t , equipotent with E , of elements of g .

.

.

.

THEORY OF RELATIONS

54

8.2. (1) Let E be an i n f i n i t e s e t , I a s e t of injective functions f i ( i € I ) from E i n t o E , where I i s subpotent w i t h E , and each f i s a t i s f i e s f i ( x ) # x on a subset of E equipotent w i t h E Then there -e x i s t s a subset C of - E , equipotent w i t h E , w i t h the non-inclusion f i o ( C ) q C f o r each i (notation f " from ch.1 5 1 . 2 ) . ( 2 ) Under the same hypotheses, _______ there e x i s t two subsets C , D Df E which are d i s j o i n t and equipotent w i t h E , w i t h the-_?on-inclusion f i o ( C ) F C W D and similarly by interchanging C and D (uses axiom of choice; SIERPINSKI 1950; see also ROSENSTEIN 1982). 0 (1) Well-order E by i t s c a r d i n a l i t y a ; well-order I f i r s t by i t s cardinalit y , then w i t h repetitions according t o the ordinal a , in the case where Card I < Card E . P u t the f i r s t xo of E s a t i s f y i n g f o ( x O ) # xo into C , and p u t

.

xb = f o ( x O ) into E - C . I n general, f o r each i < a , l e t xi be the f i r s t e l e ment of E such t h a t f i ( x i ) # xi , and where xi and x(i = f i ( x i ) are d i s t i n c t from a l l x and x! ( j < i ) . Such an x i e x i s t s , because the number of j < i j J i s s t r i c t l y l e s s t h a n the cardinal a , and there are ( c a r d i n a l i t y a)-many x which are d i f f e r e n t from f i ( x ) , as f i i s i n j e c t i v e . Finally p u t xi i n t o C and x i into E - C . 0 ( 2 ) The preceding proof modified as follows. F i r s t l y , p u t the f i r s t xo # fo(xO) i n t o C and xb = fo(xO) i n t o E - ( C u 0 ) . Secondly, p u t the f i r s t element

0

yo # f o ( y o ) , where yo and

fo(yo)ar%othd i s t i n c t from xo

and

xb , into D ,

and yb = fo(yo) i n t o E - ( C u D) . I n general, f o r each i < a , f i r s t l y find the f i r s t element xi # f i ( x i ) where both x i and x; = f i ( x i ) a r e d i s t i n c t from ( j< i )

x j , x!, y . , y j

all

J

J

. Then

put

xi

into

C

Secondly consider the f i r s t yi # f i ( y i ) with yi xj,

xi,

yj, y j ( j < i )

y! = f . ( y . ) 1

1

1

into

E

-

as well as from x i , x i ( C u D)

.

and and

xi

E

into

-

( C uD)

.

f i ( y i ) d i s t i n c t from a l l

Then p u t yi

into

D

, and

put

.0

8.3. The reader i s familiar w i t h the notion o f increasing o r decreasing ( s t r i c t l y or otherwise) function on the r e a l s . Let E be the s e t of r e a l s and R t h e i r t o t a l ordering, F a subset of E and Ft the smallest interval including F ( w i t h endpoints Inf F and Sup F modulo R ) For each increasing (mod R ) function f from F into E , there e x i s t s a t l e a s t one increasing function f + which i s an extension o f f t o the domain Ft If F i s dense f o r the t o t a l ordering R/F+ , then the extension f' of f is unique. Moreover, i f f i s s t r i c t l y increasing and F *for R/F' , then f' i s s t r i c t l y increasing and hence i s an isomorphism from R/F+ i n t o R . 0 For each element u of F+-F , l e t f t ( u ) be the infimum of the f ( x ) f o r a l l x in F such t h a t x > u . 0

.

.

