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Chapter 5 Embeddability between Partial or Total Ordering
131
5
CHAPTER
EMBEDDABILITY BETWEEN PARTIAL OR TOTAL ORDERING
§
1 - EMBEDDABILITY,
IMMEDIATE
FAITHFUL
EXTENSION,
EXTENSION
EMBEDDABILITY, EQUIMORPHISM Let
be two r e l a t i o n s o f t h e same a r i t y . We say t h a t
R, S
R
i s embeddable
S o r i s s m a l l e r t h a n S under e m b e d d a b i l i t y , o r t h a t S admits an embedding o f R o r i s g r e a t e r t h a n R , i f f t h e r e e x i s t s a r e s t r i c t i o n o f S in
R ; we w r i t e R + S o r S i s s t r i c t l y embeddable i n S
isomorphic w i t h We say t h a t
.
o r s t r i c t l y smaller than
R
, or
S admits a s t r i c t embedding o f R o r i s s t r i c t l y g r e a t e r t h a n R , denoted
that by
R
R
R < S
or
We say t h a t
S > R , i f f
R$S
but
i s equimorphic w i t h
R
The comparison r e l a t i o n
4
S $ R .
S , denoted R
5 S
, iff
R 6'S
and
S
6
R
i s r e f l e x i v e and t r a n s i t i v e , hence d e f i n e s a quasi-
o r d e r i n g o n each s e t o f r e l a t i o n s . Moreover equimorphism i s symmetric and hence d e f i n e s an e q u i v a l e n c e r e l a t i o n . E m b e d d a b i l i t y i s n o t a n t i s y m m e t r i c , even up t o isomorphism: see t h e f o l l o w i n g examples. Let
be t h e c h a i n o f t h e r a t i o n a l s , and
Q
a l a s t element: t h e n
Q+l t h e e x t e n s i o n o b t a i n e d by adding
.
Q I Q+l
L e t LJ be t h e c h a i n o f t h e n a t u r a l numbers, and ch.2
5 5
1.7). Then
u-.G)E
a-
t h e converse c h a i n ( s e e
1 + ( W - . w ) (the ordinal product i s defined i n
3.7). I n t h e c h a i n o f n a t u r a l numbers, r e p l a c e each even number by Z ( t h e c h a i n of p o s i t i v e and n e g a t i v e i n t e g e r s ) and each odd number by a f i n i t e chain. We o b t a i n continuum many m u t u a l l y non-isomorphic chains, a l l o f which a r e equimorphic.
ch.2
1.1. L e t
R, S
be two equimorphic r e l a t i o n s . Then t h e r e e x i s t s a p a r t i t i o n o f
I R I i n t o two d i s j o i n t subsets D, D ' , and a p a r t i t i o n o f 1st i n t o two d i s j o i n t subsets E, E ' w i t h R/D isomorphic t o S/E and R / D ' isomort h e base phic t o where
S/E'
f
and
.
Repeat t h e p r o o f o f BERNSTEIN-SCHRODER's theorem (ch.1 g
5 1.4),
become isomorphisms from one r e l a t i o n o n t o a r e s t r i c t i o n o f
the other. The converse i s f a l s e , even f o r c h a i n s . Indeed, t h e c h a i n w o f t h e n a t u r a l numbers and t h e c h a i n
w+1
g i v e r i s e t o p a r t i t i o n s s a t i s f y i n g t h e above condi-
t i o n s . S i m i l a r l y f o r t h e incomparable c h a i n s U+c.4-
and
Z = W-+
.
.
132 1.2.
THEORY OF RELATIONS IMMEDIATE EXTENSION
Given a r e l a t i o n R , we say t h a t S i s an immediate extension o f R i f S i s an extension, and furthermore i f there does n o t e x i s t any s t r i c t l y intermediate r e l a t i o n T such t h a t R < T < S w i t h respect t o embeddability. We say a l s o t h a t S , o r any r e l a t i o n equimorphic w i t h S , i s an immediate successor o f R w i t h respect t o embeddability.
.
Moreover .if R For each r e l a t i o n R , t h e r e e x i s t s an immediate extension o f R has a r i t y >/ 1 , then t h e r e e x i s t a t l e a s t two immediate extensions (HAGENDORF 1977, p r o p o s i t i o n VI.5.6). 0 Suppose f i r s t t h a t
i s a 0-ary r e l a t i o n , say
R
R = (E,+)
: then i t s u f f i c e s t o
replace t h e base E by a s e t w i t h immediately g r e a t e r c a r d i n a l i t y : see ch.2 5 3.10. Suppose t h a t R i s a unary r e l a t i o n . L e t a be t h e c a r d i n a l i t y o f the s e t o f e l e -
R
ments g i v i n g t h e value (+) t o i t s u f f i c e s t o replace e i t h e r using ch.2 5 3.10. Suppose now t h a t R
a
has a r i t y
, and or
b t h e analogous c a r d i n a l i t y f o r (-). Then b by an immediately g r e a t e r c a r d i n a l , again
n >/ 2
. Add
t o t h e base E o f
R
a set
D dis-
t o have base E u D w i t h R+/E = R and R+/D j o i n t w i t h E , and d e f i n e R' always (t), and f i n a l l y w i t h R+ t a k i n g the value (+) f o r those n-tuples contain i n g a t l e a s t one term i n D F i n a l l y choose f o r d = Card D t h e l e a s t aleph f o r which Rt i s n o t embeddable i n R , hence R+> R L e t us prove t h a t R' i s an
.
.
immediate extension o f R ; t h e r e l a t i o n R- s i m i l a r l y defined by exchanging (t) and ( - ) , being another immediate extension, obviously incomparable w i t h R' with respect t o embeddabi 1ity. Suppose f i r s t t h a t d i s an i n f i n i t e aleph, and t h a t t h e r e e x i s t s a s t r i c t l y i n t e r mediate r e l a t i o n T w i t h R < T < R' . Consider T as a r e s t r i c t i o n o f Rt Then
.
.
D n IT1 has c a r d i n a l i t y d Indeed i f i t had c a r d i n a l i t y c d , R (more p r e c i s e l y , i n a r e s t r i c t i o n o f Rt t o E increased w i t h (c:d) many elements o f D ). Now p a r t i t i o n DnITI i n t o two the i n t e r s e c t i o n
then T would be embeddable i n
d i s j o i n t subsets, each w i t h c a r d i n a l i t y d
, say
-
D'
and D"
. Then
T
i s isomor-
p h i c w i t h i t s r e s t r i c t i o n t o I T 1 D" , so t h a t R i s embeddable i n t h i s r e s t r i c t i o n , and f i n a l l y R+ i s embeddable i n T : c o n t r a d i c t i o n . Now i t remains t o consider t h e case where d = Card D i s f i n i t e . We f i r s t see t h a t d = 1 . Indeed assume t h a t d f i n i t e and >, 2 Then by hypothesis, the extension of R obtained by adding o n l y one element u t o t h e base I R 1 , w i t h value (t)
.
f o r a l l n-tuples c o n t a i n i n g u , i s embeddable i n R , thus equimorphic w i t h R I t e r a t i n g t h i s , t h e s i m i l a r extension obtained by adding 2 elements, i s s t i l l equimorphic w i t h
R , and so on u n t i l we add d elements: c o n t r a d i c t i o n .
Now examine t h e case where singleton constitutes
D
d = 1
. Call
. Consider
u the supplementary element, whose
again the intermediate r e l a t i o n T as a
.
Chapter 5
133
.
r e s t r i c t i o n o f R+ ; obviously u belongs t o t h e base I T We say t h a t an e l e ment x i n the base i s a (+)-element (mod R ) i f f every n - t u p l e which contains x gives value (+) t o R . Analogous d e f i n i t i o n f o r a (+)-element (mod T) ; i n p a r t i c u l a r , u i s a (+)-element (mod T) . Every (+)-element (mod R) belongs t o the base I T 1 and i s a (+)-element (mod T). Indeed otherwise, i f x i s a (+)-element (mod R ) and does n o t belong t o I T I , t h e n by r e p l a c i n g u by x we could embed T i n R : c o n t r a d i c t i o n . E i t h e r t h e r e e x i s t D e d e k i n d - i n f i n i t e l y many (+)-elements (mod R ) . Then R+ isomorphic w i t h R : c o n t r a d i c t i o n . O r t h e s e t o f (+)-elements (mod R ) has D e d e k i n d - f i n i t e c a r d i n a l i t y , say
h
. Then t h e r e e x i s t a t l e a s t
is
h + l many
(+)-elements (mod T) . Consider a r e s t r i c t i o n R ' o f T which i s isomorphic with R Since we have e x a c t l y h many (+)-elements (mod R ' ) and a t l e a s t h t l many (+)-elements (mod T) , t h e r e e x i s t s a t l e a s t one (+)-element (mod
.
T) ,
.
say v , which does n o t belong t o t h e base I R ' I Then the r e s t r i c t i o n o f T t o t h e base I R'I p l u s t h e element v i s isomorphic w i t h R' : c o n t r a d i c t i o n . 0
1.3. FAITHFUL EXTENSION
.
(1) L e t R , S be two n-ary r e l a t i o n s ( n >r 1) Assume t h a t S does not Then t h e r e e x i s t s a s t r i c t l y g r e a t e r extension T admit an embedding o f R
.
.
of S which does n o t _admit an embedding o f R We c a l l i t a f a i t h f u l extensio? o f S modulo R . Moreover we can choose T t o be an immediate extension o f S : see HAGENDORF 1977. The statement i s obviously f a l s e f o r a r i t y zero.
R aqd S are unary. L e t at be 'the c a r d i n a l i t y o f t h e s e t o f elements g i v i n g the value (+) t o R , and a- the analogous c a r d i n a l i t y f o r (-); s i m i l a r l y l e t b+ and b- be t h e analogous c a r d i n a l i t i e s f o r S . Since R $ S ,
0 Suppose f i r s t t h a t
.
e i t h e r b + < a+ o r b-< aSuppose t h e f i r s t case holds, t h e argument being analogous f o r t h e second case. It s u f f i c e s t o take an extension of S i n which bt i s preserved and b- i s replaced by an immediately l a r g e r c a r d i n a l . Suppose t h a t R and S have a r i t y n 3 2 Add t o t h e base E o f S a s e t '0
.
which i s d i s j o i n t from
E
, and
d e f i n e the extension T+
of
S w i t h base
EvD',
THEORY OF RELATIONS
134
t a k i n g t h e v a l u e (+) f o r those choose
w i t h c a r d i n a l ( a l e p h ) s u f f i c i e n t l y l a r g e t o have
D+
+
w i t h t h e value ( - ) , t h u s o b t a i n i n g
and
0-
T->
.
S
.
Do t h e same
R
4:'
T+> S
We c l a i m t h a t
. Also
D
n - t u p l e s c o n t a i n i n g a t l e a s t one t e r m o f
or
, which y i e l d s o u r c o n c l u s i o n . Indeed suopose t h e c o n t r a r y , and c o n s i d e r R as a r e s t r i c t i o n o f T+ , The base o f R i s n o t a subset o f E , s i n c e R $ R $T-
hence t h e r e e x i s t s an element
u+
R
such t h a t
u+
n - t u p l e c o n t a i n i n g a t l e a s t one t e r m equal t o
S,
takes t h e v a l u e (+) f o r each
. There
e x i s t s an analogous e l e -
ment f o r t h e v a l u e ( - ) : c o n t r a d i c t i o n . 0 be a r e l a t i o n w i t h a r i t y 5 2
R
(2) Let
e x i s t s a common e x t e n s i o n o f The c o n t r a p o s i t i v e i s : i f X
3 S2 , t h e n
or
6 S1
R
and
S1
and
S1&
R
R,< X
f o r every
R 6 S2
.
X
and
El
.
E2
p a r t i t i o n o f t h e base o f e v e r y element =
R
have d i s j o i n t
S2
taking
E2
and t h e o t h e r i n
El
R.
and . ,Sh.
.
E2
or
S+
contrary; then there e x i s t s a
i n t o two non-empty d i s j o i n t subsets such t h a t , f o r
i n one subset and
u
. Suopose t h e
R
.
S1,.
f o r t h e value ( - ) . It s u f f i c e s t o see t h a t e i t h e r
S-
does n o t a d m i t an embedding o f
S-
and
S1
St be t h e common e x t e n s i o n w i t h base El
Let
there
X >,S1
Extend t h i s t o any f i n i t e sequence
t h e value (+) f o r those o r d e r e d p a i r s w i t h one t e r m i n Analogously d e f i n e
. Then
S2 & R
which s a t i s f i e s b o t h
0 Take t h e case o f a b i n a r y r e l a t i o n , and suppose t h a t
bases
and
which does n o t a d m i t an embedding o f
S2
v
i n t h e o t h e r , we have
R(u,v) = R(v,u)
=
+ ; same c o n c l u s i o n w i t h t h e v a l u e ( - ) . Note t h a t , g i v e n two p a r t i t i o n s o f t h e u, v
base, each w i t h two non-empty d i s j o i n t s e t s , t h e r e e x i s t two elements
in
t h e base which a r e separated b o t h by t h e f i r s t p a r t i t i o n and by t h e second. Thus R(u,v)
=
+ and - :
2 and non-empty base. Suppose t h a t
S
& R1
u, v
t h e r e e x i s t two elements
g i v i n g simultaneously
contradiction. 0 The p r o p o s i t i o n i s o b v i o u s l y f a l s e f o r unary r e l a t i o n s . (3) Let
S
. Then
S%R3
>/
have a r i t y
there e x i s t s a proper extension
embeddabilities
R1 ,
S+*
S+$
R2
and
S+ f S
S+$ R3
.
element Let
S2
a
and s e t t i n g
S,(a,x)
of
= S (x,a)
. Let
S =
+
S1
, say
r o l e of
. Hence
i n t h e base o f t h i s
t o f i x t h e ideas, p l a y i n g t h e r o l e o f a
and
R2
x
in
S3
. Suppose f i r s t l y
a
t h a t t h e base
in
.
Si
(i
IS1
i n t h e base S3
=
.
be o b t a i n e d w i t h
S (x,a) = - f o r e v e r y x i n I S ( , and moreover 3 w i t h t h e same c o n d i t i o n s , except t h a t S4(a,a) = - .
R
and
be o b t a i n e d by a d d i n g a new
f o r every
Suppose o u r c o n c l u s i o n i s f a l s e . Then t h e r e e x i s t two al
R1
and
an embedding o f a same
$ R2
which r e s p e c t s t h e non-
1 be s i m i l a r l y o b t a i n e d w i t h ( - ) ' i n s t e a d o f ( + ) . L e t
S (a,x) = + 3 F i n a l l y S4
S
F o r t h e a r i t y 1 o r f o r empty
base, t h e p r o p o s i t i o n i s o b v i o u s l y f a l s e , even w i t h o n l y 0 Consider t h e 4 f o l l o w i n g e x t e n s i o n s
,
S3(a,a)
1,2,3,4)
= +
.
which a d m i t
R , t h e r e e x i s t s an element
S1 and an a3 p l a y i n g t h e
I R I has c a r d i n a l i t y
32 .
Chapter 5 Then
al
and
and
R(x,a3)
a3 =
-
are d i s t i n c t , since x # a3
f o r every
.
135
R(x,al)
+
=
R(al,a3)
takes s i m u l t a n e o u s l y t h e
f o r every
value (+) and t h e v a l u e ( - ) : c o n t r a d i c t i o n . Analogous argument f o r and
S1
,
S4
S2
and
S3
,
and
S2
Suppose now t h a t t h e base I R I takes t h e v a l u e ( - ) . Since S >
,
S4
x # al
= R(al,x)
Moreover
and
S3
and
S1
,
S2
.
S4
has c a r d i n a l i t y 1, and t o f i x t h e ideas, t h a t R R and by h y p o t h e s i s S non-empty, n e c e s s a r i l y S
i s r e f l e x i v e . As p r e v i o u s l y d e f i n e e x t e n s i o n s S1, S2, S3 which now a r e a l l t h r e e r e f l e x i v e . E i t h e r R1 = R 2 = R3 = R and t h e n o u r c o n c l u s i o n h o l d s . O r R1 and possibly
R , t h u s have c a r d i n a l i t i e s
are d i s t i n c t from
R2
our conclusion i s false: then
, and
( i = 1,2,3)
R1
1.4. Consider an 13 -sequence o f r e l a t i o n s
Ria A
i
f o r every
, then
R+
suppose Si
A
Ri
( i integer)
3
o f common a r i t y
A
2
be a r e l a t i o n o f t h e same a r i t y .
t h e r e e x i s t s a comnon e x t e n s i o n o f a l l t h e
which does n o t a d m i t an embedding o f 0 Let
. Again
t h e argument t e r m i n a t e s as p r e v i o u s l y . 0
and w i t h m u t u a l l y d i s j o i n t bases. L e t
.-have
2
f o r i n s t a n c e i s embeddable i n a t l e a s t two
Ri
.
denote t h e common e x t e n s i o n o f t h e
Ri
on t h e u n i o n o f t h e bases, which
takes t h e v a l u e (+) f o r a l l t h o s e n - t u p l e s ( n = a r i t y ) c o n t a i n i n g a t l e a s t two terms t a k e n f r o m two d i s t i n c t bases. Analogously d e f i n e t h e e x t e n s i o n that
A
i s embeddable b o t h i n
, there
Ri
R+
and
R-
n e c e s s a r i i y e x i s t two elements
. Since x, y
t r a n s f o r m e d i n t o two elements i n two d i s t i n c t
A
, simultaneously
and second embedding. tience f o r an n - t u p l e c o n t a i n i n g b o t h for
I A I , which a r e
i n t h e base
lRil
. Suppose
R-
i s n o t embeddable i n any
x
and
f o r the f i r s t
, we
y
have
t h e v a l u e (+) and t h e v a l u e ( - ) : c o n t r a d i c t i o n . 0
A
5 2 - EMBEDDABILITY
BETWEEN PARTIAL
(KRUSKAL); CANTOR'S GLEASON); TOURNAMENT
OF FINITE
TREES
(DILWORTH,
ORDERINGS;
WELL PARTIAL ORDERING ORDERINGS
THEOREM FOR PARTIAL
2.1. There e x i s t i n f i n i t e l y many f i n i t e p a r t i a l o r d e r i n g s which a r e m u t u a l l y incom-parable w i t h r e s p e c t t o e m b e d d a b i l i t y . 0 Let
a'< i
be t h e p a r t i a l o r d e r i n g on 5 elements
A1 b'<
,let
v Ai
until
ui-l
with
be t h e p a r t i a l o r d e r i n g based on 2i+3 elements
U ~ ~ ~ , V ~ , ~ , V ~ , ~ , . w. i .t h, Va ~<~b ~ , ~
bilities
a,b,a',b',v
a(
b
,
and i n c o m p a r a b i l i t y elsewhere. Now more g e n e r a l l y f o r each i n t e g e r
u1 < v ~ , and ~
and
and
a'<
u1 < v ~ , and ~ t h e n u2 < v1,2 ui-1
< vi-l,i
a,b,a',b',u
b;< and
vi-l,i
ui
l,..., and
comparaand so
; and i n c o m p a r a b i l i t y elsewhere.
The p a r t i a l o r d e r i n g s t h u s d e f i n e d a r e m u t u a l l y incomparable w i t h r e s p e c t t o emb e d d a b i l i t y (example comnunicated i n 1969 by JULLIEN). 0
I36
THEORY OF RELATIONS
2.2.
( 1 ) There e x i s t s a s t r i c t l y decreasing (under e m b e d d a b i l i t y ) w - s e q u e n c e
o f denumerable p a r t i a l o r d e r i n g s . 0
To each s e t
I o f integers, associate the p a r t i a l ordering
extension o f the f i n i t e p a r t i a l orderings
Ai
(i
e
AI
o b t a i n e d as an
I) , taken t o be m u t u a l l y
incomparable. Then t a k e an i n f i n i t e s t r i c t l y d e c r e a s i n g sequence o f s e t s
I
.0
( 2 ) There e x i s t continuum many denumerable p a r t i a l o r d e r i n g s which a r e m u t u a l l y incomparable under e m b e d d a b i l i t y . 0
Take t h e p r e c e d i n g
AI
I, J
and n o t e t h a t , f o r two s e t s
o f which i n c l u d e s t h e o t h e r , t h e n
AI
and
AJ
o f integers neither
a r e incomparable. 0
Problem posed by HAGENDORF. E x i s t e n c e o f a s t r i c t l y d e c r e a s i n g G1-sequence o f denumerable p a r t i a l o r d e r i n g s . Same problem f o r r e l a t i o n s , posed by POUZET. 2.3. THEOREM OF THE WELL PARTIAL ORDERING
OF FINITE TREES
Embeddability between f i n i t e t r e e s i s a w e l l p a r t i a l o r d e r i n g (KRUSKAL 1960). 0
We can always assume t h a t each o f t h e c o n s i d e r e d f i n i t e t r e e s has a minimum
element: i t s u f f i c e s t o add a minimum, and even a new minimum i f t h e r e a l r e a d y e x i s t s a minimum, t o each f i n i t e t r e e ; t h e n e m b e d d a b i l i t y o r non-embeddability i s preserved. Suppose t h a t e m b e d d a b i l i t y between f i n i t e t r e e s w i t h a minimum, i s n o t a w e l l p a r -
w -sequence o f such t r e e s , which i s bad w i t h
t i a l o r d e r i n g . Then t h e r e e x i s t s an r e s p e c t t o e m b e d d a b i l i t y : see ch.4
5
s t r o n g l y minimal bad w-sequence:
ch.4
3 . 2 . ( 2 ) c o u n t a b l e case. Hence t h e r e e x i s t s a
5
2.iO.
I n t h i s sequence, t h e terms a r e
m u t u a l l y incomparable w i t h r e s p e c t t o e m b e d d a b i l i t y : ch.4 5 2.3 and 2.8. L e t U denote t h i s sequence, and Ui ( i i n t e g e r ) each term. The s e t o f f i n i t e t r e e s w i t h a minimum, which a r e s t r i c t l y embeddable i n a
5
see ch.4 Let
H
Ui
,
i s a well p a r t i a l ordering:
4.3 and 2.8.
denote t h i s c o u n t a b l e s e t (up t o isomorphism) o f f i n i t e t r e e s w i t h a m i n i -
mum. Embeddability between words, o r f i n i t e sequences o f elements o f w e l l p a r t i a l o r d e r i n g : ch.4
5
4.4 (HIGMAN). To each t r e e
H
,
is a
a s s o c i a t e one o f t h e
Ui f i n i t e sequences o b t a i n e d by t o t a l l y o r d e r i n g i n an a r b i t r a r y manner t h e immediate successors o f t h e minimum, t h e n by r e p l a c i n g each o f them by t h e s u b - t r e e o f t h o s e g r e a t e r o r equal elements. Then t h e p r e c e d i n g sequence
words i n Ui Ui
H
, hence
U
becomes a sequence o f
i s good. Moreover, i f t h e word t h u s s u b s t i t u t e d f o r t h e t r e e
i s embeddable i n t h e word s u b s t i t u t e d f o r UJ. (i,j i n t e g e r s ) , t h e n t h e t r e e i t s e l f i s embeddable i n U . . Hence t h e sequence U i s good: c o n t r a d i c t i o n . 0 3
2.4. A b i n a r y r e l a t i o n
A
i s a p a r t i a l ordering i f f
A
does n o t a d m i t an embed-
ding o f the following f i n i t e relations: t h e b i n a r y r e l a t i o n w i t h c a r d i n a l i t y 1 and v a l u e ( - ) ( t h i s e n s u r e s r e f l e x i v i t y ) ; t h e r e l a t i o n always (+) w i t h c a r d i n a l i t y 2 ( a n t i s y m n e t r y ) ; t h e r e f l e x i v e b i n a r y cycle with cardinality 3
, and
f i n a l l y t h e c o n s e c u t i v i t y r e l a t i o n on 3 elements
Chapter 5
137
associated w i t h t h e c h a i n o f c a r d i n a l i t y 3 : see ch.2
5
8.6 ( t h e s e two non-embed-
d a b i l i t i e s ensure t r a n s i t i v i t y ) .
2.5.
or
Every denumerable p a r t i a l o r d e r i n g admits an embedding o f t h e o r d i n a l w
i t s converse
(J-
o r t h e denumerable f r e e p a r t i a l o r d e r i n g .
I n p a r t i c u l a r , e v e r y denumerable f i n i t e l y f r e e p a r t i a l o r d e r i n g , hence e v e r y denu-
.
merable chain, admits an embedding o f w o r L,I-
5
WiXh ch.4
0 Enumerate t h e elements
integers ai < a ch.3
5
j
T h i s i s KONIG's lemma: compare
4.5 i n t h e case 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, j
ai
o f t h e base, and p a r t i t i o n t h e p a i r s o f
( i integer)
where we assume
i< j
,
i n t o 3 c o l o r s , a c c o r d i n g t o whether
o r l a . (modulo t h e g i v e n p a r t i a l o r d e r i n g ) . By RAMSEY's theorem or 7 a j J 1.1, t h e r e e x i s t s a denumerable s e t o f i n t e g e r i n d i c e s i n which a l l p a i r s
have t h e same c o l o r . According t o whether i t i s t h e f i r s t , second o r t h i r d c o l o r , the given p a r t i a l o r d e r i n g admits an embedding o f
cc)
or
w - or
a denumerable
relation o f identity. 0 Modulo t h e denumerable subset axiom (ch.1
w
admits an embedding o f
A
or w -
5
2.6),
every i n f i n i t e p a r t i a l ordering
o r t h e denumerable f r e e p a r t i a l o r d e r i n g .
well-founded p a r t i a l o r d e r i n g does n o t a d m i t an embedding o f
w- .
