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).
'pe
y(r)=(rn, y < E > ,
[rl=
:
( $(r))
'L
d 6 f i n i e par
21
de.s r e l a t i o n s i b o m u k p h e . 4 B r , La C L a b b e d ' i b o m a k p h i e d e r e s t
: y e O ( E ) I e t , pour t o u t e r e l a t i o n r ' k r r l , on r'.
%
S i r'= y(r),
E'= y<E>, p'= y
, la bijection
e s t un ibomokphibme e n t r e r=(m,E,p) e t r ' = ( m , E ' , P ' ) , 9 q u ' o n n o t e e n c o r e : r 'L r ' . E
-f
ce
Pour rERm, s 6 R m , l o r s q u e r e s t isomorphe 2 une r e s t r i c t i o n
1.2.3 r'
E'
d e s , o n d i t q u e r b'abki-te danb s , o u que s a b h i t e r , e t l ' o n
n o t e : rss. Dans l e c a s c o n t r a i r e r q s , o n p e u t d i r e q u e s P v i Z e r et
( p l u s g6n6ralement)
lorsque s
Pvite chacune des r e l a t i o n s appar-
d e @Lm, o n p e u t d i r e q u e s B v i t e
t e n a n t 1 une s o u s - c l a s s e
L a formule rss d 6 f i n i t dans chacune d e s c l a s s e s k
m
d'abfiitement m-aire.
l e pkfio4dhe
21.
un p r 6 o r d r e :
On a p p e l l e z g e d e s l a c l a s s e B d e s
-3 e s t
r e l a t io n s r de c a r dinalit6 f i n i e v 6 r i f i a n t r<s. S i
l'ensern-
b l e d e s p a r t i e s f i n i e s du s u p p o r t de s , a l o r s E e s t l a r e u n i o n d e s classes d'isomorphie [SIX]
l'on note s(n)
(pour n e w )
n-restrictions
de s (grossisrement
-
on d 6 f i n i t une s u i t e s
: w
.
A cet egard, s i 9 s l e nombre d e s c l a s s e s d ' i s o m o r p h i e d e s
lorsque X parcourt
-f
w
-
: s ( n ) < 2 (nm) pour
appel6e
pkodid d e
tout
sE Q L ~ ) ,
s
Une r e l a t i o n s t e l l e q u e s ( n ) < l e s t d i t e n - t t t o n o m o t p h e ( c e q u i r e v i e n t 1 d i r e q u e , p o u r t o u t c o u p l e (X,Y) d e n - p a r t i e s de s : SIX dit
'L
sly).
L o r s q u e s e s t n-monomorphe
( b r i s v e m e n t ) q u e s e s t manomo4phe. S i l ' o n n o t e
des r e l a t i o n s m-aires d'exemple)
que
monomorphes, on p e u t rernarquer
]\t2c o n t i e n t
du s u p p o r t
pour t o u t n e w ,
]\^em
(5 t i t r e
l a classe des relations binaires
(ou ChaintA).
d'ordre
total
1.2.4
E t a n t donne une r e l a t i o n r = ( k , E , P ) , t o u t isomorphisme
rlX
'L
local
on
la classe
r l Y e n t r e d e u x r e s t r i c t i o n s d e r e s t a p p e l 6 un i o o m o 4 p h i ~ m e d e r. P o u r d e u x r e l a t i o n s
lorsque tout
et
s e R S md e mEme s u p p o r t E ,
isomorphisme l o c a l d e r e s t un isomorphisme l o c a l d e s ,
on d i t que s e s t Libfie-intehpkZtabLe p a r r .
Relations enchaina bles, rongements et pseudo-rangements a)
l o r s q u e r est u n e
Par e x e m p l e ,
~ € ( 8E )
libre-interprgtablr
un
IXI=lYl,
l'injection
croissante
isomorphisme e n t r e S I X e t sly". p r o p o s i t i o n 5.1.)
( [ I l l ,
les couples
pour b)
r e s t d i t e C ~ i c h u i x a b ~ (Cp l u s
par
2 d i r e : " P o u r t o u t c o u p l e (X,Y) d e p a r t i e s f i n i e s d e
ce13 revient que
c h a f n e d e s u p p o r t E, u n e r e l a t i o n
C ~ I C ~ I C ( T I I G pC a r r e t o n m o n t r e f a c i l e m e n t q u e
pr@cis&rnent, e l l e est
E tel
(X,Y)
enchaTnables,
n a b l e est m o n o m o r p h e ) .
dSmontr6 "Pour
18,
§
il rGsulte En
lY64,
r)
de vsrifier
cette condition
de E).
rm
la classe des
:
3
t m ]I[,
relations
(toute r e l a t i o n enchaf-
l'instigation
99 - v o i r a u s s i I l l 1 ,
p.
de X s u r Y est
I E [ > m + l , on p e u t p r o u v e r
partielle
de
R. F R A T S S E ( q u i
: "Toute
de cardinalit6 i n f i n i e est enchafnable"
monomorphe annonci.
suffit
1954 une r g c i p r o q u e
dGs
a v a i t demontre
qu'il
(mod.
Lorsque
de m-parties
s i l'on n o t e
De c c s r e m a r q u c s ,
m-aires
243
proposition
(r41,
relation r6sultat
I l.2),
le tti6orGme s u i v a n t ( n o t e [ I 0 1 e t t h s s e 1 1 1 1 , t h .
tout
m e o, i l e x i s t e u n c o u p l e ( n , p ) e
r e l a t i o n m-aire
n-monomorphe
UXU
pour
nous avons 12.1.1.):
lequel
toute
2p est encha4nable
de cardinalit6
(donc
le plus p e t i t des n a t u r e l s n apparaism ( n , p ) , nous disons que d est le d@ghc m < ~ l ~ t i m c cde e m o n o m o r p h i e m - a i r e . P l u s p r S c i s G m e n t , on p e u t m o n t r e r
S i l'on n o t e d
monomorphe)". sant dans de
qu'il
couples
e x i s t e deux s u i t e s de n a t u r e l s
: "Pour
que
tels
t o u t n a t u r e 1 n2d
de c a r d i n a l i t 6
enctiafnable)"
>pn (resp.
(P,),,~
m' de c a r d i n a l i t 6 si p
(bien entendu,
sont
: d =0, d = I , I
c)
tout
n a t u r e 1 m22,
Pour
phes
d
2
('n'n2d
m-aire
telles
m
n-monomorphe
>q ) e s t monomorphe
e t qn s o n t p r i s
male : pn.
et
toute relatio!
(resp.
de manisre mini-
l e s d e g r s s o p t i m a u x d e mo n o -
= 3 , d = 4 , d =5 ou 6 ( ? ) . 3 4
i l e x i s t e d e s r e l a t i o n s m-aires
monomor-
non e n c h a T n a b l e s e t , c o m m e c o r o l l a i r e d u t h s o r c m e p r G c S d e n t ,
obtient
( [ I l l ,
m-aires
monomorphes
proposition
12.2.1)
: "Pour
m22,
Ies r e l a t i o n s
n o n e n c h a T n a b l e s o n t u n e c a r d i n a l i t s maximum
" ( I 1 e n r s s u l t e un r e n f o r c e m e n t p o u r l e s c a r d i n a l i t s s m f i n i e s d u t h 6 o r S m e d e R. F R A T S S E c i t 6 p l u s h a u t , 2 s a v o i r : " T o u t e
finie k
r e l a t i o n m-aire
Par e x e m p l e connue).
monomorphe
2
m
> km e s t e n c h a E n a b l e " ) . (mais dEjS l a v a l e u r k3 n e semble p a s
de cardinalit:
: k =3 e t k > m + l
on
244
C. Frasnay
1.3
Si
&.
est u n e s o u s - c l a s s e de
(autrement dit : (vu)('dr)
c l o s e par a b r i t e m e n t
((uCr et rs?t)
=j
--
o u encore :
(ue-A)),
est une r 6 u n i o n de c l a s s e s d ' i s o m o r p h i e t e l l e q u e , pour t o u t e r e l a t i o n r de s u p p o r t E et t o u t X c E :(re&)
.h
d i s o n s , pour a b r g g e r , q u e
(rlXc2k ) ) , nous
est u n e c l a s s e
i n 4 t i U 8 t
DSs l o r s , n a t a n t X c Y l'inclusion s t r i c t e ( X & Y
n o u s a p p e l o n s p J ~ u d ~ - P l ? $ m e fdlet
ft. ( o u
"bu4ne" de
de r e l a t i o n s et X # Y ) ,
.k ,
a u s e n s de
R. F R A ' L S S E ) t u u t e r e l a t i o n m-aire s d e s u p p o r t fini Y t e l l e qut? ( s + . f t ) et
(1x1
:
( s l x t ~ ) .
X I Y Bien entendu, si
s
s'
et s i s est p s e u d o - 6 l C m e n t d e
a l o r s s'est a u s s i p s e u d o - 6 1 6 m e n t de
h .
S i b(h)
s u p 6 r i e u r e d e s c a r d i n a l i t s s d e s p s e u d o - 6 l g m e n t s de b(h)
( o u b(k) E
id)
des pseudo-616ments
h
h
, l a condition
Q q u i v a u t h d i r e q u e l c s c l a s s e s d'isomorphie de
sont e n n o m b r e f i n i ( o u e n c o r e q u e
n'admet, 2 l'isomorphie p r S s , qu'un n o m b r e f i n i d e "bornes") dit a l o r s q u e l a c l a s s e i n i t i a l e 1.3.1
,
d 6 s i g n e la b o r n e
est
4c : on
{iMd-b(J&RBt.
S i r est u n e r e l a t i o n m - a i r e d e c a r d i n a l i t 6 p, s o n S g e E; est
u n e c l a s s e i n i t i a l c et l e c a r d i n a l b(P)
e x p r i m e une p r o p r i 6 t 6 d e l a
f a m i l l e d e s r e s t r i c t i o n s "finies" d e r . B i e n e n t e n d u , s i p est f i n i , a l o r s : b(^r),
(et o n peut
pour t o u t e r e l a t i o n e n c h a T n a b l e
De m a n i P r e g G n g r a l e , l o r s q u e b(T)
(autrement dit : s i l'sge F e s t fini-bornE),
I
est f i n i
o n dit simplement que
la r e l a t i o n r est d i n i - b o t n C t . a) En 1 9 6 3 ,
n o u s a v i o n s o b t e n u u n e proposition(d6not6e
Cll1)donnant
13.1.2
dans
c o m m e c o r o l l a i r e i m m 6 d i a t : "Toute r e l a t i o n e n c h a P n a -
b l e est fini-born6e". P l u s p r G c i s E m e n t , s i r est u n e r e l a t i o n m-aire encha?nable de c a r d i n a l i t 6 p(finie
o u non),
o n peut lui
a s s o c i e r les pseudo-616ments d e ^r a y a n t u n e c a r d i n a l i t 6 < p , p u i s n o t e r v(r)
la b o r n e s u p 6 r i e u r e des c a r d i n a l i t 6 s de c e s pseudo-
6 1 6 m e n t s (bien e n t e n d u v(r)
=
b(?)
s i p e s t i n f i n i , et v(r)
s i p est fini) : n o u s d i s o n s q u e v(r) e n c h a 9 n a b l e r . En p o s a n t : v v
m
E iii
(pour tout m t u ) .
m
est la
= s u p v(r),
re
't:m
W U ~ ~ M Cde C
la r e l a t i o n
n o u s obtenions :
245
Relations enchaina bles, rangements et pseudo-rangements v =3, v = 4 , v = 6 et a u - d e l 5 , n o u s n e 1 2 3 4 p o u v i o n s a v a n c e r (5 c e t t e 6 p o q u e ) qu'une m a j o r a t i o n a p p a r e m m e n t
P a r e x e m p l e : v =0, v = I ,
trss g r o s s i s r e v < 9 m 2 p r o v e n a n t de l'utilisation d u th6orSrne d e s 'm "bichail141" (P. E R D B S et G . S Z E K E R E S , 1 3 1 ) : " P o u r n e o , s i d e u x cha'ines s , t o n t u n &me une (n+l)-partie
2
s u p p o r t E tel q u e I E l > n , a l o r s i l e x i s t e
X d e E pour l a q u e l l e les cha'ines r e s t r e i n t e s S I X ,
tlX s o n t i d e n t i q u e s o u o p p o s 6 e s " . b) A l o r s q u e l a f o r m u l e v mE O ( e x p r i m a n t le c a r a c t s r e f i n i - b o r n 6 d e t o u t e r e l a t i o n m-aire encha?nable) n ' o c c u p a i t d a n s n o t r e t h s s e d e 1965qu'une p o s i t i o n m a r g i n a l e , u n r e n v e r s e m e n t d e t e n d a n c e ( d e p u i s 1970) a r e n f o r c 6 le r c l e de c e t t e p r o p r i E t 6 : c e t t e n o u v e l l e
o r i e n t a t i o n est a p p a r u e d a n s les t r a v a u x d e D. C L A R K et P .
KRAUSS
d'une p a r t , R. F R A T S S i et M. P O U Z E T ( C 8 1 C 2 0 1 ) d'autre p a r t
(C21C191)
( t r a v a u x d o n t t i e n t c o m p t e la n o t e C 1 3 1 s u r les v a l e n c e s m - a i r e s m a x i m a l e s v ) . A i n s i e n 1972, d a n s u n e a p p r o c h e p u r e m e n t q u a l i t a t i v e m (note r 2 0 1 , s a n s i n c i d e n c e s u r les v a l e u r s v m ) , M. P O U Z E T a d g f i n i (pour tout m s o) la c l a s s e
firn d e s
b e P P ~ 0h e t a t s o n 0 r n - u L h e i ,
X420
de telle sorte que : ( I )
P o u r tout
m g IW,
s i re
(2) P o u r tout ( k , m ) t o u o , - i n t e r p r & t a b l e par u , Comme
(2) :
q)2
rm,/Rm
bm,a l o r s si
~
6
r est fini-born6e. est libreet % s i~ re'% m
alors r t 6 m .
c o n t i e n t la c l a s s e d e s cha'ines, il r 6 s u l t e d e ( I )
et
et toute r e l a t i o n encha'inable est f i n i - b o r n 6 e .
c) L'6tude q u a n t i t a t i v e e f f e c t u 6 e e n 1963-65 d a n s l e s n o t e s [ 9 l [ l O l et la thSse [ I l l u t i l i s a i t la " t h 6 o r i e d e s cha'ines p e r m u t 6 e s " et l e
.
I' des sous-groupes indicatifs de S Elle fournissait m m (pour le d e g r 6 o p t i m a l d m d e m o n o m o r p h i e m-aire et l a v a l e n c e m-aire
demi-treillis
m a x i m a l e v ) les i n 6 g a l i t 6 s : d m < v m (pour tout m e w ) et m+l,
1161.
E t a n t d o n n 6 ( m , n ) ~W X W , o n a p p e l l e o p P 4 a t c u t L i b f i e
1.3.2
les a r i t s s m , n )
une application f
r o l a t i o n m-aire r d e s u p p o r t E ) , ( I )
f(r)
:
Rm +%
(pour
v 6 r i f i a n t (pour t o u t e
les t r o i s c o n d i t i o n s s u i v a n t e s :
est u n e r e l a t i o n n - a i r e d e s u p p o r t E.
246
C. Frasnay
(2) P o u r tout (3)
P o u r tout
X E
P(E)
Y=@(E)
: f ( r J ~ )= f(r)
IX
%
f(j(r))
:
=
y(f(r))
a) Etant d o n n e u n tel o p 6 r a t e u r l i b r e f (pour les a r i t 6 s m,n), s o i t
h
la c l a s s e des r e l a t i o n s r=(m.E, p)
k
On dit q u e
t e l l e s q u e f(r)=(n,k.,E").
est la c e u 5 5 e a n c u e 4 6 e e f c (au s e n s de A .
h e p h P n e n t P e par l ' o p e r a t e u r libre f . Le d c g r l g de
petit d e s n a t u r e l s n tels q u e libre f :
fLm
h
e s t le plus
s o i t r e p r 6 s e n t a b l e par un o p G r a t e u r
(Par e x e m p l e , la c l a s s e des c h a i n e s est u n i v e r -
!?L
+
h
T A R S K I 1231)
s e l l e de d e g r E 3 ) . T o u t e c l a s s e u n i v e r s e l l e est i n i t i a l e (close par abritement),
et on peut m o n t r e r q u e la c o n c e p t i o n de A .
TARSKI
e q u i v a u t 2 d e u x a u t r e s c o n c e p t i o n s q u i n e n e c e s s i t e n t pas de f a i r e appel 2 la n o t i o n d'opgrateur l i b r e : ( I )
de
Au s e n s e q u i v a l e n t d e R . L .
Rm est
V A U G H T 1241, une s o u s - c l a s s e
u n i v e r s e l l e s i e l l e est i n i t i a l e et s'il e x i s t e n t
(I
v E r i f i a n t l a c o n d i t i o n s u i v a n t e : " L o r s q u e r est une r e l a t i o n m - a i r e d o n t t o u t e s les r e s t r i c t i o n s d e c a r d i n a l i t 6 alors r appartient 2
.k
"
Lk
,
est le p l u s petit n a t u r e 1
(Le d e g r e de
n v G r i f i a n t c e t t e condition). (2)
A u s e n s e q u i v a l e n t de R . F R A Y S S E 1 5 1 , u n e s o u s - c l a s s e
A
Lr
de fL e s t u n i v e r s e l l e s'il e x i s t e u n e n s e m b l e fini de r e l a t i o n s m est l a c l a s s e d e s m-aires de c a r d i n a l i t e s f i n i e s tel q u e : "
ft
r e l a t i o n s m-aires q u i G v i t e n t et l e d e g r 6 de
$.
pseudo-Glements d e
lr
"
(Alors
.k
est u n e c l a s s e i n i t i a l e ,
e s t la b o r n e s u p e r i e u r e des c a r d i n a l i t e s d e s
.&
).
b) E n u t i l i s a n t n o t r e t h e o r s m e s u r le degrg o p t i m a l d e m o n o m o r p h i e m-aire, M. J E A N a o b t e n u e n 1967 C 1 7 1 ,
pour chaque m e
(11,
les d e u x
p r e m i e r s e x e m p l e s non t r i v i a u x de c l a s s e u n i v e r s e l l e , e n d e m o n t r a n t : "La c l a s s e
cm
d e s r e l a t i o n s m - a i r e s e n c h a i n a b l e s et l a c l a s s e
des r e l a t i o n s m - a i r e s m o n o m o r p h e s s o n t d e u x c l a s s e s u n i v e r s e l l e s " . c) S i g n a l o n s ici u n l i e n e n t r e l a n o t i o n d e l i b r e - i n t e r p r 6 t a b i l i t 6 et c e l l e d'opgrateur l i b r e
151
:
"Pour qu'une r e l a t i o n s e !J,n
l i b r e - i n t e r p r e t a b l e par u n e r e l a t i o n r e
a
am,il
soit
f a u t et il s u f f i t
an
qu'il e x i s t e un o p E r a t e u r l i b r e f : + tel q u e f(r)=s". m Comme l e n o m b r e d e s o p s r a t e u r s libres f (pour d e s a r i t G s m,n donnges) est f i n i , il e n r E s u l t e q u e (pour r et n donn6s) l e n o m b r e d e s
Relations enchaina bles, rangements et pseudo-rangements
247
relations n-aires libre-interprgtables par la relation r est fini. P a r e x e m p l e , s i r est u n e cha'ine de s u p p o r t E tel q u e ( E / > n ,o n p e u t m o n t r e r q u e le n o m b r e d e s r e l a t i o n s n - a i r e s de s u p p o r t E e n c h a f n k e s par r est 2 " i ( n ) , oC v(n)
est le n o m b r e d e s p r E o r d r e s t o t a u x s u r
~ = { o , I .,. . , n - I I . 1.3.3 E,
L o r s q u e r et s s o n t d e u x r e l a t i o n s m-aires de m@me s u p p o r t d i s o n s qu'elles sont une d h d o h m u t i o h l l'une d e l'autre (ou
nous
q u e c h a c u n e est d ' ? < o 4 w l U b t e e n l'autre)
si :
( I )
r et s n e s o n t p a s i s o m o r p h e s ,
(2)
pour tout
X C
E , les r e s t r i c t i o n s s t r i c t e s r
/ et~
SIX
sont
isomorphes. a) Si m = I , une t e l l e s i t u a t i o n e x i g e I E I C 1 .