-

Chapter 2

55

8.4. Let E be t h e s e t of r e a l s , R t h e i r t o t a l ordering, F a non-empty subset of E The s e t of increasing (mod R ) functions from F into E i s equipotent with the continuum. 0 Assume f i r s t of a l l t h a t F i s an i n t e r v a l . An increasing function from F i n t o E has countably many points of d i s c o n t i n u i t y , s i n c e t o t h e s e points correspond non-singleton mutually d i s j o i n t i n t e r v a l s ( s e e ch.1 5 4 . 5 ) . Hence an increasing function i s defined by i t s values on the r a t i o n a l s f o r example, plus i t s values on the points of d i s c o n t i n u i t y : a l t o g e t h e r making a countable sequence of real values. I n the case of an a r b i t r a r y F , l e t Ft be t h e s m a l l e s t i n t e r v a l including F . To each increasing function f from F i n t o E , a s s o c i a t e the function f + from F+ i n t o E which i s increasing and an extension of f , obtained by t h e previous proposition. The set of f f i s equipotent w i t h the continuum, hence so i s the s e t of r e s t r i c t i o n s f of f t t o t h e domain F 0

.

.

8.5. Let R be the t o t a l ordering of t h e r e a l s , and F a s e t of r e a l s . I f f o r each automorphism h of R which i s d i f f e r e n t from t h e i d e n t i t y , t h e r e exists an element x f F with h ( x ) # x , then F is dense f o r R

.

Let a < b be r e a l s . There e x i s t s a t l e a s t one automorphism, o r s t r i c t l y increasing function f from R onto R , s a t i s f y i n g f ( x ) = x f o r a l l x 6 a and a l l x ) b , and f ( x ) # x f o r a l l x ( a 4 x G b ) . Hence F must have an e l e ment between a and b . 0

0

8.6. INTERMEDIACY, CYCLIC RELATION, CONSECUTIVITY, BINARY CYCLE We f i n i s h t h i s chapter by introducing several r e l a t i o n s frequently associated with p a r t i a l and t o t a l o r d e r i n g s , and used i n l a t e r c h a p t e r s .

Given a p a r t i a l ordering A , we c a l l intermediacy associated with A , the ternary r e l a t i o n “z i s between x and y (mod A ) ” ; in o t h e r words x & z 6 y o r y x and t h e r e does not e x i s t t s a t i s f y i n g x < t < y . Given a t o t a l ordering A , t h e binary c y c l e , o r cycle o f consecutivity associated with A , i s t h e r e l a t i o n of c o n s e c u t i v i t y w i t h the following possible modification: i f t h e r e e x i s t s a minimum u and a maximum v of A , then we p u t C ( v , u ) = + .

THEORY OF RELATIONS

56

EXERCISE 1 - THE GENERALIZED CONTINUUM HYPOTHESIS IMPLIES THE AXIOM OF CHOICE (SIERPINSKI 1947; see also COHEN 1966) 1 - Let A , B be two s e t s . If A u B i s equipotent with the s e t ? ( 2 x A ) , then T(A) i s subpotent with B . 0 Let f be a bijection from AuB onto y(2 x A ) = T(A) x ? ( A ) . Each element x of A i s transformed into the ordered pair f ( x ) = (y,z ) of elements of T(A) . To each x associate the f i r s t term y of t h i s ordered pair. By CANTOR'S lemma ch.1 5 1.5, there ex i s t s an element u of T(A ) f o r which none of the pairs (u,z) i s the image of any x of A Hence there e xists a subset of B which i s bijectively transformed by f into the s e t ? ( A ) of the second terms z of the ordered pairs (u,z) . 0 2 - For each s e t A , l e t To(A)= A ; Tl(A) = T(A) , the s e t of subsets of A ; f o r each integer i , l e t Ti+l(A)= ( 9i(A)) If A has a denumerable subset, then Pi(A)= 2 x T i ( A ) f o r each positive integer i 0 By hypothesis A i s equipotent with A augmented by an element. Hence we have

.

.

.