Conversely,
a denumerable p a r t i a l o r d e r i n g ( o r more g e n e r a l l y a p a r t i a l o r d e r i n g w i t h w e l l orderable base) which does n o t a d m i t an embedding o f
w - , i s well-founded. With
the axiom o f dependent choice, e v e r y p a r t i a l o r d e r i n g which does n o t admit an embedding o f
w - i s well-founded: see ch.2 5 2.4.
2.6. CANTOR'S THEOREM FOR PARTIAL ORDERINGS
, the
Given a p a r t i a l o r d e r i n g A
, admits
A
vals o f
I n particular, i f embedding o f 0 Let
A g B
A
p a r t i a l o r d e r i n g o f i n c l u s i o n among i n i t i a l i n t e r -
a s t r i c t embedding o f A
A
(DILWORTH, GLEASON 1964).
i s a chain, t h e n t h e c h a i n o f c u t s o f
A
B denote t h e p a r t i a l o r d e r i n g o f i n i t i a l i n t e r v a l s o f
, it
admits a s t r i c t
.
s u f f i c e s t o a s s o c i a t e t o each element
x
of
A
. To
see t h a t
I A I the i n i t i a l interval
.
6 x (mod A ) NOW suppose t h a t BG A and l e t f be an isomorphism o f B i n t o A Some i n i t i a l i n t e r v a l s X o f A s a t i s f y t h e r e l a t i o n f ( X ) E X : f o r example t h e e n t i r e
o f elements
x
f(U)
< f(U)
Phhm f ( V ) < Thus
. L e t U be t h e i n t e r s e c t i o n o f a l l t h e s e i n t e r v a l s . We s h a l l 4 U . Indeed i f f ( U ) E U , t h e n t h e i n t e r v a l V o f those e l e -
X = \A[
interval prove t h a t ments
.
f(U)
+U
there e x i s t s
(mod A)
satisfies
f ( U ) (mod A ) and s i n c e
one o f t h e s e
, hence U
VcU
( s t r i c t i n c l u s i o n ) . Hence by t h e isomor-
f(V) E V
and so
Vz
i s t h e i n t e r s e c t i o n o f those X
, say
Xo
, which
U : contradiction. X
such t h a t
f ( X ) i5 X,
does n o t c o n t a i n t h e element
f(U).
THEORY OF RELATIONS
138
Thus we have Moreover 2.7.
Xoz U
f(U)
+
Xo
and
# U , hence by isomorphism f ( U ) < f(Xo) (mod A)
and so
f(Xo)
$
Xo : c o n t r a d i c t i o n .
TOURNAMENT
A binary r e l a t i o n A
i s c a l l e d a tournament i f f i t i s r e f l e x i v e , antisymmetric by a n t i and comparable: f o r a l l x, y e i t h e r A(x,y) = + (and thus A(y,x) =
-
-
symnetry) o r A(y,x) = t (and A(x,y) = ) . I n the f i r s t case we say t h a t f o l l o w s x (mod A) ; i n t h e second case y precedes x (mod A)
y
. E - u
For each element u o f t h e base E o f A , t h e elements i n are divided i n t o two complementary sets, formed r e s p e c t i v e l y o f those elements which f o l l o w
u and those which precede u
.
Every r e s t r i c t i o n o f a tournament i s a tournament.
A tournament i s a chain i f f i t i s t r a n s i t i v e . The b i n a r y c y c l e on 3 elements (see ch.2 5 8.6), obviously m o d i f i e d t o be r e f l e x i ve, i s a tournament. Yet a b i n a r y r e f l e x i v e c y c l e w i t h c a r d i n a l i t y 3 4 i s not antisymmetric, thus i s n o t a tournament.
A tournament i s a chain i f f i t does n o t admit an embedding o f the b i n a r y r e f l e x i v e c y c l e w i t h c a r d i n a l i t y 3.
A b i n a r y r e l a t i o n i s a tournament i f f i t admits no embedding o f t h e b i n a r y r e l a t i o n w i t h c a r d i n a l i t y 1 and value ( - ) (ensures r e f l e x i v i t y ) ; no embedding o f t h e r e l a t i o n always (+) w i t h c a r d i n a l i t y 2 (ensures antisymmetry); f i n a l l y no embedding o f the i d e n t i t y r e l a t i o n w i t h c a r d i n a l i t y 2 (ensures c o m p a r a b i l i t y ) .
§
3
- DENSE CHAIN:
A-DENSE CHAIN, FOR AN I N F I N I T E CARDINAL A
3.1. DENSE CHAIN A chain i s s a i d t o be dense i f f i t s base i s i n f i n i t e , and i f between any two d i s t i n c t elements x < y t h e r e e x i s t s an element z : x c z < y Hence between any two elements x and y 7 x t h e r e e x i s t i n f i n i t e l y many e l e ments.
.
To each z ( x < z c y ) associate the i n t e r v a l (x,z) : t h e r e i s no minimal interval
(x,z)
w i t h respect t o i n c l u s i o n (use ch.1 tj 1.1). 0
Every denumerable dense chain w i t h o u t any minimum o r maximum, i s isomorphic to_ t h e chain
Q o f the ratj.1..
Every countable chain i s embeddable - -- i n__ Q
.
Every dense chain admits an embedding o f Q (uses dependent choice; i f t h e base i s denumerable, o r o n l y w e l l - o r d e r a b l e ) .
ZF s u f f i c e s
Chapter 5
U
139
Consider t h e s e t o f f i n i t e s t r i c t l y i n c r e a s i n g sequences o f elements o f t h e chain,
say u o c u1 < ... < uh ( h i n t e g e r ) sequences w i t h a r b i t r a r y l e n g t h h t h e form
<
vo
the r e l a t i o n 3.2. L e t
,u, R
A
, and t h e r e l a t i o n R which t o each , a s s o c i a t e s any sequence w i t h l e n g t h
2h
+ 1of
c v1 c u 1 < . . . < vh < uh< v ~ .+ Then ~ a p p l y dependent c h o i c e t o . The o b t a i n e d w - s e q u e n c e y i e l d s a r e s t r i c t i o n isomorphic w i t h Q.
be a denumerable c h a i n . I f
ordinal, then
o f these
A
A
0
admits an embedding o f e v e r y c o u n t a b l e
Q
admits an embedding o f t h e c h a i n
(uses c o u n t a b l e axiom o f
choice). 0
To each c o u n t a b l e o r d i n a l
embedded i n
A
@
, a s s o c i a t e 4 +1+a
. Hence a s s o c i a t e
a(& )
an element
i n t h e l o w e r i n t e r v a l and i n t h e i n t e r v a l above ble, y e t there are element
a
wl-many
which by h y p o t h e s i s can be such t h a t 4
i s embeddable
) . The c h a i n
a(*
A
i s denumera-
c o u n t a b l e o r d i n a l s 6 .So t h e r e e x i s t s a t l e a s t one
such t h a t e v e r y c o u n t a b l e o r d i n a l i s embeddable b o t h below and above
a
( c o u n t a b l e axiom o f choice: we use t h e f a c t t h a t e v e r y denumerable u n i o n o f denume-
5
r a b l e o r d i n a l s i s a denumerable o r d i n a l : see c h . 1 a restriction o f
3.3. L e t
A
A
be a
which i s isomorphic w i t h
2 . 5 ) . By i t e r a t i o n , we o b t a i l i
.0
Q
non-empty c h a i n w i t h an i n i t i a l i n t e r v a l and a f i n a l i n t e r v a l ,
b o t h d i s j o i n t and i n each of which
A
i s embeddable. Then
(uses dependent choice; ZF s u f f i c e s i f
A
Q
i s embeddable i n A
i s denumerable o r w i t h w e l l - o r d e r a b l e
base). 0 By i t e r a t i o n , t h e h y p o t h e s i s a l l o w s one t o d i v i d e
v a l s , i n each o f which that
A
A
i n t o three d i s j o i n t i n t e r -
A
i s embeddable. Hence t h e r e e x i s t s an element
i s embeddable b o t h b e f o r e and a f t e r A
choice, we o b t a i n a r e s t r i c t i o n o f
a
.
a
such
By i t e r a t i o n u s i n g dependent
which i s isomorphic w i t h
Q
.
0
3 . 4 . a-DENSE CHAIN Given an i n f i n i t e c a r d i n a l
a
, we
say t h a t a c h a i n i s
n i t e and between any two d i s t i n c t elements equal t o
a
Sense o f 3.1,
. Analogous is
(
d e f i n i t i o n f o r a chain
y
a--
, the
( 3 a)--.
i f i t s base i s i n f i i n t e r v a l has c a r d i n a l i t y A dense chain, i n t h e
a
, there
exists a chain with cardinal
a
which
a-dense (uses axiom o f c h o i c e ) .
c3 S t a r t w i t h t h e o r d i n a l p r o d u c t
a
and
w )-dense (uses denumerable subset axiom).
Given an i n f i n i t e c a r d i n a l
is
x
Q.a
where
Q
i s t h e c h a i n o f t h e r a t i o n a l s , and
i s an o r d i n a l , more p r e c i s e l y an a l e p h ( r e c a l l t h a t , w i t h choice axiom, e v e r y
c a r d i n a l i s an aleph: c h . 1
0
6 . 1 ) . The c h a i n
Q.a
has c a r d i n a l i t y
a
and i s ob-
v i o u s l y dense w i t h o u t any minimum or maximum. A l s o t h e s e t o f f i n i t e sequences o f elements i n
Q.a
has c a r d i n a l i t y
a
. We
o r d e r t h i s s e t l e x i c o g r a p h i c a l l y . Now i t
THEORY OF RELATIONS
140
a-dense o r d e r i n g . Indeed, c o n s i d e r two f i n i t e se-
s u f f i c e s t o see t h a t i t i s an
u
quences
and
Denote by
i with
index ui
ui
and
strictly after
u
and
v
vi
the
ui
< vi
, and
vi
( i integer)
w i t h r e s p e c t t o l e x i c o g r a p h i c a l comparison. i - t h terms. E i t h e r t h e r e e x i s t s a l e a s t
wi
(mod Q . a ) ; then we t a k e
we have
which i s s t r i c t l y between
a-many f i n i t e sequences b e g i n n i n g w i t h
t h e y a l l a r e s i t u a t e d between
u
and
.
v
u
Or
U ~ , . . . , U ~ - ~ , W ~:
i s an i n i t i a l i n t e r v a l o f
3.5.
< vh
wh
an element
(h = length o f
v
,
f o l l o w e d by
u
and we have t h e same c o n c l u s i o n by t a k i n g sequences b e g i n n i n g w i t h
u ). 0
L e t w , be an i n f i n i t e r e g u l a r a l e p h . Every c h a i n w i t h c a r d i n a l i t y
admits an embedding e i t h e r o f t h e o r d i n a l
o r i t s converse
a , ,
w,
,
,
CJ,
W , -dense c h a i n .
Let
0
A
be a c h a i n w i t h c a r d i n a l i t y w 4 , i n which n e i t h e r ad n o r i t s conver-
se i s embeddable. L e t
x, y
I A l ; we p u t
be elements o f
<
i f t h e i n t e r v a l between them has c a r d i n a l i t y
.
LJ&
x
equivalent w i t h
y
The e q u i v a l e n c e c l a s s e s a r e
.
Every e q u i v a l e n c e c l a s s has c a r d i n a l i t y < L d M . Indeed l e t T be an e q u i v a l e n c e c l a s s . Take an o r d i n a l indexed sequence o f elements o f T which i s
intervals o f
A
s t r i c t l y i n c r e a s i n g (mod A) and c o f i n a l i n less than
~3~
,
f i r s t t e r m o f t h e sequence, i s a u n i o n o f nality than
<
ui
. This
T
s i n c e w , i s n o t embeddable i n
.
By r e g u l a r i t y o f w ,
a d .Same
<
sequence has l e n g t h s t r i c t l y
.
The subset o f
w,-many
, this
argument f o r t h e subset o f
A
T
above t h e
i n t e r v a l s , each w i t h c a r d i -
u n i o n has c a r d i n a l i t y s t r i c t l y l e s s T
b e f o r e t h e f i r s t t e r m o f t h e se-
quence, by c o n s t r u c t i n g a d e c r e a s i n g o r d i n a l - i n d e x e d sequence. Since t h e e q u i v a l e n c e c l a s s e s a l l have c a r d i n a l i t y s t r i c t l y l e s s t h a n
ad
i s regular, there e x i s t
wd
and
C d d -many e q u i v a l e n c e c l a s s e s . Take a r e p r e s e n t a t i -
ye element f r o m each c l a s s . Again by r e g u l a r i t y , between two r e p r e s e n t a t i v e s t h e r e necessarily e x i s t
LJ, -many e q u i v a l e n c e c l a s s e s , hence as many r e p r e s e n t a t i v e s .
Finally, the r e s t r i c t i o n o f
A
t o t h e s e t o f r e p r e s e n t a t i v e s i s an W, -dense
chain. 0 T h i s p r o p o s i t i o n i s no l o n g e r t r u e i f a d i s a s i n g u l a r a l e p h . 0
Take t h e sum
cc,- + LJ
+ ... +
i~ .1
+ ...
where
i i s an a r b i t r a r y i n t e g e r
and (Ji designates t h e r e t r o - o r d i n a l converse o f o i . The sum i s t h u s a c h a i n with cardinality W . F i r s t l y , t h e o n l y w e l l - o r d e r e d r e s t r i c t i o n s of t h i s sum a r e t h e f i n i t e c h a i n s and u
. No dense
c h a i n i s embeddable i n t h i s sum.
Secondly, a r e t r o - o r d i n a l i s embeddable i n t h i s sum o n l y i f i t i s embeddable i n one o f the
wi
3.6. L e t aleph
,
A
w,
i s n o t embeddable. 0
be a p a r t i a l o r d e r i n g whose base has c a r d i n a l i t y an i n f i n i t e r e q u l a r
. Suppose t h a t ,
(dw
elements
so
f o r each element u o f t h e base I A l , t h e s e t o f I u has c a r d i n a l i t y s t r i c t l y l e s s t h a n c d K . Then t h e r e e x i s t s
141
Chapter 5
a t o t a l l y ordered r e s t r i c t i o n of
A
which i s isomorphic with the ordinal L J .~
a Well-order the base I A I i n order-type and l e t C denote the chain thus obtained. P a r t i t i o n the p a i r s of elements x, y of the base i n t o two c o l o r s . I f x < y (mod C ) , we say t h a t the p a i r i s ( + ) i f f x < y (mod A ) and ( - ) i f f x > y or xly (mod A ) . For a given element a i n the base, t h e r e a r e s t r i c t l y l e s s than wA-many elements x f o r which the p a i r { a , x ) has c o l o r ( - ) : so t h a t e i t h e r x < a (mod C ) o r x < a o r l a (mod A ) . Now apply ch.3 5 3.2: there e x i s t s a subset equipotent t o the base, i n which a l l p a i r s have color (+) . This y i e l d s a r e s t r i c t i o n isomorphic t o the ordinal d d 0 The proposition i s f a l s e f o r a s i n g u l a r aleph: For each i n t e g e r i , take a s e t Ei with c a r d i n a l i t y oi , the Ei being mutually d i s j o i n t . Define a p a r t i a l ordering on the union, which has c a r d i n a l i t y a W, by taking each Ei a s a f r e e s e t , each element i n Ei being l e s s than each element of E f o r any two i n t e g e r s j > i Then every t o t a l l y ordered j r e s t r i c t i o n has order-type a t most G)
.
.
.
3.7. (1) Let W , be a r e g u l a r l i m i t aleph. Then each ad-dense chain admits e i t h e r an embedding of wd o r i t s converse, o r an embedding of a l l o r d i n a l s s t r i c t l y l e s s than ad and t h e i r converses. ( 2 ) Let Ud be a r e g u l a r aleph. Then L3 d + l -dense chain admits e i t h e r an embeddinq of o r i t s converse, o r an embedding of a l l o r d i n a l s and their converses,(ERDOS, RADO 1953; (1) and ( 2 ) use equipotent w i t h the generalized continuum hypothesis; f o r o( = 0 , ZF p l u s choice s u f f i c e s ; f o r d = 1 , ZF plus choice plus continuum hypothesis s u f f i c e s ) . (1) Let A be an -dense chain. By r e s t r i c t i n g t o one o f i t s i n t e r v a l s , we can assume A t o have c a r d i n a l i t y a&. Let C denote a well-ordering with base 1 A I and order-type a0(. P a r t i t i o n the p a i r s { x,y} i n the base i n t o two c o l o r s : assuming t h a t x < y (mod C ) , the p a i r will have c o l o r (+) i f f x ( y (mod A ) and ( - ) i f f x > y (mod A ) . By ch.3 5 3 . 5 . ( 1 ) (gen. continuum
each
hypothesis), e i t h e r t h e r e e x i s t s a (-)-monochromatic subset equipotent t o the Or f o r each base: thus A admits an embedding of t h e retro-ordinal cardinal b < W, , t h e r e e x i s t s a (+)-monochromatic subset w i t h c a r d i n a l i t y b : thus A admits an embedding of every ordinal s t r i c t l y l e s s than C C ) ~. The
.
proof ends by interchanging the c o l o r s . ( 2 ) Let A be an Wbc+l -dense chain. Since u d i s regular by hypothesis, the l e a s t cardinal b s a t i s f y i n g b ( G), ) md i s b = &IM : see ch.2 5 6.5(1) u s i n g gen. continuum hypothesis. In o r d e r t o prove ( 2 ) , we now denote by A those oc+1 -dense chains which embed n e i t h e r the ordinal W o(+l nor i t s converse. P a r t i t i o n the p a i r s of elements of I A I i n t o two c o l o r s , e x a c t l y a s in (1) above.
>
142
THEORY OF RELATIONS
By ch.3 § 3 . 5 . ( 2 ) , e i t h e r t h e r e e x i s t s a
(-)-monochromatic subset equipotent w i t h
.
O r a (+)-monochromatic subthe base: thus A admits an embedding of @ N + l s e t w i t h c a r d i n a l i t y Cdd : thus A admits an embedding o f C d M Hence i n every case, A admits an embedding o f t h e o r d i n a l ae and i t s converse. ai ( i < wo( ) be a s t r i c t l y i n c r e a s i n g (mod A ) a,-sequence. L e t ao,al
.
,..., ,...
w , + ~-dense chain which 3 w O c . Assume, i n order
L e t C be t h e l e a s t o r d i n a l f o r which there e x i s t s an admits no embedding o f c , By the preceding we have c
.
Then c i s the l i m i t t o o b t a i n a c o n t r a d i c t i o n , t h a t c has c a r d i n a l i t y u d o f a s t r i c t l y i n c r e a s i n g sequence o f o r d i n a l s ci < c , indexed by i running through a t most
(30(
. Each
ci
aH+l -dense
i s embeddable i n every
chain. For
each i , the o r d i n a l , o r t h e aleph equal t o ci , i s embeddable i n the i n t e r v a l Hence the sum o f t h e ci i s embeddable i n A and so c i s embeddable (ai,ai+l) i n A . This c o n t r a d i c t i o n shows t h a t A admits an embedding o f every o r d i n a l equipotent w i t h W M . The same argument proves t h a t A admits as w e l l an embedding of t h e corresponding converse o r d i n a l s . 0
.
5 4 - IMMEDIATE An o r d i n a l
U
EXTENSION OF A CHAIN: FAITHFUL EXTENSION OF A CHAIN
i s indecomposable i f f ever.y o r d i n a l
(indecomposable o r d i n a l i s defined i n ch.1
5
V< U
satisfies
V.2
3.6).
.
.
U be an indecomposable o r d i n a l and \I < U Then V i U = U so V . 2 < U Conversely, suppose t h a t U i s decomposable: t h e r e e x i s t V < U and W < U
0 Let
with
V+W = U
. Let
T = Max(V,W)
. Then
T
and
T.Z&V+W
I n next chapter, indecomposability w i l l be extended t o chains (ch.6 preceding p r o p o s i t i o n must t h e r e be m o d i f i e d (ch.6 4.1.
Then
5
.0
= U
5
3 ) . The
3.4).
L e t A, A ' be two chains and U t h e l e a s t ordi-nal such t h a t U i s indecomposable (HAGENDORF 1971).
A+U+A'4
A+A'
L e t Y < U , so t h a t A+V+A' I A+A'(equimorphism L defined i n 5 1 above). Take an isomorphism from A+V+A' i n t o A+A' L e t V = W+W' w i t h W embeddable i n A and W ' embeddable i n A ' I n o r d e r t o f i x our ideas, suppose t h a t W $ W ' .
0
.
.
Then
W'+A',(
A'
(W' .2)+A',< A '
so
4
and so
(W' .4)+A'
< A'
, hence
(V.2)+A1
A ' and so A+(V.2)+A',< A+A' and f i n a l l y V.2< U . Argue analogously t o o b t a i n t h e same conclusion i f W>, W' Thus U i s indecomposable, by t h e prece-
.
ding proposition. 0 4.2.
IMMEDIATE EXTENSION OF A CHAIN
We say t h a t a chain
B
i s an immediate extension o f
i s immediately q r e a t e r than A
A
(up t o isomorphism)
( w i t h respect t o embeddability) i f
B >A
or
and
.
Chapter 5 t h e r e does n o t e x i s t any c h a i n
X
satisfying
be a c h a i n and
U
the
immediate e x t e n s i o n o f
A
(HAGENDORF 1972).
Let
A
143
A
.
B
l e a s t o r d i n a l such t h a t
N o t i c e t h a t t h e p r o p o s i t i o n n o l o n g e r h o l d s i f we r e p l a c e F o r example, l e t and 0
A =
A+U = IJ - . 2
The o r d i n a l
U
B
tion of
AtU
we have
converse o f w
,
1+ W -
2+ W
.
Then
U
satisfying
, which
an o r d i n a l . I f
A
B
< A+d .
t h u s decomposes
,
8" < U
the d e f i n i t i o n o f
by a r e t r o - o r d i n a l .
U = W - ; y e t between
B"
Consider an isomorphism o f B
B
i n t o an i n i t i a l i n t e r v a l
such t h a t
B ' G A
and
B'
. Hence
, which
and a complementary f i n a l i n t e r v a l
U
.
decomposable we have
A'
The c h a i n
A"+U = U
A"
5
A
U
B = B'+U
i s an o r d i n a l . I f
is
B
.
Hence
hence
A" = U
and
the existence o f a f i n a l i n t e r v a l o f
A+1
A
such t h a t U
i s in-
B'+U = B
A+U,<
,
.
A = A'+U
i s an i n f i n i t e indecomposable o r d i n a l , i n which case
(equimorphy). The preceding argument, w i t h
decomposes
A"
< U , since
A"
A+U = A ' t A " + U = A ' + U
so
thus c o n t r a d i c t i n g t h e d e f i n i t i o n o f Then e i t h e r
B"
B = B ' + B " , C A+B" 6 A t h u s c o n t r a d i c t i n g B" = U and B = B'+U (by = we mean "isomorphic t o " ) . onto a r e s t r i c t i o n o f
A",(
and a com-
chain
i t follows that
i n t o an i n i t i a l i n t e r v a l and
onto a r e s t r i c -
< U . The
B"
A
A+1
A
- , etc.
Consider an isomorphism f r o m A
6 B'
iLaL
o f t h e p r o p o s i t i o n i s non-zero and by t h e p r e c e d i n g 4.1 i t i s i n -
plementary f i n a l i n t e r v a l
A'
A+U
I n o r d e r t o o b t a i n a c o n t r a d i c t i o n , l e t us suppose t h a t t h e r e e x i s t s
decomposable. a chain
, the
U-
. Then
A+U $ A
so
1
A+l , proves
r e p l a c e d by
which i s isomorphic w i t h
U : contra-
diction. U = 1 and so
Or
A'
.
A'+1
A = A'+1
B = B ' + 1 , hence A ' + 1 (
and
I t e r a t e t h e p r e c e d i n g argument: we see t h a t
c e r t a i n i n i t i a l i n t e r v a l and o f t h e f i n a l i n t e r v a l (.d phic with
A
, which
contradicts the hypothesis t h a t
- .
, so
B'+l
Hence
that
i s t h e sum o f a A+1
i s isomor-
.0
U = 1
4.3. Every i n f i n i t e c h a i n admits a t l e a s t two immediately g r e a t e r c h a i n s ( w i t h r e s p e c t t o e m b e d d a b i l i t y ) , which a r e m u t u a l l y incomparable (HAGENDORF 1972). 0
Let
A
be an i n f i n i t e c h a i n and
embeddable i n
A ; similarly l e t
i s n o t embeddable i n than
A
,
.