O n trouvera dans " 2 1 1
les r 6 f s r e n c e s a u x t r a v a u x de G. L O P E Z (1972, p o u r m=2) et de
M. P O U Z E T (1975, p o u r m>3) s e l o n l e s q u e l s : l e s r e l a t i o n s b i n a i r e s "finies" d6formable.s o n t u n e c a r d i n a l i t 6 m a x i m u m G g a l e B 6 - t a n d i s q u e , pour t o u t m > 3 et tout p e
'0,
d g f o r m a b l e s de c a r d i n a l i t 6 f i n i e
i l existe des relations m-aires >p.
b) N o t r e p r o p o s i t i o n 13.1.2 de [ I l l s ' e x p r i m e suivante : "Pour tout m e w ,
e n c o r e d e la m a n i s r e
les r e l a t i o n s m - a i r e s e n c h a f n a b l e s de
cardinalit6 finie > v
sont indsformables". Bien entendu, puisque m les r e l a t i o n s m - a i r e s m o n o m o r p h e s n o n e n c h a P n a b l e s o n t u n e c a r d i n a l i t 6 m a x i m u m f i n i e k m , il e n r 6 s u l t e q u e "toute r e l a t i o n m o n o m o r p h e
est f i n i - b o r n 6 e " e t e n c o r e : " P o u r t o u t m e w ,
il e x i s t e p c w
lequel toute r e l a t i o n m-aire m o n o m o r p h e d e c a r d i n a l i t 6 f i n i e
pour >p
es t indEf o r m a b le" .
1.4
Etant donn6 u n ensemble I et une famille
u=(mi)itI
r e l s , o n a p p e l l e b t ' Z u c - t u h ( ! h d u , t i u n n t L L e u - u i h e de
[muLtihei!ution
ou
de natu-
Auppohf E
k-tclatiu~ p-aire d e s u p p o r t E s i I est f i n i ,
/ I / = k ) t o u t e f a m i l l e (ri)ie I t e l l e q u e , p o u r tout i t I , r i
soit
u n e r e l a t i o n d'arit6 m i et d e s u p p o r t E. S i n e w e s t t e l q u e mi$" pour t o u t i r I , o n dit a u s s i q u e l'arit6 p
e s t b v t t n a e et q u e
(ri)ie I est u n e s t r u c t u r e r e l a t i o n n e l l e u u p l u h n - u i h e .
1.4.1
T o u t e l a t e r m i n o l o g i e i n t r o d u i t e B p r o p o s d e s r e l a t i o n s (en
1.2 et 1.3) s'Btend a i s 6 m e n t a u x s t r u c t u r e s r e l a t i o n n e l l e s . P a r
248
C. Frasnay et tout y c Q(E),
exemple, pour tout X t P(E)
chaque structure rela-
de support E admet une restriction i it1 % 'L L'isoet une image isomorphique y(R)=(y(ri))ik I.
tionnelle p-aire R=(r ) RI X=(ri 1 X)
E I
morphisme explicite (R
?,
R') ou implicite ( R
'L
R'),
l'abritement
l'zge R , de m t m e que les propri6tGs
R
(pour une structure R) de n-monomorphie, monomorphie, enchafnabilitd, dgformabilit6, . . . se conqoivent aisdment. Par exemple, pour
. . . rk)
qu'une k-relation ( r l , r 2 ,
de support E soit libre-
interergtable par une chaPne r
de support E , i l faut et i l suffit
( r o , r l , r 2 ,. . . , rk) soit monomorphe. Une
que la (k+l)-relation
m u t t i c h u i n e de support E est une famille finie
de chafnes
t i de m2me support E.
1.4.2
D e nombreux r6sultats 6tablis pour les relations m-aires
s'6tendent (par une juste adaptation) aux multirelations et aux structures relationnelles d'arit6 p born6e. Donnons quelques exemples : a) T h Z o h P m e d e b m u e t i c h a i n e b ( [ 9 l [ l 11) gdngralisant le th6orSme des bichafnes (k=l) de P. ERDUS et G . SZEKERES C31: Pour (k,n)&
w
XW,
le naturel n (2k) est le plus petit des
naturels p vgrifiant la condition suivante : "Pour toute(k+l)-chafne
...,
t ) de support E tel que / E l > p , il existe une (n+l)-partie (to,tl, k X de E pour laquelle les chafnes restreintes t IX, t,lX , . . . , tklX
sont deux B deux identiques o u oppos6es". b) ThEofiZme d e monomohphi,c o p t i m a t e (I I 2 1 I 2 1 1) :
I une arit6 born6e.
Soit p=(rni);
I1 existe alors u n couple ( n , p ) e
vdrifiant la condition
0x0
suivante : "Toute structure relationnelle p-aire n-monomorphe de cardinalit6 p)
est enchafnable (donc monomorphe)".
Si l'on note d
v
le plus petit des naturels n apparaissant dans
de tels couples (n,p),
o n peut dire que d
est le d e g t t o p t i m a ! . de
1-I
monomorphie p-aire. Corollairement, i l existe u n naturel k
(mini11 mal) tel que : "Toute structure relationnelle p-aire monomorphe de cardinalit6 >k
LJ
est enchafnable".
c) ThEafiPmed d ' i n d E ~ a h m u b i t i t E ([20l-C221) A toute arit6 bornde p=(m;)ie
:
I , o n peut encore associer deux
Relations enchaina bles, rangements et pseudo-rangements
249
( v u k e n c e U-uihe maximale) e t w tels que : "Toute U P structure relationnelle U-aire enchaTnable (resp. monomorphe) de
naturels v
cardinalit6 finie >v
(resp. > w ) est ind6formable".
u
u
Ces 6nonc6s (pour toute arit6 P born6e) imposent un caractSre finiB toute structure relationnelle p-aire enchaPnable o u
born6
g6 n 6 r a 1 eme n t ) mo nomo rp h e Remarques : ( I )
(plus
.
Signalons aussi ( G . LOPEZ, 1 9 7 6 ) que les multirela-
tions au plus binaires et de cardinalit6 finie ,212
sont ind6for-
mables.
(2) Les m6thodes adopt6es dans C l l l , C 1 2 1
et 1 2 2 1 , pour
une relation o u une multirelation, valent aussi pour une structure relationnelle d'arit6 born6e. Pour toutes l e s arit6s p des structures relationnelles au plus m-aires, o n obtient ainsi les in6galit6s : dP 6 vu 6 vm. La dgtermination de la valence m-aire
v
m
prend donc une importance accrue : lorsqu'une relation m-aire "finie" r , enchainable et dGformable, aura la cardinalit6 maximum v
nous dirons que r est une relation enchainable c h i t i q u e .
N'
C'est par une dtude combinatoire du treillis C m des sous-groupes S m que nous obtiendrons'un exemple de telles relations m-aires ainsi que la valeur v m = 2m-2 (pour tout m>3).
G du groupe sym6trique
2.
L A T H ~ O R I ED E S CHAINES
PERMUTEIES.
Un groupe de permutations s'associe tout naturellement 1 une relation r=(m,E,p) (1)
dans deux circonstances :
Si S E d6signe le groupe des permutations de E , o n note Aut(r)
(et o n appelle g h o u p e d t n u u t o m o h p h i n r n t A
y~
S E telles que
%
(p(r)=r.
de r) le groupe des permu-
P l u s ggngralement, lorsque X parcourt
les g h o u p e n d'automokphiomen e a c a u x de r sont les groupes
P(E),
Aut (rlX) associ6s aux restrictions r ( X de r. Par exemple, une relation r de support w admet (parrni ses groupes (Gn)npw Si
B:E
+
est un isomorphisme entre deux ?elations r et r'
E'
de supports respectifs E et E ' ) , tion Aut(r)
locaux) une suite
de groupes G n € En d6finie par : G =Aut(rl n).
9-0
o
Y
o
en Aut(r').
0-I)
i l lui correspond (par transmuta-
un isomorphisme de S E sur S E l qui transforme
250
C. Frasnay
(2) L e s p e r m u t a t i o n s o ez S m c o m p o s d e s a v e c les m-uples x c E d e s m-uples x o x
0
m
e E
0
0 6
donnent
( p l u s e x p l i c i t e m e n t , p o u r x=(xO,x l , . . . , ~ m - , ) : )).
=(XO(o)’xa(l)’.”’xo(m-l)
permutations
m
S i Inv(r)
S m t e l l e s q u e : (\dx)(x&p
d6signe l e groupe des
-y x o
(?
C
o), o n d i t
q u e I n v ( r ) ~ Zm est le gkaupe d ’ i i i u a k i a n c e m a x i m u m d e la r e l a t i o n r=(m,E,p).
Tout isomorphisme r
.?,
r ’ i m p l i q u e : Inv(r)
=
Inv(r’).
P l u s g B n g r a l e m e n t , o n peut a s s o c i e r B tout g r o u p e G 6 C m et 2 toute relation re&, l a r e l a t i o n G x r( ddfinie comme suit :
km
si r=(m,E,p),
alors G
x
r = (m,E,p’)
avec p ’ = { x o o : x ~ pet
0 6 GI.
P u i s q u e r et G x r o n t m 8 m e a r i t 6 m , m 8 m e s u p p o r t E , et q u e l e u r s e n s e m b l e s caract6ristique.s p . G
x
p’
vdrifient
p g p ’ ,
o n peut d i r e que
r e s t u n e e x p u n h i O n de r : plus p r B c i s E m e n t , G x r est la
G - e x p a n b i o n d e r. B i e n e n t e n d u (si E
m d s s i g n e la p e r m u t a t i o n i d e n t i q u e , B l g m e n t n e u t r e de S ) , le g r o u p e t r i v i a l I = { E ~ } est u n a g e n t m m n e u t r e : I~ x r = r (pour t o u t r e P l u s g B n B r a l e m e n t , p o u r G E Cm ’
Rm).
la condition G x r=r Bquivaut 1 la condition G
C_
Inv(r)
et (dans ce
cas) o n dit q u e l a r e l a t i o n r est G - i n u a k i a n t e , o u e n c o r e q u e G est u n g h a u p e d’inuakiance de r. L e s r e l a t i o n s m-aires hymEtkiqucA s o n t les r e l a t i o n s r E @(E),
Rm telles
q u e Inv(r)
=
sm .
p o u r rt(R,(~),
X ~ ZP(E),
les f o r m u l e s :
G x r E (?Lm(E),
G
X
(r(X)
que la G-expansion r
I--+
=
(G
G
X
X
r)(X, G x
y% (r)=
%
~ ( G x r ) , montrent
r d 6 f i n i t u n o p 6 r a t e u r l i b r e de
(k m
d a n s lui-m8me. E n f i n , p o u r tout G t C m , o n p r o u v e a i s s m e n t q u e l a c l a s s e d e s r e l a t i o n s m-aires G - i n v a r i a n t e s e s t u n e c l a s s e u n i v e r selle. N o u s a l l o n s a d a p t e r ces c o n s i d 6 r a t i o n s g 6 n B r a l e s ( p o r t a n t s u r
les r e l a t i o n s et l e s permutations) a u x c h a f n e s et a u x r e l a t i o n s enchafnables,
2.1
+ E t a n t d o n n 6 u n n a t u r e 1 m (support d e l a c h a f n e m) et une c h a f n e
t de s u p p o r t E , l ’ e n s e m b l e p d e s i n j e c t i o n s c r o i s s a n t e s de
m={O,l,
. . . ,m - 1 )
+ (ordonn6 p a r m) d a n s E ( o r d o n n 6 par t ) d s f i n i t u n e
r e l a t i o n m-aire tCm)=(m,E,p) uples x=(xo,xI, (mod. t). t(m)
. . . ,xm- 1 )
d e s u p p o r t E , s a t i s f a i t e par les m-
d ’ B l 6 m e n t s d e E t e l s q u e : x < x C...<x 0
1
e s t le h u n g e m e n t m - a i k e t k i u i a k ? s u s c i t B (dans E) par l a
c h a f n e t.
m- I
P a r a l l u s i o n a u g r o u p e t r i v i a l I m = { ~ m } , o n peut d i r e q u e
Relations enchainables, rangements et pseudo-rangementr
25 1
Le premier acte de la thgorie "des chafnes permut6es"
2.1.1
consiste 2 introduire, pour chaque groupe de permutations G E C le G - h a n g c m e n t t G tG =
=
est donc la G-expansion du rangement trivial t(m)
(m,E,D')
s o n ensemble caractgristique
('a(o)
9
UGG
'n(l)
*
1
e t x <x 0
m'
G X ~ ' ~ suscit6 ) (dans E) par l a chafne t :
1
'Xo(m-~)) C...<x
p'
e Em
vsrifiant les deux conditions :
(mod. t).
m- I
et
est l'ensemble des m-uples
Bien entendu, si / E l C m , les
G-rangements suscit6s dans E coincident avec la relation m-aire "vide" : t G = (m,E,d) (qui admet S comme groupe d'invariance). Par m contre, si IEl a m , les G-rangements suscit6s dans E sont des relations m-aires t G " n o n vides" dont l e groupe d'invariance maximum est prgcis6ment
: Inv (t
G
)=G.
Lorsqu'un G-rangement t G abrite une relation m-aire
2.1.2
alors s est a u s s i u n G-rangement
s,
(la preuve s'effectue d'abord dans
le cas trivial G = I m , puis dans le cas gEn6ral G E E ) . Autrement m la classe des G-rangements est une m' classe initiale de relations m-aires". Si l'on appelle p h e u d u - G -
d i t : "Pour tout groupe G e i :
h a n g e m e n f tout pseudo-616ment de cette classe initiale, et si l'on tient compte des deux remarques suivantes : ( I )
tout ensemble E supporte u n G-rangement,
( 2 ) deux G-rangements tG,th de m8me cardinalit6 finie sont
implique t'=%(t G Y
isomorphes (puisque t'=;(t)
alors i l r6sulte de la terminologie g6n6rale ( 1 . 3 )
G) ) .
qu'un pseudo-
G-rangement est une relation m-aire r de cardinalits finie n+l verifiant l'une o u l'autre des d e u x conditions suivantes : (I)
r n'est pas un G-rangement, mais toute n-restriction de r est un G-rangement.
(2)
r est une dsformation d'un G-rangement.
Par ailleurs, tout G-rangement t
G
est enchalnable ( p l u s pr6ci-
sEment : t G est libre-interprgtable par la chafne t qui le suscite) et tout pseudo-G-rangement 2.2
est monomorphe.
Etant donn6 u n groupe G c C
respectifs A,B, on dit que
s
m
et deux chafnes s,t de supports
et t sont G - c o m p a t i b L Q d d S s que les
G-rangements s G et t G ont la mEme restriction de support A n B : s
G/ A n B
= t
G
[ A n B. Cette condition est trivialement r6alis6e
C. Frasnay
252
lorsque ( A n B ( c m e t , dans le cas ggn6ra1, elle revient 5 dire que les deux chafnes s ( A n B , t l A n B (de support commun A n B )
suscitent
l e mgme G-rangement. Si (s,t) est une bichafne de support E , la G-compatibilitG
2.2.1
des chaPnes s,t peut s'exprimer plus directement en associant 1 toute m-partie X de E une permutation
x
=
0
e
X ~ , . . . , X ~ - ~x} ,< x <...<x
cx
0
1
Sm
m- 1
de la faGon suivante : (mod.
s)
(mod. t). Une telle permutation est et x o(m-I) u ( 0 )< x a ( 1 ) < - * unique, d'oG une application X e P ( E ) k u E S (on dit parfois que m X m ox est la permutation permettant le p a o h a g e de la chafne S I X 1 la chafne tlX. Dhs lors, pour que les cha'ines
s
et t soient G-compati-
bles, i l faut et i l suffit que u X E G pour tout Xe Pm(E).
Ou encore,
si H m ( s , t ) ~ C m d6signe le groupe engendr6 par les permutations lorsque X parcourt Pm(E),
0
X
la G-compatibilit6 de s,t Gquivaut 1
l'inclusion H ( s , t ) ~ G (on peut dire que H (s,t) est le ghoupe d t m m
compatibiLitP minLrnum des cha'ines s,t pour le degr6 m). Si
T,
ddsigne l'ensemble des cha'ines de support E , il e n
rdsulte (pour tout groupe G E E m ) une relation binaire d'gquivalence de support dite h e L a t i o n d e G-compntibiLitP, encore not6e :
f,,
s E t
(mod. G) (formule logiquement Squivalente 1 l'identit6
s G = t G des G-rangements
suscit6s par les chafnes s,t).
2.2.2
Pour tout groupe G E Z le G-rangement suscit6 par la m' + m-I}) n'est autre que la chafne usuelle m (de support m={O,l,
...,
relation (m,m,G).
Pour simplifier, nous noterons G le G-rangement + de support w suscit6 par la chafne usuelle w : ce G-hangeme-nt
6
natuheL
+ par w,
est l'unique relation m-aire de support w ,
de restriction ilm=(rn,m,G).
bijective G H
6
enchafn6e
I1 en r6sulte une correspondance
entre le treillis C
m
et l'ensemble des rangements
m-aires naturels. Pour n e w , une permutation
~
E
lorsque la chafne associ6e (appartenant h
S est ~ +dite ~ G-admionibLe
<
e(O)<e(l)<
et pour laquelle : m+n avec la chafne usuelle
(pour
( 8 7.1.),
. . .
nous avons
8 e S m c n constituent un montr6 que les permutations G-admissibles groupe G (n)E 'm+n : ce groupe est caractsrisd par le fait que, pour
Relations enchaina bles, rangements et pseudo-rangements t o u t e b i c h a i n e ( s , t ) de c a r d i n a l i t 6 N o u s d i s o n s q u e G(n)
entendu
est le
ghOurJP
e t , p o u r (p,q)&wXw
G(O)=G
253
m+n, ~~t(mod.G)~s~t(rnod.G(~)), d i B U t E k?-i?tnc d u g r o u p e G . B i e n
, G(p)(q)=G(p+q).
E n r e v e n a n t a u C - r a n g e m e n t n a t u r e 1 G , o n peut m o n t r e r est a u s s i la s u i t e d e s g r o u p e s d ' a u t o m o r p h i s m e s
q u e (G("))nEw
l o c a u x : G(n)=Aut(hlm+n). 2.2.3
A
t o u t e b i c h a E n e ( s , t ) de c a r d i n a l i t 6 i n f i n i e , a s s o c i o n s la
suite ( H ) d o n t c h a q u e t e r m e H e 1 est le g r o u p e d e c o m p a t i b i m mCW m m l i t 6 m i n i m u m de s,t p o u r le d e g r 6 m . N o u s d i s o n s q u e c e t t e s u i t e H est une 5 u i t c i n d i c n f i v e et q u e t o u t g r o u p e H
figurant dans une m t e l l e s u i t e est u n g h v u p e i k t d ~ c a t i f i . N o u s n o t o n s 1; l ' e n s e m b l e d e s
s o u s - g r o u p e s i n d i c a t i f s de S
m
.
a) D a n s [ I l l , n o u s a v o n s r s p e r t o r i 6 t o u t e s l e s s u i t e s i n d i c a t i ves. O n peut les n o t e r : S,T,D,IP'q,Jr >I).
p,q,r
sont des naturels
L a s u i t e S est c e l l e d e s g r o u p e s s y m g t r i q u e s Sm (on peut d i r e
qu'elle est de hang
0 ) .
So,SI,...,Sh-I, puis H m)h
(oG
m
Toute suite indicative H # S commence par est un s o u s - g r o u p e s t r i c t de S
(on p e u t d i r e a l o r s q u e le r a n g de H est h). La suite J
I
m
p o u r tout
Plus pr6cissment :
e s t d e r a n g 3 e t , p o u r m 2 3 , J/ est le g r o u p e
(d'ordre 2) e n g e n d r g par l e r e t o u r n e m e n t (0,m-I)
(l,m-2)
...
L a s u i t e T est de r a n g 3 e t , p o u r m 2 3 , T m est l e g r o u p e (d'ordre m) e n g e n d r s par la p e r m u t a t i o n c i r c u l a i r e ( O , l , 2 , . . . , m-I).
La suite D
est d e r a n g 4 e t , p o u r m 2 4 , D m est l e g r o u p e p r o d u i t d i r e c t (d'ordre
2m) : D m = J A Tm . P o u r p k l , q a l , l a s u i t e "'1
est d e r a n g p+q e t , p o u r m)p+q,
est le g r o u p e (d'ordre p!q!) vant l'intervalle initial {m-q,
. . . , m-2,m-I}
c o n s t i t u g p a r les p e r m u t a t i o n s c o n s e r -
{ o , I , ...,p-
I } , l'intervalle final
e t c h a c u n des n a t u r e l s p , p + I ,
l a s u i t e J r e s t d e r a n g 2r e t , p o u r m > 2 r , J: d i r e c t (d'ordre 2(r!) b)
2
)
: J:
=
. . . , m-q-I.
Pour r22,
e s t le g r o u p e p r o d u i t
J A I:'r.
On p e u t c o n s t a t e r q u e t o u t g r o u p e i n d i c a t i f H
m
# Sm appar-
t i e n t 2 u n e s u i t e i n d i c a t i v e u n i q u e et q u e 1' e s t un s u p . d e m i m il t r e i l l i s (si G ' e t G " s o n t d e u x s o u s - g r o u p e s i n d i c a t i f s de S m' e n e s t d e m t m e p o u r le g r o u p e G ' B G " , t a n d i s q u e G' A G " = G ' n G" n'est pas n 6 c e s s a i r e m e n t un g r o u p e indicatif).