P(A)= A2 equipotent with 2 x A2 . I t follows tha t ? ( A ) i s equipotent with i t s e l f augmented by an element: then we i t e r a t e . 0 3 - Let A be an i n f i n i t e s e t , assumed t o be equipotent with 2 x A . If f o r i = 0,1,2,3, the sets T i ( A ) and (j)i+l(A) do not have any s e t which i s s t r i c t ly intermediate with respect t o subpotence, then there e xists a well-ordering of the s e t A . 0 The s e t A i s equipotent with a proper subset of i t s e l f , hence has a denumerable subset: ch.1 5 2 . 6 . Hence Ti(A) i s equipotent with 2 x Ti(A) for each integer i : see ( 2 ) above. Consider the well-orderings based on subsets of A , and the isomorphism cl.asses of these well-orderings. From t h i s point on, denote Ti(A) by Pi . The pairs of elements of A belong t o P1 The ordered pairs belong t o P2 : see ch.1 5 1.2. The well-orderings based on subsets of A belong t o P3 Finally the isomorphism classes belong t o P4 The s e t H of isomorphism classes i s equipotent with the Hartogs aleph o f A : see ch.1 5 6 . 2 ; and i t i s included i n P4 . The union HuP3 is intermediate, under subpotence, between P3 and P3 u P4 , the l a t t e r equipotent with P4 , since P3 i s equiBy hypothesis, H u P3 potent with a subset of P4 as well as with 2 x P3 i s equipotent e i t h e r with P4 or with P3 I n the f i r s t case, i t i s equipote n t with ( 2 x P3) , and by the previous (1) P4 i s subpotent w i t h H , hence well-orderable, so A i s well-orderable as well. Then H and P2 are Consider the second case: H v P3 i s equipotent with P3 both subpotent with P3 , hence H u P2 subpotent with 2 x P3 and hence with P3 By hypothesis H u P 2 i s equipotent ei t h er t o PJ or t o P2 I n the f i r s t case, by the previous ( l ) , the s e t P3 i s subpotent with H ,

.

.

.

.

.

9

.

.

.

Chapter 2

hence well-orderable,

so t h a t

i s subpotent w i t h

. By

hence

H

I J

A

P2

A

51

i s well-orderable.

I n t h e second case,

i t e r a t i n g , we o b t a i n t h a t

.

H u P1 P1 ,

i s equipotent w i t h A o r w i t h P1 The f i r s t case i s excluded, The previous (1) shows A i s n o t subpotent w i t h A

.

since the Hartogs aleph o f that

i s subpotent w i t h

H

i s subpotent w i t h

, hence

well-orderable, and consequently A

itself i s well-orderable. 0 4 - L e t A be an i n f i n i t e s e t . Take the union o f A w i t h JL. and l e t B = F ( A u w ) . Then by ( 2 ) , we have 2 x B equipotent w i t h B . Using the gener a l i z e d continuum hypothesis, the statement (3) shows t h a t B i s well-orderable, P1

so A u W and hence EXERCISE 2

-

A

H

as w e l l .

THE CARDINAL OF THE SET OF FILTERS

be an i n f i n i t e s e t w i t h c a r d i n a l a . We s h a l l prove t h a t the cardinal o f the s e t o f f i l t e r s , o r as w e l l t h e c a r d i n a l o f t h e s e t o f u l t r a f i l t e r s on E Let

E

equals

2

t o t h e power

(a2)

(TARSKI; see BELL, SLOMSON 1969 p . 108; uses the

axiom o f choice). Since each f i l t e r i s a s e t of subsets o f

E

, the

s e t o f f i l t e r s has a t most the

above s t a t e d c a r d i n a l i t y . Hence i t s u f f i c e s t o c o n s t r u c t a s e t o f u l t r a f i l t e r s

E having t h i s c a r d i n a l . For each X we denote by (X) the s e t o f f i n i t e X . By the axiom o f choice, F ( X ) is equipotent w i t h X f o r each i n f i n i t e s e t X . I n the f o l l o w i n g , we o b t a i n u l t r a f i l t e r s on F ( ( E ) ) which i s equipotent w i t h E 1 Divide E i n t o two complementary subsets A, B o f t h e same c a r d i n a l a , and l e t f be a b i j e c t i v e mapping from A onto B To each subset X o f A associate X+ = the union o f X and (B-fo(X)) (notation, O from ch.1 5 1.2).

on

subsets o f

.