A
U U'
t h e l e a s t o r d i n a l such t h a t
A+U
By t h e p r e c e d i n g 4.2,
It remains t o prove t h a t t h e y a r e m u t u a l l y incomparable w i t h r e s p e c t t o
Take an isomorphism f r o m interval
U'+A
t h e s e sums a r e immediately g r e a t e r
e m b e d d a b i l i t y . We s h a l l argue ad absurdum: t o f i x t h e ideas, suppose n o t embed
i s not
be t h e l e a s t r e t r o - o r d i n a l such t h a t
U'+A of
morphic w i t h
A
in
A
U'+A
, nor
in
onto a r e s t r i c t i o n o f A+V
f o r any
which i s c o f i n a l l y embeddable
U : o t h e r w i s e t h e r e would e x i s t
which c o n t r a d i c t s t h e d e f i n i t i o n o f
U'
.
A+U
< U . Thus i n U . This
.
V
V< U
A+U
.
there exists a f i n a l final interval i s iso-
such t h a t
F i n a l l y by 4.1,
U'+A,(
T h i s isomorphism can-
U
U'+A$
A+V$
A
i s indecomposable.
,
144
THEORY OF RELATIONS
If U is an infinite indecomposable ordinal, then 1 < U so A+1 I A . The preceding argument, where A is replaced by A+l , proves the existence of a final interval of A+1 which is isomorphic with U : contradiction. Hence U = 1 and U'+A d A + 1 . Thus every isomorphism of U'+A into A+l has in its range the maximum element of A+1 ; for otherwise U'+A would be embeddable in A . Thus A has a maximum. By iteration, we see that A has a final interval which is isomorphic with the retro-ordinal w - . Thus A+1 is isomorphic with A , contradicting the inequality A+l = A+U > A . 0 An infinite chain can have 3, 4, ... immediately greater chains. has the 3 immediately greater chains For example the product A = b -. B = A+l , C = W + A , D = ( w 2 ) - + A , where ( W 2 ) - is the converse of the
4.4.
ordinal
W
.
These are the only immediate extensions of A
.
The reader easily see that there does not exist any chain strictly intermediate between A and B , A and C , A and 0 . Let us see that every chain strictly greater than A admits an embedding of B or C or 0 . Indeed, every chain which is an extension of A can be obtained as follows. Either by adding new elements at the end, so admitting an embedding of B . Or by adding a chain at the beginning, in which w is embeddable (embedding of C ) , or in which the converse of an ordinal is embeddable (from the ordinal w 2 we get an embedding of D ) . Or by partitioning the new elements into a finite number of components W - of A , which is the same as adding them at the beginning. Or by modifying infinitely many components, by adding an ordinal CC, (embedding of C ) , or by replacing these components by a retro-ordinal ( s o getting an embedding of D from the converse of u 2) . 0
0
Another example: the chain W1 + Q , where Q is the chain of the rationals, has the 5 following immediate extensions: the addition of &J1 or its converse at the end or between k, and Q , the addition of W1- at the beginning. For each integer p 3 2 , the chain sions (among chains).
G,
.(p-1)
has exactly p immediate exten-
There exist chains having infinitely many immediately greater chains. For example, every chain A which has no strict restriction isomorphic with A : see the method of SIERPINSKI 1950, used in 5.3 below. FAITHFUL EXTENSION BETWEEN CHAINS Let A be an infinite chain, B a chain in which A is not embeddable. lhen there exists a chain, immediate extension of B , in which A is not embeddable (uses axiom of choice; HAGENDORF 1972; the case for scattered chains .e. chains
4.5.
145
Chapter 5
without any embedding of 0 If
, was
Q
a l r e a d y proved by JULLIEN 1969).
B i s a f i n i t e chain, t h e n i t s u f f i c e s t o t a k e
i n f i n i t e . We d i s t i n g u i s h two cases:
B
B IA
and
.
B+l
Suppose t h a t
B
is
( w i t h r e s p e c t t o embeddabi-
lity).
I n the case
A ,let
B
diately greater than o r A,(
.
B"
B'
and
be two m u t u a l l y incomparable chains, each imme-
B"
(see 4 . 3 ) . Suppose t h e p r o p o s i t i o n i s f a l s e . Then
B
since B ' l B" . Hence A < B ' f o r i n s t a n c e , and so between B and B ' : c o n t r a d i c t i o n .
It remains t o examine t h e case where
A
dum by supposing t h a t
B
B
into
t h e complementary f i n a l i n t e r v a l . L e t B ; similarly
.
UA
6U
,
V
and t h e converse o f
A
i n t o three intervals
.
B
If
UA<
U
D
then A A
morphic t o t h e r e t r o - o r d i n a l
V
i s included i n A
, then
E
\< B , c o n t r a d i c t i n g
U i n E'
, then
U+D',<
,
.
D'
with
Thus
UA = U
>
U
B
.
By 4.1,
\< E gC
Hence
U = 1
D = E
and
, and
0' =
E'4
C'
so
,
V so
preceding, we have
,
and
. Similar-
and an i n t e r v a l i s o E,<
C
,
E'Q
C'
.
A . I f the interC+C' = B : i n o t h e r
intervals o f
D+U+D',(
A = D+U+D'&
B , contradicting the , isomorphic t o an indecom-
C+C' =
U
V , isomorphic t o a r e t r o - o r d i n a l .
V = 1 : the intervals
U = V = 1 a r e t h e same, t h u s
.
be an a r b i t r a r y element o f t h e base
of those elements
0 4 C
the hypothesis. I f there e x i s t s a f i n a l i n t e r v a l o f
similarly
E'
and
A = D+U+D'
E , E'
A = E+V+E' and
A = D+UA+D'
and t h e n
h y p o t h e s i s . Thus we must suppose t h a t t h e i n t e r v a l
x
C+V+C'
B , c o n t r a d i c t i n g t h e hypothe-
into intervals
posable o r d i n a l , i s i n c l u d e d i n t h e i n t e r v a l
Let
.
B
i s an i n i t i a l i n t e r v a l
C
be t h e l e a s t o r d i n a l such t h a t
C+UA+C',<
such t h a t D+U
,
UA
A<
Envisage a l l t h e r e l a t i v e p o s i t i o n s o f U
We argue ad absur-
a r e indecomposable o r d i n a l s . By h y p o t h e s i s Consider an isomorphism o f k o n t o a r e s t r i c t i o n o f t h i s sum: t h i s
i s incomparable w i t h
val
U
B'
,
V
l y we d e f i n e t h e decomposition o f
words
IA .
t h e l e a s t r e t r o - o r d i n a l such t h a t
U
0 ' 4 C'
sis that
B
, where
C+C'
C+U+C'>
<
A,<
A z B"
i s a s t r i c t l y intermediate
i s i n f i n i t e and
and C '
A C+U+C' decomposes
A
and
i s embeddable i n e v e r y c h a i n s t r i c t l y g r e a t e r t h a n
I n an a r b i t r a r y manner decompose
the o r d i n a l
A E B'
I t i s i m p o s s i b l e t o have b o t h equimorphisms
6X
(mod 6 )
Bx+l+B;
>B
and
IS1
.
Let
Bx
be t h e i n i t i a l i n t e r v a l
>
the final interval
B; hence > / A
. Moreover,
a s s o c i a t e t o each x an element f x o f t h e base d e s i g n a t e t h e i n i t i a l i n t e r v a l of t h o s e elements
x (mod B)
. By
the
by t h e axiom of choice,
I A I , such t h a t i f we l e t Ax (mod A ) , t h e n we have
< fx
A = Ax+l+A;( w i t h A x < Bx and A<,; B; . We s h a l l prove t h a t t h e f u n c t i o n f which t o each e:ement x o f t h e base I B I a s s o c i a t e s f x , an element o f A i s s t r i c t l y i n c r e a s i n g . From t h i s we w i l l deduce t h a t t h e h y p o t h e s i s o f i n c o m p a r a b i l i t y o f A and B . TO do t h i s , we argue ad absurdum by supposing t h a t t h u s t h e r e e x i s t two elements x, y o f I S ( w i t h
B,(
A
, which
,
contradicts
f i s not s t r i c t l y increasing: x < y (mod B) , and so w i t h
THEORY OF RELATIONS
146
<
. Yet
B
Bxtl&
Y
Bx t 1
\
4.6.
+
fy
4 BY
B' Y
A
and
,
B
A
and
Obvious f o r
A 4B
.
than
B
A i d A ' 6 B;I
and so
,
B' = B Y
A
thus
Y
+
A = Ax
1 + A;
c o n t r a d i c t i n g the hypothesis. 0
have t h e same s t r i c t l y g r e a t e r c h a i n s ( w i t h r e s p e c t t o B a r e equimorphic.
a r e i n f i n i t e , t h e n i t even s u f f i c e s t h a t e v e r y c h a i n s t r i c t l y qrea-
be
B
t e r than 0
B
+
(mod A)
then
I f two c h a i n s
embeddability), If
(only) greater than
A
finite or
B
A
, and
conversely.
f i n i t e . Suppose t h a t
,
A
B a r e i n f i n i t e and t h a t
Then by t h e p r e c e d i n g p r o p o s i t i o n , t h e r e e x i s t s a c h a i n s t r i c t l y g r e a t e r and n o t g r e a t e r t h a n
A
.0
4.7. N o t i c e (by JULLIEN 1969) t h a t t h e f a i t h f u l e x t e n s i o n no l o n g e r h o l d s i f we
A
replace 0
by two c h a i n s
For example n e i t h e r
A1
w
A1 =
and
A2
+1 n o r
. A2
=
However, e v e r y c h a i n which i s s t r i c t l y g r e a t e r t h a n
Al
or of
A2
.
-+ W i s embeddable i n W w admits an embedding o f
Z = W
.0
The e x i s t e n c e o f a f a i t h f u l e x t e n s i o n between chains, no l o n g e r h o l d s i f we r e p l a c e B 0
by two c h a i n s
B1
and
For example A = W + B2 = w . W -
converse and
B2
c3
.
B2
.
-
i s n e i t h e r embeddable i n
Yet
A
B1 = U - .
nor i n i t s
i s embeddable i n e v e r y c h a i n i n which b o t h
a r e embeddable: t h i s w i l l be proved i n ch.6
B1
3.7. 0
We w i l l have an o p p o s i t e r e s u l t i f we a l l o w e x t e n s i o n s by a r b i t r a r y b i n a r y r e l a t i o n s , i n s t e a d o f r e s t r i c t i n g o u r s e l v e s t o c h a i n s : see e x e r c . 2 below. Problem posed by SABBAGH i n 1975. E x i s t e n c e o f two chains
A, B which a r e incom-
p a r a b l e w i t h r e s p e c t t o e m b e d d a b i l i t y , and have a supremum c h a i n
C
.
More p r e c i s e -
X > / A and X B, . I t has been proved by HAGENDORF 1977 t h a t t h i s supremum c h a i n does n o t e x i s t f o r
ly, f o r every chain
A
or
B
X
we would have
X >/C
iff
s c a t t e r e d ; t h e general case remains unsolved.
For t h e dual o f t h e above statement, i . e . t h e e x i s t e n c e o f t h e infimum, we have t h e easy example
A = w + 1 and
5 5 - DECREASING SEQUENCES DUSHN I K, MILLER, S IERP INSKI 5.1.
Let
A, B
B = Z
w i t h t h e infimum
C =
AND SETS OF INCOMPARABLE
w . CHAINS
OF REALS:
be two chains, each o f which i s embeddable i n t h e r e a l s . Then
t h e r e a r e a t most continuum many r e s t r i c t i o n s o f
B
isomorphic w i t h
0 Consider t h e base
E
o f r e a l s , and l e t
lBl
as a subset o f t h e s e t
A
. F be
Chapter 5
IBI
a subset o f
B/F
such t h a t
For e v e r y subset
X
i s isomorphic w i t h
1 B l such t h a t
of
147
A
( i f t h e r e e x i s t s such).
i s isomorphic w i t h
B/X
f X f r o m F o n t o X , hence f r o m F t h e r e a r e continuum many s t r i c t l y i n c r e a s i n g maps f r o m F i n t o
5
8.4;
and f i n a l l y
5.2. L e t
A
X = fXo(F)
i s determined by
f X( n o t a t i o n
A
, there
into
a s t r i c t l y i n c r e a s i n g map
E
exists
. Now
E : see c h . 2 O
i n ch.1
5
1.2). U
embeddable i n t h e
be a c h a i n o f continuum c a r d i n a l i t y , which i s
chain o f the reals. (1) There e x i s t s a s t r i c t l y s m a l l e r ( w i t h r e s p e c t t o e m b e d d a b i l i t y ) r e s t r i c t i o n A which has continuum c a r d i n a l i t y (DUSHNIK, MILLER 1940; uses t h e axiom o f
of
choice). ( 2 ) Even s t r o n g e r , t h e r e e x i s t s a r e s t r i c t i o n l i t y , such t h a t no i n t e r v a l o f
,of
A
B
of
A
w i t h continuum c a r d i n a -
continuum c a r d i n a l i t y , i s ernbeddable i n
B
(HAGENDORF 1977, u n p u b l i s h e d ) . ( 3 ) F o r e v e r y denumerable c h a i n preceding ( 2 ) ) . 0 (2) Let
X
U
we have
be any subset o f t h e base
an i n t e r v a l o f
A
A $ B.U
(where
I A 1 such t h a t A/X
B
satisfies the
i s isomorphic w i t h
o f continuum c a r d i n a l i t y . For each such i n t e r v a l , by t h e p r e -
c e d i n g p r o p o s i t i o n , t h e r e a r e a t most continuum many corresponding s e t s
X
. More-
R o f r e a l s , each i n t e r v a l o f A i s t h e r e s t r i c t i o n t o I A l o f an i n t e r v a l o f R , which i s i t s e l f
over, i f we c o n s i d e r
A
as a r e s t r i c t i o n o f t h e c h a i n
d e f i n e d by i t s e n d p o i n t s . Consequently t h e r e a r e continuum many such i n t e r v a l s . X
Finally, the set o f a l l the
5
Apply ch.2 where
C
has a t most continuum c a r d i n a l i t y .
8.1 (axiom o f c h o i c e ) . There e x i s t s a s e t
and
D
=
IAl
a l l the intersections thus the r e s t r i c t i o n
-C
C
included i n
IAI ,
b o t h a r e e q u i p o t e n t w i t h t h e continuum, as w e l l as
.
C n X and D n X Consequently no X i s i n c l u d e d i n C B = A/C a d m i t s no embedding o f any i n t e r v a l o f A which
has continuum c a r d i n a l i t y . 0 0
( 3 ) Suppose
A 6B.U
.
Then t h e base
I A I i s p a r t i t i o n e d i n t o c o u n t a b l y many
i n t e r v a l s , each c o r r e s p o n d i n g w i t h an element o f
U
. At
t e r v a l s i s e q u i p o t e n t w i t h t h e continuum: see c h . 1 § 4.3, o v e r i t i s embeddable i n
B
, contradicting
l e a s t one o f t h e s e i n axiom o f c h o i c e . More-
the preceding (2). 0
N o t i c e t h a t (1) i m p l i e s t h e e x i s t e n c e , s t a r t i n g f r o m t h e c h a i n o f t h e r e a l s , o f a s t r i c t l y decreasing w -sequence o f c h a i n s . We s h a l l see t h a t such a sequence does n o t e x i s t f o r s c a t t e r e d c h a i n s , i . e . those i n which t h e c h a i n
Q
o f the
r a t i o n a l s i s n o t embeddable: see ch.8
5
5.3.
R t h e c h a i n o f t h e r e a l s . There e x i s t two
Let
subsets
E C,
be t h e s e t o f r e a l s and
D
of
E
4.4.
which a r e d i s j o i n t , e q u i p o t e n t w i t h t h e continuum, dense
,
148
THEORY OF RELATIONS
in
R and such t h a t , f o r each subset X f D , every r e s t r i c t i o n of R isomorphic w i t h R / ( C u X ) , e i t h e r has base C u X , o r i t s base contains a t l e a s t one
element o f E
-
(C u D)
.
Consequently f o r Y C X cD , we have the s t r i c t embeddability R / ( C uY)< R/(CuX). For X and Y s u b s e t s of D which a r e incomparable with r e s p e c t t o i n c l u s i o n ,
the preceding two restrictions are incomparable with respect to embeddability (SIERPINSKI 1950; see a l s o ROSENSTEIN 1982; uses axiom of c h o i c e ) . Take the s e t s C , D in ch.2 5 8.2.(2) (axiom of c h o i c e ) , where t h e f i designate a l l isomorphisms from R i n t o R , d i s t i n c t from t h e i d e n t i t y . For such an isomorphism f , i f a real x i s mapped t o f x # x , f o r example i f f x > x (mod R ) , then every real i n t h e i n t e r v a l ( x , f x ) i s mapped t o a s t r i c t l y g r e a t e r r e a l ,
0
hence f y # y f o r continuum many r e a l s y . Notice t h a t C and 0 a r e d i s j o i n t and each equipotent with t h e continuum. Moreover by t h e same proposition we have f " ( C ) # C f o r each considered isomorphism f ; s i m i l a r l y with D T h u s by ch.2 5 8 . 5 , t h e s e t s C and D a r e both dense (mod R ) . Take an a r b i t r a r y subset X of D and an isomorphism g of R / ( C u X ) i n t o R , which i s d i s t i n c t from t h e i d e n t i t y . Since C , hence C u X , i s dense, t h e r e
.
e x i s t s an isomorphism gt of R i n t o R , which extends g t o t h e domain E of a l l r e a l s : see ch.2 5 8.3. Hence g+ i s one of t h e previously considered i s o morphisms f . By ch.2 5 8.2, t h e s e t of images g"(C) = ( g + ) " ( C ) i s not i n c l u ded in C u D . 0 This immediately implies t h e e x i s t e n c e of a s t r i c t l y decreasing (with r e s p e c t t o embeddability) sequence, indexed by t h e continuum, of chains of r e a l s . Also t h e e x i s t e n c e of a s e t , equipotent w i t h t h e continuum, of mutually incompar a b l e chains of r e a l s (SIERPINSKI 1950).
§
6 - SUSLINC H A I N
AND
SUSLINTREE
in A : Given a chain A , t h e reader i s acquainted w i t h t h e notion of a s e t a subset D of t h e base f o r which, given any two elements x < y (mod A ) , t h e r e e x i s t s an element t of D w i t h x st d y (mod A ) The chain of r e a l s , and more g e n e r a l l y any chain A which i s embeddable in the chain of r e a l s , s a t i s f i e s t h e two following conditions: (1) t h e r e e x i s t s a countable s e t which i s dense in A ; (2) every s e t of mutually d i s j o i n t i n t e r v a l s of A , none o f which a r e s i n g l e t o n s , i s countable.
The condition (2) follows from ( 1 ) . I f D i s countable and dense i n A , then every non-singleton i n t e r v a l contains a t l e a s t one element of D . Two d i s j o i n t i n t e r v a l s cannot contain a same element
0
Chapter 5
of
D , so there are countably many i n t e r v a l s .
149 0
SUSLIN'S HYPOTHESIS (see SUSLIN 1920) The axiom called S u s l i n ' s hypothesis, a s s e r t s that the preceding condition (2) implies ( l ) , hence t h a t ( 1 ) & (2) are equivalent. This axiom i s neither provable nor refutable in ZF, even with the axiom of choice and even with the generalized continuum hypothesis. More precisely JECH and TENNENBAUM have proved the consistency of the existence o f a Suslin t r e e ( i . e . the negation of the axiom) with ZF (modulo the consistency of Z F ) . Whereas SOLOVAY and TENNENBAUM have proved the relative consistency of the axiom: see JECH 1978. For a detailed discussion of Suslin chains and Suslin t r e e s , as well as f o r the advanced r e s u l t s o f JENSEN, see for example DEVLIN, JOHNSBRiTEN 1974.
6.1. SUSLIN CHAIN I t i s more convenient t o work with the negation of Suslin's hypothesis, rather than the hypothesis i t s e l f . We say t h a t a chain i s a Suslin chain i f i t s a t i s f i e s ( 2 ) and n o t ( l ) , i . e . i f every s e t of non-singleton mutually d i s j o i n t intervals i s countable, y e t there e x i s t s no countable s e t which i s dense in the chain. A Suslin chain i s uncountable; moreover i t admits an embedding of t h e c h a i n Q rationals (uses axiom of choice). 0 The inexistence of any countable dense s e t implies t h a t the chain i t s e l f be
of
uncountable: i t s cardinality i s a t l e a s t W1 (axiom of choice). If Q i s n o t embeddable in i t , then e i t h e r the ordinal W 1 or i t s converse i s embeddable in i t : see 3.5. Hence there e x i s t uncountably many non-singleton mutually d i s j o i n t intervals. 0
6 . 2 . Every Suslin chain has cardinality exactly w 1 (uses axiom of choice and the continuum hypothesis). 0 Let A be a Suslin chain; we already know t h a t A i s uncountably i n f i n i t e , so has cardinality a t l e a s t w 1 (axiom of choice). Suppose t h a t A has cardinality a t least w . Replace A by a r e s t r i c t i o n o f cardinality O 2 and l e t B be a well-ordering of type o 2 on the same base. Partition the pairs of elements x, y of the base into two colors: l e t t i n g x < y (mod A ) , we say t h a t the pair has color (+) i f x < y (mod B ) , and color (-) i f x ) y (mod B ) By the ERDOS partition lemma (ch.3 5 3.4) f o r d = 0 (hence using only the continuum hypothesis), there e x i s t s a subset of the base of cardinality G, , a l l of whose pairs have a same color. Hence there e x i s t s a s t r i c t l y increasing or a s t r i c t l y decreasing 0 l-sequence, and hence wl-many non-singleton mutually d i s j o i n t intervals: contradiction. 0
.
THEORY OF RELATIONS
150
6.3. SUSLIN TREE We say t h a t a t r e e i s a S u s l i n t r e e i f i t has c a r d i n a l i t y
w1
and i f every chain
( o r t o t a l l y ordered r e s t r i c t i o n ) and every a n t i c h a i n i s countable. The existence o f a S u s l i n chain o f c a r d i n a l i t y
W1 S u s l i n t r e e ( t h e a d d i t i o n a l assumption o f c a r d i n a l i t y
i m p l i e s t h e existence o f a
w 1 allows us t o avoid
using the axiom o f choice: ZF s u f f i c e s ) . 0
Let
be a
A
S u s l i n chain o f c a r d i n a l i t y
w1 .
i ,
To each countable o r d i n a l
defined by i t s two endpoints ui < vi (mod A) , where Ai an a r b i t r a r y are d i s t i n c t . To do t h i s , begin w i t h A. = (uo,vo)
associate an i n t e r v a l all
ui
and
vi
interval. Let
i be a non-zero countable o r d i n a l , and suppose t h a t t h e
A
for
j j < i have already been defined so t h a t they are m u t u a l l y e i t h e r d i s j o i n t o r one
contained i n the other. The s e t o f endpoints uj, v j ( j < i ) i s countable: by hypothesis i t i s n o t dense i n A , hence there e x i s t two elements u, v i n t h e base o f
, between
A
.
v. ( j < i ) L e t Ai = J vi = v : t h i s i n t e r v a l must be e i t h e r d i s j o i n t o r included
which there i s no endpoint
uj
or
ui = u and A. ( j < i ) J thus obtained has c a r d i n a l i t y w 1 Reverse i n c l u s i o n The s e t o f i n t e r v a l s Ai defines a t r e e on the s e t o f these Ai . Every antichain, i . e . every s e t o f i n t e r which are m u t u a l l y d i s j o i n t , i s countable. F i n a l l y , a chain, o r s e t o f v a l s Ai (u,v)
so
.
i n each
Ai
intervals
.
which are mutually comparable w i t h respect t o i n c l u s i o n , i s w e l l f o r every p a i r o f countable o r d i n a l s A . c Ai J Such a chain i s countable; f o r i f i t had c a r d i n a l i t y w 1 , then
ordered by the o r d i n a l i n d i c e s w i t h
i, j ( i
using the endpoints o f preceding i n t e r v a l s , we could o b t a i n W l-many
mutually
d i s j o i n t intervals. 0 6.4. The existence o f a S u s l i n t r e e i m p l i e s , and hence i s e q u i v a l e n t w i t h t h e existence o f a S u s l i n chain (here we use the r e g u l a r i t y o f
W1
, thus
f o r example
the countable axiom o f choice). 0 Let
be a S u s l i n t r e e ; we can assume t h a t
A
A
i s a well-founded p a r t i a l orde-
A by a c o f i n a l well-founded p a r t i a l ordering. To 5.1 w h i l e n o t i n g t h a t , by hypothesis, t h e base o f A i s
r i n g , i f necessary by r e p l a c i n g see t h i s , apply ch.2
5
.
well-orderable w i t h order type w 1 More p r e c i s e l y l e t ui ( i < td 1) be an indexation o f the base; then remove each u f o r which t h e r e e x i s t s an i
>
.
i , the removed elements u are a l l < ui (mod A) j there are countably many such; thus t h e r e remain w l-many elements.
f o r each index
We can f u r t h e r assume t h a t every non-empty f i n a l i n t e r v a l o f W
. To
see t h i s , l e t
many successors (mod A )
x
A
, and
hence
has c a r d i n a l i t y
be an element o f the base which has o n l y countably
. Those
x
o f minimal h e i g h t are m u t u a l l y incomparable,
Chapter 5
151
x and the ir successors. Neither A nor any f in al interval of A i s f i n i t e l y fre e . For otherwise, by ch.4 5 3.1 (using the regularity of LJ there would e x i s t a t o t a l l y ordered re stric contradicting our hypotheses. tion of A with cardinality w W e can assume t h a t A has a minimum, whose singleton will be denoted by Eo For each countable non-zero ordinal i , l e t Ei be the denumerable s e t of elements of height i : we require t h at t h i s s e t be i n f i n i t e . For each element x of Ei we require that there e x i s t denumerably many elements which are immediate successops of x (mod A ) . This s e t i s denoted Eitl ,X and the union of these se ts must be E i t l . For each countable l i mi t ordinal i and each x of height < i , we require t h a t there e x i s t i n f i n i t e l y many elements in Ei which are 7 x (mod A ) . Finally, for each countable l i mi t ordinal i and each t o t a l l y ordered re stric tion X of A containing elements of a l l heights < i , we require tha t there e xist either a unique element of Ei which i s a successor of a l l elements of X , or none such. These requirements are easy t o s a t i s f y . For example f o r the minimum, take an arbitrary element having W l-many successors. Then having obtained Ei , the s e t of elements x with height i , note t h at f o r each x in Ei , the final interval > x has cardinality w 1 and i s not f i n i t e l y f r e e . Hence take a denumerable free subset as E i + l , x and among the successors of x , retain only those which Finally, f o r each counare identical t o or successors of an element of Ei+l ,X table limit ordinal i and each chain X containing elements of a l l heights 4 i , i f there e x i s t elements above X , then decide t o retain one such plus the wl-many successors of t h i s element. For each element x of the base IAl with height i , t o t a l l y order the denumeNow consider the with the order type of a dense chain C i + l , x rable s e t Ei+l ,X set of a l l maximal t o t a l l y ordered r e s t r i c t i o n s , or maximal chains of A . This set i s t o t a l l y ordered by the preceding dense chains. Indeed, given two distinc t maximal chains U and V : none of the two bases i s included in the other. Moreover there e x i st s a l e a s t element u among those elements of I U 1 which do not belong t o I V I , and a l e a s t element v among those elements of I V I which do not belong t o I U I By the preceding, there ex i s t s a l a s t element x whose height will be denoted i , common t o b o t h bases of U and V , and having u and v as immediate successors. Let U < V i f u < v (mod C i + l , x ) : t h i s tota lly orders the s e t of maximal chains. Let H be the chain thus obtained. We shall prove tha t H i s a Suslin chain. =i-r+ -11 a rnt n irrhirh i c h n c o i n H rannot be countable. For i f i t were, and so there are countably many such: i t suffices t o remove these
.