Tout groupe G e C
m
contient donc un groupe indicatif maximum H E C ' qui dgtermine sans m m
C. Frasnay
254 ambiguit6 la diche
indicative H de G .
Les dilatss successifs d'un groupe indicatif sont des groupes indicatifs de msme fiche. Notamment, p o u r les groupes symgtriques (de fiche S )
:
(Sm)(n)=Sm+n.
Pour toute suite indicative H P S , de
rang h , o n d6montre sgalement : (Hm)(n)
Hm+n p o u r tout m>h.
=
sont indicatifs : c e m (pour le degr6 I ) , S 2 et 1;" (pour
c) Pour m<3, tous les sous-groupes de S sent S
(pour le degr6 0), S l
le degre 2), S3, I:*', 11", I:*', J 31 , T 3 (pour le degr6 3). 3 Le nrouae . S . contient 30 sous-groupes dont I I sont indicatifs, 2 l , 1 1,2 12,1 - 2 , 2 1 , 3 13,1 1 2 savoir : S 4 ' I 4 9 I4 , 4 , 14 I4 3 4 , J 4 , T4, D4* 4m2 groupes indicatifs. Plus ggnsralement, pour m24, Sm contient 3 . 1 7 1 9
2.3.
9
9
L'un des rEsultats-cl6s de la thsorie des chaines permutges
(note C 9 1 et thPse [ I l l )
a 6t6 la mise e n evidence, pour chaque
sous-groupe G de S
d'un nature1 minimum y ( G ) qu'on peut appeler m' la c L a b h e i n d i c a t i v e de G pour les raisons suivantes : si H est la fiche de G (et H m le groupe indicatif maximum contenu dans G ) ,
alors la suite ( G ( ~ )
)nt w
des dilatgs de G v6rifie H m + n ~ G(n)
(inclusion stricte) pour O < n < y ( ~ )et H
m+n
= G ( ~ ) pour n
Corollairement (pour tout n e u et tout G
2 y(G).
: n > y ( ~ ) M G ( ~ )est
indicatif. 2.3.1.
En 1 9 6 5 ,
n o u s avions explicit6 [ I l l
les classes indicatives
Em).
AprSs y= ,O
y(G)
les calculs concernant
et leur maximum y m (quand G parcourt
pour m63, n o u s obtenions y4=2 en calculant y ( G )
pour l e s 1 9 sous-groupes non indicatifs de S 4
: y(A
4
) = 2 pour le
groupe altern6 et y ( G ) = l pour les 18 autres groupes. Pour m 5 5 , la technique employ6e (utilisant le th6oreme des bichafnes C31) donnant une majoration assez grossisre y
m
6 am'+bm+c
dont les
coefficients ( a = 9 par exemple) importent peu.
2.3.2
Plus rgcemment, 1 l'occasion de recherches concernant le
degrs d'homog6nEitE et le degr6 de transitivits des groupes de permutations, certains mathsmaticiens (P.J. CAMERON [ I ] ,
C l 5 1 , W. HODGES
-
A.H. LACHLAN
-
S.
G.
HIGMAN
SHELAH 1 1 6 1 ) ont retrouv6 ind6-
pendamment de n o s travaux ant6rieui-s (et, de ce fait, dans leur terminologie propre) la notion de "gnoupe indicatif" et s o n rSle
Relations enchainables, rangements e t pseudo-rangements
255
d a n s 1 ' g t u d e d e s r e l a t i o n s c n c h a l n a b l e s . On d o i t a i n s i a u x t r o i s d e r n i e r s c i t P s 1 1 6 1 une a m g l i o r a t i o n d g c i s i v e d e l a m a j o r a t i o n d e s
ob t e n i r c e t e m a j o r a t i o n , concernant
r de support E ,
m-aire CLI 12 5
P cu i c 6 .s
(mod.
:
~ ( a ) = be t
d'une
(mod.
t)
y(x)= x pour tout x E
rl(E-{a})
9:
la bijection
e t a,<x&b ( m o d . t ) ,
ble par r , modulo r
alors
(E-{a,b?).
On & t e n d a i n s i
DCs
(h,h+l,.
lors,
et
. . ,k) e
,
O
si r
notons c i , j m,n
Sm t e l s q u e O
pour t o u t groupe G c 1
l e lemme d e c o n s C -
m'
c u t i v i t g p e u t s ' g n o n c e r de l a faqon s u i v a n t e
:
i { i , j l e s t une
"
;In,
alors
Gqq.
Une r e l e c t u r e d e l ' a r t i c l e c l a s s e i n d i c a t i v e y(G)
1161 f a i t a l o r s comprendre que l a
peut dspendre de l a f i c h e H de G ( c 'e s t - l -
d i r e d u g r o u p e i n d i c a t i f maximum H p o u r H=T o u D ,
y(G)&Max(2,m-p-q)
Dans l e c a s d e l a f i c h e
H=Jr).
mieux p o u r m25
: y(G)$m-3.
par W.
A.H.
6
(si
s i s est libre-interprgta-
p a i r e d'ElCments c o n s 6 c u t i f s modulo l e G-rangement
m
telle
{a,b}eP2(E)) l a cons6cutivitg
(pour toute paire
des cycles
O
m >, 5
l'image de (E-{a})
i m p l i q u e l a c o n s P c u t i v i t 6 modulo s .
l'ensemble
-
est
(E-{b})+
a l o r s x = a o u x = b ) . En o u t r e ,
E t a n t d o n n P d e s n a t u r e l s n>m>,l
m,n
fin
{ a , b ? t P 2 ( E ) e s t formCe d ' C l 6 r n e n t s
e t s s o n t deux r e l a t i o n s de support E e t
c i ,j
trss
relation
r e l a t i o n une notion c l a s s i q u e pour une chafne t E f E
B toute
acb
une p a i r e
par
Pour
I e s a u t e u r s u t i l i s e n t u n lemme
lorsque l a relation
r)
l a relation rl(E-{b}) que
6 m-3 p o u r t o u t m 3 5 .
c ~ n ~ ~ c u f i w i f c V? i.s - s - v i s
n o t i o n de
a
y,
2 savoir :
classes indicatives,
HODGES,
C G ) , p a r exemple : y(G),<2
m-
pour H=Ip'q
( e t pareillement pour
triviale H=I"',
o n o b t i e n t &me
Finalement,
LACHLAN e t
S.
l e b i l a n de l a mgthode s u i v i e
SHELAH e s t
:
y,<
m-3
pour tout
( i n f o r m a t i o n 2 complCter p a r nos v a l e u r s y4=2 e t y =O pour
m
3).
2.3.3.
DCbut
de f i c h e
1983,
I"',
obtenue en
nous avons trouvE
t e l que y(G)=m-3
1976-77
( p o u r m > 3 ) un g r o u p e G c Z m ,
(d'oii i l r 6 s u l t e que l a majoration
par l e s auteurs prCcit6s 6 t a i t
l a meilleure
possible). P o u r m=2p o u m = 2 p + l , e t n > l , n o t o n s Sn l e s o u s - g r o u p e
m
d 6 f i n i de l a faqon s u i v a n t e ( I )
Si
I < n < p , S:
est
de S
m
:
l e g r o u p e (non i n d i c a t i f )
des permutations
256
C. Frasnay
conservant chacun des Elfments O , l , 2 , (au t o t a l
2n GlEments s u r m,
. . . , n-I
e t m-n,
m-n+l,
. . . ,m - 1
c e q u i l a i s s e s u b s i s t e r au moins
2 o u 3 ClSments i n t e r c a l a i r e s ) . S i nDp,
(2)
S t (d'ordre
S:
=
11"
I
=
m
(groupe i n d i c a t i f
et
l a fiche triviale II'I,
f a c i l e (s:)(')
=
Sn+ 1 m+l
'
I
trivial).
s i 2n)m)
i r i t e h c a t a i k e i d e Sm
appelEs l e s hOuh-ghoupe4 ve e s t
m
s i 2n&m, e t d ' o r d r e
(m-2n)!
Les groupcs
peuvent g t r e
: leur
fiche indicati-
ils s e d i l a t e n t s u i v a n t l a r 6 g l e
DBs l o r s , portons n o t r e a t t e n t i o n s u r l e (pour m>3)
premier groupe i n t e r c a l a i r e S'
m
:
a)
S i m = 3 , a l o r s S 1 = 1;" (groupe i n d i c a t i f t r i v i a l ) e t 3 1 S i m=4, a l o r s S 4 n ' e s t p a s i n d i c a t i f , m a i s s o n p r e m i e r y(S3)=0. I
dilatE (Si)(l)=S:
=
b ) P l u s gGnGralement, dilatEs successifs sont
(S;)(n)
=
I donc y ( S ) = I
est indicatif,
1;"
p o u r m 2 5 , S;
4
(mais y 4 = 2 ) .
n ' e s t pas i n d i c a t i f e t s e s
:
( S m1 ) (n) = S m - 3 2m-4
Sn+'. A i n s i , p o u r n = m - 4 , m+n
(laissant
s u b s i s t e r 2 ElEments i n t e r c a l a i r e s ) n ' e s t i n d i c a t i f . P a r c o n t r e , p o u r n=m-3 indicatif et
(finalement)
I
pas encore un groupe Sm-2 I,1 est : (SA)(n) = 2m-3 - '2m-3
y(Sm)=m-3.
En t e n a n t c o m p t e d e c e g r o u p e p a r t i c u l i e r t i o n obtenue dans [16],
S 1 e t de l a majora-
m
nous o b t e n o n s donc b i e n
: y,=m-3
comme
c l a s s e i n d i c a t i v e maximum p o u r t o u t m 2 5 .
2.4
De 1 9 6 3 1 1 9 7 6 , u n a u t r e r E s u l t a t - e l 6
c h a i n e s permutEes G-recollement
(prEsentC i n i t i a l e m e n t
des ordres totaux")
de l a t h 6 o r i e des
1 9 1 comme " t h E o r B m e d e
e s t apparu sous d i v e r s e s formes
S q u i v a l e n t e s . La f o r m e l a p l u s r E c e n t e de l a SociEtE f r a n q a i s e de Logique,
(annoncEe a u Colloque
S c i e n c e s , mais d o n t l a p u b l i c a t i o n a Ct6 r e t a r d s e problBmes d ' E d i t i o n ) G&Cm,
e s t a u s s i l a plus brPve
l a c l a s s e d e s G-rangements
1976
Philosophie e t MEthodologie des
1141 par des
: "Pour
est universelle".
tout
groupe
Le d e g r 6 d e
c e t t e c l a s s e ( c a r d i n a l i t 6 maximum d e s p s e u d o - G - r a n g e m e n t s )
e s t Egal
1 l a v a l e n c e v(G) du G-rangement n a t u r e 1 ( s u s c i t C p a r l a c h a ? n e -+
u s u e l l e a) : p a r a b u s d ' 6 c r i t u r e e t d e l a n g a g e , n o u s n o t e r o n s s i m -
Relations enchahables, rangements et pseudo-rangements
p l e m e n t v(G) y4OUpC
2.4.1
251
cette valence et nous l'appellerons la w u ! e ~ c e du
G. E x p l i q u o n s c o m m e n t c e t t e v a l e n c e v(G)
i n t e r v i e n t d a n s le
thGorSme de G-recollement. E t a n t d o n n 6 u n n a t u r e 1 n et u n e n s e m b l e E , o n dit qu'une f a m i l l e (Ai)icI d ' e n s e m b l e s e s t u n ~ - 4 C C 0 u w h e m e n t d e E l o r s q u e , il e x i s t e u n i n d i c e i t 1 tel q u e X & A i
p o u r tout X,Pn(E),
n o t i o n u s u e l l e de r e c o u v r e m e n t c o r r e s p o n d a u c a s n=l).
DSs
la c o n j o n c t i o n d e s d i f f 6 r e n t s t r a v a u x e f f e c t u g s d e 1 9 6 3 - 6 5
1 1972-73
(121C131)
f a i t a p p a r a i t r e l a v a l e n c e v(G)
(la lors,
(C911111)
c o m m e G t a n t le
plus p e t i t d e s n a t u r e l s n v g r i f i a n t la c o n d i t i o n s u i v a n t e : "Pour tout e n s e m b l e E d e c a r d i n a l 1El2n et p o u r t o u t e f a m i l l e d e c h a l n e s d e u x 1 d e u x G - c o m p a t i b l e s d o n t les s u p p o r t s f o r m e n t u n n-recouvrement de E ,
il e x i s t e u n e c h a f n e s d e s u p p o r t E q u i e s t
G - c o m p a t i b l e a v e c c h a c u n e d e s c h a f n e s t . ( p o u r i eI)". S i deux relations m-aires r , r '
d e s u p p o r t r e s p e c t i f s A,A',
sont d i t e s c o r n p a t i 6 L e h d S s q u e rl ( A n A')=r'l ( A n A'),
o n obtient une
v a r i a n t e ( t r s s proche) d e l a c o n d i t i o n p r 6 c G d e n t e e n p r e n a n t u n e f a m i l l e (ri)it I d e G - r a n g e m e n t s d e u x 1 d e u x c o m p a t i b l e s , c e q u i p e r m e t d ' i n t r o d u i r e u n e r e l a t i o n r l e s a d m e t t a n t c o m m e restrictions: r
i
rlAi.
=
D S s lors, l a valence v ( G )
est le p l u s petit des naturels
n t e l s q u e : "Si les n - r e s t r i c t i o n s d'une r e l a t i o n m-aire r (de c a r d i n a l i t s )n) 2.4.2
s o n t d e s G - r a n g e m e n t s , a l o r s r e s t u n G-rangement".
L e s t r a v a u x c i t e s prec6dernment ( n o t a m m e n t C 1 1 1 1 1 3 1 )
rnontrent a u s s i q u e la v a l e n c e m - a i r e m a x i m a l e v m ( c a r d i n a l i t s m a x i m u m d e s r e l a t i o n s m-aires "finies" e n c h a l n a b l e s et d g f o r m a b l e s ) e s t l a p l u s g r a n d e d e s v a l e n c e v(G)
l o r s q u e G p a r c o u r t Cm. Autrement
d i t , p a r m i l e s r e l a t i o n s m-aires e n c h a l n a b l e s et d s f o r m a b l e s d e cardinalits finie maximum v o n est assurg de trouver certains m' G - r a n g e m e n t s . N o u s d i r o n s qu'un g r o u p e G c 1m e s t u n g4Oupe C 4 i t i g u e s'il e x i s t e u n p s e u d o - G - r a n g e m e n t d e c a r d i n a l i t s m a x i m u m v m (autrement d i t : s i v(G) = vm). a) Nos c a l c u l s 1111 et c e u x d e P. J U L L I E N [ I 8 1 o n t m o n t r 6 q u e l a v a l e n c e d e s g r o u p e s i n d i c a t i f s G c C m' e s t e n g e n 6 r a l v(G)=m+l. S e u l s f o n t e x c e p t i o n les g r o u p e s s y m s t r i q u e s S m et l e g r o u p e 2
J 2 : v(S )=m et v(J4)=6. 4 m
C Frasnay
258 b) L o r s q u ' u n g r o u p e G c E
m rnontrg [ I l l que s a v a l e n c e v(G)
n ' e s t pas i n d i c a t i f , n o u s a v o n s e s t li6e 2 s a c l a s s e i n d i c a t i v e
p a r l'insgalitg : v(G)<m+y(G)tI.
C e t t e i n 6 g a l i t E (valable 2 e n c o r e p o u r les g r o u p e s i n d i c a t i f s a u t r e s q u e J 4 ) , j o i n t e B la y(G)
m a j o r a t i o n d e y m (HODGES tion : v
m
<
-
LACHLAN - S H E L A H ) ,
c o n d u i t 5 la m a j o r a -
2m-2 p o u r tout m 3 5 .
c) P a r a i l l e u r s , n o s c a l c u l s d i r e c t s r l I I a v a i e n t fait conna'itre tous les g r o u p e s c r i t i q u e s d e d e g r 6 m C 4 et les v a l e u r s : 2 v =0, v l = l , v 2 =3, v 3 = 4 , v 4 =6. P a r e x e m p l e , le g r o u p e i n d i c a t i f J 4 et le g r o u p e a l t e r n 6 A 4 s o n t d e s groupes c r i t i q u e s d e d e g r 6 4 . F i n a l e m e n t , la m a j o r a t i o n v < 2 m - 2 v a u t p o u r tout m23. 'm 2.5
Nous allons dsmontrer l e th6orSme suivant (nouveau, 2 notre
connaissance)
Lai4e SA 2.5.1 11, I
v(s,)
I
:
e h t un
' ' P O U ~t o u t natu4e.l m > 3 , L e p4emiefi gficrupe i n t e . f i c a g h o u p ~c f i i t i q u e . de. v a l e n c e v m - 2 m - 2 " . O n s a i t (2.3.3) q u e S mI e s t un g r o u p e d e f i c h e
Preuve :
,
I
d e c l a s s e i n d i c a t i v e y ( S )=m-3 ( m a x i m u m s i m2,5), d e v a l e n c e m I I m+y(Sm)tl = 2m-2. P u i s q u e v(S ) v 6 2m-2 ( 2 . 4 . 2 ) , i l m m
<
s u f f i t d e p r o u v e r q u e 2m-3 < v(S,)
1
: n o u s a l l o n s le f a i r e e n u t i -
l i s a n t le t h 6 o r b m e d e r e c o l l e m e n t des cha'ines et e n r a i s o n n a n t par l'absurde. S i v(SA)<2m-3,
o n pourrait appliquer la condition de
1
S - r e c o l l e m e n t ( g n o n c 6 e e n 2.4.1)
m
E
=
2m-2
=
{O,l,2,
. . . ,2111-31
a u x n a t u r e l s n = 2 m - 3 et -f
=
ntl. Soit r=wlE la chaine usuelle :
vis-5-vis d u g r o u p e i n t e r c a l a i r e G r Z (G=Szrn-2 m-2
=
lit6 d e r d a n s
(m-Z,m-I)}),
cE.
d e degri? 2m-2 et d ' o r d r e 2 ntl s o i t {r,sl la c l a s s e d e G - c o m p a t i b i -
I n t u i t i v e m e n t , o n peut d i r e q u e l a c h a i n e s se d E d u i t d e l a cha'ine u s u e l l e r e n i n t e r v e r t i s s a n t les d e u x 6 1 6 m e n t s m g d i a n s m-2, m-l
:
. . <m-3<m-2<m-l<m< . . . < O
O<1 <.
Sur chacune des n-parties Ai u n e cha'ine t i (de s u p p o r t A . )
2m-3
(mod. r)
2m-3
(mod. s)
=
E-{i}
e n posant :
d e E (OCiCn), d g f i n i s s o n s
259
Relations enchaina bles, rangernents et pseudo-rangenzer~ts pour 0 6
i : m-3
t.
=
rl A;
L.
=
r l A i = s j ~ .p o u r i = m - 2 et i=m-l
t. = s ( A i pour
m 6 i 6 2m-3
O n o b t i e n t d e s p a i r e s i t . , t . ! dt. c h a l n e s c o m p a t i b l e s (donc 1 . 1
S I - c o m p a t i b l e s ! ) p o u r : 0 6 i f j < m - l et m - 2 < i c j < 2 m - 3 ( n o t a m m e n t (i,j)
=
(ni-2,m-I)
d a n s c h a c u n de c e s d e u x cas).
avec t . = r l A i , t . = s l A J j' A i n A . = E - { i , j 'i, I A . fl A . = 2 m - 4 , et d G s l o r s t i ( A . f ) A , ) , t . ( A i (1 A . ) 1 l J 1 1 J J m- 3 ni- 3 e s t aussi le dilate sont S 2 m - 4 - c o m p a t i b l c s . Comiiie le g r o u p e S2m-4
iieste le cas
: OciGm-3 e t
m;ji2m-3,
I
I
(s
i1)
)
(
1
, o n e n d 6 d u i t e n c o r e q u c t 1. , tJ. s o n t S mI - c o m p a t i b l e s . I
n (et p u i s q u e ( A 1. ) o
D a n s l ' h y p o t h c s e v(Sm)
n - r e c o u v r e m e n t "exact" d e E=n+l),
le thi?ori!me
est un
d e r e c o l l e m e n t des
d'une c h a f n e 0 d e s u p p o r t E q u i
c h a l n e s i m p l i q u e r a i t l'existence
s e r a i t S I - c o m p a t i b l e a v e c cliacune d e s c h a f n e s t i . O n a u r a i t : m 1 elAi t i (mod. S ) p o u r O s i G n , a v e c I A i / = 2 m - 3 . C o m m e l e dilate d e m ~
et les c h a l n e s t i s e r a i e n t d e u x 5
(pour Oci(n),
aurait : t. = B I A i
d e u x c o m p a t i b l e s , e n c o n t r a d i c t i o n a v e c m-2rm-I
(mod. t . )
et
m-lcm-2 (mod. t.) p o u r 0 6 i C m - 3 , m < j < 2 m - 3 . J
I 1 e n r e s u l t e : 2m-3Cv(S bien apparaftre
2.5.2 dente),
I
1
)<2m-2, d o n c 2m-2=v(S )
Corollairement
( d a n s les n o t a t i o n s d e la p r e u v e p r 6 c 6 -
p u i s q u e l e s cha?nes t . s o n t S ' - c o m p a t i b l e s , e l l e s s u s c i t e n t m I
s u r l e s n - p a r t i e s A; d e E d e s l S - r a n g e m e n t s c o m p a t i b l e s SLxtjm).
m E u n e et u n e s e u l e r e l a t i o n m-aire 11 t e l l e q u e
I1 en rgsulte SUP I $ ( A i = Smxtim) ( p o u r Osil'n).