-

.

X, Y

Show t h a t f o r two d i s t i n c t subsets

X+, Y+ i s included i n the other. 2 To the s e t E associate t h e s e t

-

E'

, neither

A

of

= F(E)

-

o f the two images

o f a l l f i n i t e subsets o f

E

.

.

A associate X ' = E ' F(X+) Note t h a t the t r a n s f o r mation from X i n t o X ' i s i n j e c t i v e , hence t h e c a r d i n a l o f t h e s e t o f X ' i s a2 Moreover, f o r X, Y d i s t i n c t subsets o f A , we have X ' , Y ' n e i t h e r included i n t h e other. To each subset

X

of

.

3

-

Let

X

be a subset o f

A

Y1,

and

...,Yn .

( n integer)

be a f i n i t e s e t o f

Y o f A , a l l d i s t i n c t from X Then t h e i n t e r s e c t i o n o._ fI-It h e images " Y' = E' -3(Y') o f t h e Y i s n o t included i n the image X ' f X

subsets

Indeed t o each index

i = 1, ...,n g(i)

The f i n i t e s e t o f t h e Hence i t does n o t belong t o

Y; = E '

- F(Yf)

associate an element

i s included i n X ' = E'

-

3(X+)

X+

g(i)

of

X+

. -

y e t n o t included i n any y e t does belong t o each

, hence belongs t o t h e i n t e r s e c t i o n o f t h e

Y'

.

(YtnX')

Yt .

.

THEORY OF RELATIONS

58

t h a t E " has c a r d i n a l a . Denote by X ' o f subsets X o f A . Hence has c a r d i n a l a2 i s the set o f a l l F ( E ' - F ( X + ) ) L e t f denote t h e s e t o f a l l F ( X ' ) Hence where X i s an a r b i t r a r y subset o f A . Thus i s a l s o o f c a r d i n a l a2 . Moreover, each element o f i s a subset o f E" L e t H, K be two f i n i t e non-empty d i s j o i n t subsets o f . Then t h e i n t e r s e c t i o n 4

-

Let

E" = F ( E ' ) =

T(3 ( E ) ) , so

t h e s e t o f p r e c e d i n g images

.

.

K i s not included i n the union

A

Indeed

i s a set o f

H

F(X;)

U H

(i= 1,

.

...,m)

and

T(Y!) J

i s a set of

K

...,n )

w i t h m, n i n t e g e r s . Each X; and Y ! i s t h e image o f a subset J Y . o f A , a l l d i s t i n c t . F i x an i n d e x i 6 m : by ( 3 ) above, t h e i n t e r s e c J t i o n o f t h e Y ! ( j = 1, ..., n ) i s n o t i n c l u d e d i n X; . Hence t h e r e e x i s t s an J element h ( i ) which belongs t o t h i s i n t e r s e c t i o n and n o t t o X i . The s e t of h ( i ) (i= 1, ...,m) i s a f i n i t e subset o f each Y ' , hence an element o f each j ( Y j ) , hence an element o f t h e i n t e r s e c t i o n n K . However t h i s element i s n o t i n c l u d e d i n any o f t h e X; , hence belongs t o no F ( X ; ) , and so i s n o t an e l e ( j = 1,

or

Xi

uH

ment o f t h e u n i o n

. i s non-

By t h e p r e v i o u s statement, each f i n i t e i n t e r s e c t i o n o f elements of

-

empty. I n p a r t i c u l a r , two complementary subsets o f

5 - Let

, where

E"-S

. Associate

S

e.

U+ =, U p l u s a l l complements belongs t o t h e d i f f e r e n c e s e t 't'- U F o r two d i s t i n c t U ,

be a subset o f

U

cannot b o t h belong t o

E"

to it

.