.
.
.
THEORY OF RELATIONS
152
the i n t e r v a l o f maximal chains passing through of D
z
does n o t c o n t a i n any element
.
Now suppose t h a t t h e r e e x i s t o l-many non-singleton m u t u a l l y d i s j o i n t i n t e r v a l s I n each i n t e r v a l take two elements, o r maximal chains U and V . As of H before, take an element x whose h e i g h t i s denoted i and t h e elements u, v
.
immediate successors o f x (mod A) ; and take w between u and v modulo the chain Ci+,lx Then these w thus associated w i t h our d i s j o i n t i n t e r v a l s o f H ,
.
are m u t u a l l y incomparable (mod A): they must be countably many; c o n t r a d i c t i o n . 0
§
7 - ARONSZAJN
TREE,
SPECKER CHAIN
7.1. ARONSZAJN TREE
This i s a well-founded t r e e o f c a r d i n a l i t y ~3~ whose chains and h e i g h t l e v e l s are countable. I t i s n o t r e q u i r e d t h a t every a n t i c h a i n be countable. Hence every well-founded S u s l i n t r e e i s an Aronszajn tree; b u t the converse p o s s i b l y depends on s e t - t h e o r e t i c axioms: see the problem a t the end o f 7.4. The f o l l o w i n g c o n s t r u c t i o n o f an Aronszajn t r e e , using ZF p l u s choice, goes back t o KUREPA 1935 p . 96, c i t i n g a l e t t e r from ARONSZAJN i n 1934. The elements o f the t r e e w i l l be o r d i n a l sequences o f i n t e g e r s ai ( i < ) y w i t h o u t r e p e t i t i o n , where o( v a r i e s over a l l countable o r d i n a l s . We say t h a t such a sequence u precedes v o r t h a t v f o l l o w s u , i f u i s an i n i t i a l i n t e r v a l o f v . Moreover we r e q u i r e t h e f o l l o w i n g c o n d i t i o n s o f convergence and denumerability. Convergence. For each sequence ai (i< o ( ) , the sum o f the inverses l/ai is f i n i t e . Furthermore for each sequence u w i t h l e n g t h o( , each countable o r d i n a l and each p o s i t i v e r e a l number r , t h e r e must e x i s t , i n our set, a sequence w i t h l e n g t h o i + /s , f o l l o w i n g u , and such t h a t the sum of the inverses l/ai f o r o( i
.
denumerably many o( -sequences (uses axiom o f choice). Using t h e preceding convergence c o n d i t i o n , we see t h a t , given a l i m i t countable o r d i n a l 4
we can r e t a i n
only o( -sequences u such t h a t , f o r each i ( 4 , t h e i n i t i a l i n t e r v a l o f u w i t h l e n g t h i a l r e a d y belongs t o our set; more e x a c t l y , we r e t a i n denumerably many such o( -sequences. F i n a l l y f o r each sequence u i n our set, every i n i t i a l i n t e r v a l o f u w i l l belong t o our set. These c o n d i t i o n s a l l o w us, f o r each o( , t o c o n s t r u c t a " d e f i n i t i v e " denumerable s e t o f o( -sequences, which w i l l n o t be unduly increased by the u l t e r i o r construct i o n o f longer sequences. Because t h e n o n - r e p e t i t i o n of values
ai
, every
t o t a l l y ordered r e s t r i c t i o n , o r
Chapter 5
153
chain, of the preceding Aronszajn t r e e , i s countable. In the preceding t r e e , f o r each countable ordinal o( and each o( -sequence u , there e x i s t u1-many sequences which are successors of u However t h i s i s not true for a l l Aronszajn t r e e s ; f o r instance we can add t o the preceding tree new ordinal sequences without successors, or with countably many successors.
.
7.2. A chain A i s embeddable in the chain of r e a l s , i f f there e x i s t s a countable subset which i s dense in A (uses countable axiom of chsice). 0
If
A
i s a r e s t r i c t i o n o f the chain of r e a l s , then f o r each integer
n
take a
partition o f the reals into intervals of length l / n , and choose an element of I A l in each interval (countable axiom o f choice). Conversely, i f there i s a r e s t r i c t i o n I o f A , which i s denumerable and dense in A , then i f necessary complete I i n t o J , the l a t t e r which i s isomorphic with the chain of rationals. Then i n each cut o f J , we take the unique element o f I A I ( i f i t exists) defined by i t s position with respect t o the elements of I I I . Thus we embed A into the reals. 0
7.3. SPECKER CHAIN This i s an uncountable chain A such t h a t neither w nor i t s converse are embeddable in A ; moreover any chain which i s embeddable both in A and in the chain of reals, i s countable. A Specker chain has no countable dense subset. For i f s o , then by the preceding, i t would be embeddable in the chain of r e a l s as well as in i t s e l f , and consequently i t would be countable. Every Specker chain has cardinality W (uses choice plus continuum hypothesis). Suppose on the contrary t h a t there e x i s t s a Specker chain of cardinality 3 w 2 By 3.5, t h i s chain admits an embedding e i t h e r of w 2 , or i t s converse: contradiction with the definition o f Specker chains. Or our chain admits an embedding of an W *-dense chain. By 3.7 ( 2 ) (choice and continuum hypothesis), the l a t t e r admits an embedding o f w 1 : contradiction. 0
0
.
7.4. Construct as follows a Specker chain, s t a r t i n g with the Aronszajn t r e e of 7.1, hence on a s e t of ordinal sequences of integers, without repetition. Apply ch.4 5 6.2 (using u l t r a f i l t e r axiom) which associates t o the given t r e e A , a chain C with the same base. For each element u (ordinal sequence of integers without repet i t i o n ) , the final interval 3 u (mod A ) becomes an interval (mod C ) with minimum u .
THEORY OF RELATIONS
154
We s h a l l show f i r s t o f a l l t h a t t h e c h a i n C embedding o f
t h u s defined does n o t admit an
u1, n o r o f i t s converse,
C can be decomposed i n a denumerable sum o f d i s j o i n t i n t e r v a l s , By ch.4 5 6.2, each corresponding t o an i n t e g e r a , and formed from some sequences which begin
with
a
.
Consequently, given a s t r i c t l y i n c r e a s i n g (mod C)
W1-sequence whose
terms are denoted ui ( i countable o r d i n a l ) , t h e r e e x i s t s an i n t e g e r al such t h a t , from a c e r t a i n o r d i n a l on, t h e sequence ui begins w i t h al By i t e r a t i o n ,
.
f o r every countable o r d i n a l k , t h e r e e x i s t s a k-sequence o f i n t e g e r s such t h a t , from a c e r t a i n o r d i n a l index on, t h e sequence a j' * * ( j < k ) begins w i t h the above k-sequence. F i n a l l y , these al, a2, ... , aj, a s t r i c t l y i n c r e a s i n g wl-sequence o f elements o f the t r e e A , thus an
..
al,
a*,
...
ui form o l-se-
quence o f i n t e g e r s w i t h o u t r e p e t i t i o n : c o n t r a d i c t i o n . 0 L e t us now show t h a t if t r i c t i o n D dense i n H
H jA_an uncountable chai_n-_which has a denumerable res,then H i s n o t embeddable i n C .
D i s denumerable, i t i s formed o f ord'nal sequences o f i n t e g e r s w i t h o u t r e p e t i t i o n , a l l o f l e n g t h l e s s than a c e r t a i n countable o r d i n a l . L e t o( be a countable o r d i n a l s t r i c t i y g r e a t e r than a l l these lengths. Since H i s uncounta-
0 Since
ble, t h e r e e x i s t s a t l e a s t one sequence u w i t h l e n g t h o( having ul-many sequences i n I H I which extend u . By ch.4 4 6.2, t h e r e e x i s t s an i n t e r v a l (mod C) formed o f a l l those elements, o r sequences extending u Thus any two o f them are n o t separated (mod C) by any sequence belonging t o I D l : t h i s contra-
.
d i c t s the density o f
D
in H
.0
Problem. I n 7.1 we constructed a well-founded Aronszajn t r e e , based on c e r t a i n o r d i n a l sequences o f i n t e g e r s ( w i t h o u t r e p e t i t i o n ) , u s i n g ZF p l u s choice. Whereas a S u s l i n t r e e can o n l y be constructed by u s i n g t h e negation o f S u s l i n ' s hypothes i s . Consequently, w i t h ZF p l u s choice p l u s t h e axiom c a l l e d S u s l i n ' s hypothesis, we are insured t h a t t h e r e e x i s t s an Aronszajn t r e e which i s n o t a S u s l i n t r e e . However i f we admit ZF p l u s choice p l u s t h e negation o f S u s l i n ' s hypothesis, then we do n o t know whether the Aronszajn t r e e i n 7.1 i s o r n o t a S u s l i n t r e e . And more g e n e r a l l y w i t h these s e t - t h e o r e t i c axioms, we do n o t know whether t h e r e e x i s t Aronszajn t r e e s which are n o t S u s l i n t r e e s .
Chapter 5
5 8 - FAITHFUL
INFINITE
155
MALITZ'
EXTENSION:
AND
LOPEZ' COUNTEREXAMPLES
8.1. FAITHFUL INFINITE EXTENSION Let A1,
...,Ah
be a f i n i t e s e t o f f i n i t e r e l a t i o n s w i t h common a r i t y , and
be a f i n i t e r e l a t i o n w i t h
R
I f there e x i s t e x t e n s i o n s o f
2A1
and ... and 3 Ah - 3 A1 are
and
...
R
.
and $ A h
w i t h a r b i t r a r y l a r g e f i n i t e c a r d i n a l i t i e s , which
R
, t h e n t h e r e e x i s t s a denumerable e x t e n s i o n o f
R
which respects t h e same c o n d i t i o n s .
0 We can assume t h a t
1, ... ,p
i s d e f i n e d on t h e i n t e g e r s
R
and t h a t , f o r each
i , t h e r e e x i s t s an e x t e n s i o n Ri o f R based on t h e i n t e g e r s 1 t o p+i and r e s p e c t i n g t h e c o n d i t i o n s . F o r i n f i n i t e l y many i n t e g e r s i , t h e Ri have a same r e s t r i c t i o n S1 t o 1,. . . ,p+1 . Among these, t h e r e a r e i n f i n i t e l y many i n t e g e r s i f o r which t h e Ri have a same r e s t r i c t i o n S2 t o 1, ...,p+ 2 integer
I t e r a t i n g t h i s , we t h u s d e f i n e
S
j
t a k e t h e common e x t e n s i o n o f t h e
...,Ah
A1,
8.2. L e t
and
common a r i t y , and l e t and
...
and
>/
Bk
R
B1,
u
...,Bk
satisfy
f o r each i n t e g e r j . I t now s u f f i c e s t o , based on t h e s e t o f a l l i n t e g e r s . 0 j be two f i n i t e s e t s o f f i n i t e r e l a t i o n s o f
R3AA1
and
... and
#Ah
R >/El
as w e l l as
.
Then t h e r e e x i s t s an i n t e g e r equal t o
S
u
such t h a t
every
R
with cardinality _ a t_ l e_ a s-t .
s a t i s f y i n g t h e p r e c e d i n g c o n d i t i o n s has a r e s t r i c t i o n
t i n g t h e same c o n d i t i o n s , and such t h a t
R'
R'
respec-
has a denumerable e x t e n s i o n s t i l l
respecting the conditions. 0
Let
v
F o r each
be t h e sum o f t h e c a r d i n a l i t i e s of t h e r e l a t i o n s
B1
through
s a t i s f y i n g the conditions, there e x i s t s a r e s t r i c t i o n
R
R'
.
Bk
of
R
v , which s a t i s f i e s t h e c o n d i t i o n s . Consider , which a r e o n l y f i n i t e l y many, up t o isomorphism. F o r each,
w i t h c a r d i n a l i t y a t most equal t o a l l these
R'
.
e i t h e r t h e r e e x i s t s a denumerable e x t e n s i o n s a t i s f y i n g t h e c o n d i t i o n s . O r t h e r e e x i s t s an i n t e g e r
u(R')
a l l extensions o f
R'
which i s s t r i c t l y g r e a t e r t h a n t h e c a r d i n a l i t i e s o f
r e s p e c t i n g t h e c o n d i t i o n s . Then i t s u f f i c e s t o s e t
t o be t h e maximum o f t h e s e
u(R')
.0
u
THEORY OF RELATIONS
156
8.3. MALITZ' COUNTEREXAMPLE Can we require t h a t R ' = R ; in other words, does there e x i s t an integer u such t h a t , i f R has cardinality greater than or equal t o u and s a t i s f i e s the conditions, then there e x i s t s a denumerable extension of R satisfying them. A negative answer i s due t o MALITZ 1967. 0 Take the base of integers from 0 t o n-1 Let I n be the usual chain of these integers; l e t Cn be the consecutivity r e l a t i o n ( y = x + l ) ; l e t O n be the unary relation called the singleton of zero, i . e . the relation taking (+) f o r 0 and ( - ) elsewhere; and l e t U n be the relation singleton of n-1 . Finally l e t Rn be the quadrirelation (In,Cn,On,Un) . From n = 7 on, a l l the R have the n same r e s t r i c t i o n s of c a r d i n a l i t i e s 1, 2 and 3 , up t o isomorphism. Let A1, ..., Ah be those quadrirelations of the same a r i t y and c a r d i n a l i t i e s 1, 2 , 3 which are n o t embeddable in R, , and hence in Rn ( n 3 7 ) . We see t h a t every extension of an Rn t o a new element added t o i t s base admits an embedding of one of the A1,...,Ah . An analogous b u t rather complicated counterexample i s obtained f o r binary relations by LOPEZ 1973. 0
.
8.4. LOPEZ ' COUNTEREXAMPLE Given the f i n i t e relations A1, ...,Ah and B1,...,Bk , one can ask whether there e x i s t two integers u , v such t h a t , f o r every R with cardinality greater t h a n or equal t o u , there e x i s t v elements of the base, such t h a t the r e s t r i c t i o n of R t o i t s base with these v elements removed respects the embedding inequal i t i e s in the 6 ' s and has an extension of a r b i t r a r y large cardinality respecting the non-embedding inequalities in the A's . Negative answer by LOPEZ 1973. 0 For the base, take the s e t of points, or ordered pairs of integers called the abscissa and ordinate, and which vary from 0 t o n-1 Let Rn be the multirelation on t h i s base, which i s composed of the following 4 unary relations and 6 binary relations. The unary relation O n takes the value (+) f o r the points with abscissa 0 The relation U n takes (+) f o r the points with abscissa n-1 Similarly 01; and U,', are defined by interchanging the abscissas and ordinates. The s t r a t i f i e d partial ordering I n takes the value (+) f o r each ordered p a i r of points ( i , x ) , ( j , y ) whose abscissas s a t i s f y i -c j < n , with a r b i t r a r y ordinates x , y ; moreover I n i s reflexive. The equivalence relation E n takes (+) for any two points with a same abscissa and a r b i t r a r y ordinates. The equivalence classes of t h i s relation are thus the classes of elements which a r e pairwise incomparable modulo I n . The binary relation Cn , which by abuse of notation we shall c a l l a consecutivity, takes the value (t) f o r each ordered pair of points
.
.
.
157
Chapter 5
( i , x ) , ( i + l , y ) whose abscissas are consecutive. Finally, the s t r a t i f i e d partial ordering I,', , the equivalence relation E,', and the consecutivity C,', are obtained from the preceding by interchanging abscissas and ordinates. From n = 7 on, every Rn has the same r es t r i ct i o ns B1, ...,B k with cardinalities 1, 2 , 3 ( u p t o isomorphism). Let A1, ...,Ah be the other multirelations of the same a r i t y and car d i n al i t i es 1, 2 , 3 . We see t h a t every proper extension of Rn ( n 3 7 ) admits an embedding of a t l e a s t one of the A's Indeed, add a new element t t o the base of Rn . Consider the case where e i t h e r O n or U n or 0,', or U,', takes the value (+) f o r t , and reduce t h i s t o the preceding 5 8.3. Now consider the case where a l l the preceding unary relations take the value ( - ) f o r t Then e i t h e r there ex i s t s an equivalence class of En t o which t belongs: again reduce t o 5 8.3. Or t occurs between two consecutive equivalence classes of En I n t h i s case, use the consecutivity Cn t o see t h a t the extension of Rn t h u s obtained admits an embedding of one o f the A's . Now suppose the existence of u and v satisfying our hypothesis; take n > u and > v . Let Sn be a r es t r i ct i o n of Rn in which the B's are embeddable, and which i s obtained by removing v points. Then in each equivalence class of Add a En , there remains a t l e a s t one element of I S n l ; similarly f o r E,', new element t t o the base o f Sn , and attempt t o require t h a t the extension of Sn t o i t s base with t added admit only embeddings of the B's and not of the A's . This leads us t o s i t u a t e t in the chain of the equivalence classes By using Cn , one sees t h at t necessarily belongs t o one o f the equiof En valence classes: t cannot be situated between two consecutive classes. Thus we obtain an element in the base of Sn , which i s equivalent with t (mod E n ) , and another element equivalent with t (mod E,',) From t h i s , we deduce tha t t is the unique element common t o both equivalence classes. Thus,we have again a restriction of Rn obtained by removing v-1 points: t h i s i s our extension of Sn Iterating t h i s , we obtain Rn i t s e l f , and a t the following step we obtain a proper extension o f Rn , in which necessarily one of the A's i s embeddable. 0
.
.
.
.
.
.
.
EXERCISE 1 - RESTRICTIONS OBTAINED BY REMOVING ONE OR TWO ELEMENTS 1 Existence of a relation R with base E and two elements a , b o f E such t h a t R z R / ( E - j a } ) I R / ( E - {b}) y et R > R / ( E - { a , b ) ) Let R be the binary relation based on the s e t formed with negative integers -1, -2, and ordered pairs of natural numbers i , j = 0 , 1, 2 , .. , taking the value (+) in only following cases. R(u-1,u) = + f o r each negative u ; and R ( - l , ( O , O ) ) = t ; and R ( ( i , j ) , ( i + l , j ) ) = R ( ( i , j ) , ( i , j + l ) ) = + f o r a l l natural numbers i , j Show t h a t R s a t i s f i e s the stated conditions with a = ( 0 , l ) and b = ( 1 , O ) (communicated by POUZET in 1975).
-
.
...
.
2
- Similarly with
R
> R/(E -
\aj )
>
R/(E
-
{a,b})
ye t
R
L
R/(E
- { b)) .
158
Let
THEORY OF RELATIONS R
be 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 o f p o s i t i v e and n e g a t i v e i n t e g e r s ,
which we denote by
O,l,-1,2,-2,..
and n a t u r a l numbers, which a r e assumed t o be 0 ' ,1' ,2'
d i s t i n c t from t h e i n t e g e r s and denoted by o n l y i n f o l l o w i n g cases:
+
R ( i ,i')=
i3 0
f o r every i n t e g e r
value; f i n a l l y
R(i',(i+l)')
. Show
= R(i,i+2)
R(i,i+l)
that
(communicated by LOPEZ i n 1977).
For
Similarly with R
R
R
> R/(E
i ' w i t h equal
i ' and i t s immediate suc-
satisfies the stated conditions with
(i+l)'
b = 0'
-
v a l u e (+) i s t a k e n on
and f o r t h e n a t u r a l number
f o r e v e r y n a t u r a l number
cessor 3
,. . . The
+ f o r a l l i n t e g e r s i ; and
=
- { a } ) :R/(E - { b ) )
- {a,b}
:R/(E
a = 0
and
.
)
t a k e t h e c o n s e c u t i v i t y on p o s i t i v e and n e g a t i v e i n t e g e r s , w i t h
a, b
two
c o n s e c u t i v e i n t e g e r s (communicated by HAGENDORF i n 1977).
-
EXERCISE 2
Val o f
Let
A, B, C
B satisfying
Defining
by
D
be t h r e e c h a i n s such t h a t
Let
he an isomorphism f r o m
C+D 5 C
A+B
t h e non-empty f i n a l i n t e r v a l o f f"(Do)
i s a t t h e l e f t of
which a r e a t l e f t o f
Do I t e r a t i n g t h i s , we o b t a i n
Di
Either a l l o f the elements o f
B into
B
and
B
C<
i n i t i a l inter-
C
(see HAGENDORF 1977 lemma 1.10).
A
if
A+C+D
,then
A+C
3
in
Do
, and l e t
A+C
onto a r e s t r i c t i o n o f
B which i s t h e complement o f
. Let
A+B
. The
C
be t h e i n t e r v a l o f those elements
D1
and more g e n e r a l l y
D2
Di
.
B , i n which case t h e i d e n t i t y on those , when completed by f , g i v e s an isomorphism o f
Di
h y p o t h e s i s . O r a t l e a s t one o f t h e
hence a l l t h e
times i s an isomorphism o f
included i n
A+B
A
.
I n t h i s case,
onto a r e s t r i c t i o n o f
iterated
, hence A+B
A
. Or
take
, even
B
if
C<
B
. For
instance take
A =
B
=
.
A = U - + W1
-+
Prove t h a t
Q +
";+
0
= Q
i s an
C
l + w - and
compare w i t h
has 3 immediate e x t e n s i o n s : 1+A
, A+1
5
+
; and
has 5 immediate e x t e n s i o n s :
Q
+ W - . W1
, not
B+1,
wl+B
4.4).
and a l s o
which i s n o t o b t a i n e d by s i m p l y a d d i n g an i n t e r v a l t o
B
.
2 A
.
EXERCISE 3 - IMMEDIATE EXTENSIONS (HAGENDORF 1983, unp.; 0 W
(i+l)
A = B = Q+O1+Q and C = W1+Q , where Q i s t h e c h a i n o f A = converse o f W2 and B = Z and C = 0
the r a t i o n a l s . O r again
Prove t h a t
,
A
hits
Di
f
Note t h a t o u r statement no l o n g e r h o l d s i f we remove t h e h y p o t h e s i s t h a t C = CJ-
.
f"(Do) i
f o r each i n t e g e r
are included i n
preceding the
i n i t i a l interval o f
be
Do
image s e t
y e t g r e a t e r t h a n o r equal t o some element of
, contradicting the D . (j > i) a r e J
C
1
.
A+C I A
f
A+B
, we have e q u i v a l e n t l y t h a t
B = C+D
either
.o- r
then
A+B 5 A+C ;
,w
o b t a i n e d by a d d i n g an i n t e r v a l .