P o u r O<,k*n, t o u t e s l e s k - r e s t r i c t i o n s
1
d e UJ s o n t d e s S - r a n g e m e n t s : p a r c o n t r e $=(m,E,p) (de c a r d i n a l i t 6 m n + l = 2 m - 2 ) n ' e s t p a s un S I - r a n g e m e n t , p u i s q u ' a u c u n e c h a l n e 0 (sur m E) n'est S 1 - c o m p a t i b l e a v e c c h a q u e cha'ine t i . D o n c $ e s t un p s e u d o m r a n g e m e n t m - a i r e d e c a r d i n a l i t 6 m a x i m u m v m = 2m-2 ( p o u r m>3). S o n g r o u p e d ' i n v a r i a n c e e s t Inv((l)=Sm caractgristique est
101
I
=(m+i)(m+2)
et l e c a r d i n a l d e s o n e n s e m b l e
. . . (2m-2).
260
C. Frasnay
2.5.3
La proposition selon laquelle des rangements m-aires
(notamment des S1-rangements) figurent toujours parmi les relations m m-aires enchafnables critiques se laisse prgciser, pour tout groupe G € C m , de la manisre suivante ( 1 2 1 1 1 4 1 )
:
a) Soit (t,r) une birelation monomorphe de support E , de cardinalit6 ( E l >/ m, comprenant une chaine t et une relation m-aire r (autrement dit, r est une relation encha?nable et t est l'une des chafnes par lesquelles r est libre-interpretable).
ex
partie X de E , soit t / X ) sur m={O,l,
l'injection croissante de X (ordonnge par +
. . . ,m-I}
(ordonng par la chafne usuelle m ) .
Lorsque
'L
X parcourt P,(E), r'
Pour toute m-
la relation €I (rlX) garde une valeur constante X e x : X + m est l'unique isomorphisme
(m,m,p) et la bijection
=
-f
de la birelation (tlX, rlX) vers la birelation (m, r'). groupe d'automorphismes G=Aut(r')
est un sous-groupe de S m dont la
connaissance redonne le groupe d'automorphismes Aut(r/X)
{zi o
=
u
8x
o
*
a
u
E
Puisque le
locaux :
G j , o n peut dire (par abus de langage)
que G est le " g t r a u p e l?acaf?l' (mod. t) de la relation enchainable r. La relation r' est l'une des 2v(m) m={O,l,
. . . , m-I})
relations m-aires (de support +
libre-interprgtables par la chaine usuelle m ( u ( m )
d6signe le nombre des preordres totaux de support m : v(O)=u(l)=I, v(2)=3,
u(3)=13,
v ( 4 ) = 7 5 , u(5)=541,
G d Cm, le G-rangement r=Gxt(m)
. . .) .
Par exemple, pour tout
suscit6 par la chafne t admet G
comme "groupe local" (mod. t) puisque r'=(m,m,G). b) D S s lors, l a preuve de ( [ I l l , v(G)
13.1.2)
montre que la valence
est la cardinalit6 maximum des relations m-aires "finies'',
enchafnables et dgformables, admettant G comme "groupe local". Le th6orSme 2 . 5 concerne les cas oii ce "groupe local" est le premier groupe intercalaire S;
v(S
= S l ,
1
3
I
S2
=
I:",
s: . Pour mi3, ce groupe est indicatif S:
=
I;",
:
1 1 de valence v ( S 1 ) = l = v l , v(S 2 )=3=v2,
)=4=v3. D ' o C les deux propositions : (I)
Pour
1 LOU^ m>,l, Sm est un groupe critique (autrement dit, sa
1
valence v ( S )=v est la cardinalit5 maximum des relations m-aires m m "finies" enchaynables et dEformables). (2) v = I , 1
v = 3 et v =2m-2 2 m
pour tout m a 3 .
Relations enchainables, rangements et pseudo-rangements c) B i e n e n t e n d u , les g r o u p e s S ;
n e s o n t pas n E c e s s a i r e m e n t les
seuls groupes critiques. En dehors de S
I
l
l
S 2 , S 3 , il y a a u s s i 5 1 2 , I32,l , T 3 , .J3, .I4.
autres groupes indicatifs critiques : I3 le d e g r 6 4 , le g r o u p e [ c 4 , (1,3)1
261
Pour
(de f i c h e i n d i c a t i v e I"')
et
1
le g r o u p e a l t e r n e A 4 (de f i c h e i n d i c a t i v e J ) s o n t d e s g r o u p e s criI
tiques non indicatifs autres que S 4
=
{ t 4 , ( l , 2 ) } .
Pour m > 4 , 1'inE-
\< m+y(G)+I
g a l i t 6 v(G)
e t 1 ' 6 g a l i t E v = 2 m - 2 m o n t r e n t qu'un g r o u p e m c r i t i q u e G n o n i n d i c a t i f doit a v o i r la c l a s s e i n d i c a t i v e m a x i m u m y(G)=m-3,
m a i s c e t t e c o n d i t i o n n k c e s s a i r e n'est p a s s u f f i s a n t e
(par e x e m p l e , p o u r le g r o u p e G v(G)=5).
=
{r,,
1
m
)=2m-2,
de d e g r E m=4:y(G)=l
et
I
- y ( S )=m-3 n ' i m p l i q u a i t pas m une d s m o n s t r a t i o n d u t h G o r E m e 2 . 5
Pour cette raison, puisque
n g c e s s a i r e m e n t v(S
(O,3)}
s ' i m p o s a i t !.. D e m z m e , l e s d e u x e x e m p l e s d e H O D G E S - L A C H L A N - S H E L A H [I61
(montrant q u e l e u r m a j o r a t i o n y m 6 m - 3 e s t "pointue", s o i t -clans I 1 d o n n e n t seulement:v(A ),<8,v(S & l o . 5 6 En reprenant la mgthode de L 1 6 1 donnant pour y(G) une majoration qui depend de
n o s n o t a t i o n s : y(A5)=2,y(S6)=3)
la f i c h e i n d i c a t i v e H de G , o n p e u t c o n j e c t u r e r q u e H=I'"
ou J
1
pour t o u t g r o u p e c r i t i q u e G de d e g r E m > 6 . Les g r o u p e s a l t e r n E s A =I1", 2
2
A =T3 e t A 4 s o n t d e s g r o u p e s c r i t i q u e s (et i l e s t p o s s i b l e 3
q u e A 5 le s o i t Ggalement). C o m m e l a f i c h e i n d i c a t i v e de A v a u t m I J , D , 1'" , T r e s p e c t i v e m e n t p o u r m F O , 1 , 2 , 3 (mod. 4 ) , il e n ) ~ l 2 orsque m est impair : ainsi A n'est pas u n g r o u p e m m c r i t i q u e p o u r m i m p a i r > 7 , p a r c o n t r e i l est p o s s i b l e ( ? ) q u e A m
r 6 s u l t e y(A
s o i t u n g r o u p e c r i t i q u e p o u r tout m p a i r > 2 .
2.6
P o u r t e r m i n e r c e t t e E t u d e , n o u s a l l o n s examiner(d'apr2s
M. P O U Z E T , 1221) c o m m e n t l a t h e o r i e d e s c h a f n e s p e r m u t g e s p e u t c o n t r i b u e r 1 l a d g t e r m i n a t i o n d e c e r t a i n s g r o u p e s de p e r m u t a t i o n s d'un e n s e m b l e E , d i t s g r o u p e s h o m o g P n e n . D a n s l a d g m o n s t r a t i o n d'un t h 6 o r S m e 1 c e s u j e t p o u r E inf i n i , d G 5 P . J . C A M E R O N (II ] ,
§
6 ) , les
1
, J m , T m , D m (ainsi q u e les c o m p a t i b i g r o u p e s i n d i c a t i f s S m' 1''' m l i t g s a s s o c i E e s d a n s l'ensemble des chafnes sur E ) intervien-
XE
nent d a n s u n e t e r m i n o l o g i e v o i s i n e d e l a n 6 t r e - t o u t e f o i s l'auteur n ' u t i l i s e p a s l a p r o p o s i t i o n 8.3.2. de r i l l r s p e r t o r i a n t t o u t e s l e s suites indicatives ( S ,
IP q , J r , T , D) e n g e n d r g e s p a r les b i c h a f n e s 9
(s,t) d e s u p p o r t E i n f i n i .
C. Frasnay
262 2.6.1
l e s 8 r e l a t i o n s binaircls
I'armi
et
l e s 8192 r e l a t i o n s
"
un r 6 l e t e r n a i r e s l i b r e - i n t e r p r 6 t a b l e s p a r une chaEne t c 'C' (dont l'ensemble
p r i v i l6 g i 6 e s t accord6 2 l a chafne Upl~uSCct ' c a r a c t 6 ri s t i q u e e s t celui des
(x , x , ) e E2 t c l s q u e x 1 6 x o ( m o d .
e t 2 l a r e l a t i o n C ~ h c u ~ ! u c h ce d I 5 r i v g e d e t
x 1 <, x 2 6 x o o u x 2
<
xo
<
la relation circulaire c' (1)
y
(t)
t
(2)
4'
y(t)=t
Pour 2 $ m < l E l ,
6 SE transforme
E
(t)
On l e u r a d j o i n t g g a l e m e n t
de t e s t
-2 E'
l a chaene t e n une c h a r n e
l a c l a s s e de I 1 ' l - c o m p a t i b i l i t G de t e s t { t i .
m
'f
l e groupe des permutations
: ces permutations
y
(ou
de t e s t { t , t ' ) .
yc
( t ) l e groupe des permutations
SE t e l l e s que
E 'L t r ( t ) = t ' , ce groupe e s t a u s s i c e l u i des permutations
%
y ( t ) = t ou
t-monotonen
que
Aut(t).
=
(3) P o u r 3 < m < l E I , l a c l a s s e d e J ' - c o m p a t i b i l i t 6 m note J
SE t e l l e s
s o n t l e s automorphismes de t
t - ~ K o i n o n n f e o ) ,d o n c IE(t)
permutations
S i l'on
011
( m o d . Sm).
On n o t e e n c o r e I rv
t)).
x2
dSriv6e de l a chafne t ' .
I<m$lEl, l a c l a s s e d e S - c o r n p a t i h i l i t 6 m
Pour
e t toute permutation
-.
x 1 (mod.
<
xI
: xo
( x o , x I , x 2 ) e L3 v g r l f i a n t
t g r i s t i q u e e s t c e l u i des
t))
(dont l'ensemble cnrac-
(autrement d i t : t-croissantes
ou t-dEchol46nntel).
( 4 ) Pour 3<mhlEl, l a c l a s s e de T - c o m p a t i b i l i t G d e t e s t consm t i t u g e p a r l e s c h a ' i n e s s u r E a d m e t t a n t c comme r e l a t i o n c i r c u l a i r e
ye
d6rivGe. Pour t o u t e permutation
SE,
la relation circulaire
Y
%
~ ( t n) ' e s t a u t r e q u e
d6riv6e de l a c h a i n e
l e groupe des permutations
T E ( t ) = A u t( c ) .
y
(c).
S i TE(t) d6signe
'L
Y
sE
que
y,(c)=c,
alors
:
(5) Pour 4,<m$lEl, l a c l a s s e de D - c o r n p a t i h i l i t & de t e s t consm t i t u s e p a r l e s c h a f n e s s u r E a d m e t t a n t c o u c ' comme r e l a t i o n c i r c u l a i r e d 6 r i v 6 e . On n o t e DE(t)
y
g
SE t e l l e s que
-
'L
~ ( c ) = co u
Par abus d ' c c r i t u r e ,
'L
y
l e groupe des permutations (c)=c'.
lorsque l a chayne t e t l a r e l a t i o n
c i r c u l a i r e dGriv6e c s o n t f i x s e s une f o i s p o u r t o u t e s , on n o t e s i r n p l e m e n t I E , J E , T E , DE l e s g r o u p e s p r 6 c 6 d e n t s .
On p e u t d i r e a l o r s
q u e I E ( r e s p . T E ) e s t l e g r o u p e d e s p e r m u t a t i o n s d e E q u i C(JnA@KW C I ? ~l a c h a i ' n e t ( r e s p . l a r e l a t i o n c i r c u l a i r e c ) , e t q u e J ( r e s p .
E
D ) e s t l e g r o u p e d e s p e r m u t a t i o n s d e C q u h conneKwen-t E ntnt la chaIne t (resp. l a relation circulaire c ) .
o u Ketouh-
Relations enchainables, rangements et pseudo-rangemento 2.6.2
So1t
c
sb . Pour n
a)
treillis des sous-groupes
$1,
un g r o u p e G e
~ o u p l e( X , Y )
tout
'f
le
de n-parties
y
2 G t c l l c que
on d i t
simplement
X'=
que G est
permutation e t
iil
(ce qui
i
I!J
sE
de
E,
S i G
dit
est
E
du g r o u p e s y m 6 t r i q u e
~i-/ioniug?uie d S s
que,
pour
tout
infini),
on d i t qu'une
e s t ccdhPtrcntc 2 u n g r o u p e ~t z E s i , p o u r m x t 1; , i l e x i s t e y 6 G t e l q u e \)ox=
2 d i r e que
neiri,
hort~cigCnp.
t o u t m-uple
revient
pour
i l e x i s t e une p e r m u t a t i o n e s ~t n - ~h o m o g s n e
(essentiellement pour E
Par a i l l e u r s
m
Y.
il:
263
y
e t
h/J
o n t mzme
tout
Y
Ox
sur la
restriction
...,
p a r t i e f i n i e X={xo,xl, x 1 de E ) . L'ensemble d e s p e r m u t a t i o n s m- I d e 1: a d h 6 r e n t e s a u g r o u p e G e ), e s t a l o r s un g r o u p e G e IE ( q u ' o n E p e u t a p p e l e r ( c t n ~ c t u k cde G ) c t , l o r s q u e G=G, on p e u t d i r e q u e l e
$etitni'.
groupe G e s t b)
Lorsque
l'ensemble
des chaTnes s u r E c o n t i e n t
E est
i n f i n i dgnombrable,
un s o u s - e n s e m b l e
d e s c h a E n e s d Q n b c 5 sans m i n i m u m n i m a x i m u m chafne
u s u e l l e du corps ordonn6 Q des
peut Gtablir G
soit
&
IiomogPne,
et une
t e l telles
quc Aut(R)
-
= G
indicative
(H
DSs
: "Pour
-
Lemme
Sc A
a P u e 5 La elllYle
A
U
(s,t) )
de support
infini E,
on peut est
c t t A u n t d c u x c h a i n e s d c mPnic ~ ~ p p o f i .Et e t ~
n4
iCn d t c a t ( w e
dc4 5 Ju4tt5 En v e r t u d e
et
suppose que t
s'il
pour
identiques
m>2r,
s, r 1 J
H e n g e n d k h e pati La J , J 1 , T , D.
la proposition 8.3.2
( p a r exeniple)
montrons qu'alors
ou opposEes
par
t
associer
le groupe
5 t
L'une
maxcmum ( c e q u i impose que E e s t i n f i n i ) , bcchaine
de [ I l l ,
(4,X)
ebt
la preuve consiste
I p ' q ou J r p o u r r = M a x ( p , q ) 3 2 .
n'a
n i minimum n i maximum d a n s
e x i s t e un n a t u r e 1 r < 2 t e l q u e s e t
tout
on
groupe
e x i s t e une chazne
dont chaque
2 Gcarter toute suite indicative l'on
lors, qu'un
libre-interprGtable
m m e 111 tfrme H m G m ' d e c o r n p a t i h i l i t 6 minimum d e s , t p o u r l e d e g r E m .
d ' e t ! c ) A n ' a nc mcnmuni
e,
: celui
."
c ) A toute bichafne la suite
"22 I
il f a u t e t il s u f f i t q u ' i l
structure relationnelle R
'
+
isomorphes 1 l a
(toutes
rationnels).
la caractcrisation suivante
l'ensemble
remarquable
t
soient
Si E,
Jr-compatibles
m 1 s e t t sont J3-compntihles
: s = t ou s = t ' ) . P r e n o n s m=2r+3
: pour
(donc tout
264
C. Frasnay
triplet t n'a
u
( x ~ , x ~ , x ~t e)l ~q uEe ~x < x < x 2 ( m o d . 0
n i minimum n i maximum)
= xo,
sait (d'aprss
Jr m
I
=
Jm
0
t e l l e que u
(7
1
I r , r) que
* '
t).
< x,
< x2 o u x 2 < x
l o J3), c e qui gquivaut
< x
1
(d'aprPs
m
s ) , on
(r,r+l,r+2)
:
s ) . Finalement 6.2.3)
[ I l l ,
.Jr
IrPr e t m
le triplet (mod.
t e l que
u e
Si (mod.
.<' o(m-I)
"conserve ou retourne"
0
m
(mod,
(mod.
m- 1
(o)
,. . . , u m- 1 ) e E m
l a d6finition des groupes i n d i c a t i f s
il e n r h s u l t e x s :t
(uo , u l
~ + ~ e= t xu ~< u < . . . < u
U ~ + ~ = X ] u ,
la permutation
est
u n m-uple
il existe (puisqe
t ) ,
1
1 : s = t
ou s = t ' .
Remarque
: En m o d i f i a n t
lgggrement l a r6daction de la preuve,
nous
a u r i o n s pu prendre m = 2 r + l . d ) L o r s q u e E e s t un e n s e m b l e i n f i n i d g n o m b r a b l e , structure relationnelle chafne t t
2,
(de support E)
( c h a f n e d e n s e sans m i n i m u m n i m a x i m u m ) .
c o u r t l e groupe des automorphismes de R, prGc6dent 1 l a bicha7ne c e t t e technique,
: "Lorsqu'une
support E i n f i n i dgnombrable,
,2E,a l o r s
5 groupes SE, I E ( t ) , J E ( t ) ,
2.6.3
Rappelons qu'un
tout couple tation O
ye :
=
yi).
'p
par-
~ ( t ) .E n u t i l i s a n t la propo-
structure relationnelle
(de ses automorphismes)
R,
de
p a r u n e chai'ne est l'un
des
TE(t), DE(t)".
groupe G
(x,y) de n-uples G t e l l e que
'f(xi)
=
est libre-interprgtable
l e groupe Aut(R)
y
Quand
21
t e l l e que s
(s,t)
p a r une
o n p e u t a p p l i q u e r l e lemme
POUZET d s m o n t r e a l o r s t r e s s i m p l e m e n t
M.
sition suivante r22J
t t
s o i t K une
libre-interprgtable
e s t d i t tz-Zfiun4ibi6 s i , p o u r
):E
i n j e c t i f s dans E, o x = y
il e x i s t e une permu-
(autrement d i t ,
pour
Un t e l g r o u p e e s t 6 v i d e m m e n t n - h o m o g s n e ,
mais p a r e x e m p l e ( l o r s q u e l e c o r p s Q d e s r a t i o n n e l s e s t o r d o n n 6 p a r sa cha'ine u s u e l l e )
I-transitif 2-transitifs 3-transitif
: l e g r o u p e homogsne
s a n s Gtre 2 - t r a n s i t i f ,
s a n s letre 3 - t r a n s i t i f s , sans S t r e 4-transitif.
q u e SQ e s t n - t r a n s i t i f Rgciproquement, relationnelles
I
Q
est
les groupes homogenes J
l e g r o u p e homog6ne D Bien entendu,
Q
Q
et T
Q
sont
est
le groupe symstri-
pour t o u t n t w.
l a p r o p o s i t i o n de M.