V , t h e r e e x i s t s f o r example an element S o f which belongs t o U , hence t o U+ , and whose complement E"-S belongs t o V+ . I t f o l l o w s t h a t U+ and V+ a r e d i s t i n c t . F o r o t h e r w i s e , two complementary elements would belong t o U+ , these elements b e i n g o b t a i n e d f r o m two complementary elements o f f . say

U

and

Thus t h e s e t o f a l l power

(a2)

U+

, as

the set o f a l l

Beginning w i t h an a r b i t r a r y non-empty s u b s e t p o s i t i v e f i n i t e number o f elements o f

S1,. .. ,Sm

of

U and t h e elements

the difference s e t

if - U .

E"-T

cardinality

U+

E"-T1,.

of

U

, by

$ , c o n s i d e r an a r b i t r a r y

. . ,E"-Tn

T

, hence

where t h e s e

U

T

belong t o

the intersection o f the E"

S S and

extending

U+

.

correspond two d i s t i n c t u l t r a f i l t e r s , because o f t h e e x i s -

tence of complementary elements ( w i t h r e s p e c t t o

E" ) b e l o n g i n g t o t h e c o r r e s -

.

, j u s t as t h e s e t o f

ponding s e t s U+ Thus t h e s e t of u l t r a f i l t e r s on E" subsets o f f , has c a r d i n a l i t y 2 t o t h e power (a2)

-

t o the

d i s t i n g u i s h i n g t h e elements

i s non-empty. Thus t h e r e e x i s t s an u l t r a f i l t e r on

TO two d i s t i n c t

6

2

By t h e p r e v i o u s ( 4 ) , t h e i n t e r s e c t i o n o f t h e

i s n o t included i n t h e union o f the the

, has

U

.

.

Modulo t h e g e n e r a l i z e d continuum h y p o t h e s i s , we have a much s i m p l e r p r o o f

of t h e preceding, y i e l d i n g , f o r a s e t f i l t e r s on

E has c a r d i n a l i t y

2

E o f cardinal

t o t h e power

(a2)

a

.

, t h a t the set o f u l t r a -

.

Chapter 2

59

To see t h i s , consider a l l p a r t i t i o n s of E into two d i s j o i n t a , and t o t a l l y order t h i s s e t of p a r t i t i o n s by i t s cardinal Let uo be the f i r s t p a r t i t i o n , whose two associated subsets Eo(+) (associated t o the 1-sequence t ) and E o ( - ) . For each

subsets of cardinal b = a2 . shall be denoted of these subsets, Eo ( + ) f o r example, take the f i r s t p a r t i t i o n ul(+) such t h a t the intersection of Eo ( + ) with each of the subsets of E given by u l ( + ) has cardinal a Denote by El(++) and E l ( + - ) these two intersections. Do the same thing f o r the I n general, f o r each ordinal i s t r i c t l y l e s s t h a n 2-sequences (-+) and (--) b , note t h a t Card i 6 a by the generalized continuum hypothesis. We have t o Fix i and f i x an a r b i t r a r y i-sequence x with values (+) and ( - ) define the p a r t i t i o n u i ( x ) . Assume t h a t we have obtained, f o r each j < i , the sequence of the E . ( x . ) where xj+l i s the i n i t i a l interval of x with J J+1 length j+l By 8.1, the s e t of the E . ( x ) generates a f i l t e r which i s n o t J . j+l an u l t r a f i l t e r . So there e x i s t s a p a r t i t i o n u i ( x ) yielding f o r every intersection of f i n i t e l y many of the E . ( x . ) a s e t of cardinal a J J+1 Finally, for each b-sequence of values (+) and ( - ) , we obtain a f i l t e r such t h a t two d i s t i n c t b-sequences give two d i s t i n c t f i l t e r s . Hence there are b2 many such f i l t e r s .

.

.

.

.

.