.
A
+
B
,
5
CHAPTER
EMBEDDABILITY BETWEEN PARTIAL OR TOTAL ORDERING
§
1 - EMBEDDABILITY,
IMMEDIATE
FAITHFUL
EXTENSION,
EXTENSION
EMBEDDABILITY, EQUIMORPHISM Let
be two r e l a t i o n s o f t h e same a r i t y . We say t h a t
R, S
R
i s embeddable
S o r i s s m a l l e r t h a n S under e m b e d d a b i l i t y , o r t h a t S admits an embedding o f R o r i s g r e a t e r t h a n R , i f f t h e r e e x i s t s a r e s t r i c t i o n o f S in
R ; we w r i t e R + S o r S i s s t r i c t l y embeddable i n S
isomorphic w i t h We say t h a t
.
o r s t r i c t l y smaller than
R
, or
S admits a s t r i c t embedding o f R o r i s s t r i c t l y g r e a t e r t h a n R , denoted
that by
R
R
R < S
or
We say t h a t
S > R , i f f
R$S
but
i s equimorphic w i t h
R
The comparison r e l a t i o n
4
S $ R .
S , denoted R
5 S
, iff
R 6'S
and
S
6
R
i s r e f l e x i v e and t r a n s i t i v e , hence d e f i n e s a quasi-
o r d e r i n g o n each s e t o f r e l a t i o n s . Moreover equimorphism i s symmetric and hence d e f i n e s an e q u i v a l e n c e r e l a t i o n . E m b e d d a b i l i t y i s n o t a n t i s y m m e t r i c , even up t o isomorphism: see t h e f o l l o w i n g examples. Let
be t h e c h a i n o f t h e r a t i o n a l s , and
Q
a l a s t element: t h e n
Q+l t h e e x t e n s i o n o b t a i n e d by adding
.
Q I Q+l
L e t LJ be t h e c h a i n o f t h e n a t u r a l numbers, and ch.2
5 5
1.7). Then
u-.G)E
a-
t h e converse c h a i n ( s e e
1 + ( W - . w ) (the ordinal product i s defined i n
3.7). I n t h e c h a i n o f n a t u r a l numbers, r e p l a c e each even number by Z ( t h e c h a i n of p o s i t i v e and n e g a t i v e i n t e g e r s ) and each odd number by a f i n i t e chain. We o b t a i n continuum many m u t u a l l y non-isomorphic chains, a l l o f which a r e equimorphic.
ch.2
1.1. L e t
R, S
be two equimorphic r e l a t i o n s . Then t h e r e e x i s t s a p a r t i t i o n o f
I R I i n t o two d i s j o i n t subsets D, D ' , and a p a r t i t i o n o f 1st i n t o two d i s j o i n t subsets E, E ' w i t h R/D isomorphic t o S/E and R / D ' isomort h e base phic t o where
S/E'
f
and
.
Repeat t h e p r o o f o f BERNSTEIN-SCHRODER's theorem (ch.1 g
5 1.4),
become isomorphisms from one r e l a t i o n o n t o a r e s t r i c t i o n o f
the other. The converse i s f a l s e , even f o r c h a i n s . Indeed, t h e c h a i n w o f t h e n a t u r a l numbers and t h e c h a i n
w+1
g i v e r i s e t o p a r t i t i o n s s a t i s f y i n g t h e above condi-
t i o n s . S i m i l a r l y f o r t h e incomparable c h a i n s U+c.4-
and
Z = W-+
.
.
132 1.2.
THEORY OF RELATIONS IMMEDIATE EXTENSION
Given a r e l a t i o n R , we say t h a t S i s an immediate extension o f R i f S i s an extension, and furthermore i f there does n o t e x i s t any s t r i c t l y intermediate r e l a t i o n T such t h a t R < T < S w i t h respect t o embeddability. We say a l s o t h a t S , o r any r e l a t i o n equimorphic w i t h S , i s an immediate successor o f R w i t h respect t o embeddability.
.
Moreover .if R For each r e l a t i o n R , t h e r e e x i s t s an immediate extension o f R has a r i t y >/ 1 , then t h e r e e x i s t a t l e a s t two immediate extensions (HAGENDORF 1977, p r o p o s i t i o n VI.5.6). 0 Suppose f i r s t t h a t
i s a 0-ary r e l a t i o n , say
R
R = (E,+)
: then i t s u f f i c e s t o
replace t h e base E by a s e t w i t h immediately g r e a t e r c a r d i n a l i t y : see ch.2 5 3.10. Suppose t h a t R i s a unary r e l a t i o n . L e t a be t h e c a r d i n a l i t y o f the s e t o f e l e -
R
ments g i v i n g t h e value (+) t o i t s u f f i c e s t o replace e i t h e r using ch.2 5 3.10. Suppose now t h a t R
a
has a r i t y
, and or
b t h e analogous c a r d i n a l i t y f o r (-). Then b by an immediately g r e a t e r c a r d i n a l , again
n >/ 2
. Add
t o t h e base E o f
R
a set
D dis-
t o have base E u D w i t h R+/E = R and R+/D j o i n t w i t h E , and d e f i n e R' always (t), and f i n a l l y w i t h R+ t a k i n g the value (+) f o r those n-tuples contain i n g a t l e a s t one term i n D F i n a l l y choose f o r d = Card D t h e l e a s t aleph f o r which Rt i s n o t embeddable i n R , hence R+> R L e t us prove t h a t R' i s an
.
.
immediate extension o f R ; t h e r e l a t i o n R- s i m i l a r l y defined by exchanging (t) and ( - ) , being another immediate extension, obviously incomparable w i t h R' with respect t o embeddabi 1ity. Suppose f i r s t t h a t d i s an i n f i n i t e aleph, and t h a t t h e r e e x i s t s a s t r i c t l y i n t e r mediate r e l a t i o n T w i t h R < T < R' . Consider T as a r e s t r i c t i o n o f Rt Then
.
.
D n IT1 has c a r d i n a l i t y d Indeed i f i t had c a r d i n a l i t y c d , R (more p r e c i s e l y , i n a r e s t r i c t i o n o f Rt t o E increased w i t h (c:d) many elements o f D ). Now p a r t i t i o n DnITI i n t o two the i n t e r s e c t i o n
then T would be embeddable i n
d i s j o i n t subsets, each w i t h c a r d i n a l i t y d
, say
-
D'
and D"
. Then
T
i s isomor-
p h i c w i t h i t s r e s t r i c t i o n t o I T 1 D" , so t h a t R i s embeddable i n t h i s r e s t r i c t i o n , and f i n a l l y R+ i s embeddable i n T : c o n t r a d i c t i o n . Now i t remains t o consider t h e case where d = Card D i s f i n i t e . We f i r s t see t h a t d = 1 . Indeed assume t h a t d f i n i t e and >, 2 Then by hypothesis, the extension of R obtained by adding o n l y one element u t o t h e base I R 1 , w i t h value (t)
.
f o r a l l n-tuples c o n t a i n i n g u , i s embeddable i n R , thus equimorphic w i t h R I t e r a t i n g t h i s , t h e s i m i l a r extension obtained by adding 2 elements, i s s t i l l equimorphic w i t h
R , and so on u n t i l we add d elements: c o n t r a d i c t i o n .
Now examine t h e case where singleton constitutes
D
d = 1
. Call
. Consider
u the supplementary element, whose
again the intermediate r e l a t i o n T as a
.
Chapter 5
133
.
r e s t r i c t i o n o f R+ ; obviously u belongs t o t h e base I T We say t h a t an e l e ment x i n the base i s a (+)-element (mod R ) i f f every n - t u p l e which contains x gives value (+) t o R . Analogous d e f i n i t i o n f o r a (+)-element (mod T) ; i n p a r t i c u l a r , u i s a (+)-element (mod T) . Every (+)-element (mod R) belongs t o the base I T 1 and i s a (+)-element (mod T). Indeed otherwise, i f x i s a (+)-element (mod R ) and does n o t belong t o I T I , t h e n by r e p l a c i n g u by x we could embed T i n R : c o n t r a d i c t i o n . E i t h e r t h e r e e x i s t D e d e k i n d - i n f i n i t e l y many (+)-elements (mod R ) . Then R+ isomorphic w i t h R : c o n t r a d i c t i o n . O r t h e s e t o f (+)-elements (mod R ) has D e d e k i n d - f i n i t e c a r d i n a l i t y , say
h
. Then t h e r e e x i s t a t l e a s t
is
h + l many
(+)-elements (mod T) . Consider a r e s t r i c t i o n R ' o f T which i s isomorphic with R Since we have e x a c t l y h many (+)-elements (mod R ' ) and a t l e a s t h t l many (+)-elements (mod T) , t h e r e e x i s t s a t l e a s t one (+)-element (mod
.
T) ,
.
say v , which does n o t belong t o t h e base I R ' I Then the r e s t r i c t i o n o f T t o t h e base I R'I p l u s t h e element v i s isomorphic w i t h R' : c o n t r a d i c t i o n . 0
1.3. FAITHFUL EXTENSION
.
(1) L e t R , S be two n-ary r e l a t i o n s ( n >r 1) Assume t h a t S does not Then t h e r e e x i s t s a s t r i c t l y g r e a t e r extension T admit an embedding o f R
.
.
of S which does n o t _admit an embedding o f R We c a l l i t a f a i t h f u l extensio? o f S modulo R . Moreover we can choose T t o be an immediate extension o f S : see HAGENDORF 1977. The statement i s obviously f a l s e f o r a r i t y zero.
R aqd S are unary. L e t at be 'the c a r d i n a l i t y o f t h e s e t o f elements g i v i n g the value (+) t o R , and a- the analogous c a r d i n a l i t y f o r (-); s i m i l a r l y l e t b+ and b- be t h e analogous c a r d i n a l i t i e s f o r S . Since R $ S ,
0 Suppose f i r s t t h a t
.
e i t h e r b + < a+ o r b-< aSuppose t h e f i r s t case holds, t h e argument being analogous f o r t h e second case. It s u f f i c e s t o take an extension of S i n which bt i s preserved and b- i s replaced by an immediately l a r g e r c a r d i n a l . Suppose t h a t R and S have a r i t y n 3 2 Add t o t h e base E o f S a s e t '0
.
which i s d i s j o i n t from
E
, and
d e f i n e the extension T+
of
S w i t h base
EvD',
THEORY OF RELATIONS
134
t a k i n g t h e v a l u e (+) f o r those choose
w i t h c a r d i n a l ( a l e p h ) s u f f i c i e n t l y l a r g e t o have
D+
+
w i t h t h e value ( - ) , t h u s o b t a i n i n g
and
0-
T->
.
S
.
Do t h e same
R
4:'
T+> S
We c l a i m t h a t
. Also
D
n - t u p l e s c o n t a i n i n g a t l e a s t one t e r m o f
or
, which y i e l d s o u r c o n c l u s i o n . Indeed suopose t h e c o n t r a r y , and c o n s i d e r R as a r e s t r i c t i o n o f T+ , The base o f R i s n o t a subset o f E , s i n c e R $ R $T-
hence t h e r e e x i s t s an element
u+
R
such t h a t
u+
n - t u p l e c o n t a i n i n g a t l e a s t one t e r m equal t o
S,
takes t h e v a l u e (+) f o r each
. There
e x i s t s an analogous e l e -
ment f o r t h e v a l u e ( - ) : c o n t r a d i c t i o n . 0 be a r e l a t i o n w i t h a r i t y 5 2
R
(2) Let
e x i s t s a common e x t e n s i o n o f The c o n t r a p o s i t i v e i s : i f X
3 S2 , t h e n
or
6 S1
R
and
S1
and
S1&
R
R,< X
f o r every
R 6 S2
.
X
and
El
.
E2
p a r t i t i o n o f t h e base o f e v e r y element =
R
have d i s j o i n t
S2
taking
E2
and t h e o t h e r i n
El
R.
and . ,Sh.
.
E2
or
S+
contrary; then there e x i s t s a
i n t o two non-empty d i s j o i n t subsets such t h a t , f o r
i n one subset and
u
. Suopose t h e
R
.
S1,.
f o r t h e value ( - ) . It s u f f i c e s t o see t h a t e i t h e r
S-
does n o t a d m i t an embedding o f
S-
and
S1
St be t h e common e x t e n s i o n w i t h base El
Let
there
X >,S1
Extend t h i s t o any f i n i t e sequence
t h e value (+) f o r those o r d e r e d p a i r s w i t h one t e r m i n Analogously d e f i n e
. Then
S2 & R
which s a t i s f i e s b o t h
0 Take t h e case o f a b i n a r y r e l a t i o n , and suppose t h a t
bases
and
which does n o t a d m i t an embedding o f
S2
v
i n t h e o t h e r , we have
R(u,v) = R(v,u)
=
+ ; same c o n c l u s i o n w i t h t h e v a l u e ( - ) . Note t h a t , g i v e n two p a r t i t i o n s o f t h e u, v
base, each w i t h two non-empty d i s j o i n t s e t s , t h e r e e x i s t two elements
in
t h e base which a r e separated b o t h by t h e f i r s t p a r t i t i o n and by t h e second. Thus R(u,v)
=
+ and - :
2 and non-empty base. Suppose t h a t
S
& R1
u, v
t h e r e e x i s t two elements
g i v i n g simultaneously
contradiction. 0 The p r o p o s i t i o n i s o b v i o u s l y f a l s e f o r unary r e l a t i o n s . (3) Let
S
. Then
S%R3
>/
have a r i t y
there e x i s t s a proper extension
embeddabilities
R1 ,
S+*
S+$
R2
and
S+ f S
S+$ R3
.
element Let
S2
a
and s e t t i n g
S,(a,x)
of
= S (x,a)
. Let
S =
+
S1
, say
r o l e of
. Hence
i n t h e base o f t h i s
t o f i x t h e ideas, p l a y i n g t h e r o l e o f a
and
R2
x
in
S3
. Suppose f i r s t l y
a
t h a t t h e base
in
.
Si
(i
IS1
i n t h e base S3
=
.
be o b t a i n e d w i t h
S (x,a) = - f o r e v e r y x i n I S ( , and moreover 3 w i t h t h e same c o n d i t i o n s , except t h a t S4(a,a) = - .
R
and
be o b t a i n e d by a d d i n g a new
f o r every
Suppose o u r c o n c l u s i o n i s f a l s e . Then t h e r e e x i s t two al
R1
and
an embedding o f a same
$ R2
which r e s p e c t s t h e non-
1 be s i m i l a r l y o b t a i n e d w i t h ( - ) ' i n s t e a d o f ( + ) . L e t
S (a,x) = + 3 F i n a l l y S4
S
F o r t h e a r i t y 1 o r f o r empty
base, t h e p r o p o s i t i o n i s o b v i o u s l y f a l s e , even w i t h o n l y 0 Consider t h e 4 f o l l o w i n g e x t e n s i o n s
,
S3(a,a)
1,2,3,4)
= +
.
which a d m i t
R , t h e r e e x i s t s an element
S1 and an a3 p l a y i n g t h e
I R I has c a r d i n a l i t y
32 .
Chapter 5 Then
al
and
and
R(x,a3)
a3 =
-
are d i s t i n c t , since x # a3
f o r every
.
135
R(x,al)
+
=
R(al,a3)
takes s i m u l t a n e o u s l y t h e
f o r every
value (+) and t h e v a l u e ( - ) : c o n t r a d i c t i o n . Analogous argument f o r and
S1
,
S4
S2
and
S3
,
and
S2
Suppose now t h a t t h e base I R I takes t h e v a l u e ( - ) . Since S >
,
S4
x # al
= R(al,x)
Moreover
and
S3
and
S1
,
S2
.
S4
has c a r d i n a l i t y 1, and t o f i x t h e ideas, t h a t R R and by h y p o t h e s i s S non-empty, n e c e s s a r i l y S
i s r e f l e x i v e . As p r e v i o u s l y d e f i n e e x t e n s i o n s S1, S2, S3 which now a r e a l l t h r e e r e f l e x i v e . E i t h e r R1 = R 2 = R3 = R and t h e n o u r c o n c l u s i o n h o l d s . O r R1 and possibly
R , t h u s have c a r d i n a l i t i e s
are d i s t i n c t from
R2
our conclusion i s false: then
, and
( i = 1,2,3)
R1
1.4. Consider an 13 -sequence o f r e l a t i o n s
Ria A
i
f o r every
, then
R+
suppose Si
A
Ri
( i integer)
3
o f common a r i t y
A
2
be a r e l a t i o n o f t h e same a r i t y .
t h e r e e x i s t s a comnon e x t e n s i o n o f a l l t h e
which does n o t a d m i t an embedding o f 0 Let
. Again
t h e argument t e r m i n a t e s as p r e v i o u s l y . 0
and w i t h m u t u a l l y d i s j o i n t bases. L e t
.-have
2
f o r i n s t a n c e i s embeddable i n a t l e a s t two
Ri
.
denote t h e common e x t e n s i o n o f t h e
Ri
on t h e u n i o n o f t h e bases, which
takes t h e v a l u e (+) f o r a l l t h o s e n - t u p l e s ( n = a r i t y ) c o n t a i n i n g a t l e a s t two terms t a k e n f r o m two d i s t i n c t bases. Analogously d e f i n e t h e e x t e n s i o n that
A
i s embeddable b o t h i n
, there
Ri
R+
and
R-
n e c e s s a r i i y e x i s t two elements
. Since x, y
t r a n s f o r m e d i n t o two elements i n two d i s t i n c t
A
, simultaneously
and second embedding. tience f o r an n - t u p l e c o n t a i n i n g b o t h for
I A I , which a r e
i n t h e base
lRil
. Suppose
R-
i s n o t embeddable i n any
x
and
f o r the f i r s t
, we
y
have
t h e v a l u e (+) and t h e v a l u e ( - ) : c o n t r a d i c t i o n . 0
A
5 2 - EMBEDDABILITY
BETWEEN PARTIAL
(KRUSKAL); CANTOR'S GLEASON); TOURNAMENT
OF FINITE
TREES
(DILWORTH,
ORDERINGS;
WELL PARTIAL ORDERING ORDERINGS
THEOREM FOR PARTIAL
2.1. There e x i s t i n f i n i t e l y many f i n i t e p a r t i a l o r d e r i n g s which a r e m u t u a l l y incom-parable w i t h r e s p e c t t o e m b e d d a b i l i t y . 0 Let
a'< i
be t h e p a r t i a l o r d e r i n g on 5 elements
A1 b'<
,let
v Ai
until
ui-l
with
be t h e p a r t i a l o r d e r i n g based on 2i+3 elements
U ~ ~ ~ , V ~ , ~ , V ~ , ~ , . w. i .t h, Va ~<~b ~ , ~
bilities
a,b,a',b',v
a(
b
,
and i n c o m p a r a b i l i t y elsewhere. Now more g e n e r a l l y f o r each i n t e g e r
u1 < v ~ , and ~
and
and
a'<
u1 < v ~ , and ~ t h e n u2 < v1,2 ui-1
< vi-l,i
a,b,a',b',u
b;< and
vi-l,i
ui
l,..., and
comparaand so
; and i n c o m p a r a b i l i t y elsewhere.
The p a r t i a l o r d e r i n g s t h u s d e f i n e d a r e m u t u a l l y incomparable w i t h r e s p e c t t o emb e d d a b i l i t y (example comnunicated i n 1969 by JULLIEN). 0
I36
THEORY OF RELATIONS
2.2.
( 1 ) There e x i s t s a s t r i c t l y decreasing (under e m b e d d a b i l i t y ) w - s e q u e n c e
o f denumerable p a r t i a l o r d e r i n g s . 0
To each s e t
I o f integers, associate the p a r t i a l ordering
extension o f the f i n i t e p a r t i a l orderings
Ai
(i
e
AI
o b t a i n e d as an
I) , taken t o be m u t u a l l y
incomparable. Then t a k e an i n f i n i t e s t r i c t l y d e c r e a s i n g sequence o f s e t s
I
.0
( 2 ) There e x i s t continuum many denumerable p a r t i a l o r d e r i n g s which a r e m u t u a l l y incomparable under e m b e d d a b i l i t y . 0
Take t h e p r e c e d i n g
AI
I, J
and n o t e t h a t , f o r two s e t s
o f which i n c l u d e s t h e o t h e r , t h e n
AI
and
AJ
o f integers neither
a r e incomparable. 0
Problem posed by HAGENDORF. E x i s t e n c e o f a s t r i c t l y d e c r e a s i n g G1-sequence o f denumerable p a r t i a l o r d e r i n g s . Same problem f o r r e l a t i o n s , posed by POUZET. 2.3. THEOREM OF THE WELL PARTIAL ORDERING
OF FINITE TREES
Embeddability between f i n i t e t r e e s i s a w e l l p a r t i a l o r d e r i n g (KRUSKAL 1960). 0
We can always assume t h a t each o f t h e c o n s i d e r e d f i n i t e t r e e s has a minimum
element: i t s u f f i c e s t o add a minimum, and even a new minimum i f t h e r e a l r e a d y e x i s t s a minimum, t o each f i n i t e t r e e ; t h e n e m b e d d a b i l i t y o r non-embeddability i s preserved. Suppose t h a t e m b e d d a b i l i t y between f i n i t e t r e e s w i t h a minimum, i s n o t a w e l l p a r -
w -sequence o f such t r e e s , which i s bad w i t h
t i a l o r d e r i n g . Then t h e r e e x i s t s an r e s p e c t t o e m b e d d a b i l i t y : see ch.4
5
s t r o n g l y minimal bad w-sequence:
ch.4
3 . 2 . ( 2 ) c o u n t a b l e case. Hence t h e r e e x i s t s a
5
2.iO.
I n t h i s sequence, t h e terms a r e
m u t u a l l y incomparable w i t h r e s p e c t t o e m b e d d a b i l i t y : ch.4 5 2.3 and 2.8. L e t U denote t h i s sequence, and Ui ( i i n t e g e r ) each term. The s e t o f f i n i t e t r e e s w i t h a minimum, which a r e s t r i c t l y embeddable i n a
5
see ch.4 Let
H
Ui
,
i s a well p a r t i a l ordering:
4.3 and 2.8.
denote t h i s c o u n t a b l e s e t (up t o isomorphism) o f f i n i t e t r e e s w i t h a m i n i -
mum. Embeddability between words, o r f i n i t e sequences o f elements o f w e l l p a r t i a l o r d e r i n g : ch.4
5
4.4 (HIGMAN). To each t r e e
H
,
is a
a s s o c i a t e one o f t h e
Ui f i n i t e sequences o b t a i n e d by t o t a l l y o r d e r i n g i n an a r b i t r a r y manner t h e immediate successors o f t h e minimum, t h e n by r e p l a c i n g each o f them by t h e s u b - t r e e o f t h o s e g r e a t e r o r equal elements. Then t h e p r e c e d i n g sequence
words i n Ui Ui
H
, hence
U
becomes a sequence o f
i s good. Moreover, i f t h e word t h u s s u b s t i t u t e d f o r t h e t r e e
i s embeddable i n t h e word s u b s t i t u t e d f o r UJ. (i,j i n t e g e r s ) , t h e n t h e t r e e i t s e l f i s embeddable i n U . . Hence t h e sequence U i s good: c o n t r a d i c t i o n . 0 3
2.4. A b i n a r y r e l a t i o n
A
i s a p a r t i a l ordering i f f
A
does n o t a d m i t an embed-
ding o f the following f i n i t e relations: t h e b i n a r y r e l a t i o n w i t h c a r d i n a l i t y 1 and v a l u e ( - ) ( t h i s e n s u r e s r e f l e x i v i t y ) ; t h e r e l a t i o n always (+) w i t h c a r d i n a l i t y 2 ( a n t i s y m n e t r y ) ; t h e r e f l e x i v e b i n a r y cycle with cardinality 3
, and
f i n a l l y t h e c o n s e c u t i v i t y r e l a t i o n on 3 elements
Chapter 5
137
associated w i t h t h e c h a i n o f c a r d i n a l i t y 3 : see ch.2
5
8.6 ( t h e s e two non-embed-
d a b i l i t i e s ensure t r a n s i t i v i t y ) .
2.5.
or
Every denumerable p a r t i a l o r d e r i n g admits an embedding o f t h e o r d i n a l w
i t s converse
(J-
o r t h e denumerable f r e e p a r t i a l o r d e r i n g .
I n p a r t i c u l a r , e v e r y denumerable f i n i t e l y f r e e p a r t i a l o r d e r i n g , hence e v e r y denu-
.
merable chain, admits an embedding o f w o r L,I-
5
WiXh ch.4
0 Enumerate t h e elements
integers ai < a ch.3
5
j
T h i s i s KONIG's lemma: compare
4.5 i n t h e case 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, j
ai
o f t h e base, and p a r t i t i o n t h e p a i r s o f
( i integer)
where we assume
i< j
,
i n t o 3 c o l o r s , a c c o r d i n g t o whether
o r l a . (modulo t h e g i v e n p a r t i a l o r d e r i n g ) . By RAMSEY's theorem or 7 a j J 1.1, t h e r e e x i s t s a denumerable s e t o f i n t e g e r i n d i c e s i n which a l l p a i r s
have t h e same c o l o r . According t o whether i t i s t h e f i r s t , second o r t h i r d c o l o r , the given p a r t i a l o r d e r i n g admits an embedding o f
cc)
or
w - or
a denumerable
relation o f identity. 0 Modulo t h e denumerable subset axiom (ch.1
w
admits an embedding o f
A
or w -
5
2.6),
every i n f i n i t e p a r t i a l ordering
o r t h e denumerable f r e e p a r t i a l o r d e r i n g .
well-founded p a r t i a l o r d e r i n g does n o t a d m i t an embedding o f
w- .