POUZET s u r l e s s t r u c t u r e s
de s u p p o r t i n f i n i dhnombrable ( s u s c e p t i b l e d ' u n e
preuve simple u t i l i s a n t
l e lemme de l a t h 6 o r i e d e s cha'ines
permutges
Relations enchainables, rangements et pseudo-rangements
265
cit6 prGc6demment) donne comme corollaire une version du beau t h 6 o r S m e de P.J. C A M E R O N ( r l l ,
6) : So4f G u n g k u u p e AekrnP de
§
pe+tmutatcon5 d l u n e n b e m b t e E c n d i n c dPnornbkable. l o k d q u e G e d t n-tfian4ct4d p o u k t v u t n e u , n l o f i b G = S E .
e t n-tfian54t4d
ban5
e x 4 5 t e une c h a i n e t e tetPc que : ( I )
o u bien
(2)
OU
(3)
ou b i e n
2.7
2,
G e b t hornog@ne
i c h a i n e d e n J e nanb m-cn-cmum n c max-cmum)
et G
n
=
1,
b ~ e n n
=
2, e t G
S u r le m g m e
Lokdrjue
C t a e [n+l)-tfianbct4d, a t o m n = 1 , 2 , 3 e t it
n = 3,
et G
=
IE(t),
= J E ( t ) 011 =
T,(t),
DE(t)."
thsme, signalons 6galement un article d e
G. H I G M A N 1 1 5 1 d a n s l e q u e l l'auteur r e t r o u v e (dans u n e t e r m i n o l o g i e propre) p l u s i e u r s t h 6 o r S m e s i m p o r t a n t s c o n c e r n a n t : d'une p a r t les r e l a t i o n s m o n o m o r p h e s (qu'il
a p p e l l e r e l a t i o n s homogcneb),
d'autre
part l e s g r o u p e s d ' a u t o m o r p h i s m e s l o c a u x d e s r e l a t i o n s e n c h a f n a b l e s (la t h s o r i e d e s c h a i n e s p e r m u t g e s e s t , e n f i n d e c o m p t e , l'6tude d e c e s groupes). 2.7.1.
P a r e x e m p l e les p r o p o s i t i o n s 2.1 et 2 . 3 de 1 1 5 1 s o n t les
v e r s i o n s i n f i n i t i s t e s des p r o p o s i t i o n s 12.2.1 et 5 . 4
d e 1111 s e l o n
l e s q u e l l e s ( 2 p a r t i r d e s e u i l s f i n i s c o n v e n a b l e s n ( m ) et 5 ( n ) m a s s o c i 6 s a u x c a r d i n a u x m < w , m
e s t e n c h a f n a b l e " et (2) ( l e m m e
d ' e n c h a f n a b i l i t 6 r e s t r e i n t e ) "toute b i r e l a t i o n de c a r d i n a l i t 6 >/c,(n),
f o r m g e d'une c h a f n e et d'une
n - r e s t r i c t i o n monomorphe".
relation m-aire, admet une
L a m 6 t h o d o l o g i e de la p r e u v e d e ( 1 ) ( 2 )
s e m b l e a u s s i s'imposer a v e c u n e s o r t e d e n 6 c e s s i t 6 (puisqu'il s'agit de p r o b l s m e s c o m b i n a t o i r e s "1 s e u i L s num6riques")
: l'utRka6iLtna-
t i o n , c o m m e t e c h n i q u e d e p a s s a g e d u f i n i 2 l'infini e t r g c i p r o q u e m e n t ( [ I l l , 1.3 et 3 . 4 ) , qu'on a i t d 6 m o n t r 6 ( 2 )
intervient dans la preuve de ( I )
k
r e l a t i f a u x a p p l i c a t i o n s f : P,(E)
(nornb4en de R A M S E Y
aprSs
e n f a i s a n t a p p e l a u t h 6 o r S m e d e F.P. R A M S E Y +
F p o u r (F(,
: il e x i s t e a l o r s u n e n - p a r t i e X d e E p o u r
l a q u e l l e la r e s t r i c t i o n d e f 1 P (X) est constante). m 2.7.2.
P a r a i l l e u r s (sous les n o t a t o n s
retrouve les suites indicatives H
=
s,
Hm)m 6 w
A"~,
B',c,D)
G. H I G M A N
d o n t il c o n f i r m e
266
C. Frasnay
e n montrant: que l e u r s
l'intGri5t
apparaissent
toujours,
c e r t a i n rang)
de manicre asymptotique
dans des
"order-coherent
2 nos s u i t e s d e d i l a t a t i o n
titre
quc
(G("))n,
travaux de M.
les
"pour
toute
nalitG
Ji,
TE, D E "
Pour m <
seuils [inis
de
Aut(r)
de
c a r d i n a l 2\J(m))
est
par
son " g r o u p e
on peut
dire aussi
t o t a u x d e s u p p o r t m=[O,l,
quc H e s t
l a "fiche
Pour m
r e rm(n)
<-I
-' r
' t
HIGMAN,
r
(n')
une "classc
indicative"
( v n ) (rn+C'j(r)
(maximum d e s indicative
4m
r m ( i L i ) ) . I1
est
: 'A4
l'6galiti.
(qui
indicative"
a(r)
de
la
:
relation
canonique l ~ m ( n ) ,r m ( n ' ) )
En v e r t u d u t l i G o r S m c d e
(de fiche
qui est
indicative
H)
admct
aussi
l e n a t u r e l minimum t e l q u c
-.:+ A u t ( r / n ) = H n ) naturel
G,
on r e t r o u v e
y,
facile d'obtenir
% 2 e t =
rs
15
1
=
ii2
= 0,
i n a i s nous i g n o r o n s
( c l a s s e i n d i c a t i v e maximum d e s 8 1 9 2 r e l a t i o n s
encha9nables
f,
fm ( f o )
l'inGgalit6
d G j 2 l a v a l e u r <S3 ternaires
r i m
m e n t r e l a c l a s s e i n d i c a t i v e y, y(G) l o r s q u e l e g r o u p e G p a r c o u r t 1. ) c t l a c l a s s e m (maximum des 6(r) lorsquc la relation r parcourt
d'oti
y(C)=tS(d),
. . . ,m - 1 1 ) .
sa r e s t r i c t i o n
bijective
: r=r'In.
s i r e s t l e G-rangement
Bien entendu,
par
( e n t r e les dcux ensembles
relation r e
toute
(on r a p p e l l e que .'(m)
11
1 o c a 1 " G = ~ u t ( r J m ) ,d e f i c h e i n d i c a t i v e tl
est definic par restriction
ment
usuelle
m
fournit
indieatif
CAMERON ! I ] ,
notons I '
I' (n) e s t d G t e r m i n P e
6
de cardi-
(n) l'cnsemble f i n i de m 1 e s r e l a t i o n s m - a i r e s ( d c s u p p o r t n)
111,
l a cha'ine
l e nombre d e s p r e o r d r e s
Toute r e l a t i o n r
r).
C
constitug par
libre-interprgtables
t c l s clue
!I
SE e s t d e t y p e
(dans
e t m ,C n
[,I
I15 I renuu-
l a t h e o r i e d e s chaEncs permu-
l e "main theorem" d e P . J . r: I , 1 s e u l s i n t e r v i e n n e n t l e s t y p e s S E , Tli , J E1 , T E , D E ) .
2.7.3.
A u InSme
57 et 8 ) .
enchaPnable r de s u p p o r t E c t
le sous-groupe
m'
d'un
un peu a n a l o g u e s
d e 6 . HlGMAN
travail
le
l'existence
r e l a t i o n m-aire
1El>m+6
SE, I S ' t ,
(I I I I ,
st!
v e l l e e t approfondit avantageusemcnt : il montre cn e f f e t
(2 partir
sequences"
POUZET ( 1 2 1 1 1 2 2 1 ) e t d e W . H O D G E S -
A.H. L A C H L A N - S. S H E L A H 1 1 6 1 , tGes
(les groupes i n d i c a t i f s )
tcrmes H
r t 1 7 3 ( t ~ ~ ) ) . Notre calcul de
f i m >, m - 3 ?
pour
tout m 3 5.
A-t-on
y ni d o n n e s e u l e (ou non)
:
Relations enchaina bles, rangernents et pseudo-rangernents
267
B I B L LOGRAPH I E
I I 1 CAMERON, ordered
I:!I C L A R K ,
I).
EKDOS, try,
sets,
P.,
and
R.,
relations,
I 5 I FRATSSE, formule
R.,
des
par
R.,
(1971),
1 9 i FRASNAY,
Alger,
et
e t
1974),
M.,
POUZET,
C.R.
Acad.
I'OUZET,
Sc.
Pf.,
Paris,
Sur une
C.R.
de permutations
finies
C.,
Groupes
de cornpatihilit;
thGorie
A-259
Quelques
totaux et
(Grenoble,
1 1 2 I PRASNAY, C . , tntaux
(1964),
les
problcmes
1624-1627.
relations
Sc.
e t
Paris,
faniilles
relations,
de deux o r d r e s
d'orcires
Acad.
C.R.
r e l a t i o n m-aire,
combinatoires
d e G-recollcnient
: Permutations
in
Villars
1974),
mcnt d ' u n 865-868.
(1971),
Acad.
monomorphes,
; application aux relations
C.,
d'une relation
Sc.
totaux C.R.
;
Acad.
concernant
Ann.
Tnst.
les
Fourier,
1965) 415-524.
multirelations,
I I 3 I FRASNAY,
Gauthier-
3910-3913.
relations
'I'h6orGme
(Paris,
des
d'une
a p p l i c a t i o n a u x n-morphismes
15-2
A-272
f i n i de bornes,
2944-2947.
C.,
en th6o-
1972),
275-278.
(1963),
ordres
et
102-113.
relations
(Paris,
c l a s s e de
A-257
I I I FRASNAY,
(Relation
1971),
Interpr6tabilitC
Paris,
Paris,
t. I
35-182.
211-228.
; a p p l i c a t i o n '2 l a
Sc.
,
les
: Permutations
in
systsmes d e
(1954),
(Paris,
totaux
1 1 0 1 FRASNAY,
des
A-I
P r o b l 6 m a t i q u e a p p o r t 6 e par
Groupes
C.,
classifications Univ.
Gauthier-Villars
q u ' u n nombre
n'ayant
A-273
Scient.
i n geome-
463-470.
Cours de Logique math6matique
u n e ctia'ine,
I 8 1 FKATSSE,
A combinatorial problem
G.,
Sur quelques
(Paris,
I FKAYSSE, R . ,
127-139.
(1976),
( I 935).
2
permutations,
Villars
I 7
R.,
on un-
groups
, Monomorphic r e l a t i o n a l s y s t e m s ,
SZEKERES,
Publ.
permutation
229-235.
Math.,
logique),
r 6 1 FRATSSE,
148
Z.,
, a n d KKAUSS, i'.
Compos.
1 4 1 FRATSSi,
r i e
Math.
Z . , 18 ( 1 9 7 2 ) ,
Math. r 3 l
T r a n s i t i v i t y of
P.J.,
d'une
famille d'ordres
monomorphes, (Paris,
1972),
extension aux Gauthier-
229-237.
InterprGtation
groupe nnturel,
relationniste
C.R.
Acad.
Sc.
du seuil
Paris,
de
A-277
recolle-
(1973),
2 68
C. Frasnay
[ I 4 1 F R A S N A Y , C., S u r q u e l q u e s c l a s s e s u n i v e r s e l l e s d e r e l a t i o n s
m - a i r e s , Publ. D E p . M a t h . (Lyon), 1 6 ( 1 9 7 9 ) , n o 3 - 4 ,
21-32.
1 1 5 1 H I G M A N , G . , H o m o g e n e o u s r e l a t i o n s , Q u a r t . J . M a t h . Oxford
S e r . ( 2 ) , 28 ( 1 9 7 7 ) , 1 1 6 1 HODGES, W . ,
31-39.
L A C H L A N , A . H . , and S H E L A H , S . , P o s s i b l e o r d e r i n g s
o f a n i n d i s c e r n i b l e s e q u e n c e , Bull. L o n d o n Math.
SOC., 9 ( l 9 7 7 ) ,
21 2-2 15. 1 1 7 1 J E A N , M., R e l a t i o n s m o n o m o r p h e s et c l a s s e s u n i v e r s e l l e s , C . R .
A c a d . Sc. P a r i s , A - 2 6 4
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[ I S ] J U L L I E N , P . , S u r q u e l q u e s p r o b l s m e s d e l a t h g o r i e des c h a T n e s
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(1966),
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L o g i k G r u n d l a g . M a t h . , 17 ( 1 9 7 1 ) , r 2 0 1 P O U Z E T , M.,
351-370.
Un belordre d'abritement et ses r a p p o r t s avec les
b o r n e s d'une m u l t i r e l a t i o n , C . R . A c a d . S c . P a r i s , A - 2 7 4
(1972),
1677-1 680.
[ 2 1 1 P O U Z E T , M . , A p p l i c a t i o n d'une p r o p r i E t 6 c o m b i n a t o i r e des part i e s d'un e n s e m b l e a u x g r o u p e s et a u x r e l a t i o n s , Math. Z . , 150 (1976),
117-134.
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e n c h a l n a b l e a u d s n o m b r e m e n t d e s r e s t r i c t i o n s f i n i e s d'une relation, 2 . Math. Logik Grundlag. Math., 2 7 ( 1 9 8 1 ) ,
289-332.
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390-391
and C o n t r i b u -
t i o n s to the theory of m o d e l s , I n d a g . M a t h . , 1 6 ( 1 9 5 4 ) , 588,
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***
3 9 1 a n d Indag. M a t h . ,
Annals of Discrete Mathematics 23 (1984) 269-286 0 Elsevier Science Publishers B.V. (North-Holland)
269
H,-CATEGORICAL PARTIALLY ORDERED SETS James H. Schmerl Department of Mathematics U n i v e r s i t y of C o n n e c t i c u t S t o r r s , CT 06268, U . S . A .
To Professor Emzest Corominas RCsumC: Un ensemble dBnombrable p a r t i e l l e m e n t ordonnB (A; < ) e s t .-cat&orique l o r s q u e , pour t o u t ensemble g dcnombrable q u i e s t p a r t i e l l e m e n t ordonne e t q u i p o s s l d e l a mcme t h B o r i e du p r e m i e r o r d r e que A ( i . e . Th(A) = T h ( B ) ) , A 8 . Nous c o n s i d g r o n s l a q u e s t i o n g 6 n 6 r a l e s u i v a n t e :
A
=
Q u e l l e complexit; d o i t a v o i r un ensemble A k o - c a t 6 g o r i q u e p a r t i e l l e m e n t ordonng pour que Th(A) s o i t complexe? Les deux emplois du terme "complexe" d a n s l a q u e s t i o n d o i v e n t e t r e vagues e t d i f f 6 r e n t s . L e p r e m i e r "complexe" f a i t r'efgrence aux p r o p r i g t 6 s s t r u c t u r e l l e s de A t e l l e s que l a h a u t e u r , l a l a r g e u r , l a dimension ou l e plongement d e c e r t a i n e s c o n f i g u r a t i o n s . Le second emploi du mot f a i t r g f g r e n c e aux a t t r i b u t s l o g i q u e s t e l s que l a d B d c i d a b i l i t 6 ou l ' a x i o m a t i s a t i o n f i n i e . Cet a r t i c l e , q u i s ' a d r e s s e au l e c t e u r a y a n t peu d e c o n n a i s s a n c e s dans l e domaine d e l a l o g i q u e , p r & e n t e un risum; d e s r g s u l t a t s e t l e s t e c h n i q u e s p e r m e t t a n t d e les o b t e n i r , c e q u i a p p o r t e une rgponse l a q u e s t i o n posge e t aux q u e s t i o n s a d j a c e n t e s . Un exemple t y p i q u e d e r ' e s u l t a t partiellement e s t c e l u i d e t o u t ensemble H .-cat;gorique ordonng de l a r g e u r f i n i e d o n t l a t h B o r i e n ' a q u ' u n nombre f i n i d'axiomes. Quelques thBorlmes nouveaux, complets e t accompagnBs de p r e u v e s , s o n t e'galement p r g s e n t b s . Par exemple, chaque ensemble K .-cat&gorique p a r t i e l l e m e n t ordonn; de dimension 2 e t de h a u t e u r f i n i e p o s s l d e une t h g o r i e d g c i d a b l e . Par c o n t r e , il e x i s t e d e s ensembles H .-cat&orique p a r t i e l l e m e n t ordonnBs, de dimension 3 e t de h a u t e u r 2. L ' a r t i c l e c o n t i e n t Bgalement d e s q u e s t i o n s o u v e r t e s .
e
1.
INTRODUCTION
Probably t h e e a r l i e s t theorem on K . - c a t e g o r i c a l
p a r t i a l l y o r d e r e d sets ( o r itself, i s C a n t o r ' s theorem c h a r a c t e r i z i n g t h e o r d e r t y p e of t h e r a t i o n a l numbers.
p o s e t s ) , proved l o n g b e f o r e t h e i s o l a t i o n of t h e n o t i o n of H . - c a t e g o r i c i t y THEOREM 1.1. (A; R ) is order-isomorphic t o t h e r a t i o n u l s foZlowing two conditions hold:
endpoints.
I&;
<)
i f f the
is countable; is a dense, Linearly ordered s e t without f i r s t and l a s t
(1)
A
(2)
(A; R )
J. H. Schtiterl
270
Let u s e x a m i n e ( 2 ) more c a r e f u l l y . The p r o p e r t i e s of ( A ; R) w h i c h a r e specified i n ( 2 ) a r e a l l first-order properties; that is, there a r e sentences has t h e s e o f f i r s t - o r d e r l o g i c w h i c h a r e t r u e i n ( A ; R) j u s t i n c a s e ( A ; K ) properties. I t i s a s i m p l e m a t t e r t o w r i t e down t h e f i r s t - o r d e r s e n t e n c e s i n volved. transitivity:
Vx Vy Vz (xRy
antisymmetry:
Vx Vy (xRy
c o n n e c t e d n e s s : Vx Vy ( x
# y
denseness:
Vx Vy (xRy
endpointless:
Vx 3y (yRx)
A
-
yRz
xRz);
yRx);
+
--t
xRy
A
yKx);
V
3z(xRz
+
A
zRy));
Vx 3y ( x K y ) .
(A; R) is c o u n t a b l e and i f e v e r y f i r s t - o r d e r s e n t e n c e T h u s , w e see t h a t i f t r u e i n ( Q ; <) i s t r u e i n ( A ; R ) , t h e n (9; < ) a n d ( A ; R ) are i s o m o r p h i c . T h i s i s j u s t what is r e q u i r e d f o r (0; < ) t o be .-categorical; we now g i v e a p r e c i s e d e f i n i t i o n of t h i s n o t i o n . F o r o u r p u r p o s e s , a ~ i r ~ u c t u rics a s e t A t o g e t h e r w i t h f i n i t e l y many r e l a t i o n s R 1 , Rq, Rk o n A , w h e r e i < i i s n i - a r y ( i . e . R i E Ani) f o r some itk). p o s i t i v e i n t e g e r n i . We w i l l w r i t e a s t r u c t u r e A as ( A ; R 1 , R 2 , F a m i l i a r e x a m p l e s o f s t r u c t u r e s a r e p o s e t s and g r a p h s . 'The t/iccira:j o f A , d e n o t e d by Th(A) , i s t h e s e t of f i r s t - o r d e r s e n t e n c e s t r u e i n A . When we s a y t h a t A i s c o u n t a b l e , w e w i l l mean t h a t i t s u n ~ l t 9 r ~ $ sz ek t~ ~A i s c o u n t a b l e , i n c l u d i n g t h e p o s s i b i l i t y t h a t A i s f i n i t e . S i m i l a r l y , A is f i n i t e i f A is f i n i t e . We c a n now d e f i n e t h e n o t i o n o f i-4 . - c a t e g o r i c i t y .
...,
...,
DEFINITION 1 . 2 . A structure c o n d i t i o n s are m e t :
(1) A (2)
A
is
Ko-onti'~i"r'i('"2 p r o v i d e d t h e f o l l o w i n g t w o
is countable;
whenever
6
i s c o u n t a b l e and
Th(6) = Th(A), t h e n
6
:A
.
(9io')
ti us, is H,-categorical. I t i s v e r y e a s y t o see t h a t e v e r y f i n i t e structure is categorical. In a sense, No-categoricity generalizes f i n i t e n e s s ; Theorem 2 . 1 w i l l make t h i s e v e n more e v i d e n t . We now p r e s e n t some o t h e r e x a m p l e s o f K o - c a t e g o r i c a l p o s e t s . Boolean a l g e I t is w e l l known t h a t a n y two c o u n t b r a s a n d l a t t i c e s w i l l be v i e w e d as p o s e t s . a b l e , atomless B o o l e a n a l g e b r a s a r e i s o m o r p h i c , a n d t h a t a n y two f i n i t e B o o l e a n It e a s i l y follows t h a t a l g e b r a s w i t h t h e same number o f a t o m s a r e i s o m o r p h i c . a n y two c o u n t a b l y i n f i n i t e B o o l e a n a l g e b r a s w i t h t h e same f i n i t e number of a t o m s a r e i s o m o r p h i c , t h e r e b y i m p l y i n g t h e f o l l o w i n g t h e o r e m from t h e f o l k l o r e . THEOREM 1 . 3 .
E v e q c o u n t u b k BooLenn ulgebr?~w i t h o n l y J i n i t e l y mung izlorns i::
Uo-cnti!~or~?:cuI. We e n d t h i s i n t r o d u c t i o n w i t h some m o d e l - t h e o r e t i c r e m a r k s o n c a t e g o r i c i t y . The r e a d e r who i s s e e k i n g more i n f o r m a t i o n s h o u l d c o n s u l t t h e s t a n d a r d t e x t 111 by Chang and K e i s l e r . A f i n i t e s t r u c t u r e A i s n o t o n l y X o - c a t e g o r i c a l , i t i s cutcgorical ( m e a n i n g t h i s : € o r a n y s t r u c t u r e 8, i f T h ( B ) = T h ( A ) , t h e n B z A ) The c o n v e r s e i s a l s o t r u e : i f A is c a t e g o r i c a l , then A i s f i n i t e . The r e a s o n f o r t h i s i s o n e o f t h e most f u n d a m e n t a l t h e o r e m s o f model t h e o r y , t h e I.~wenheim-Skolem Theorem.