EXERCISE 3 - A CLASSICAL PROOF OF HAUSDORFF-ZORN WITHOUT ORDINALS With the notions of partial and t o t a l ordering, b u t without using ordinals, we outline as follows the classical proof, using the axiom of choice, of the existence of a maximal chain: HAUSDORFF-ZORN axiom. Let A be a partial ordering, C a t o t a l l y ordered r e s t r i c t i o n of A We denote by X any chain which i s simultaneously a r e s t r i c t i o n of A and an extension of C Suppose t h a t no X i s maximal under inclusion, and l e t f be a choice function which associates t o each X a s t r i c t extension f(X) of X , again a r e s t r i c t i o n of A and an extension of C . We shall obtain a contradiction as follows. Denote by & every s e t of chains X satisfying the following conditions: (1) ‘6 i s t o t a l l y ordered under r e s t r i c t i o n (or extension); admits (2) i s closed with respect t o taking the union of the bases; hence a maximum element, which i s the common extension of a l l the chains in t o the union of t h e i r bases; (3) f o r each X in 6 other than the maximum chain, f(X) belongs t o and no s t r i c t intermediate chain between X and f(X) belongs t o There e x i s t such : f o r instance the singleton of C and the pair {C,f(C)} . Given two such E , say and E ’ , we shall prove t h a t one i s an i n i t i a l interval of the other ( f o r the ordering of extension defined between the chains X ) .

.

.

e

6

.

8,

THEORY OF RELATIONS

60 Let

1

be t h e s e t o f

&

elements o f

X

belonging t o the i n t e r s e c t i o n

which a r e r e s t r i c t i o n s o f

restrictions o f

, are

X

t h e same.

3

X

E

fi

E'

and t h e elements o f

, f o r which t h e E' which a r e

i s non-empty s i n c e t h e element

C

belongs

t o it. Define the chain

U

3 .

as t h e common e x t e n s i o n o f a l l elements o f

By ( 2 ) , t h e

U belongs t o t h e i n t e r s e c t i o n $ , A I ' . Moreover, e v e r y element X o f E which i s a r e s t r i c t i o n o f U , e i t h e r i s i d e n t i c a l t o U , o r i s a r e s t r i c t i o n o f a c h a i n b e l o n g i n g t o 9 : i n t h i s case X belongs t o 3 . Thus U belongs chain

to 9 . Either U

i s t h e maximum o f

f',

i s t h e maximum o f chain

f(U)

, so

&'

that

&

?,

U

and

i s an i n i t i a l segment o f

i s an i n i t i a l segment o f

n

belongs t o t h e i n t e r s e c t i o n

mediate c h a i n between longs t o

E

so t h a t

f(U)

belongs t o

which c o n t r a d i c t s t h e m a x i m a l i t y

We s h a l l prove t h a t t h e r e e x i s t s a maximum

.

& , and

( w i t h r e s p e c t t o i n c l u s i o n ) ; we

&' V

ment t h e c h a i n

V

f(V) denote t h e u n i o n o f a l l t h e

&

,

is t h a t no t o which we add as l a s t e l e -

.

e k i s indeed an

I n o t h e r words, t h a t

t i s f i e s ( l ) , ( Z ) , ( 3 ) . C o n d i t i o n (1) i s obvious, as w e l l as ( 2 ) s i n c e as l e a s t element. To o b t a i n ( 3 ) : i f

X

l y intermediate

f(X)

Y

between

X

and

c o n t a i n i n g as elements

f(X)

and

belong t o

belongs t o

X, f ( X )

and

one i s an i n i t i a l segment o f t h e o t h e r ) . Then t h i s contradiction.

E.

which i s t h e common e x t e n s i o n o f a l l t h e elements o f a l l t h e

It i s s u f f i c i e n t t o prove t h a t

l e a s t one

o f t h e maxi-

. So suppose

then adding the s t r i c t extension

maximum and l e t

U

or

E' . Moreover no s t r i c t l y i n t e r € , n o r t o & ' ; t h u s f ( U ) beo f U . Hence o u r c l a i m i s proved.

s h a l l t h e n o b t a i n a c o n t r a d i c t i o n by c o n s i d e r i n g t h e maximum c h a i n mum

%';

O r by ( 3 ) , t h e

Y

€*.

& * , and

&'

E"

sa-

has

V

if a strict-

then there e x i s t s a t

( s i n c e , f o r any two

E

,

i t s e l f does n o t s a t i s f y ( 3 ) :