Conversely,
a denumerable p a r t i a l o r d e r i n g ( o r more g e n e r a l l y a p a r t i a l o r d e r i n g w i t h w e l l orderable base) which does n o t a d m i t an embedding o f
w - , i s well-founded. With
the axiom o f dependent choice, e v e r y p a r t i a l o r d e r i n g which does n o t admit an embedding o f
w - i s well-founded: see ch.2 5 2.4.
2.6. CANTOR'S THEOREM FOR PARTIAL ORDERINGS
, the
Given a p a r t i a l o r d e r i n g A
, admits
A
vals o f
I n particular, i f embedding o f 0 Let
A g B
A
p a r t i a l o r d e r i n g o f i n c l u s i o n among i n i t i a l i n t e r -
a s t r i c t embedding o f A
A
(DILWORTH, GLEASON 1964).
i s a chain, t h e n t h e c h a i n o f c u t s o f
A
B denote t h e p a r t i a l o r d e r i n g o f i n i t i a l i n t e r v a l s o f
, it
admits a s t r i c t
.
s u f f i c e s t o a s s o c i a t e t o each element
x
of
A
. To
see t h a t
I A I the i n i t i a l interval
.
6 x (mod A ) NOW suppose t h a t BG A and l e t f be an isomorphism o f B i n t o A Some i n i t i a l i n t e r v a l s X o f A s a t i s f y t h e r e l a t i o n f ( X ) E X : f o r example t h e e n t i r e
o f elements
x
f(U)
< f(U)
Phhm f ( V ) < Thus
. L e t U be t h e i n t e r s e c t i o n o f a l l t h e s e i n t e r v a l s . We s h a l l 4 U . Indeed i f f ( U ) E U , t h e n t h e i n t e r v a l V o f those e l e -
X = \A[
interval prove t h a t ments
.
f(U)
+U
there e x i s t s
(mod A)
satisfies
f ( U ) (mod A ) and s i n c e
one o f t h e s e
, hence U
VcU
( s t r i c t i n c l u s i o n ) . Hence by t h e isomor-
f(V) E V
and so
Vz
i s t h e i n t e r s e c t i o n o f those X
, say
Xo
, which
U : contradiction. X
such t h a t
f ( X ) i5 X,
does n o t c o n t a i n t h e element
f(U).
THEORY OF RELATIONS
138
Thus we have Moreover 2.7.
Xoz U
f(U)
+
Xo
and
# U , hence by isomorphism f ( U ) < f(Xo) (mod A)
and so
f(Xo)
$
Xo : c o n t r a d i c t i o n .
TOURNAMENT
A binary r e l a t i o n A
i s c a l l e d a tournament i f f i t i s r e f l e x i v e , antisymmetric by a n t i and comparable: f o r a l l x, y e i t h e r A(x,y) = + (and thus A(y,x) =
-
-
symnetry) o r A(y,x) = t (and A(x,y) = ) . I n the f i r s t case we say t h a t f o l l o w s x (mod A) ; i n t h e second case y precedes x (mod A)
y
. E - u
For each element u o f t h e base E o f A , t h e elements i n are divided i n t o two complementary sets, formed r e s p e c t i v e l y o f those elements which f o l l o w
u and those which precede u
.
Every r e s t r i c t i o n o f a tournament i s a tournament.
A tournament i s a chain i f f i t i s t r a n s i t i v e . The b i n a r y c y c l e on 3 elements (see ch.2 5 8.6), obviously m o d i f i e d t o be r e f l e x i ve, i s a tournament. Yet a b i n a r y r e f l e x i v e c y c l e w i t h c a r d i n a l i t y 3 4 i s not antisymmetric, thus i s n o t a tournament.
A tournament i s a chain i f f i t does n o t admit an embedding o f the b i n a r y r e f l e x i v e c y c l e w i t h c a r d i n a l i t y 3.
A b i n a r y r e l a t i o n i s a tournament i f f i t admits no embedding o f t h e b i n a r y r e l a t i o n w i t h c a r d i n a l i t y 1 and value ( - ) (ensures r e f l e x i v i t y ) ; no embedding o f t h e r e l a t i o n always (+) w i t h c a r d i n a l i t y 2 (ensures antisymmetry); f i n a l l y no embedding o f the i d e n t i t y r e l a t i o n w i t h c a r d i n a l i t y 2 (ensures c o m p a r a b i l i t y ) .
§
3
- DENSE CHAIN:
A-DENSE CHAIN, FOR AN I N F I N I T E CARDINAL A
3.1. DENSE CHAIN A chain i s s a i d t o be dense i f f i t s base i s i n f i n i t e , and i f between any two d i s t i n c t elements x < y t h e r e e x i s t s an element z : x c z < y Hence between any two elements x and y 7 x t h e r e e x i s t i n f i n i t e l y many e l e ments.
.
To each z ( x < z c y ) associate the i n t e r v a l (x,z) : t h e r e i s no minimal interval
(x,z)
w i t h respect t o i n c l u s i o n (use ch.1 tj 1.1). 0
Every denumerable dense chain w i t h o u t any minimum o r maximum, i s isomorphic to_ t h e chain
Q o f the ratj.1..
Every countable chain i s embeddable - -- i n__ Q
.
Every dense chain admits an embedding o f Q (uses dependent choice; i f t h e base i s denumerable, o r o n l y w e l l - o r d e r a b l e ) .
ZF s u f f i c e s
Chapter 5
U
139
Consider t h e s e t o f f i n i t e s t r i c t l y i n c r e a s i n g sequences o f elements o f t h e chain,
say u o c u1 < ... < uh ( h i n t e g e r ) sequences w i t h a r b i t r a r y l e n g t h h t h e form
<
vo
the r e l a t i o n 3.2. L e t
,u, R
A
, and t h e r e l a t i o n R which t o each , a s s o c i a t e s any sequence w i t h l e n g t h
2h
+ 1of
c v1 c u 1 < . . . < vh < uh< v ~ .+ Then ~ a p p l y dependent c h o i c e t o . The o b t a i n e d w - s e q u e n c e y i e l d s a r e s t r i c t i o n isomorphic w i t h Q.
be a denumerable c h a i n . I f
ordinal, then
o f these
A
A
0
admits an embedding o f e v e r y c o u n t a b l e
Q
admits an embedding o f t h e c h a i n
(uses c o u n t a b l e axiom o f
choice). 0
To each c o u n t a b l e o r d i n a l
embedded i n
A
@
, a s s o c i a t e 4 +1+a
. Hence a s s o c i a t e
a(& )
an element
i n t h e l o w e r i n t e r v a l and i n t h e i n t e r v a l above ble, y e t there are element
a
wl-many
which by h y p o t h e s i s can be such t h a t 4
i s embeddable
) . The c h a i n
a(*
A
i s denumera-
c o u n t a b l e o r d i n a l s 6 .So t h e r e e x i s t s a t l e a s t one
such t h a t e v e r y c o u n t a b l e o r d i n a l i s embeddable b o t h below and above
a
( c o u n t a b l e axiom o f choice: we use t h e f a c t t h a t e v e r y denumerable u n i o n o f denume-
5
r a b l e o r d i n a l s i s a denumerable o r d i n a l : see c h . 1 a restriction o f
3.3. L e t
A
A
be a
which i s isomorphic w i t h
2 . 5 ) . By i t e r a t i o n , we o b t a i l i
.0
Q
non-empty c h a i n w i t h an i n i t i a l i n t e r v a l and a f i n a l i n t e r v a l ,
b o t h d i s j o i n t and i n each of which
A
i s embeddable. Then
(uses dependent choice; ZF s u f f i c e s i f
A
Q
i s embeddable i n A
i s denumerable o r w i t h w e l l - o r d e r a b l e
base). 0 By i t e r a t i o n , t h e h y p o t h e s i s a l l o w s one t o d i v i d e
v a l s , i n each o f which that
A
A
i n t o three d i s j o i n t i n t e r -
A
i s embeddable. Hence t h e r e e x i s t s an element
i s embeddable b o t h b e f o r e and a f t e r A
choice, we o b t a i n a r e s t r i c t i o n o f
a
.
a
such
By i t e r a t i o n u s i n g dependent
which i s isomorphic w i t h
Q
.
0
3 . 4 . a-DENSE CHAIN Given an i n f i n i t e c a r d i n a l
a
, we
say t h a t a c h a i n i s
n i t e and between any two d i s t i n c t elements equal t o
a
Sense o f 3.1,
. Analogous is
(
d e f i n i t i o n f o r a chain
y
a--
, the
( 3 a)--.
i f i t s base i s i n f i i n t e r v a l has c a r d i n a l i t y A dense chain, i n t h e
a
, there
exists a chain with cardinal
a
which
a-dense (uses axiom o f c h o i c e ) .
c3 S t a r t w i t h t h e o r d i n a l p r o d u c t
a
and
w )-dense (uses denumerable subset axiom).
Given an i n f i n i t e c a r d i n a l
is
x
Q.a
where
Q
i s t h e c h a i n o f t h e r a t i o n a l s , and
i s an o r d i n a l , more p r e c i s e l y an a l e p h ( r e c a l l t h a t , w i t h choice axiom, e v e r y
c a r d i n a l i s an aleph: c h . 1
0
6 . 1 ) . The c h a i n
Q.a
has c a r d i n a l i t y
a
and i s ob-
v i o u s l y dense w i t h o u t any minimum or maximum. A l s o t h e s e t o f f i n i t e sequences o f elements i n
Q.a
has c a r d i n a l i t y
a
. We
o r d e r t h i s s e t l e x i c o g r a p h i c a l l y . Now i t
THEORY OF RELATIONS
140
a-dense o r d e r i n g . Indeed, c o n s i d e r two f i n i t e se-
s u f f i c e s t o see t h a t i t i s an
u
quences
and
Denote by
i with
index ui
ui
and
strictly after
u
and
v
vi
the
ui
< vi
, and
vi
( i integer)
w i t h r e s p e c t t o l e x i c o g r a p h i c a l comparison. i - t h terms. E i t h e r t h e r e e x i s t s a l e a s t
wi
(mod Q . a ) ; then we t a k e
we have
which i s s t r i c t l y between
a-many f i n i t e sequences b e g i n n i n g w i t h
t h e y a l l a r e s i t u a t e d between
u
and
.
v
u
Or
U ~ , . . . , U ~ - ~ , W ~:
i s an i n i t i a l i n t e r v a l o f
3.5.
< vh
wh
an element
(h = length o f
v
,
f o l l o w e d by
u
and we have t h e same c o n c l u s i o n by t a k i n g sequences b e g i n n i n g w i t h
u ). 0
L e t w , be an i n f i n i t e r e g u l a r a l e p h . Every c h a i n w i t h c a r d i n a l i t y
admits an embedding e i t h e r o f t h e o r d i n a l
o r i t s converse
a , ,
w,
,
,
CJ,
W , -dense c h a i n .
Let
0
A
be a c h a i n w i t h c a r d i n a l i t y w 4 , i n which n e i t h e r ad n o r i t s conver-
se i s embeddable. L e t
x, y
I A l ; we p u t
be elements o f
<
i f t h e i n t e r v a l between them has c a r d i n a l i t y
.
LJ&
x
equivalent w i t h
y
The e q u i v a l e n c e c l a s s e s a r e
.
Every e q u i v a l e n c e c l a s s has c a r d i n a l i t y < L d M . Indeed l e t T be an e q u i v a l e n c e c l a s s . Take an o r d i n a l indexed sequence o f elements o f T which i s
intervals o f
A
s t r i c t l y i n c r e a s i n g (mod A) and c o f i n a l i n less than
~3~
,
f i r s t t e r m o f t h e sequence, i s a u n i o n o f nality than
<
ui
. This
T
s i n c e w , i s n o t embeddable i n
.
By r e g u l a r i t y o f w ,
a d .Same
<
sequence has l e n g t h s t r i c t l y
.
The subset o f
w,-many
, this
argument f o r t h e subset o f
A
T
above t h e
i n t e r v a l s , each w i t h c a r d i -
u n i o n has c a r d i n a l i t y s t r i c t l y l e s s T
b e f o r e t h e f i r s t t e r m o f t h e se-
quence, by c o n s t r u c t i n g a d e c r e a s i n g o r d i n a l - i n d e x e d sequence. Since t h e e q u i v a l e n c e c l a s s e s a l l have c a r d i n a l i t y s t r i c t l y l e s s t h a n
ad
i s regular, there e x i s t
wd
and
C d d -many e q u i v a l e n c e c l a s s e s . Take a r e p r e s e n t a t i -
ye element f r o m each c l a s s . Again by r e g u l a r i t y , between two r e p r e s e n t a t i v e s t h e r e necessarily e x i s t
LJ, -many e q u i v a l e n c e c l a s s e s , hence as many r e p r e s e n t a t i v e s .
Finally, the r e s t r i c t i o n o f
A
t o t h e s e t o f r e p r e s e n t a t i v e s i s an W, -dense
chain. 0 T h i s p r o p o s i t i o n i s no l o n g e r t r u e i f a d i s a s i n g u l a r a l e p h . 0
Take t h e sum
cc,- + LJ
+ ... +
i~ .1
+ ...
where
i i s an a r b i t r a r y i n t e g e r
and (Ji designates t h e r e t r o - o r d i n a l converse o f o i . The sum i s t h u s a c h a i n with cardinality W . F i r s t l y , t h e o n l y w e l l - o r d e r e d r e s t r i c t i o n s of t h i s sum a r e t h e f i n i t e c h a i n s and u
. No dense
c h a i n i s embeddable i n t h i s sum.
Secondly, a r e t r o - o r d i n a l i s embeddable i n t h i s sum o n l y i f i t i s embeddable i n one o f the
wi
3.6. L e t aleph
,
A
w,
i s n o t embeddable. 0
be a p a r t i a l o r d e r i n g whose base has c a r d i n a l i t y an i n f i n i t e r e q u l a r
. Suppose t h a t ,
(dw
elements
so
f o r each element u o f t h e base I A l , t h e s e t o f I u has c a r d i n a l i t y s t r i c t l y l e s s t h a n c d K . Then t h e r e e x i s t s
141
Chapter 5
a t o t a l l y ordered r e s t r i c t i o n of
A
which i s isomorphic with the ordinal L J .~
a Well-order the base I A I i n order-type and l e t C denote the chain thus obtained. P a r t i t i o n the p a i r s of elements x, y of the base i n t o two c o l o r s . I f x < y (mod C ) , we say t h a t the p a i r i s ( + ) i f f x < y (mod A ) and ( - ) i f f x > y or xly (mod A ) . For a given element a i n the base, t h e r e a r e s t r i c t l y l e s s than wA-many elements x f o r which the p a i r { a , x ) has c o l o r ( - ) : so t h a t e i t h e r x < a (mod C ) o r x < a o r l a (mod A ) . Now apply ch.3 5 3.2: there e x i s t s a subset equipotent t o the base, i n which a l l p a i r s have color (+) . This y i e l d s a r e s t r i c t i o n isomorphic t o the ordinal d d 0 The proposition i s f a l s e f o r a s i n g u l a r aleph: For each i n t e g e r i , take a s e t Ei with c a r d i n a l i t y oi , the Ei being mutually d i s j o i n t . Define a p a r t i a l ordering on the union, which has c a r d i n a l i t y a W, by taking each Ei a s a f r e e s e t , each element i n Ei being l e s s than each element of E f o r any two i n t e g e r s j > i Then every t o t a l l y ordered j r e s t r i c t i o n has order-type a t most G)
.
.
.
3.7. (1) Let W , be a r e g u l a r l i m i t aleph. Then each ad-dense chain admits e i t h e r an embedding of wd o r i t s converse, o r an embedding of a l l o r d i n a l s s t r i c t l y l e s s than ad and t h e i r converses. ( 2 ) Let Ud be a r e g u l a r aleph. Then L3 d + l -dense chain admits e i t h e r an embeddinq of o r i t s converse, o r an embedding of a l l o r d i n a l s and their converses,(ERDOS, RADO 1953; (1) and ( 2 ) use equipotent w i t h the generalized continuum hypothesis; f o r o( = 0 , ZF p l u s choice s u f f i c e s ; f o r d = 1 , ZF plus choice plus continuum hypothesis s u f f i c e s ) . (1) Let A be an -dense chain. By r e s t r i c t i n g t o one o f i t s i n t e r v a l s , we can assume A t o have c a r d i n a l i t y a&. Let C denote a well-ordering with base 1 A I and order-type a0(. P a r t i t i o n the p a i r s { x,y} i n the base i n t o two c o l o r s : assuming t h a t x < y (mod C ) , the p a i r will have c o l o r (+) i f f x ( y (mod A ) and ( - ) i f f x > y (mod A ) . By ch.3 5 3 . 5 . ( 1 ) (gen. continuum
each
hypothesis), e i t h e r t h e r e e x i s t s a (-)-monochromatic subset equipotent t o the Or f o r each base: thus A admits an embedding of t h e retro-ordinal cardinal b < W, , t h e r e e x i s t s a (+)-monochromatic subset w i t h c a r d i n a l i t y b : thus A admits an embedding of every ordinal s t r i c t l y l e s s than C C ) ~. The
.
proof ends by interchanging the c o l o r s . ( 2 ) Let A be an Wbc+l -dense chain. Since u d i s regular by hypothesis, the l e a s t cardinal b s a t i s f y i n g b ( G), ) md i s b = &IM : see ch.2 5 6.5(1) u s i n g gen. continuum hypothesis. In o r d e r t o prove ( 2 ) , we now denote by A those oc+1 -dense chains which embed n e i t h e r the ordinal W o(+l nor i t s converse. P a r t i t i o n the p a i r s of elements of I A I i n t o two c o l o r s , e x a c t l y a s in (1) above.
>
142
THEORY OF RELATIONS
By ch.3 § 3 . 5 . ( 2 ) , e i t h e r t h e r e e x i s t s a
(-)-monochromatic subset equipotent w i t h
.
O r a (+)-monochromatic subthe base: thus A admits an embedding of @ N + l s e t w i t h c a r d i n a l i t y Cdd : thus A admits an embedding o f C d M Hence i n every case, A admits an embedding o f t h e o r d i n a l ae and i t s converse. ai ( i < wo( ) be a s t r i c t l y i n c r e a s i n g (mod A ) a,-sequence. L e t ao,al
.
,..., ,...
w , + ~-dense chain which 3 w O c . Assume, i n order
L e t C be t h e l e a s t o r d i n a l f o r which there e x i s t s an admits no embedding o f c , By the preceding we have c
.
Then c i s the l i m i t t o o b t a i n a c o n t r a d i c t i o n , t h a t c has c a r d i n a l i t y u d o f a s t r i c t l y i n c r e a s i n g sequence o f o r d i n a l s ci < c , indexed by i running through a t most
(30(
. Each
ci
aH+l -dense
i s embeddable i n every
chain. For
each i , the o r d i n a l , o r t h e aleph equal t o ci , i s embeddable i n the i n t e r v a l Hence the sum o f t h e ci i s embeddable i n A and so c i s embeddable (ai,ai+l) i n A . This c o n t r a d i c t i o n shows t h a t A admits an embedding o f every o r d i n a l equipotent w i t h W M . The same argument proves t h a t A admits as w e l l an embedding of t h e corresponding converse o r d i n a l s . 0
.
5 4 - IMMEDIATE An o r d i n a l
U
EXTENSION OF A CHAIN: FAITHFUL EXTENSION OF A CHAIN
i s indecomposable i f f ever.y o r d i n a l
(indecomposable o r d i n a l i s defined i n ch.1
5
V< U
satisfies
V.2
3.6).
.
.
U be an indecomposable o r d i n a l and \I < U Then V i U = U so V . 2 < U Conversely, suppose t h a t U i s decomposable: t h e r e e x i s t V < U and W < U
0 Let
with
V+W = U
. Let
T = Max(V,W)
. Then
T
and
T.Z&V+W
I n next chapter, indecomposability w i l l be extended t o chains (ch.6 preceding p r o p o s i t i o n must t h e r e be m o d i f i e d (ch.6 4.1.
Then
5
.0
= U
5
3 ) . The
3.4).
L e t A, A ' be two chains and U t h e l e a s t ordi-nal such t h a t U i s indecomposable (HAGENDORF 1971).
A+U+A'4
A+A'
L e t Y < U , so t h a t A+V+A' I A+A'(equimorphism L defined i n 5 1 above). Take an isomorphism from A+V+A' i n t o A+A' L e t V = W+W' w i t h W embeddable i n A and W ' embeddable i n A ' I n o r d e r t o f i x our ideas, suppose t h a t W $ W ' .
0
.
.
Then
W'+A',(
A'
(W' .2)+A',< A '
so
4
and so
(W' .4)+A'
< A'
, hence
(V.2)+A1
A ' and so A+(V.2)+A',< A+A' and f i n a l l y V.2< U . Argue analogously t o o b t a i n t h e same conclusion i f W>, W' Thus U i s indecomposable, by t h e prece-
.
ding proposition. 0 4.2.
IMMEDIATE EXTENSION OF A CHAIN
We say t h a t a chain
B
i s an immediate extension o f
i s immediately q r e a t e r than A
A
(up t o isomorphism)
( w i t h respect t o embeddability) i f
B >A
or
and
.
Chapter 5 t h e r e does n o t e x i s t any c h a i n
X
satisfying
be a c h a i n and
U
the
immediate e x t e n s i o n o f
A
(HAGENDORF 1972).
Let
A
143
A
.
B
l e a s t o r d i n a l such t h a t
N o t i c e t h a t t h e p r o p o s i t i o n n o l o n g e r h o l d s i f we r e p l a c e F o r example, l e t and 0
A =
A+U = IJ - . 2
The o r d i n a l
U
B
tion of
AtU
we have
converse o f w
,
1+ W -
2+ W
.
Then
U
satisfying
, which
an o r d i n a l . I f
A
B
< A+d .
t h u s decomposes
,
8" < U
the d e f i n i t i o n o f
by a r e t r o - o r d i n a l .
U = W - ; y e t between
B"
Consider an isomorphism o f B
B
i n t o an i n i t i a l i n t e r v a l
such t h a t
B ' G A
and
B'
. Hence
, which
and a complementary f i n a l i n t e r v a l
U
.
decomposable we have
A'
The c h a i n
A"+U = U
A"
5
A
U
B = B'+U
i s an o r d i n a l . I f
is
B
.
Hence
hence
A" = U
and
the existence o f a f i n a l i n t e r v a l o f
A+1
A
such t h a t U
i s in-
B'+U = B
A+U,<
,
.
A = A'+U
i s an i n f i n i t e indecomposable o r d i n a l , i n which case
(equimorphy). The preceding argument, w i t h
decomposes
A"
< U , since
A"
A+U = A ' t A " + U = A ' + U
so
thus c o n t r a d i c t i n g t h e d e f i n i t i o n o f Then e i t h e r
B"
B = B ' + B " , C A+B" 6 A t h u s c o n t r a d i c t i n g B" = U and B = B'+U (by = we mean "isomorphic t o " ) . onto a r e s t r i c t i o n o f
A",(
and a com-
chain
i t follows that
i n t o an i n i t i a l i n t e r v a l and
onto a r e s t r i c -
< U . The
B"
A
A+1
A
- , etc.
Consider an isomorphism f r o m A
6 B'
iLaL
o f t h e p r o p o s i t i o n i s non-zero and by t h e p r e c e d i n g 4.1 i t i s i n -
plementary f i n a l i n t e r v a l
A'
A+U
I n o r d e r t o o b t a i n a c o n t r a d i c t i o n , l e t us suppose t h a t t h e r e e x i s t s
decomposable. a chain
, the
U-
. Then
A+U $ A
so
1
A+l , proves
r e p l a c e d by
which i s isomorphic w i t h
U : contra-
diction. U = 1 and so
Or
A'
.
A'+1
A = A'+1
B = B ' + 1 , hence A ' + 1 (
and
I t e r a t e t h e p r e c e d i n g argument: we see t h a t
c e r t a i n i n i t i a l i n t e r v a l and o f t h e f i n a l i n t e r v a l (.d phic with
A
, which
contradicts the hypothesis t h a t
- .
, so
B'+l
Hence
that
i s t h e sum o f a A+1
i s isomor-
.0
U = 1
4.3. Every i n f i n i t e c h a i n admits a t l e a s t two immediately g r e a t e r c h a i n s ( w i t h r e s p e c t t o e m b e d d a b i l i t y ) , which a r e m u t u a l l y incomparable (HAGENDORF 1972). 0
Let
A
be an i n f i n i t e c h a i n and
embeddable i n
A ; similarly l e t
i s n o t embeddable i n than
A
,
.