.
THEOREM 1 . 4 .
wid i/,y
K K
(The LEwenheim-Skolem Theorem)
If A is t m g i n f i n i t e structure ( I structur2t. B 01' cnudiricil-
nn!j i n f i n i t e cnrd'inal nurribc:rd, t h e n thi'rs is such LIut l'hiBI = Th(AI.
No a l e g o r i c a l partially ordered set
27 1
The f o l l o w i n g d e f i n i t i o n , g e n e r a l i z i n g D e f i n i t i o n 1 . 2 , becomes n a t u r a l i n l i g h t of t h e p r e v i o u s theorem. L)l~I~lN17’I0N 1.5.
is ~ , 6 - A .
Let
K
b e an i n f i n i t e c a r d i n a l and 8 cardinality K
- ~ ~ ~ I ~ , ~ i ~f f[ ,whcnever ’ ~ , / ~ ~ ~ ~ ~has ~
A and
a s t r u c t u r e . Then A T h ( 6 ) = T ~ ( A ) ,t h e n
S i n c e f o r each i n f i n i t e c a r d i n a l K t h e r e is a correspondi ng n o t i o n of c a t e g o r i c i t y , t h e s i t u a t i o n seems c h a o t i c . However, t h e f o l l o w i n g b e a u t i f u l t h e o r e m of M. Morlcy [ 7 ] h a s t u r n e d t h i s c h a o s i n t o o r d e r , a t l e a s t p a r t i a l l y . THEORDI 1 . 6 .
untob7i- /-.:riyiinuIs, tiicit A t l n l Th(A) = T h l B ) . Then
L
8
ilnj
A
is
M o r l e y ’ s t h e o r e m r e d u c e s t h e number o f t y p e s o f c a t e g o r i c i t y t o j u s t two: & - c a t e g o r i c i t y and H 1 - c a t e g o r i c i t y . The s t u d y o f H 1 - c a t e g o r i c i t y h a s l e d t o some of t h e d e e p e s t and most r e v e a l i n g work i n model t h e o r y , a s w i t n e s s [ 9 ] . Howe v e r , H l - c a t e g o r i c a l p o s e t s a r e n o t s o i n t e r e s t i n g a s p o s e t s , s i n c e f o r any H lc a t e g o r i c a l p o s e t t h e r e i s a f i n i t e u p p e r bound on t h e l e n g t h s o f i t s c h a i n s .
2 . ORBITS Why s h o u l d a n y o n e , o t h e r t h a n a l o g i c i a n , b e i n t e r e s t e d i n H 0 - c a t e g o r i c a l p o s e t s ? One o f t h e most c o n v i n c i n g a r g u m e n t s i s t h a t t h e r e e x i s t s a n a l g e b r a i c c h a r a c t e r i z a t i o n of n o - c a t e g o r i c i t y . For a s t r u c t u r e A = (A; R 1 , R p , ..., R k ) , l e t Aut(A) b e i t s g r o u p o f a u t o m o r p h i s m s . For e e c h n a t u r a l number n , a n nor.bit o f A i s a s e t o f t h e form { (a(al),
a(a2),
...,
.(an)>
:
CY
E
Aut(A)I
f o r some a l , a 2 , ..., a n E A . The f o l l o w i n g t h e o r e m i s o n e form o f t h e fundament a l c h a r a c t e r i z a t i o n of H . - c a t e g o r i c i t y p r o v e d by R y l l - N a r d z e w s k i [16] i n 1 9 5 9 .
Supposc A i s muntabLe. Then t h e r e are o n l y f i n i t e l j mang n-orJbits.
THEOREM 2 . 1 .
n
A
i s H o-cntegoriraZ iff for tach
H e r e i s a n a p p l i c a t i o n o f Theorem 2 . 1 . We m e n t i o n e d i n Theorem 1 . 3 t h a t e a c h c o u n t a b l e B o o l e a n a l g e b r a w i t h a t most f i n i t e l y many a t o m s i s H . - c a t e g o r i c a l . Theorem 2 . 1 c a n b e u s e d t o d e m o n s t r a t e t h e c o n v e r s e t o t h i s : I f a Boolean a l g e b r a i s H o - c a t e g o r i c a l , t h e n i t h a s o n l y f i n i t e l y many a t o m s . F o r , s u p p o s e t h e r e w e r e i n f i n i t e l y many a t o m s . Then f o r e a c h n t h e r e i s a n e l e m e n t a n which i s t h e j o i n o f n + l a t o m s b u t i s n o t t h e j o i n o f n a t o m s . Then d i s t i n c t an and am a r e i n d i f f e r e n t 1 - o r b i t s , s o t h a t t h e r e a r e i n f i n i t e l y many 1 - o r b i t s .
...,
A r e d u c t o f a s t r u c t u r e A = ( A ; R 1 , R2, Rk) i s a s t r u c t u r e o b t a i n e d by i g n o r i n g some of t h e r e l a t i o n s of A . For e x a m p l e , i f m < k , t h e n (A; R 1 , R2, R,,,)), i t follows G) i s a r e d u c t o f A . S i n c e Aut(A) c A u t ( ( A ; R 1 , R 2 , from Theorem 2 . 1 t h a t i f A i s P o - c a t e g o r i c a l , t h e n so a r e a l l o f i t s r e d u c t s .
...,
...,
Let u s s a y t h a t a s t r u c t u r e i s n-determined i f f t h e f o l l o w i n g h o l d s f o r each p a i r < a l , a2, , am> , < b l , b g , bm> o f m - t u p l e s :
...,
THEOREM 2 . 2 .
. ..
...,
...,
...,
For each
but w i t h i n f i n i t p l y many
n
..., ...
t h e r e is a countable p o s e t w i t h f i n i t e Z 3 many n-orbits in+LI-orbits.
J.H. Schmerl
212
I t i s a l s o p o s s i b l e t o c o n s t r u c t an Ho-categorical poset which i s not n-determined f o r any c h o i c e of n. A d d i t i o n a l l y , f o r each n t h e r e i s an H o - c a t e g o r i c a l p o s e t which i s n o t n-determined b u t i s (n+l)-determined. One more t y p i c a l a p p l i c a t i o n of Theorem 2 . 1 w i l l now be p r e s e n t e d . Given a p o s e t A , l e t i t s chain s p e c t r m be t h e set of a l l p o s i t i v e i n t e g e r s n such t h a t A h a s a maximal c h a i n of l e n g t h n. The f o l l o w i n g theorem c h a r a c t e r i z e s t h o s e sets which can occur a s c h a i n s p e c t r a of N o - c a t e g o r i c a l p o s e t s . THEOREM 2.3.
(I)
M
(2)
m
2
b, m
For a s e t
M
of p o s i t i v e i n t e g e w , the foLZowing are e q u i v a l e n t :
there i s an Ho-categorical poset whose chain s i J e c t r m i s M; E
M
i s eventuaZly p e r i o d i c ( i . e . there are iff m + p E Ml.
b
PROOF: Only t h e proof of (1) => ( 2 ) w i l l be g i v e n now. d i r e c t i o n w i l l b e d e l a y e d u n t i l a f t e r Theorem 3.4.
and
rl
suck t h a t for. a l l
The proof of t h e converse
What w e w i l l prove i s t h e f o l l o w i n g , which by Theorem 2 . 1 s u f f i c e s : Every p o s e t w i t h only f i n i t e l y many 1 - o r b i t s h a s an e v e n t u a l l y p e r i o d i c c h a i n spectrum. Suppose A h a s e x a c t l y k 1 - o r b i t s , where k is f i n i t e . We w i l l show t h a t we can t a k e p = k ! , b u t w e w i l l d e t e r m i n e o n l y t h e e x i s t e n c e of b by proving t h a t whenever k < m E M , t h e n m k! E M.
+
...
So, suppose k < m E M , and l e t a 1 < a2< < a,,, be a maximal c h a i n . Since m > k , t h e r e are i < j 5 m such t h a t a i and a j are i n t h e same 1 - o r b i t . Let a be an automorphism of A such t h a t a ( a i ) = a j . Then a1 < a 2 < < aj < a(ai+l) < < .(am) i s a maximal c h a i n of l e n g t h j i m - i , s o t h a t m ( j - i ) E M. elements of t h i s c h a i n , Also, n o t i c e t h a t a j and a ( a j ) , t h e j - t h and ( 2 j - i ) - t h are i n t h e same o r b i t . Thus, w e can s i m i l a r l y show t h a t (m+(j-i)) + ( @ j - i ) - j ) = m + 2 ( j - i ) E M. Proceeding i n d u c t i v e l y , f o r any n a t u r a l number r , m + r ( j - i ) E M. S i n c e 1 I j - i 2 k w e g e t , i n p a r t i c u l a r , t h a t mfk! E M. 0
... +
...
Corresponding t o c h a i n s p e c t r a i s t h e analogous n o t i o n f o r a n t i c h a i n s . Given a p o s e t A , l e t i t s a n t i c h a i n s p e c t r m be t h e s e t of a l l p o s i t i v e i n t e g e r s n such t h a t A has a maximal a n t i c h a i n of c a r d i n a l i t y n . We know of no p o s s i b i l i t i e s f o r a n t i c h a i n s p e c t r a o f )4 o - c a t e g o r i c a l p o s e t s o t h e r t h a n f i n i t e s e t s of p o s i t i v e i n t e g e r s , and t h e s e can e a s i l y be r e a l i z e d i n f i n i t e p o s e t s . On t h e o t h e r hand, w e know of no s e t which cannot be a n a n t i c h a i n spectrum of an H o - c a t e g o r i c a l poset. QUESTION 2.4. For which s e t s c h a i n spectrum M ?
3.
M
i s t h e r e an
H
.-categorical
p o s e t which h a s a n t i -
HOMOGENEITY
Theorems 1.1 and 2 . 1 t o g e t h e r imply t h a t t h e r a t i o n a l s ( 9 ; <) have o n l y f i n i t e l y many n - o r b i t s f o r each n - h a r d l y a s u r p r i s i n g f a c t . Moreover, t h e numb e r of o r b i t s i s i n some s e n s e as small a s p o s s i b l e . T h i s w i l l b e made more prec i s e a f t e r f i r s t s t a t i n g a d e f i n i t i o n . Two n - t u p l e s ( a l , a 2 , . , an) and < b l , b2 bn) from A are isomorphic i f {(ai, b ) : i = 1, 2 n l i s an isomorphism from t h e s u b s t r u c t u r e o f A induced by t a l , a 2 , . , an} o n t o t h e one induced by { b l , b2, bn}. C l e a r l y , i f two n - t u p l e s a r e i n t h e same o r b i t , t h e n t h e y are isomorphic. I n t h e c a s e of (Q; <), t h e proof of C a n t o r ’ s theorem a c t u a l l y shows, c o n v e r s e l y , t h a t isomorphic n - t u p l e s a r e i n t h e same o r b i t . This suggests a method of o b t a i n i n g H o - c a t e g o r i c a l s t r u c t u r e s f o r which a d e f i n i t i o n i s needed.
..
,...,
..
...,
A countable s t r u c t u r e DEFINITION 3.1. t u p l e s are i n t h e same n - o r b i t .
A
,...,
i s homogeneous i f any two isomorphic n-
Ho -categorical partially ordered set
273
C l e a r l y , isomorphism of n - t u p l e s i s an e q u i v a l e n c e r e l a t i o n w i t h o n l y f i n i t e l y many e q u i v a l e n c e c l a s s e s . Applying Theorem 2 . 1 , w e t h u s o b t a i n t h e followi n g consequence.
If
COROLLARY 3.2.
A
i s homogeneous, then
A
i s Ho-categoricaZ.
The t e c h n i q u e f o r c o n s t r u c t i n g homogeneous s t r u c t u r e s evolved o u t of work of Fra’iss;? 1 2 1 . I f A = (A; R 1 , R2 ,..., Rk) and 8 = ( B ; S 1 , S2 ,..., Sk) are s t r u c t u r e s , and a: A + B is an isomorphism of A w i t h a s u b s t r u c t u r e of B, t h e n we say t h a t a i s an embedding of A i n t o B, o r t h a t A i s embeddable i n B. The s t r a t e g y f o r c o n s t r u c t i n g homogeneous s t r u c t u r e s i s t o f o c u s on t h e c l a s s of f i n i t e s t r u c t u r e s which a r e embeddable i n t h e homogeneous s t r u c t u r e , r a t h e r t h a n on t h e homogeneous s t r u c t u r e i t s e l f . I t i s c o n v e n i e n t h e r e t o p e r m i t s t r u c t u r e s t o have u n d e r l y i n g s e t s which are empty. DEFINITION 3.3.
(1) K beddable i n (2) K a l : A, + A1
6 1 : A,
+
8
Let
K
be a c l a s s of f i n i t e s t r u c t u r e s .
A
has t h e hereditary property ( H F ) i f whenever
E
B
and
K
i s em-
A , then
B E K. h a s t h e amaZgamation property ( A F ) i f whenever and and
“2:
A0
+
A2
62: A2
+
B
A,,
a r e embeddings, t h e n t h e r e a r e 8 such t h a t B l a l = B2a2.
E
A 1 , A2 E K and K and embeddings
For a s t r u c t u r e A w e l e t H(A) d e n o t e t h e c l a s s of f i n i t e s t r u c t u r e s which a r e embeddable i n A . The c o n n e c t i o n between homogeneous s t r u c t u r e s and t h e prop e r t i e s i n D e f i n i t i o n 3 . 3 i s t h e c o n t e n t of t h e n e x t theorem, which is t h e method of Frays&.
THEOREM^.^.
(1)
If
A
is a homogeneous s t r u c t u r e , then
HlA)
has b o t h
HF
and
AP.
(2) If K is a c l a s s o f f i n i t e s t r u c t u r e s which has both HP and AP, then there i s a unique (up t o isomorphism) countable homogeneous A such t h a t H ( A I = K. Here i s a s i m p l e example of how Theorem 3.4 i s implemented. Let K be t h e c l a s s of a l l f i n i t e p o s e t s . Then i t i s q u i t e c l e a r t h a t K h a s b o t h HP and A P , so t h a t t h e r e i s a u n i q u e homogeneous p o s e t A such t h a t H(A) = K. T h i s p o s e t i s c a l l e d t h e universaZ homogeneous p o s e t . I n g e n e r a l , i f A i s any homogeneous s t r u c t u r e and B t u r e such t h a t H ( B ) C H ( A ) , t h e n 8 i s embeddable i n A .
is any c o u n t a b l e s t r u e
A s a n o t h e r d e m o n s t r a t i o n of t h e u s e of Theorem 3 . 4 , w e w i l l complete t h e proof of Theorem 2 . 3 , by p r o v i n g (2) => (1). We w i l l c o n s i d e r o n l y t h e c a s e t h a t M consists o f a l l p o s i t i v e i n t e g e r s ; f o r a r b i t r a r y e v e n t u a l l y p e r i o d i c M, e i t h e r t h e f o l l o w i n g proof o r t h e r e s u l t i n g s t r u c t u r e can b e a p p r o p r i a t e l y m o d i f i e d .
Consider t h e c l a s s K of f i n i t e s t r u c t u r e s ( A ; <, B , T , I ) , where ( A ; < ) i s a p o s e t , B and T a r e 1-ary r e l a t i o n s , I i s a 2-ary r e l a t i o n , and t h e f o l lowing s e n t e n c e s are i n Th((A; <, B , T, I ) ) : Vx Vy (B(x) Vx VY (T(x) VX
+
-f
vy ( I ( X , y )
l y < x), i x < Y). -f
X
y
A VZ l ( X
<
2 A 2 <
y)).
The i n t e n t i s , of c o u r s e , t h a t o n l y minimal e l e m e n t s are i n B , o n l y maximal e l e m e n t s a r e i n T , and ( a , b ) E I i m p l i e s t h a t b i s a cover of a It i s e a s i l y s e e n t h a t K h a s HP, and o n l y s l i g h t l y l e s s e a s i l y s e e n t h a t K h a s AP. By Theorem 3 . 4 ( 2 ) t h e r e i s a homogeneous B s u c h t h a t H ( B ) = K , and C o r o l l a r y 3 . 2 implies t h a t B is No-categorical. Let A be t h e p o s e t which i s t h e r e d u c t of 8; an o b s e r v a t i o n made a f t e r Theorem 2 . 1 i m p l i e s t h a t A i s H o - c a t e g o r i c a l . To
.
J.H. Schnierl
214
see t h a t that for B = {l], dable i n
t h e c h a i n s p e c t r u m o f A i s t h e s e t of a l l p o s i t i v e i n t e g e r s , j u s t n o t i c e e a c h n , t h e s t r u c t u r e (Cn; < , B , T , I ) i s i n K , w h e r e Cn= 1 1 , 2 ....,n t , T = { n ] and I = { 1 , 2 , 2 , 3 n - l , n 1. T h i s s t r u c t u r e i s embed8 a s a ma x i mal c h a i n of l e n g t h n .
,...,
I t i s o f t e n i n t e r e s t i n g and u s e f u l t o c h a r a c t e r i z e t h e homogeneous s t r u c t u r e s o f a p a r t i c u l a r k i n d ; however, i t may n o t a l w a y s b e p o s s i b l e t o do s o , e s p e c i a l l y when t h e r e are u n c o u n t a b l y many. Henson [ 3 ] showed t h i s i s j u s t t h e s i t u a t i o n f o r d i r e c t e d graphs. THEOREM 3 . 5.
Them
U Y J ~
homvgicni:ous (i%rv:r.tc,Ll ~ ~ r u p l ~ s .
2 "I
H From t h i s t h e o r e m , a s Henson n o t e d , t h e e x i s t e n c e of 2 graphs and p o s e t s c a n b e o b t a i n e d . COROLLARY 3 . 6 .
There an? 2
K
No-categorical
~ o - c o t i : ~ i "1 ~L ~~ ir ~~ ~ u ~~ ~i h s .
PROOF. From a n y g i v e n d i r e c t e d g r a p h w e c a n c o n s t r u c t a g r a p h i n t h e f o l l o w i n g m an n er. I f a a n d b a r e a r b i t r a r y v e r t i c e s of t h e d i r e c t e d g r a p h w i t h a d i r e c t e d e d g e from a t o b , t h e n r e p l a c e t h a t d i r e c t e d e d g e by t h e f o l l o w i n g c onf i g u r a t i o n , a d d i n g 7 new v e r t i c e s .
I t i s c l e a r t h a t n o n i s o m o r p h i c g r a p h s a r e c o n s t r u c t e d from n o n i s o m o r p h i c d i r e c t e d graphs. F u r t h e r m o r e , i t i s e a s y t o see from R y l l - N a r d z e w s k i ' s Theorem 2 . 1 t h a t e a c h H o - c a t e g o r i c a l d i r e c t e d g r a p h p r o d u c e s a n H o - c a t e g o r i c a l g r a p h , S i n c e , by Theorem 3.5 t h e r e a r e 2 ' 0 r( 0 - c a t e g o r i c a l d i r e c t e d g r a p h s , w e ha ve j u s t r o n s t r u c t e d 2 ' 0 nonisomorphic N o - c a t e g o r i c a l g r a p h s . 0 COROLLARY 3 . 7 .
There are
2
H
Ho-cateyorical p o se ts.
PROOF: From a n y g i v e n g r a p h , w e c a n c o n s t r u c t a p o s e t i n t h e f o l l o w i n g m a n n e r . L e t (A; E ) b e a g r a p h . Then l e t F = { { a , b ) : a , b and l e t B = A u F. D e f i n e < on B so t h a t x < y i f f x E A , y E F an: y . Then i t f o l l o w s fro m Theorem 2 . 1 t h a t (B; <) i s K o - c a t e g o r i c a l i f (A; E) i s . A l s o , n o n i s o (A c u r i o u s a nd u n i q u e morphic p o s e t s are c o n s t r u c t e d from nonisomorphic g r a p h s . e x c e p t i o n t o t h i s l a s t s e n t e n c e is t h a t t h e complete graph on 2 e l e m e n t s and t h e empty g r a p h on 3 e l e m e n t s y i e l d i s o m o r p h i c p o s e t s . ) 0
['"x
The p r e c e d i n g c o n s t r u c t i o n of p o s e t s i s f r o m [ 2 3 ] . a d d i t i o n a l p r o p e r t i e s whi ch w i l l b e c o n s i d e r e d l a t e r . COROLLARY 3.8.
There m e
2
H 0
T h e s e p o s e t s h a v e some
Ho-cateyorical l a t t i c e s .