A
U U'
t h e l e a s t o r d i n a l such t h a t
A+U
By t h e p r e c e d i n g 4.2,
It remains t o prove t h a t t h e y a r e m u t u a l l y incomparable w i t h r e s p e c t t o
Take an isomorphism f r o m interval
U'+A
t h e s e sums a r e immediately g r e a t e r
e m b e d d a b i l i t y . We s h a l l argue ad absurdum: t o f i x t h e ideas, suppose n o t embed
i s not
be t h e l e a s t r e t r o - o r d i n a l such t h a t
U'+A of
morphic w i t h
A
in
A
U'+A
, nor
in
onto a r e s t r i c t i o n o f A+V
f o r any
which i s c o f i n a l l y embeddable
U : o t h e r w i s e t h e r e would e x i s t
which c o n t r a d i c t s t h e d e f i n i t i o n o f
U'
.
A+U
< U . Thus i n U . This
.
V
V< U
A+U
.
there exists a f i n a l final interval i s iso-
such t h a t
F i n a l l y by 4.1,
U'+A,(
T h i s isomorphism can-
U
U'+A$
A+V$
A
i s indecomposable.
,
144
THEORY OF RELATIONS
If U is an infinite indecomposable ordinal, then 1 < U so A+1 I A . The preceding argument, where A is replaced by A+l , proves the existence of a final interval of A+1 which is isomorphic with U : contradiction. Hence U = 1 and U'+A d A + 1 . Thus every isomorphism of U'+A into A+l has in its range the maximum element of A+1 ; for otherwise U'+A would be embeddable in A . Thus A has a maximum. By iteration, we see that A has a final interval which is isomorphic with the retro-ordinal w - . Thus A+1 is isomorphic with A , contradicting the inequality A+l = A+U > A . 0 An infinite chain can have 3, 4, ... immediately greater chains. has the 3 immediately greater chains For example the product A = b -. B = A+l , C = W + A , D = ( w 2 ) - + A , where ( W 2 ) - is the converse of the
4.4.
ordinal
W
.
These are the only immediate extensions of A
.
The reader easily see that there does not exist any chain strictly intermediate between A and B , A and C , A and 0 . Let us see that every chain strictly greater than A admits an embedding of B or C or 0 . Indeed, every chain which is an extension of A can be obtained as follows. Either by adding new elements at the end, so admitting an embedding of B . Or by adding a chain at the beginning, in which w is embeddable (embedding of C ) , or in which the converse of an ordinal is embeddable (from the ordinal w 2 we get an embedding of D ) . Or by partitioning the new elements into a finite number of components W - of A , which is the same as adding them at the beginning. Or by modifying infinitely many components, by adding an ordinal CC, (embedding of C ) , or by replacing these components by a retro-ordinal ( s o getting an embedding of D from the converse of u 2) . 0
0
Another example: the chain W1 + Q , where Q is the chain of the rationals, has the 5 following immediate extensions: the addition of &J1 or its converse at the end or between k, and Q , the addition of W1- at the beginning. For each integer p 3 2 , the chain sions (among chains).
G,
.(p-1)
has exactly p immediate exten-
There exist chains having infinitely many immediately greater chains. For example, every chain A which has no strict restriction isomorphic with A : see the method of SIERPINSKI 1950, used in 5.3 below. FAITHFUL EXTENSION BETWEEN CHAINS Let A be an infinite chain, B a chain in which A is not embeddable. lhen there exists a chain, immediate extension of B , in which A is not embeddable (uses axiom of choice; HAGENDORF 1972; the case for scattered chains .e. chains
4.5.
145
Chapter 5
without any embedding of 0 If
, was
Q
a l r e a d y proved by JULLIEN 1969).
B i s a f i n i t e chain, t h e n i t s u f f i c e s t o t a k e
i n f i n i t e . We d i s t i n g u i s h two cases:
B
B IA
and
.
B+l
Suppose t h a t
B
is
( w i t h r e s p e c t t o embeddabi-
lity).
I n the case
A ,let
B
diately greater than o r A,(
.
B"
B'
and
be two m u t u a l l y incomparable chains, each imme-
B"
(see 4 . 3 ) . Suppose t h e p r o p o s i t i o n i s f a l s e . Then
B
since B ' l B" . Hence A < B ' f o r i n s t a n c e , and so between B and B ' : c o n t r a d i c t i o n .
It remains t o examine t h e case where
A
dum by supposing t h a t
B
B
into
t h e complementary f i n a l i n t e r v a l . L e t B ; similarly
.
UA
6U
,
V
and t h e converse o f
A
i n t o three intervals
.
B
If
UA<
U
D
then A A
morphic t o t h e r e t r o - o r d i n a l
V
i s included i n A
, then
E
\< B , c o n t r a d i c t i n g
U i n E'
, then
U+D',<
,
.
D'
with
Thus
UA = U
>
U
B
.
By 4.1,
\< E gC
Hence
U = 1
D = E
and
, and
0' =
E'4
C'
so
,
V so
preceding, we have
,
and
. Similar-
and an i n t e r v a l i s o E,<
C
,
E'Q
C'
.
A . I f the interC+C' = B : i n o t h e r
intervals o f
D+U+D',(
A = D+U+D'&
B , contradicting the , isomorphic t o an indecom-
C+C' =
U
V , isomorphic t o a r e t r o - o r d i n a l .
V = 1 : the intervals
U = V = 1 a r e t h e same, t h u s
.
be an a r b i t r a r y element o f t h e base
of those elements
0 4 C
the hypothesis. I f there e x i s t s a f i n a l i n t e r v a l o f
similarly
E'
and
A = D+U+D'
E , E'
A = E+V+E' and
A = D+UA+D'
and t h e n
h y p o t h e s i s . Thus we must suppose t h a t t h e i n t e r v a l
x
C+V+C'
B , c o n t r a d i c t i n g t h e hypothe-
into intervals
posable o r d i n a l , i s i n c l u d e d i n t h e i n t e r v a l
Let
.
B
i s an i n i t i a l i n t e r v a l
C
be t h e l e a s t o r d i n a l such t h a t
C+UA+C',<
such t h a t D+U
,
UA
A<
Envisage a l l t h e r e l a t i v e p o s i t i o n s o f U
We argue ad absur-
a r e indecomposable o r d i n a l s . By h y p o t h e s i s Consider an isomorphism o f k o n t o a r e s t r i c t i o n o f t h i s sum: t h i s
i s incomparable w i t h
val
U
B'
,
V
l y we d e f i n e t h e decomposition o f
words
IA .
t h e l e a s t r e t r o - o r d i n a l such t h a t
U
0 ' 4 C'
sis that
B
, where
C+C'
C+U+C'>
<
A,<
A z B"
i s a s t r i c t l y intermediate
i s i n f i n i t e and
and C '
A C+U+C' decomposes
A
and
i s embeddable i n e v e r y c h a i n s t r i c t l y g r e a t e r t h a n
I n an a r b i t r a r y manner decompose
the o r d i n a l
A E B'
I t i s i m p o s s i b l e t o have b o t h equimorphisms
6X
(mod 6 )
Bx+l+B;
>B
and
IS1
.
Let
Bx
be t h e i n i t i a l i n t e r v a l
>
the final interval
B; hence > / A
. Moreover,
a s s o c i a t e t o each x an element f x o f t h e base d e s i g n a t e t h e i n i t i a l i n t e r v a l of t h o s e elements
x (mod B)
. By
the
by t h e axiom of choice,
I A I , such t h a t i f we l e t Ax (mod A ) , t h e n we have
< fx
A = Ax+l+A;( w i t h A x < Bx and A<,; B; . We s h a l l prove t h a t t h e f u n c t i o n f which t o each e:ement x o f t h e base I B I a s s o c i a t e s f x , an element o f A i s s t r i c t l y i n c r e a s i n g . From t h i s we w i l l deduce t h a t t h e h y p o t h e s i s o f i n c o m p a r a b i l i t y o f A and B . TO do t h i s , we argue ad absurdum by supposing t h a t t h u s t h e r e e x i s t two elements x, y o f I S ( w i t h
B,(
A
, which
,
contradicts
f i s not s t r i c t l y increasing: x < y (mod B) , and so w i t h
THEORY OF RELATIONS
146
<
. Yet
B
Bxtl&
Y
Bx t 1
\
4.6.
+
fy
4 BY
B' Y
A
and
,
B
A
and
Obvious f o r
A 4B
.
than
B
A i d A ' 6 B;I
and so
,
B' = B Y
A
thus
Y
+
A = Ax
1 + A;
c o n t r a d i c t i n g the hypothesis. 0
have t h e same s t r i c t l y g r e a t e r c h a i n s ( w i t h r e s p e c t t o B a r e equimorphic.
a r e i n f i n i t e , t h e n i t even s u f f i c e s t h a t e v e r y c h a i n s t r i c t l y qrea-
be
B
t e r than 0
B
+
(mod A)
then
I f two c h a i n s
embeddability), If
(only) greater than
A
finite or
B
A
, and
conversely.
f i n i t e . Suppose t h a t
,
A
B a r e i n f i n i t e and t h a t
Then by t h e p r e c e d i n g p r o p o s i t i o n , t h e r e e x i s t s a c h a i n s t r i c t l y g r e a t e r and n o t g r e a t e r t h a n
A
.0
4.7. N o t i c e (by JULLIEN 1969) t h a t t h e f a i t h f u l e x t e n s i o n no l o n g e r h o l d s i f we
A
replace 0
by two c h a i n s
For example n e i t h e r
A1
w
A1 =
and
A2
+1 n o r
. A2
=
However, e v e r y c h a i n which i s s t r i c t l y g r e a t e r t h a n
Al
or of
A2
.
-+ W i s embeddable i n W w admits an embedding o f
Z = W
.0
The e x i s t e n c e o f a f a i t h f u l e x t e n s i o n between chains, no l o n g e r h o l d s i f we r e p l a c e B 0
by two c h a i n s
B1
and
For example A = W + B2 = w . W -
converse and
B2
c3
.
B2
.
-
i s n e i t h e r embeddable i n
Yet
A
B1 = U - .
nor i n i t s
i s embeddable i n e v e r y c h a i n i n which b o t h
a r e embeddable: t h i s w i l l be proved i n ch.6
B1
3.7. 0
We w i l l have an o p p o s i t e r e s u l t i f we a l l o w e x t e n s i o n s by a r b i t r a r y b i n a r y r e l a t i o n s , i n s t e a d o f r e s t r i c t i n g o u r s e l v e s t o c h a i n s : see e x e r c . 2 below. Problem posed by SABBAGH i n 1975. E x i s t e n c e o f two chains
A, B which a r e incom-
p a r a b l e w i t h r e s p e c t t o e m b e d d a b i l i t y , and have a supremum c h a i n
C
.
More p r e c i s e -
X > / A and X B, . I t has been proved by HAGENDORF 1977 t h a t t h i s supremum c h a i n does n o t e x i s t f o r
ly, f o r every chain
A
or
B
X
we would have
X >/C
iff
s c a t t e r e d ; t h e general case remains unsolved.
For t h e dual o f t h e above statement, i . e . t h e e x i s t e n c e o f t h e infimum, we have t h e easy example
A = w + 1 and
5 5 - DECREASING SEQUENCES DUSHN I K, MILLER, S IERP INSKI 5.1.
Let
A, B
B = Z
w i t h t h e infimum
C =
AND SETS OF INCOMPARABLE
w . CHAINS
OF REALS:
be two chains, each o f which i s embeddable i n t h e r e a l s . Then
t h e r e a r e a t most continuum many r e s t r i c t i o n s o f
B
isomorphic w i t h
0 Consider t h e base
E
o f r e a l s , and l e t
lBl
as a subset o f t h e s e t
A
. F be
Chapter 5
IBI
a subset o f
B/F
such t h a t
For e v e r y subset
X
i s isomorphic w i t h
1 B l such t h a t
of
147
A
( i f t h e r e e x i s t s such).
i s isomorphic w i t h
B/X
f X f r o m F o n t o X , hence f r o m F t h e r e a r e continuum many s t r i c t l y i n c r e a s i n g maps f r o m F i n t o
5
8.4;
and f i n a l l y
5.2. L e t
A
X = fXo(F)
i s determined by
f X( n o t a t i o n
A
, there
into
a s t r i c t l y i n c r e a s i n g map
E
exists
. Now
E : see c h . 2 O
i n ch.1
5
1.2). U
embeddable i n t h e
be a c h a i n o f continuum c a r d i n a l i t y , which i s
chain o f the reals. (1) There e x i s t s a s t r i c t l y s m a l l e r ( w i t h r e s p e c t t o e m b e d d a b i l i t y ) r e s t r i c t i o n A which has continuum c a r d i n a l i t y (DUSHNIK, MILLER 1940; uses t h e axiom o f
of
choice). ( 2 ) Even s t r o n g e r , t h e r e e x i s t s a r e s t r i c t i o n l i t y , such t h a t no i n t e r v a l o f
,of
A
B
of
A
w i t h continuum c a r d i n a -
continuum c a r d i n a l i t y , i s ernbeddable i n
B
(HAGENDORF 1977, u n p u b l i s h e d ) . ( 3 ) F o r e v e r y denumerable c h a i n preceding ( 2 ) ) . 0 (2) Let
X
U
we have
be any subset o f t h e base
an i n t e r v a l o f
A
A $ B.U
(where
I A 1 such t h a t A/X
B
satisfies the
i s isomorphic w i t h
o f continuum c a r d i n a l i t y . For each such i n t e r v a l , by t h e p r e -
c e d i n g p r o p o s i t i o n , t h e r e a r e a t most continuum many corresponding s e t s
X
. More-
R o f r e a l s , each i n t e r v a l o f A i s t h e r e s t r i c t i o n t o I A l o f an i n t e r v a l o f R , which i s i t s e l f
over, i f we c o n s i d e r
A
as a r e s t r i c t i o n o f t h e c h a i n
d e f i n e d by i t s e n d p o i n t s . Consequently t h e r e a r e continuum many such i n t e r v a l s . X
Finally, the set o f a l l the
5
Apply ch.2 where
C
has a t most continuum c a r d i n a l i t y .
8.1 (axiom o f c h o i c e ) . There e x i s t s a s e t
and
D
=
IAl
a l l the intersections thus the r e s t r i c t i o n
-C
C
included i n
IAI ,
b o t h a r e e q u i p o t e n t w i t h t h e continuum, as w e l l as
.
C n X and D n X Consequently no X i s i n c l u d e d i n C B = A/C a d m i t s no embedding o f any i n t e r v a l o f A which
has continuum c a r d i n a l i t y . 0 0
( 3 ) Suppose
A 6B.U
.
Then t h e base
I A I i s p a r t i t i o n e d i n t o c o u n t a b l y many
i n t e r v a l s , each c o r r e s p o n d i n g w i t h an element o f
U
. At
t e r v a l s i s e q u i p o t e n t w i t h t h e continuum: see c h . 1 § 4.3, o v e r i t i s embeddable i n
B
, contradicting
l e a s t one o f t h e s e i n axiom o f c h o i c e . More-
the preceding (2). 0
N o t i c e t h a t (1) i m p l i e s t h e e x i s t e n c e , s t a r t i n g f r o m t h e c h a i n o f t h e r e a l s , o f a s t r i c t l y decreasing w -sequence o f c h a i n s . We s h a l l see t h a t such a sequence does n o t e x i s t f o r s c a t t e r e d c h a i n s , i . e . those i n which t h e c h a i n
Q
o f the
r a t i o n a l s i s n o t embeddable: see ch.8
5
5.3.
R t h e c h a i n o f t h e r e a l s . There e x i s t two
Let
subsets
E C,
be t h e s e t o f r e a l s and
D
of
E
4.4.
which a r e d i s j o i n t , e q u i p o t e n t w i t h t h e continuum, dense
,
148
THEORY OF RELATIONS
in
R and such t h a t , f o r each subset X f D , every r e s t r i c t i o n of R isomorphic w i t h R / ( C u X ) , e i t h e r has base C u X , o r i t s base contains a t l e a s t one
element o f E
-
(C u D)
.
Consequently f o r Y C X cD , we have the s t r i c t embeddability R / ( C uY)< R/(CuX). For X and Y s u b s e t s of D which a r e incomparable with r e s p e c t t o i n c l u s i o n ,
the preceding two restrictions are incomparable with respect to embeddability (SIERPINSKI 1950; see a l s o ROSENSTEIN 1982; uses axiom of c h o i c e ) . Take the s e t s C , D in ch.2 5 8.2.(2) (axiom of c h o i c e ) , where t h e f i designate a l l isomorphisms from R i n t o R , d i s t i n c t from t h e i d e n t i t y . For such an isomorphism f , i f a real x i s mapped t o f x # x , f o r example i f f x > x (mod R ) , then every real i n t h e i n t e r v a l ( x , f x ) i s mapped t o a s t r i c t l y g r e a t e r r e a l ,
0
hence f y # y f o r continuum many r e a l s y . Notice t h a t C and 0 a r e d i s j o i n t and each equipotent with t h e continuum. Moreover by t h e same proposition we have f " ( C ) # C f o r each considered isomorphism f ; s i m i l a r l y with D T h u s by ch.2 5 8 . 5 , t h e s e t s C and D a r e both dense (mod R ) . Take an a r b i t r a r y subset X of D and an isomorphism g of R / ( C u X ) i n t o R , which i s d i s t i n c t from t h e i d e n t i t y . Since C , hence C u X , i s dense, t h e r e
.
e x i s t s an isomorphism gt of R i n t o R , which extends g t o t h e domain E of a l l r e a l s : see ch.2 5 8.3. Hence g+ i s one of t h e previously considered i s o morphisms f . By ch.2 5 8.2, t h e s e t of images g"(C) = ( g + ) " ( C ) i s not i n c l u ded in C u D . 0 This immediately implies t h e e x i s t e n c e of a s t r i c t l y decreasing (with r e s p e c t t o embeddability) sequence, indexed by t h e continuum, of chains of r e a l s . Also t h e e x i s t e n c e of a s e t , equipotent w i t h t h e continuum, of mutually incompar a b l e chains of r e a l s (SIERPINSKI 1950).
§
6 - SUSLINC H A I N
AND
SUSLINTREE
in A : Given a chain A , t h e reader i s acquainted w i t h t h e notion of a s e t a subset D of t h e base f o r which, given any two elements x < y (mod A ) , t h e r e e x i s t s an element t of D w i t h x st d y (mod A ) The chain of r e a l s , and more g e n e r a l l y any chain A which i s embeddable in the chain of r e a l s , s a t i s f i e s t h e two following conditions: (1) t h e r e e x i s t s a countable s e t which i s dense in A ; (2) every s e t of mutually d i s j o i n t i n t e r v a l s of A , none o f which a r e s i n g l e t o n s , i s countable.
The condition (2) follows from ( 1 ) . I f D i s countable and dense i n A , then every non-singleton i n t e r v a l contains a t l e a s t one element of D . Two d i s j o i n t i n t e r v a l s cannot contain a same element
0
Chapter 5
of
D , so there are countably many i n t e r v a l s .
149 0
SUSLIN'S HYPOTHESIS (see SUSLIN 1920) The axiom called S u s l i n ' s hypothesis, a s s e r t s that the preceding condition (2) implies ( l ) , hence t h a t ( 1 ) & (2) are equivalent. This axiom i s neither provable nor refutable in ZF, even with the axiom of choice and even with the generalized continuum hypothesis. More precisely JECH and TENNENBAUM have proved the consistency of the existence o f a Suslin t r e e ( i . e . the negation of the axiom) with ZF (modulo the consistency of Z F ) . Whereas SOLOVAY and TENNENBAUM have proved the relative consistency of the axiom: see JECH 1978. For a detailed discussion of Suslin chains and Suslin t r e e s , as well as f o r the advanced r e s u l t s o f JENSEN, see for example DEVLIN, JOHNSBRiTEN 1974.
6.1. SUSLIN CHAIN I t i s more convenient t o work with the negation of Suslin's hypothesis, rather than the hypothesis i t s e l f . We say t h a t a chain i s a Suslin chain i f i t s a t i s f i e s ( 2 ) and n o t ( l ) , i . e . i f every s e t of non-singleton mutually d i s j o i n t intervals i s countable, y e t there e x i s t s no countable s e t which i s dense in the chain. A Suslin chain i s uncountable; moreover i t admits an embedding of t h e c h a i n Q rationals (uses axiom of choice). 0 The inexistence of any countable dense s e t implies t h a t the chain i t s e l f be
of
uncountable: i t s cardinality i s a t l e a s t W1 (axiom of choice). If Q i s n o t embeddable in i t , then e i t h e r the ordinal W 1 or i t s converse i s embeddable in i t : see 3.5. Hence there e x i s t uncountably many non-singleton mutually d i s j o i n t intervals. 0
6 . 2 . Every Suslin chain has cardinality exactly w 1 (uses axiom of choice and the continuum hypothesis). 0 Let A be a Suslin chain; we already know t h a t A i s uncountably i n f i n i t e , so has cardinality a t l e a s t w 1 (axiom of choice). Suppose t h a t A has cardinality a t least w . Replace A by a r e s t r i c t i o n o f cardinality O 2 and l e t B be a well-ordering of type o 2 on the same base. Partition the pairs of elements x, y of the base into two colors: l e t t i n g x < y (mod A ) , we say t h a t the pair has color (+) i f x < y (mod B ) , and color (-) i f x ) y (mod B ) By the ERDOS partition lemma (ch.3 5 3.4) f o r d = 0 (hence using only the continuum hypothesis), there e x i s t s a subset of the base of cardinality G, , a l l of whose pairs have a same color. Hence there e x i s t s a s t r i c t l y increasing or a s t r i c t l y decreasing 0 l-sequence, and hence wl-many non-singleton mutually d i s j o i n t intervals: contradiction. 0
.
THEORY OF RELATIONS
150
6.3. SUSLIN TREE We say t h a t a t r e e i s a S u s l i n t r e e i f i t has c a r d i n a l i t y
w1
and i f every chain
( o r t o t a l l y ordered r e s t r i c t i o n ) and every a n t i c h a i n i s countable. The existence o f a S u s l i n chain o f c a r d i n a l i t y
W1 S u s l i n t r e e ( t h e a d d i t i o n a l assumption o f c a r d i n a l i t y
i m p l i e s t h e existence o f a
w 1 allows us t o avoid
using the axiom o f choice: ZF s u f f i c e s ) . 0
Let
be a
A
S u s l i n chain o f c a r d i n a l i t y
w1 .
i ,
To each countable o r d i n a l
defined by i t s two endpoints ui < vi (mod A) , where Ai an a r b i t r a r y are d i s t i n c t . To do t h i s , begin w i t h A. = (uo,vo)
associate an i n t e r v a l all
ui
and
vi
interval. Let
i be a non-zero countable o r d i n a l , and suppose t h a t t h e
A
for
j j < i have already been defined so t h a t they are m u t u a l l y e i t h e r d i s j o i n t o r one
contained i n the other. The s e t o f endpoints uj, v j ( j < i ) i s countable: by hypothesis i t i s n o t dense i n A , hence there e x i s t two elements u, v i n t h e base o f
, between
A
.
v. ( j < i ) L e t Ai = J vi = v : t h i s i n t e r v a l must be e i t h e r d i s j o i n t o r included
which there i s no endpoint
uj
or
ui = u and A. ( j < i ) J thus obtained has c a r d i n a l i t y w 1 Reverse i n c l u s i o n The s e t o f i n t e r v a l s Ai defines a t r e e on the s e t o f these Ai . Every antichain, i . e . every s e t o f i n t e r which are m u t u a l l y d i s j o i n t , i s countable. F i n a l l y , a chain, o r s e t o f v a l s Ai (u,v)
so
.
i n each
Ai
intervals
.
which are mutually comparable w i t h respect t o i n c l u s i o n , i s w e l l f o r every p a i r o f countable o r d i n a l s A . c Ai J Such a chain i s countable; f o r i f i t had c a r d i n a l i t y w 1 , then
ordered by the o r d i n a l i n d i c e s w i t h
i, j ( i
using the endpoints o f preceding i n t e r v a l s , we could o b t a i n W l-many
mutually
d i s j o i n t intervals. 0 6.4. The existence o f a S u s l i n t r e e i m p l i e s , and hence i s e q u i v a l e n t w i t h t h e existence o f a S u s l i n chain (here we use the r e g u l a r i t y o f
W1
, thus
f o r example
the countable axiom o f choice). 0 Let
be a S u s l i n t r e e ; we can assume t h a t
A
A
i s a well-founded p a r t i a l orde-
A by a c o f i n a l well-founded p a r t i a l ordering. To 5.1 w h i l e n o t i n g t h a t , by hypothesis, t h e base o f A i s
r i n g , i f necessary by r e p l a c i n g see t h i s , apply ch.2
5
.
well-orderable w i t h order type w 1 More p r e c i s e l y l e t ui ( i < td 1) be an indexation o f the base; then remove each u f o r which t h e r e e x i s t s an i
>
.
i , the removed elements u are a l l < ui (mod A) j there are countably many such; thus t h e r e remain w l-many elements.
f o r each index
We can f u r t h e r assume t h a t every non-empty f i n a l i n t e r v a l o f W
. To
see t h i s , l e t
many successors (mod A )
x
A
, and
hence
has c a r d i n a l i t y
be an element o f the base which has o n l y countably
. Those
x
o f minimal h e i g h t are m u t u a l l y incomparable,
Chapter 5
151
x and the ir successors. Neither A nor any f in al interval of A i s f i n i t e l y fre e . For otherwise, by ch.4 5 3.1 (using the regularity of LJ there would e x i s t a t o t a l l y ordered re stric contradicting our hypotheses. tion of A with cardinality w W e can assume t h a t A has a minimum, whose singleton will be denoted by Eo For each countable non-zero ordinal i , l e t Ei be the denumerable s e t of elements of height i : we require t h at t h i s s e t be i n f i n i t e . For each element x of Ei we require that there e x i s t denumerably many elements which are immediate successops of x (mod A ) . This s e t i s denoted Eitl ,X and the union of these se ts must be E i t l . For each countable l i mi t ordinal i and each x of height < i , we require t h a t there e x i s t i n f i n i t e l y many elements in Ei which are 7 x (mod A ) . Finally, for each countable l i mi t ordinal i and each t o t a l l y ordered re stric tion X of A containing elements of a l l heights < i , we require tha t there e xist either a unique element of Ei which i s a successor of a l l elements of X , or none such. These requirements are easy t o s a t i s f y . For example f o r the minimum, take an arbitrary element having W l-many successors. Then having obtained Ei , the s e t of elements x with height i , note t h at f o r each x in Ei , the final interval > x has cardinality w 1 and i s not f i n i t e l y f r e e . Hence take a denumerable free subset as E i + l , x and among the successors of x , retain only those which Finally, f o r each counare identical t o or successors of an element of Ei+l ,X table limit ordinal i and each chain X containing elements of a l l heights 4 i , i f there e x i s t elements above X , then decide t o retain one such plus the wl-many successors of t h i s element. For each element x of the base IAl with height i , t o t a l l y order the denumeNow consider the with the order type of a dense chain C i + l , x rable s e t Ei+l ,X set of a l l maximal t o t a l l y ordered r e s t r i c t i o n s , or maximal chains of A . This set i s t o t a l l y ordered by the preceding dense chains. Indeed, given two distinc t maximal chains U and V : none of the two bases i s included in the other. Moreover there e x i st s a l e a s t element u among those elements of I U 1 which do not belong t o I V I , and a l e a s t element v among those elements of I V I which do not belong t o I U I By the preceding, there ex i s t s a l a s t element x whose height will be denoted i , common t o b o t h bases of U and V , and having u and v as immediate successors. Let U < V i f u < v (mod C i + l , x ) : t h i s tota lly orders the s e t of maximal chains. Let H be the chain thus obtained. We shall prove tha t H i s a Suslin chain. =i-r+ -11 a rnt n irrhirh i c h n c o i n H rannot be countable. For i f i t were, and so there are countably many such: i t suffices t o remove these
.
.
.
.
THEORY OF RELATIONS
152
the i n t e r v a l o f maximal chains passing through of D
z
does n o t c o n t a i n any element
.
Now suppose t h a t t h e r e e x i s t o l-many non-singleton m u t u a l l y d i s j o i n t i n t e r v a l s I n each i n t e r v a l take two elements, o r maximal chains U and V . As of H before, take an element x whose h e i g h t i s denoted i and t h e elements u, v
.
immediate successors o f x (mod A) ; and take w between u and v modulo the chain Ci+,lx Then these w thus associated w i t h our d i s j o i n t i n t e r v a l s o f H ,
.
are m u t u a l l y incomparable (mod A): they must be countably many; c o n t r a d i c t i o n . 0
§
7 - ARONSZAJN
TREE,
SPECKER CHAIN
7.1. ARONSZAJN TREE
This i s a well-founded t r e e o f c a r d i n a l i t y ~3~ whose chains and h e i g h t l e v e l s are countable. I t i s n o t r e q u i r e d t h a t every a n t i c h a i n be countable. Hence every well-founded S u s l i n t r e e i s an Aronszajn tree; b u t the converse p o s s i b l y depends on s e t - t h e o r e t i c axioms: see the problem a t the end o f 7.4. The f o l l o w i n g c o n s t r u c t i o n o f an Aronszajn t r e e , using ZF p l u s choice, goes back t o KUREPA 1935 p . 96, c i t i n g a l e t t e r from ARONSZAJN i n 1934. The elements o f the t r e e w i l l be o r d i n a l sequences o f i n t e g e r s ai ( i < ) y w i t h o u t r e p e t i t i o n , where o( v a r i e s over a l l countable o r d i n a l s . We say t h a t such a sequence u precedes v o r t h a t v f o l l o w s u , i f u i s an i n i t i a l i n t e r v a l o f v . Moreover we r e q u i r e t h e f o l l o w i n g c o n d i t i o n s o f convergence and denumerability. Convergence. For each sequence ai (i< o ( ) , the sum o f the inverses l/ai is f i n i t e . Furthermore for each sequence u w i t h l e n g t h o( , each countable o r d i n a l and each p o s i t i v e r e a l number r , t h e r e must e x i s t , i n our set, a sequence w i t h l e n g t h o i + /s , f o l l o w i n g u , and such t h a t the sum of the inverses l/ai f o r o( i
.
denumerably many o( -sequences (uses axiom o f choice). Using t h e preceding convergence c o n d i t i o n , we see t h a t , given a l i m i t countable o r d i n a l 4
we can r e t a i n
only o( -sequences u such t h a t , f o r each i ( 4 , t h e i n i t i a l i n t e r v a l o f u w i t h l e n g t h i a l r e a d y belongs t o our set; more e x a c t l y , we r e t a i n denumerably many such o( -sequences. F i n a l l y f o r each sequence u i n our set, every i n i t i a l i n t e r v a l o f u w i l l belong t o our set. These c o n d i t i o n s a l l o w us, f o r each o( , t o c o n s t r u c t a " d e f i n i t i v e " denumerable s e t o f o( -sequences, which w i l l n o t be unduly increased by the u l t e r i o r construct i o n o f longer sequences. Because t h e n o n - r e p e t i t i o n of values
ai
, every
t o t a l l y ordered r e s t r i c t i o n , o r
Chapter 5
153
chain, of the preceding Aronszajn t r e e , i s countable. In the preceding t r e e , f o r each countable ordinal o( and each o( -sequence u , there e x i s t u1-many sequences which are successors of u However t h i s i s not true for a l l Aronszajn t r e e s ; f o r instance we can add t o the preceding tree new ordinal sequences without successors, or with countably many successors.
.
7.2. A chain A i s embeddable in the chain of r e a l s , i f f there e x i s t s a countable subset which i s dense in A (uses countable axiom of chsice). 0
If
A
i s a r e s t r i c t i o n o f the chain of r e a l s , then f o r each integer
n
take a
partition o f the reals into intervals of length l / n , and choose an element of I A l in each interval (countable axiom o f choice). Conversely, i f there i s a r e s t r i c t i o n I o f A , which i s denumerable and dense in A , then i f necessary complete I i n t o J , the l a t t e r which i s isomorphic with the chain of rationals. Then i n each cut o f J , we take the unique element o f I A I ( i f i t exists) defined by i t s position with respect t o the elements of I I I . Thus we embed A into the reals. 0
7.3. SPECKER CHAIN This i s an uncountable chain A such t h a t neither w nor i t s converse are embeddable in A ; moreover any chain which i s embeddable both in A and in the chain of reals, i s countable. A Specker chain has no countable dense subset. For i f s o , then by the preceding, i t would be embeddable in the chain of r e a l s as well as in i t s e l f , and consequently i t would be countable. Every Specker chain has cardinality W (uses choice plus continuum hypothesis). Suppose on the contrary t h a t there e x i s t s a Specker chain of cardinality 3 w 2 By 3.5, t h i s chain admits an embedding e i t h e r of w 2 , or i t s converse: contradiction with the definition o f Specker chains. Or our chain admits an embedding of an W *-dense chain. By 3.7 ( 2 ) (choice and continuum hypothesis), the l a t t e r admits an embedding o f w 1 : contradiction. 0
0
.
7.4. Construct as follows a Specker chain, s t a r t i n g with the Aronszajn t r e e of 7.1, hence on a s e t of ordinal sequences of integers, without repetition. Apply ch.4 5 6.2 (using u l t r a f i l t e r axiom) which associates t o the given t r e e A , a chain C with the same base. For each element u (ordinal sequence of integers without repet i t i o n ) , the final interval 3 u (mod A ) becomes an interval (mod C ) with minimum u .
THEORY OF RELATIONS
154
We s h a l l show f i r s t o f a l l t h a t t h e c h a i n C embedding o f
t h u s defined does n o t admit an
u1, n o r o f i t s converse,
C can be decomposed i n a denumerable sum o f d i s j o i n t i n t e r v a l s , By ch.4 5 6.2, each corresponding t o an i n t e g e r a , and formed from some sequences which begin
with
a
.
Consequently, given a s t r i c t l y i n c r e a s i n g (mod C)
W1-sequence whose
terms are denoted ui ( i countable o r d i n a l ) , t h e r e e x i s t s an i n t e g e r al such t h a t , from a c e r t a i n o r d i n a l on, t h e sequence ui begins w i t h al By i t e r a t i o n ,
.
f o r every countable o r d i n a l k , t h e r e e x i s t s a k-sequence o f i n t e g e r s such t h a t , from a c e r t a i n o r d i n a l index on, t h e sequence a j' * * ( j < k ) begins w i t h the above k-sequence. F i n a l l y , these al, a2, ... , aj, a s t r i c t l y i n c r e a s i n g wl-sequence o f elements o f the t r e e A , thus an
..
al,
a*,
...
ui form o l-se-
quence o f i n t e g e r s w i t h o u t r e p e t i t i o n : c o n t r a d i c t i o n . 0 L e t us now show t h a t if t r i c t i o n D dense i n H
H jA_an uncountable chai_n-_which has a denumerable res,then H i s n o t embeddable i n C .
D i s denumerable, i t i s formed o f ord'nal sequences o f i n t e g e r s w i t h o u t r e p e t i t i o n , a l l o f l e n g t h l e s s than a c e r t a i n countable o r d i n a l . L e t o( be a countable o r d i n a l s t r i c t i y g r e a t e r than a l l these lengths. Since H i s uncounta-
0 Since
ble, t h e r e e x i s t s a t l e a s t one sequence u w i t h l e n g t h o( having ul-many sequences i n I H I which extend u . By ch.4 4 6.2, t h e r e e x i s t s an i n t e r v a l (mod C) formed o f a l l those elements, o r sequences extending u Thus any two o f them are n o t separated (mod C) by any sequence belonging t o I D l : t h i s contra-
.
d i c t s the density o f
D
in H
.0
Problem. I n 7.1 we constructed a well-founded Aronszajn t r e e , based on c e r t a i n o r d i n a l sequences o f i n t e g e r s ( w i t h o u t r e p e t i t i o n ) , u s i n g ZF p l u s choice. Whereas a S u s l i n t r e e can o n l y be constructed by u s i n g t h e negation o f S u s l i n ' s hypothes i s . Consequently, w i t h ZF p l u s choice p l u s t h e axiom c a l l e d S u s l i n ' s hypothesis, we are insured t h a t t h e r e e x i s t s an Aronszajn t r e e which i s n o t a S u s l i n t r e e . However i f we admit ZF p l u s choice p l u s t h e negation o f S u s l i n ' s hypothesis, then we do n o t know whether the Aronszajn t r e e i n 7.1 i s o r n o t a S u s l i n t r e e . And more g e n e r a l l y w i t h these s e t - t h e o r e t i c axioms, we do n o t know whether t h e r e e x i s t Aronszajn t r e e s which are n o t S u s l i n t r e e s .
Chapter 5
5 8 - FAITHFUL
INFINITE
155
MALITZ'
EXTENSION:
AND
LOPEZ' COUNTEREXAMPLES
8.1. FAITHFUL INFINITE EXTENSION Let A1,
...,Ah
be a f i n i t e s e t o f f i n i t e r e l a t i o n s w i t h common a r i t y , and
be a f i n i t e r e l a t i o n w i t h
R
I f there e x i s t e x t e n s i o n s o f
2A1
and ... and 3 Ah - 3 A1 are
and
...
R
.
and $ A h
w i t h a r b i t r a r y l a r g e f i n i t e c a r d i n a l i t i e s , which
R
, t h e n t h e r e e x i s t s a denumerable e x t e n s i o n o f
R
which respects t h e same c o n d i t i o n s .
0 We can assume t h a t
1, ... ,p
i s d e f i n e d on t h e i n t e g e r s
R
and t h a t , f o r each
i , t h e r e e x i s t s an e x t e n s i o n Ri o f R based on t h e i n t e g e r s 1 t o p+i and r e s p e c t i n g t h e c o n d i t i o n s . F o r i n f i n i t e l y many i n t e g e r s i , t h e Ri have a same r e s t r i c t i o n S1 t o 1,. . . ,p+1 . Among these, t h e r e a r e i n f i n i t e l y many i n t e g e r s i f o r which t h e Ri have a same r e s t r i c t i o n S2 t o 1, ...,p+ 2 integer
I t e r a t i n g t h i s , we t h u s d e f i n e
S
j
t a k e t h e common e x t e n s i o n o f t h e
...,Ah
A1,
8.2. L e t
and
common a r i t y , and l e t and
...
and
>/
Bk
R
B1,
u
...,Bk
satisfy
f o r each i n t e g e r j . I t now s u f f i c e s t o , based on t h e s e t o f a l l i n t e g e r s . 0 j be two f i n i t e s e t s o f f i n i t e r e l a t i o n s o f
R3AA1
and
... and
#Ah
R >/El
as w e l l as
.
Then t h e r e e x i s t s an i n t e g e r equal t o
S
u
such t h a t
every
R
with cardinality _ a t_ l e_ a s-t .
s a t i s f y i n g t h e p r e c e d i n g c o n d i t i o n s has a r e s t r i c t i o n
t i n g t h e same c o n d i t i o n s , and such t h a t
R'
R'
respec-
has a denumerable e x t e n s i o n s t i l l
respecting the conditions. 0
Let
v
F o r each
be t h e sum o f t h e c a r d i n a l i t i e s of t h e r e l a t i o n s
B1
through
s a t i s f y i n g the conditions, there e x i s t s a r e s t r i c t i o n
R
R'
.
Bk
of
R
v , which s a t i s f i e s t h e c o n d i t i o n s . Consider , which a r e o n l y f i n i t e l y many, up t o isomorphism. F o r each,
w i t h c a r d i n a l i t y a t most equal t o a l l these
R'
.
e i t h e r t h e r e e x i s t s a denumerable e x t e n s i o n s a t i s f y i n g t h e c o n d i t i o n s . O r t h e r e e x i s t s an i n t e g e r
u(R')
a l l extensions o f
R'
which i s s t r i c t l y g r e a t e r t h a n t h e c a r d i n a l i t i e s o f
r e s p e c t i n g t h e c o n d i t i o n s . Then i t s u f f i c e s t o s e t
t o be t h e maximum o f t h e s e
u(R')
.0
u
THEORY OF RELATIONS
156
8.3. MALITZ' COUNTEREXAMPLE Can we require t h a t R ' = R ; in other words, does there e x i s t an integer u such t h a t , i f R has cardinality greater than or equal t o u and s a t i s f i e s the conditions, then there e x i s t s a denumerable extension of R satisfying them. A negative answer i s due t o MALITZ 1967. 0 Take the base of integers from 0 t o n-1 Let I n be the usual chain of these integers; l e t Cn be the consecutivity r e l a t i o n ( y = x + l ) ; l e t O n be the unary relation called the singleton of zero, i . e . the relation taking (+) f o r 0 and ( - ) elsewhere; and l e t U n be the relation singleton of n-1 . Finally l e t Rn be the quadrirelation (In,Cn,On,Un) . From n = 7 on, a l l the R have the n same r e s t r i c t i o n s of c a r d i n a l i t i e s 1, 2 and 3 , up t o isomorphism. Let A1, ..., Ah be those quadrirelations of the same a r i t y and c a r d i n a l i t i e s 1, 2 , 3 which are n o t embeddable in R, , and hence in Rn ( n 3 7 ) . We see t h a t every extension of an Rn t o a new element added t o i t s base admits an embedding of one of the A1,...,Ah . An analogous b u t rather complicated counterexample i s obtained f o r binary relations by LOPEZ 1973. 0
.
8.4. LOPEZ ' COUNTEREXAMPLE Given the f i n i t e relations A1, ...,Ah and B1,...,Bk , one can ask whether there e x i s t two integers u , v such t h a t , f o r every R with cardinality greater t h a n or equal t o u , there e x i s t v elements of the base, such t h a t the r e s t r i c t i o n of R t o i t s base with these v elements removed respects the embedding inequal i t i e s in the 6 ' s and has an extension of a r b i t r a r y large cardinality respecting the non-embedding inequalities in the A's . Negative answer by LOPEZ 1973. 0 For the base, take the s e t of points, or ordered pairs of integers called the abscissa and ordinate, and which vary from 0 t o n-1 Let Rn be the multirelation on t h i s base, which i s composed of the following 4 unary relations and 6 binary relations. The unary relation O n takes the value (+) f o r the points with abscissa 0 The relation U n takes (+) f o r the points with abscissa n-1 Similarly 01; and U,', are defined by interchanging the abscissas and ordinates. The s t r a t i f i e d partial ordering I n takes the value (+) f o r each ordered p a i r of points ( i , x ) , ( j , y ) whose abscissas s a t i s f y i -c j < n , with a r b i t r a r y ordinates x , y ; moreover I n i s reflexive. The equivalence relation E n takes (+) for any two points with a same abscissa and a r b i t r a r y ordinates. The equivalence classes of t h i s relation are thus the classes of elements which a r e pairwise incomparable modulo I n . The binary relation Cn , which by abuse of notation we shall c a l l a consecutivity, takes the value (t) f o r each ordered pair of points
.
.
.
157
Chapter 5
( i , x ) , ( i + l , y ) whose abscissas are consecutive. Finally, the s t r a t i f i e d partial ordering I,', , the equivalence relation E,', and the consecutivity C,', are obtained from the preceding by interchanging abscissas and ordinates. From n = 7 on, every Rn has the same r es t r i ct i o ns B1, ...,B k with cardinalities 1, 2 , 3 ( u p t o isomorphism). Let A1, ...,Ah be the other multirelations of the same a r i t y and car d i n al i t i es 1, 2 , 3 . We see t h a t every proper extension of Rn ( n 3 7 ) admits an embedding of a t l e a s t one of the A's Indeed, add a new element t t o the base of Rn . Consider the case where e i t h e r O n or U n or 0,', or U,', takes the value (+) f o r t , and reduce t h i s t o the preceding 5 8.3. Now consider the case where a l l the preceding unary relations take the value ( - ) f o r t Then e i t h e r there ex i s t s an equivalence class of En t o which t belongs: again reduce t o 5 8.3. Or t occurs between two consecutive equivalence classes of En I n t h i s case, use the consecutivity Cn t o see t h a t the extension of Rn t h u s obtained admits an embedding of one o f the A's . Now suppose the existence of u and v satisfying our hypothesis; take n > u and > v . Let Sn be a r es t r i ct i o n of Rn in which the B's are embeddable, and which i s obtained by removing v points. Then in each equivalence class of Add a En , there remains a t l e a s t one element of I S n l ; similarly f o r E,', new element t t o the base o f Sn , and attempt t o require t h a t the extension of Sn t o i t s base with t added admit only embeddings of the B's and not of the A's . This leads us t o s i t u a t e t in the chain of the equivalence classes By using Cn , one sees t h at t necessarily belongs t o one o f the equiof En valence classes: t cannot be situated between two consecutive classes. Thus we obtain an element in the base of Sn , which i s equivalent with t (mod E n ) , and another element equivalent with t (mod E,',) From t h i s , we deduce tha t t is the unique element common t o both equivalence classes. Thus,we have again a restriction of Rn obtained by removing v-1 points: t h i s i s our extension of Sn Iterating t h i s , we obtain Rn i t s e l f , and a t the following step we obtain a proper extension o f Rn , in which necessarily one of the A's i s embeddable. 0
.
.
.
.
.
.
.
EXERCISE 1 - RESTRICTIONS OBTAINED BY REMOVING ONE OR TWO ELEMENTS 1 Existence of a relation R with base E and two elements a , b o f E such t h a t R z R / ( E - j a } ) I R / ( E - {b}) y et R > R / ( E - { a , b ) ) Let R be the binary relation based on the s e t formed with negative integers -1, -2, and ordered pairs of natural numbers i , j = 0 , 1, 2 , .. , taking the value (+) in only following cases. R(u-1,u) = + f o r each negative u ; and R ( - l , ( O , O ) ) = t ; and R ( ( i , j ) , ( i + l , j ) ) = R ( ( i , j ) , ( i , j + l ) ) = + f o r a l l natural numbers i , j Show t h a t R s a t i s f i e s the stated conditions with a = ( 0 , l ) and b = ( 1 , O ) (communicated by POUZET in 1975).
-
.
...
.
2
- Similarly with
R
> R/(E -
\aj )
>
R/(E
-
{a,b})
ye t
R
L
R/(E
- { b)) .
158
Let
THEORY OF RELATIONS R
be 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 o f p o s i t i v e and n e g a t i v e i n t e g e r s ,
which we denote by
O,l,-1,2,-2,..
and n a t u r a l numbers, which a r e assumed t o be 0 ' ,1' ,2'
d i s t i n c t from t h e i n t e g e r s and denoted by o n l y i n f o l l o w i n g cases:
+
R ( i ,i')=
i3 0
f o r every i n t e g e r
value; f i n a l l y
R(i',(i+l)')
. Show
= R(i,i+2)
R(i,i+l)
that
(communicated by LOPEZ i n 1977).
For
Similarly with R
R
R
> R/(E
i ' w i t h equal
i ' and i t s immediate suc-
satisfies the stated conditions with
(i+l)'
b = 0'
-
v a l u e (+) i s t a k e n on
and f o r t h e n a t u r a l number
f o r e v e r y n a t u r a l number
cessor 3
,. . . The
+ f o r a l l i n t e g e r s i ; and
=
- { a } ) :R/(E - { b ) )
- {a,b}
:R/(E
a = 0
and
.
)
t a k e t h e c o n s e c u t i v i t y on p o s i t i v e and n e g a t i v e i n t e g e r s , w i t h
a, b
two
c o n s e c u t i v e i n t e g e r s (communicated by HAGENDORF i n 1977).
-
EXERCISE 2
Val o f
Let
A, B, C
B satisfying
Defining
by
D
be t h r e e c h a i n s such t h a t
Let
he an isomorphism f r o m
C+D 5 C
A+B
t h e non-empty f i n a l i n t e r v a l o f f"(Do)
i s a t t h e l e f t of
which a r e a t l e f t o f
Do I t e r a t i n g t h i s , we o b t a i n
Di
Either a l l o f the elements o f
B into
B
and
B
C<
i n i t i a l inter-
C
(see HAGENDORF 1977 lemma 1.10).
A
if
A+C+D
,then
A+C
3
in
Do
, and l e t
A+C
onto a r e s t r i c t i o n o f
B which i s t h e complement o f
. Let
A+B
. The
C
be t h e i n t e r v a l o f those elements
D1
and more g e n e r a l l y
D2
Di
.
B , i n which case t h e i d e n t i t y on those , when completed by f , g i v e s an isomorphism o f
Di
h y p o t h e s i s . O r a t l e a s t one o f t h e
hence a l l t h e
times i s an isomorphism o f
included i n
A+B
A
.
I n t h i s case,
onto a r e s t r i c t i o n o f
iterated
, hence A+B
A
. Or
take
, even
B
if
C<
B
. For
instance take
A =
B
=
.
A = U - + W1
-+
Prove t h a t
Q +
";+
0
= Q
i s an
C
l + w - and
compare w i t h
has 3 immediate e x t e n s i o n s : 1+A
, A+1
5
+
; and
has 5 immediate e x t e n s i o n s :
Q
+ W - . W1
, not
B+1,
wl+B
4.4).
and a l s o
which i s n o t o b t a i n e d by s i m p l y a d d i n g an i n t e r v a l t o
B
.
2 A
.
EXERCISE 3 - IMMEDIATE EXTENSIONS (HAGENDORF 1983, unp.; 0 W
(i+l)
A = B = Q+O1+Q and C = W1+Q , where Q i s t h e c h a i n o f A = converse o f W2 and B = Z and C = 0
the r a t i o n a l s . O r again
Prove t h a t
,
A
hits
Di
f
Note t h a t o u r statement no l o n g e r h o l d s i f we remove t h e h y p o t h e s i s t h a t C = CJ-
.
f"(Do) i
f o r each i n t e g e r
are included i n
preceding the
i n i t i a l interval o f
be
Do
image s e t
y e t g r e a t e r t h a n o r equal t o some element of
, contradicting the D . (j > i) a r e J
C
1
.
A+C I A
f
A+B
, we have e q u i v a l e n t l y t h a t
B = C+D
either
.o- r
then
A+B 5 A+C ;
,w
o b t a i n e d by a d d i n g an i n t e r v a l .
.
A
+
B
,