PROOF: C o n s t r u c t p o s e t s ( B ; d ) w i t h t h e same u n d e r l y i n g s e t s as t h e p o s e t s cons t r u c t e d i n t h e p r o o f of C o r o l l a r y 3 . 7 , b u t now d e f i n e x -!y i f f x E y I f we
.
275
No -categorical partially ordered set
a d j o i n a new minimal element 0 and a new maximal element 1 , t h e n w e o b t a i n a l a t t i c e which i s H,-categorical provided t h e graph ( A ; E ) was. A l s o , nonisomorp h i c l a t t i c e s a r e c o n s t r u c t e d from nonisomorphic g r a p h s . 0 In c o n t r a s t t o Theorem 3.5 a r e t h e n e x t two r e s u l t s , each of which have remarkably l o n g and i n t r i c a t e p r o o f s . THEOREM 3 . 9 .
(1)
Lachlan-Woodrow [ 4 ] )
There are only countably many homogeneous
gru+. (2) UYW
(Lachlan [ 5 ] )
The1.e a r e onZy five homogeneous tournaments, two of which
j-inite.
I n each of t h e p a r t s of Theorem 3 . 9 t h e homogeneous s t r u c t u r e s a r e e x p l i c i t l y g i v e n . There a r e no s u r p r i s i n g examples; t h e d i f f i c u l t y is i n showing t h a t t h e r e a r e none b u t t h e obvious o n e s . Much more a c c e s s i b l e i s t h e c o r r e s p o n d i n g r e s u l t f o r p o s e t s proved i n [ 2 1 ] .
There are onZy countab2y many homogeneous p o s e t s .
THEOKEM 3 . 1 0 .
Again, t h e r e i s a n e x p l i c i t l i s t of t h e homogeneous p o s e t s . There a r e t h r e e i n f i n i t e c l a s s e s which a r e q u i t e e a s i l y d e s c r i b e d . Let n r a n g e o v e r c a r d i n a l numbers, 1 5 n 5 % . The f i r s t c l a s s c o n s i s t s of a n t i c h a i n s w i t h n e l e m e n t s ; t h e second c l a s s c o n s i s t s of n d i s j o i n t c o p i e s of (Q; < ) ; and t h e t h i r d c l a s s c o n s i s t s of p o s e t s which r e s u l t by r e p l a c i n g each p o i n t i n Q w i t h a n a n t i c h a i n of n e l e m e n t s . There i s one l a s t homogeneous p o s e t - t h e u n i v e r s a l homogeneous poset.
4.
THEORIES
C a n t o r ' s theorem s a y s more t h a n j u s t t h a t (Q; <) i s N o - c a t e g o r i c a l , f o r i t a l s o g i v e s an e x p l i c i t f i n i t e set C of s e n t e n c e s (namely t h o s e f i v e s e n t e n c e s l i s t e d i n 5 1 ) which a r e i n c l u d e d i n Th((Q; < ) ) w i t h t h e p r o p e r t y t h a t whenever 8 i s c o u n t a b l e and C C Th(B), t h e n 8 (Q; < ) . Suppose A i s a s t r u c t u r e and C C Th(A). Then C i f f o r any B , Th(B) = Th(A) i f f Z C Th(B). I f n i t e a x i o m a t i z a t i o n , t h e n A i s f i n i t e 2 3 miomatizab2e. DEFINITION 4.1.
A
matization f o r
i s a n axiohas a f i -
A
F i n i t e a x i o m a t i z a b i l i t y of a s t r u c t u r e A has i m p o r t a n t consequences conc e r n i n g Th(A). We make t h e f o l l o w i n g d e f i n i t i o n which by n e c e s s i t y i s somewhat vague. A s e t T of s e n t e n c e s i s decidable i f t h e r e i s a computer program ( i . e . an e f f e c t i v e a l g o r i t h m ) which can i d e n t i f y whether o r n o t a n a r b i t r a r y s e n t e n c e i s i n T. More p r e c i s e l y , T i s d e c i d a b l e i f i t is r e c u r s i v e i n t h e s e n s e of t h e t h e o r y of r e c u r s i v e f u n c t i o n s [ 1 2 ] . One of t h e common t y p e s of problems i n l o g i c is t o d e t e r m i n e whether o r n o t a s p e c i f i c Th(A) i s d e c i d a b l e . We w i l l s a y t h a t t h e s t r u c t u r e A i s decidable whenever i t s t h e o r y Th(A) i s d e c i d a b l e . The f o l l o w i n g fundamental c o n n e c t i o n between f i n i t e a x i o m a t i z a b i l i t y and dec i d a b i l i t y i s a consequence of t h e Lowenheim-Skolem Theorem 1 . 4 and t h e Completen e s s Theorem of l o g i c . THEOREM 4 . 2 .
If
A
is finitely uxiomatizabZe, then A
is decidab2e.
Thus, C a n t o r ' s theorem i m p l i e s t h a t (Q; < ) i s d e c i d a b l e . Is i t p o s s i b l e t h a t e v e r y H o - c a t e g o r i c a l p o s e t A i s d e c i d a b l e ? Far from i t , s i n c e t h e r e are o n l y c o u n t a b l y many nonisomorphic N o - c a t e g o r i c a l d e c i d a b l e s t r u c t u r e s A , and as we have s e e n i n C o r o l l a r y 3 . 7 , t h e r e are 2 '0 K o - c a t e g o r i c a l p o s e t s . However, i t f r e q u e n t l y happens t h a t i f f u r t h e r c o n d i t i o n s a r e imposed on an N o - c a t e g o r i c a l A , t h e n A i s d e c i d a b l e . For example, e v e r y f i n i t e s t r u c t u r e i s f i n i t e l y a x i -
J.H. Schmerl
216
o m a t i z a b l e . Also, of t h e homogeneous p o s e t s ( s e e Theorem 3 . 1 0 ) , a l l a r e d e c i d a b l e and o n l y f o u r of t h e s e a r e n o t f i n i t e l y a x i o m a t i z a b l e . A l l of t h e homogeneous graphs and tournaments a r e a l s o d e c i d a b l e . I t should be a p p a r e n t t h a t e v e r y r e d u c t of a d e c i d a b l e s t r u c t u r e i s d e c i d a b l e . I n p a r t i c u l a r , every r e d u c t of a f i n i t e l y a x i o m a t i z a b l e s t r u c t u r e i s d e c i d a b l e . Many of t h e d e c i d a b l e N o - c a t e g o r i c a l s t r u c t u r e s r e f e r r e d t o l a t e r i n t h i s paper a r e d e c i d a b l e by v i r t u e of t h e i r b e i n g r e d u c t s of f i n i t e l y a x i o m a t i z a b l e s t r u c t u r e s . The r e d u c t of an u n d e c i d a b l e s t r u c t u r e may be d e c i d a b l e . However, i f A i s a r e d u c t of 8, Aut(A) = Aut(B) and A i s d e c i d a b l e and N,-categorical, No-categorical. then 8 i s a l s o d e c i d a b l e and, of c o u r s e ,
5.
EXPANDABILITY
Many c l a s s e s o f p a r t i a l l y o r d e r e d s e t s are c l a s s i f i e d by b e i n g r e d u c t s of c e r t a i n t y p e s o f s t r u c t u r e s , o r , from a n o t h e r v i e w p o i n t , by t h e i r e x p a n d a b i l i t y . For example, a p o s e t (A; <) h a s dimension a t most n i f f t h e r e a r e l i n e a r o r cn e x t e n d i n g < such t h a t < = n < 2 n . . . n
...,
We begin w i t h a n H o - c a t e g o r i c a l v e r s i o n of t h e p r i n c i p l e a s s e r t i n g t h a t e v e r y s e t can b e l i n e a r l y o r d e r e d . A s p e c i a l c a s e w a s f i r s t proved i n [ 9 ] ; t h e theorem t h a t f o l l o w s was proved i n [ 2 1 ] . THEOREM 5 . 1 . Suppose A = (A; R1, R2,. . . , R k ) is No-categorical. Then t h e r e is a linear order < o f A such that (A; R1, R2, ..., Rk, < I is N o - c a t c y o r i c u l . Moreover, if A is i n f i n i t e , then < can be chosen so t h a t ( A ; < ) 2 (Q; < ) .
The n e x t two theorems, b o t h proved i n [ M I , u t i l i z e Theorem 5 . 1 i n t h e i r proofs. THEOREM 5.2.
t h a t (A; El such t h a t :
. ., R k ) is an No-categorical s t r u c t u r e such i s a cornparubiZity graph, then t h e r e is a partia2 order < o f A If (A; E, R I , R2,.
(1)
(A; E, ~ 1 RZ , ,..., ~ k ,< I
(2)
(A; El
is
N0-categorical;
i s t h e comparability graph o f
(A;
<).
S i n c e t h e i n c o m p a r a b i l i t y graph of a 2-dlmensional p o s e t i s a l s o a comparab i l i t y g r a p h , i t f o l l o w s q u i t e e a s i l y from Theorem 5.2 t h a t H o - c a t e g o r i c a l 2d i m e n s i o n a l p o s e t s have N o - c a t e g o r i c a l r e a l i z a t i o n s . THEOREM 5 . 3 .
If
i s an
(A;
and
are l i n e a r orders (1)
(A;
(2)
< =
<,
<,I
<el i s
<2
No-categoricaZ 2-dimensionu2 poset, then there o f A such t h a t :
Ho-categorical;
n <2.
I t i s unknown whether t h e r e i s a n analogue o f Theorem 5.3 f o r a l l f i n i t e - d i mensional H o - c a t e g o r i c a l p o s e t s . Even t h e c a s e of dimension 3 i s s t i l l u n s e t tled. QUESTION 5 . 4 . For e v e r y H,. - c a t e g o r i c a l 3-dimensional p o s e t (A; < ) , are t h e r e l i n e a r o r d e r s < I , < 2 , <3 of A such t h a t (A; <,
Ho -categorical partially ordered set
277
One of t h e b a s i c theorems about p o s e t s i s D i l w o r t h ’ s theorem c h a r a c t e r i z i n g p o s e t s of f i n i t e w i d t h . The next theorem i s t h e H o - c a t e g o r i c a l v e r s i o n of D i l w o r t h ’ s theorem, which i s i m p o r t a n t i n t h e proof of Theorem 6 . 4 . I t i s proved i n [ 2 2 ] , and a l s o i n [ 2 6 ] where an e r r o r i s c o r r e c t e d . THEOREM 5.5.
thew
m e ( I )
If
ehniris ( A ; <,
(A; < I is un H,-cutcgur~cuL posct of f i n i t e width C1, C;?,.. . Cn -C A such that:
CI, C,,
. . .,
CrL) i s
n, then
H,-cutegor>iea’L;
(2) A = CI u Cz u... u C,. The Order Extension P r i n c i p l e s t a t e s t h a t e v e r y p o s e t can be extended t o a l i n e a r l y o r d e r e d s e t . Theorems 5 . 1 , 5 . 3 and 5 . 5 each imply some i n s t a n c e of an N,-categorical v e r s i o n of t h i s p r i n c i p l e . l H E O R E M 5.6. Suppose (A;
Ho-cutegorical posct which has f i n i t e Then it has a linear extension < such
No i n s t a n c e of t h e f a i l u r e of t h e H o - c a t e g o r i c a l v e r s i o n of t h e Order Ext e n s i o n P r i n c i p l e i s known, y e t Theorem 5.6 r e p r e s e n t s j u s t about t h e e x t e n t of o u r knowledge. We do n o t know i t s v a i l d i t y even f o r 3-dimensional p o s e t s . T h i s s u g g e s t s t h e f o l l o w i n g q u e s t i o n f i r s t asked i n [ 2 2 ] . QUESTION 5 . 7 . < such t h a t
Does e v e r y Ho-categorical poset (A; < , < ) remains H,-categorical?
(A;<)
have a l i n e a r e x t e n s i o n
We end t h i s s e c t i o n w i t h a r e l a t e d q u e s t i o n which does n o t e x p l i c i t l y mention ( A ; E) i s a n H o - c a t e g o r i c a l n-colp o s e t s . Theorems 5.2 and 5.5 imply t h a t i f o r a b l e graph which is t h e complement of a c o m p a r a b i l i t y g r a p h , t h e n i t c a n be nc o l o r e d w i t h c o l o r c l a s s e s C1, C 2 , Cn such t h a t (A; E , C 1 , C 2 , C,) is Ho-categorical. still The same c o n c l u s i o n can be o b t a i n e d j u s t from Theorem 5.2 i f ( A ; E) i s a n H o - c a t e g o r i c a l , n - c o l o r a b l e c o m p a r a b i l i t y graph.
...,
...,
Can e v e r y QUESTION 5.8. g o r i c a l l y n-colored?
6.
H o - c a t e g o r i c a l n - c o l o r a b l e p e r f e c t graph b e
Ko-cate-
F I N I T E AXIOMATLZABILITY
C a n t o r ’ s proof of t h e H,-categoricity of ( Q ; <) a l s o d e m o n s t r a t e s t h a t (Q; i)i s f i n i t e l y a x i o m a t i z a b l e . R o s e n s t e i n [14] h a s shown t h a t t h i s phenomenon H o - c a t e g o r i c a l l i n e a r l y o r d e r e d sets. extends t o a l l THEOREM 6.1.
Every
uo-categorical l i n e a r l y ordered s e t i s f i n i t e l y axiomatizable
.-cateA c t u a l l y R o s e n s t e i n proved more by s t r u c t u r a l l y c h a r a c t e r i z i n g t h e g o r i c a l l i n e a r l y o r d e r e d s e t s . Another i n s t a n c e of t h i s phenomenon of f i n i t e a x i o m a t i z a b i l i t y o c c u r s w i t h Boolean a l g e b r a s , as w e a l r e a d y have s e e n b u t n o t made e x p l i c i t . Macintyre and R o s e n s t e i n [ 6 ] extended t h i s to augmented Boolean a l g e b r a s , which a r e s t r u c t u r e s of t h e form ( B ; <, 11, 1 2 , I k ) w i t h (B; < ) a Boolean a l g e b r a and e a c h I i a n i d e a l .
...,
THEOREM 6.2.
mat i z a b 1e .
Every
Ho-categoricaI augmented Boolean algebra i s f i n i t e l y axio-
S i n c e , as p r e v i o u s l y n o t e d , l i n e a r l y o r d e r e d sets a r e 2-determined, Theorem 6 . 1 can be r e s t a t e d t o s a y t h a t e v e r y l i n e a r l y o r d e r e d s e t w i t h f i n i t e l y many 2o r b i t s i s f i n i t e l y a x i o m a t i z a b l e . M . Rubin [ 1 5 ] improved upon t h i s , even though t h e r e are c o u n t a b l e l i n e a r l y o r d e r e d sets which have o n l y f i n i t e l y many 1 - o r b i t s
J. H. Schm erl
278
but a r e not THEOREM 6 . 3 .
m i o m a t i z u bl e
U0-categorical.
Every ZineuYly ouder>ed S P L m‘th f i n i t e z u man$
.
I-or>bits is j’init( LIJ
Theorem 6 . 1 was improved upon i n a n o t h e r d i r e c t i o n i n [ 2 3 ] . some of t h e i d e a s o c c u r r i n g i n t h e proof of Theorem 5 . 5 . THEOREM 6 . 4 .
Eueray
The proof u s e s
i$,-categor,ic*aZpo.’eL of f i n i t e width is f i n i t e l y miornuti--
able. The proof of Theorem 6 . 4 proceeds by i n d u c t i o n on t h e w i d t h . I n o r d e r t o m a i n t a i n t h e i n d u c t i o n f o r t h e c o u r s e of t h e p r o o f , a s t r e n g t h e n e d form of Theorem 6.4 must b e proved. T h i s s t r o n g e r theorem t u r n s o u t t o be q u i t e u s e f u l i n t h e I n o r d e r t o s t a t e t h e s t r o n g e r v e r s i o n i n Theop r o o f s of Theorems 7 . 3 and 7 . 1 3 . I f ( A ; < ) i s a p o s e t and R 5 A2 a b i n a r y rem 6 . 5 , w e w i l l need a d e f i n i t i o n . i f whenever < x , y ) E R , r e l a t i o n , t h e n R i s monotone ( w i t h r e s p e c t t o (A; < ) ) x ’ 5 x and y 5 y ‘ , t h e n ( x ’ , y ’ ) E R. THEOREM 6 . 5 . Suppose A = (A; <, R I , (A; <,J has f i n i t e width and each Ri tizabZe.
R?, ..., R k l
The b a s i s s t e p of Theorem 6 . 5 , when f i r s t proved i n [ 1 9 ] .
(A; < )
.-cateyoricaZ, wher>e A is j’initclg miornu-
is
zs monotone.
Then
i s a l i n e a r l y o r d e r e d s e t , was
One might wonder what happens i f t h e r e s t r i c t i o n t o f i n i t e width on t h e posets of Theorem 6.4 i s r e l a x e d t o j u s t f o r b i d d i n g any i n f i n i t e a n t i c h a i n . An e a s y consequence of t h e Compactness Theorem of l o g i c i s t h a t f o r each p o s e t (A; < ) having a r b i t r a r i l y l a r g e f i n i t e a n t i c h a i n s , t h e r e i s a c o u n t a b l e poset (B; <) w i t h a n i n f i n i t e a n t i c h a i n such t h a t Th((B; < ) ) = Th((A; < ) ) . Thus, i f an H o - c a t e g o r i c a l p o s e t has no i n f i n i t e a n t i c h a i n s , t h e n i n f a c t i t a l r e a d y has f i n i t e width. I t i s n o t p o s s i b l e t o improve Theorem 6 . 4 i n t h e d i r e c t i o n of Theorem 6 . 3 . This i s demonstrated i n t h e n e x t two theorems, which s o l v e Problem 5 . 5 of [ 2 3 ] . THEOREM 6.6. There arc? 2 % countable posets which have pairwise d i s t i n c t t h e ories such t h a t each posst has width 3 und o n l y one 1-orbit.
PROOF: The r e s t r i c t i o n t h a t t h e p o s e t s be c o u n t a b l e can be i g n o r e d because of t h e Lowenheim-Skolem Theorem 1 . 4 . L e t R be t h e s e t of r e a l s , l e t Z3 = ( 0 , 1, 21, and l e t A = R x 2 3 . For each i r r a t i o n a l a , w i t h 1 < a < 2 , d e f i n e a p o s e t A, = ( A ;
x < y;
(y, i + l )
iff
x+l 5 y ;
( x , i )
iff
x+a 5 y.
c,
There i s o n l y 1 - o r b i t s i n c e f o r e a c h r t R and ( x , i ) I+ ( x + r , i+k) ‘is an automorphism of A,.
k
E
23, t h e f u n c t i o n
I t remains t o check t h a t whenever a and f3 a r e i r r a t i o n a l s , where 1< a < < 2 , t h e n Th(A,) # Th(A6). N o t i c e t h a t each a = (x, i ) t A h a s two L e t u s d e n o t e t h e s e by c l ( x ) c o v e r s i n A,, namely ( x + l , i + l ) and (x+a, i - 1 ) . and c 2 ( x ) r e s p e c t i v e l y . These two c o v e r s a r e d i s t i n g u i s h a b l e i n t h e f o l l o w i n g s e n s e . Independently of a , t h e r e a r e f i r s t - o r d e r formulas JI1(x,y) and J12(x,y) and y = c 2 ( x ) r e s p e c t i v e l y . To o b t a i n such which hold j u s t i n c a s e y = c l ( x ) formulas we make u s e of t h e f o l l o w i n g two formulas which are t r u e i n A, : c 1 ( c 2 ( x ) ) = c Z ( c l ( x ) ) and c l ( c l ( c l ( x ) ) ) < C ? . ( C ~ ( C ~ ( X ) ) L) .e t C(x, Y ) b e t h e f i r s t - o r d e r formula a s s e r t i n g t h a t y i s a c o v e r of x Now l e t $ l ( x , y) be t h e f o l l o w i n g formula:
.
Ho -categorical partially ordered set
3y2, Y39 21, Z2r $ ( Z l , 22) Y # Zl
A
w [ $ ( x , Y)
239
A
$(22, 23)
$(w, Y3)
A
A
A
$ ( y , y2)
$ ( y , W)
$(W, 23)
h
A
A
279
$(Y2, Y3)
$ ( Z 1 , W)
Y3 < 231
.
A
W
A
#
$(x, z1) y2
h
W
# 22
.
Then $ 2 ( x , y ) i s j u s t $ ( x , Y ) A - $ l ( x , y ) For p o s i t i v e i n t e g e r p d e f i n e $P,(x, y ) so t h a t $ & ( x , Y ) = $ e (X, Y ) and ~ i ~ + ' ( x ,Y ) = l z [ r / l E ( x , Z ) A $ e ( ~ y, ) ~ . Then, c l e a r l y , t h e s e n t e n c e
vx v Y , vY2[$y(X' Y1) A 0 i f f a > p/q
.
holds
$q(X.
Y p ) - + Y1 < Y21
THEOREM 6 . 7 . Thcr7e is a countable poset of width 2 but which is not j-ifinitely ax-iomatizable.
which has onZy one
1-orbit
We w i l l d e s c r i b e s u c h a p o s e t . L e t {Q1,Q2} b e a p a r t i t i o n of t h e set Q o f r a t i o n a l s i n t o two s e t s e a c h o f w h i c h i s d e n s e . F i x some i r r a t i o n a l a > 0 . D e f i n e 4 o n Q so t h a t x < y i f f e i t h e r x u. < y o r e l s e x < y and x , y a r e i n t h e same Q i . The p o s e t (Q;<) h a s t h e p r o p e r t i e s demanded by t h e t h e o r e m . N o t i c e t h a t t h e p o s e t i s , u p t o i s o m o r p h i s m , i n d e p e n d e n t o f t h e c h o i c e o f a. It c a n b e shown t h a t t h i s p o s e t i s n o t f i n i t e l y a x i o m a t i z a b l e , b u t t h a t i t i s decidable.
+
If w e form a p o s e t by r e p l a c i n g e a c h e l e m e n t o f (Q; < ) by a n i s o m o r p h i c copy o f ( Q ; 4 ) , t h e n t h e r e s u l t i n g p o s e t s t i l l i s c o u n t a b l e , has w i d t h 2 , h a s o n l y o n e 1 - o r b i t , and i s n o t f i n i t e l y a x i o m a t i z a b l e . T h i s p o s e t i s t h e example o f P o u z e t [lo] o f a c o u n t a b l e p o s e t o f w i d t h 2 . C u r i o u s l y , t h e p o s e t s o c c u r r i n g i n t h e p r o o f o f Theorem 6 . 6 c a n b e u s e d t o d e m o n s t r a t e t h e n o n e x i s t e n c e of a c o u n t a b l e p o s e t o f w i d t h 3 which embeds e v e r y c o u n t a b l e p o s e t o f w i d t h 3 , t h u s a n s w e r i n g i n t h e n e g a t i v e a q u e s t i o n o f P o u z e t ( 7 . 1 5 o n p a g e 833 o f [ll]). For e a c h i r r a t i o n a l a, w i t h 1 < LY < 2 , l e t B, be a countable poset such t h a t Th(B,) = T h ( ( A ;
A c o m p a r i s o n of Theorems 6 . 6 and 6.7 s u g g e s t t h a t w i d t h 2 m i g h t b e a much more s e v e r e r e s t r i c t i o n t h a n w i d t h 3 . T h i s i s t h e g i s t o f t h e f o l l o w i n g q u e s t i o n .
QUESTION 6 . 8 . Does t h e r e e x i s t a n u n d e c i d a b l e p o s e t o f w i d t h l y many 1 - o r b i t s ?
2
with only f i n i t e -
Theorem 6 . 4 h a s b e e n e x t e n d e d i n [ 2 6 ] , s t a t e d as Theorem 6 . 9 b e l o w , i n a way Theorem 6 . 9 i m p l i e s Theorem 6 . 4 which i s i n t e r e s t i n g l y compared w i t h Theorem 7.2. q u i t e r e a d i l y , b u t t h e r e a p p e a r s t o b e no s u c h r e a d y c o n v e r s e i m p l i c a t i o n . THEOREM 6 . 9 . Every Ho-categorical, finite-dimensionaz, f i n i t e l y miornatizable.
d i s t r i b u t i v e l a t t i c e is
We end t h i s s e c t i o n w i t h a v e r y i n t e r e s t i n g o p e n q u e s t i o n w h i c h i s q u i t e n a t u r a l a f t e r t h e a p p e a r a n c e o f Theorems 6 . 2 and 6 . 9 . QUESTION 6 . 1 0 . able?
Is e v e r y
vo-categorical
d i s t r i b u t i v e l a t t i c e f i n i t e l y axiomatiz-
I t i s e v e n n o t known t h a t t h e r e are n o t
2
%-categorical
d i s t r i b u t i v e lat-
tices.
7.
DECIDABILITY Theorem 6 . 2 i m p l i e s t h a t t h e r e a r e o n l y c o u n t a b l y many s , , - c a t e g o r i c a l
posets
J.H. Schmerl
280
of f i n i t e width. No such r e s u l t f o l l o w s i f e i t h e r t h e dimension o r t h e h e i g h t (which i s t h e number of elements i n t h e l o n g e s t c h a i n , i f oIie e x i s t s ) i s s i m i l a r l y r e s t r i c t e d . The f i r s t of t h e f o l l o w i n g two theorems i s a q u i t e easy consequence of Theorem 3.5; i n d e e d , a l l t h e p o s e t s c o n s t r u c t e d i n t h e proof of C o r o l l a r y 3 . 7 have h e i g h t 2 . The second theorem, which i s proved i n [22], r e q u i r e s more c a r e f o r i t s p r o o f . A somewhat d i f f e r e n t c o n s t r u c t i o n can be used t o o b t a i n 2 " r ( o c a t e g o r i c a l l a t t i c e s of dimension 2 , t h u s r e n d e r i n g as u n l i k e l y a s a t i s f a c t o r y s o l u t i o n t o Problem 2 o f [ 2 8 ] .
H
THEOREM 7 .l.
There are
2
THEOREM 7 . 2 .
There are
2
'Ko-categorical
p o s e t s o f height
Ho-cateyoricaZ p o s e t s o f dimension
Theorems 7 . 1 and 7.2 cannot be merged i n t o one theorem. which i s a new o b s e r v a t i o n , g i v e s t h e r e a s o n f o r t h i s . THEOREM 7.3.
2.
Every Ho-cateyorical poset of dimension
2
2.
The next theorem,
and f i n i t e height i s
decidable. PROOF: Let (A; <) b e an . - c a t e g o r i c a l p o s e t having dimension 2 and f i n i t e and < 2 o f A,* such t h a t h e i g h t . By Theorem 5 . 3 t h e r e are l i n e a r o r d e r s < = n < 2 and (A; <,
Therefore, is d e c i d -
I n c o n t r a s t t o t h e p r e c e d i n g theorem i s t h e c a s e of p a r t i a l l y o r d e r e d s e t s of dimension 3 . The f o l l o w i n g i s a l s o a new theorem. TFEOREM 7.4.
There are
2
H
Ho-categorical p o s e t s o f dimension
3
and height 2.
PROOF: Let (A; R) b e one of t h e 2 H o - c a t e g o r i c a l p o s e t s of h e i g h t 2 which e x i s t a c c o r d i n g t o Theorem 7 . 1 . L e t A1 b e t h e s e t of minimal elements of (A; R), and A2 t h e s e t of maximal elements We can assume w i t h o u t l o s s of gene r a l i t y t h a t A 1 n A2 = 0. By Theorem 5 . 1 t h e r e i s a l i n e a r o r d e r < of A such t h a t (A; R, <) i s s t i l l H o - c a t e g o r i c a l . We can t h e n assume t h a t A L Q n (0,l). We w i l l c o n s t r u c t a p o s e t & = ( B ; 4 ) of h e i g h t 2 . The set of maximal e l e ments of B w i l l be A, and t h e s e t of minimal e l e m e n t s w i l l be t h e s e t Bo = R u { < O , l > , < 1 , 0 > } . Define 4 on B = B o u A so t h a t <x, y> a i f f one of t h e f o l l o w i n g h o l d s :
<
a
E
A1
and
x < a;
a
E
A2
and
y < a.
It f o l l o w s from an a p p l i c a t i o n of categorical.
Ryll-Nardzewski's
Theorem 2 . 1 t h a t
8
is
We w i l l now d e m o n s t r a t e t h a t t h i s c o n s t r u c t i o n y i e l d s 2 nonisomorphic 8 by showing t h a t i f w e s t a r t w i t h p o s e t s (A; R) which are n e i t h e r isomorphic n o r a n t i - i s o m o r p h i c , t h e n w e end up w i t h nonisomorphic 8. This w i l l be done by "recovering" e i t h e r (A; R) o r i t s d u a l from &. N o t i c e t h a t { A i r A21 i s t h e unique p a r t i t i o n IP1, P 2 } of t h e set of maximal elements of B f o r which t h e r e are e l e m e n t s p i , p2 E B such t h a t P i = { x E B : p i 4 X I W e get t h i s p a r t i t i o n
.
No -categorical partially ordered set
28 1
. .
by s e t t i n g p i = <0,1> and p2 = <1,0> Define < i on A i by: x c i y i f f x The o r d e r i n g < on i Ai coincides f o r some a E B , a < y b u t n o t a with t h e o r d e r i n g < o n A i . Now, f o r x E A 1 and y E A2 t h e f o l l o w i n g h o l d s : xRy i f f f o r some a E B ,
<
{b c B : a Thus
R
<
bl = i b
E
A1 : x
b ) u {b
E
A 2 : y c 2 b}.
is recoverable.
8 h a s dimension 3. We c l a i m t h a t 4 = <1n.(2,,<3 t h e iia r e l i n e a r o r d e r s of B d e f i n e d so t h a t i f < x , y > ,
I t remains t o show t h a t
where
or
al ( 2
bl
iff
a1 < bl;
a2 42
b2
iff
a2 < b2;
< x , y > <3
iff
(x = z
and
y < w);
< x , Y > < ~a l ; < x , Y ><3
a2;
a1 <3
9;
a1 <3
bl
a2 <3
bg
iff iff
b l < al; b2 < a2.
It i s v e r y e a s y t o check t h a t
4
=
( 1
n
(2 n ( 3
.
0
Let P = ( P ; <) be a f i n i t e p o s e t , and t h e n l e t Kp b e t h e class of p o s e t s which have no s u b p o s e t s isomorphic t o 9. Thus, (A; < ) b e l o n g s t o K t j f f Z h o KO-cateP H((A; < ) ) . We w i l l c o n s i d e r t h e q u e s t i o n of whether t h e r e are g o r i c a l p o s e t s i n KP. When t h i s f a i l s f o r a p a r t i c u l a r P, i n a l l known c a s e s , i t i s because e v e r y N o - c a t e g o r i c a l p o s e t i n Kp i s d e c i d a b l e . N o t i c e , by way of comparison, t h a t i f P i s a f i n i t e a n t i c h a i n , t h e n Theorem 6 . 4 asserts t h a t e v e r y B o - c a t e g o r i c a l p o s e t i n Kp is f i n i t e l y a x i o m a t i z a b l e . Conversely, i f P i s n o t an a n t i c h a i n , t h e n t h e c o u n t a b l y i n f i n i t e a n t i c h a i n i s i n Kp, and t h i s poset is No-categorical but not f i n i t e l y axiomatizable.
4
A p o s e t ( P ; <) i s a weak a n t i c h a i n i f t h e r e i s e x a c t l y one element of P which is n o t maximal and e x a c t l y one element which i s n o t minimal. A p o s e t h a s weak width 5 n i f f i t i s i n Kp where P i s t h e weak a n t i c h a i n w i t h n+2 elements. Thus, a p o s e t h a s weak w i d t h 5 1 j u s t i n c a s e i t i s a weak o r d e r . Not i c e t h a t a c o u n t a b l e a n t i c h a i n h a s weak w i d t h 0 , y e t i t i s tZ - c a t e g o r i c a l and n o t f i n i t e l y a x i o m a t i z a b l e . The f o l l o w i n g r e s u l t i s proved i n 7231.
THEOREM 7 . 5 .
Every
&-categorical
poset w i t h f i n i t e weak width is decidable.
It w a s proved i n [ 1 8 ] , and i n [17] by somewhat d i f f e r e n t means, t h a t e v e r y c a t e g o r i c a l t r e e i s d e c i d a b l e . T h i s r e s u l t can be improved upon i n two d i f f e r e n t
J.H. Sc hmerl
282 ways.
The n e x t theorem, proved i n [ 2 3 ] , i s one way; t h e o t h e r i s C o r o l l a r y 7 . 9 .
We have d e p i c t e d below t h e Hasse diagrams of 1 0 f i n i t e p o s e t s , PO, P1, We w i l l w r i t e Ki f o r Kpi.
THEOREM 7 . 6 .
Every X3-categorical poset i n KO
DEFINITION 7 . 7 .
Let
(A; < )
..., p9.
i s decidable.
be a f i n i t e p o s e t .
(1) A s u b s e t I S A is an i n t e r v a l i f f o r every a b o t h of t h e f o l l o w i n g hold: a < b i f f a < c , and b < a (2) (A; <) i s indecomposable i f f whenever e i t h e r I = A o r ( I ( 5 1.
I S A
E
A \ I and i f f c < a.
b,c
E
I,
is an i n t e r v a l , then
The f o l l o w i n g i s a new r e s u l t which i s proved by an e x t e n s i o n of t h e t e c h n i q u e used i n [25] t o prove C o r o l l a r y 7.9 below. The p r o o f , which will n o t b e g i v e n h e r e , i s based on r e s u l t s i n a forthcoming paper [ 2 7 ] . THEOREM 7.8. For each n 2 4 , every undecidable N,-categoricaZ decomposable subposet w i t h exact Zy n elements.
poset has an i n -
An example of a n indecomposable p o s e t i s P i , which i s t h e o n l y indecomposab l e p o s e t w i t h e x a c t l y 4 e l e m e n t s . I n f a c t , it i s proved i n [24] t h a t e v e r y i n decomposable p o s e t w i t h a t l e a s t 4 elements h a s a subposet isomorphic t o Pi. The f o l l o w i n g c o r o l l a r y i s immediate. COROLLARY 7 . 9 .
Every H,-categoricaZ
poset i n K 1
i s decidable.
The n e x t theorem y i e l d s v a r i o u s t y p e s of examples of u n d e c i d a b l e Ho-categ o r i c a l p o s e t s . The f i r s t two c a s e s a r e proved i n [ 2 2 ] . The p o s e t s c o n s t r u c t e d i n t h e proof of C o r o l l a r y 3.7 can b e used t o prove t h e l a s t c a s e . THEOREM 7.10.
classes:
There are
2
H,-eategorical
p o s e t s i n each of t h e following
Kz n K3, K2 n Kg, and K 5 n K6.
I t can e a s i l y be checked f o r e a c h f i n i t e p o s e t ing t h r e e conditions holds:
P e x a c t l y one of t h e follow-
H, -categorical parlia1I.v ordered set
in
(1)
P
i s a n a n t i c h a i n o r a weak a n t i c h a i n ;
or
(2)
?
has height
p2, P 3 , P4, P t .
Pg,
P
i s citiier
23;
or
P. (3)
p 7 , P:,
7-83
P
Po, P i , o r
is
$, o r
Pg
P1;
is e m b e d d a b l e
~ 8 .
or
I t f o l l o w s i r o m l'lieorems 7 . 5 , 7 . 6 a n d C o r o l l a r y 7 . 9 t h a t i f (1) i s s a t i s f i e d , t h e n e v e r y H - c a t e g o r i c a l p o s e t i n Kp i s d e c i d a b l e . C o n v e r s e l y , i t follows from T h e 0 r c n i s ~ 7 . 1 and 7 . 1 0 t h a t i f ( 2 ) i s s a t i s f i e d , t h e n t h e r e a r e 2 NO k - ' a t e g o r i r a l p o s e t s i n Kp. T h i s l e a v e s as u n s e t t l e d w h a t t h e s i t u a t i o n i s i n t h e c a s e of ( 3 ) . T h u s , Lhe f o l l o w i n g two q u e s t i o n s , o r i g L n a l l y a s k e d i n [ 2 1 ] , a r e s t i l l unanswered.
QUESTION 7 . 1 1 .
Is t h e r e an u n d e c i d a b l e
Ho-categorica7
poset i n
QUESTION 7 . 1 2 .
Is t h e r e a n u n d e c i d a b l e
Ho-categorical
p o s e t i n Kg?
K7?
We end w i t h a new o b s e r v a t i o n . A ' i s a p o s e t t h a t c a n b e represent e d by u n i t i n t e r v a l s . Equivalently, t h e semiorders are p r e c i s e l y t h e p o s e t s i n the c l a s s K3 n Kg. THEOREPl 7 . 1 3 .
L'LICY2
is dcciJabk,
Ncl-~'Ot
PROOF: L e t ( A ; <) b e a n W o - c a t e g o r i c a l s e m i o r d e r . We w i l l u s e t h e n o t a t i o n x / y t o d e n o t e t h a t x and y a r e incomparable. F o l l o w i n g Theorem 5 . 1 , l e t < be a l i n e a r o r d e r of A such t h a t (A; < , < ) i s still Ho-categorical. Define t h e b i n a r y r e l a t i o n 4 on A so t h a t x 4 y i f f a t l e a s t one of t h e f o l l o w i n g holds :
(1) x
<
y;
(2)
XIy A 3 z ( X
<
(3)
Xly
A32(7.
< y
(4)
X/y
A
X
Z A y I Z ) ; A
ZlX);
< y AvZ[(X
2
y <
2) A (2
<
X
f-t
2
<
y)].
I t f o l l o w s f r o m R y l l - N a r d z e w s k i ' s Theorem 2 . 1 t h a t (A; < , 4 ,4 ) i s (\',-categori c a l , s i n c e A u t ( ( A ; < , < , a ) ) = Aut ( ( A ; < , < ) ) . An e a s y e x e r c i s e now i s t o show t h a t ( A ; Q ) i s a l i n e a r l y o r d e r e d s e t . It i s a l s o a n e a s y e x e r c i s e t o show t h a t t h e r e l a t i o n < i s m o n o t o n e w i t h r e s p e c t t o a. T h u s , by Theorem 6 . 5 , t h e s t r u c t u r e ( A ; a , <) is f i n i t e l y a x i o m a t i z a b l e , and t h e r e f o r e d e c i d a b l e . (A; < ) i s a l s o d e c i d a b l e . 0 Thus, i t s r e d u c t
J.H. Schmerl
284
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Co.,
[25] Schmerl, J.H., Arborescent structures, 11: Interpretability in the theory of trees. Trans. Amer. Math. SOC. 266 (1981), 629-643. [26]
Schmerl, J . H . , Ho-categorical distributive lattices of finite breadth, Proc. Amer. Math. SOC. 3 (1983), 707-713.
[27] Schmerl, J . H . and Trotter, W.T., Indecomposable Graphs, Posets, Tournaments and other Binary Relational Structures (Tentative title, in preparation). [28]
Smith, K.W., Stability and categoricity of lattices, Can. J. Math. 1380-1419.
2 (1981),
[29] Shelah, S., Classification Theory and the Number of Nonisomorphic Models, (North-Holland, Amsterdam, 1978).
This Page Intentionally Left Blank
Annals of Discrete Mathematics 23 (1984) 287-312 0 Elsevier Science PublishersB.V. (North-Holland)
THE ARITHMETICS
OF
TWO
287
AS
THEORIES
ORDERS
Denis RICHARD Universitg Claude Bernard (LYON I ) Dedie a Monsieur le Professeur COROMINAS qui crba le laboratoire d'Alg6bre ordinale de LYON.
ABSTRACT. We study properties of the natural order and the order of divisibility in arithmetic. After a brief survey of some classical results on endextensions and cofinality for non-standard models of Peanoarithmetic, we l o o k at the incompleteness of arithmetic and present the undecidability of Th(W, S, 1)and several related theories. Next we discuss saturation: in particular a model is saturatedifandonly if one of its order structures is saturated. Then, we give a survey, new proofs and new results of the definability of arithmetic as a theory of two orders and answer two questions of J . ROBINSON. RESUME. Nous ?Studions des proprigtss de l'ordre nature1 et de l'ordre de divisibilits en arithmstique. AprPs un bref survol de rssultats classiques sur les extensions finales et cofinales des modsles nonstandards de l'arithmgtique de Peano, nous passons 21 l'incomplctude de celle-ci et b l'indscidabilit.5 de Th(lN, S, 1) et de plusieurs autres thgories. Ensuite nous parlons de saturation: en particulier un modzle est satur6 si et seulement si l'une de ces deux structures d'ordre l'est. Nous terminons par une synthPse, avec de nouvelles preuves et de nouveaux r?Ssultats, sur la dsfinissabilitg de l'arithm6tique comme thCorie de deux ordres et r6pondons b deux questions de J. ROBINSON. INTRODUCTION In the theory of numbers there are many results on approximations and asymptotic behaviour of classical functions needed for the investigation of integers (see for instance [HW] or [EMF]). This involves more the study of the connection between analysis and arithmetic than the connection between theory o f orders and arithmetic. There is also a combinatorial approach (see [EG]). Our aim is to demonstrate the importance of order structures in arithmetic. We start from the following simple observations. The usual axioms of arithmetic of second order (this and other logical and mathematical terms are defined in section 0 ) can be formalized only with two types of objects (the non negative integers and the sets of non negative integers) and the successor function S. Also the Peano arithmetic PA is usually described in the language { 0 , 1 , S , < , + , 0 but, already long ago, Julia ROBINSON [JR] showed that PA can be perfectly well defined in the language I S , I ) where I is the divisibility relation. In other words, the arithmetics can be considered as theories of two orders < and I and therefore the (distributive) lattice order I linearly extended by < Firstly we give interesting examples of rich order theories. Among them Th(lN, S, 1) and obviously also Th(lN, S, I ) - where I denotes coprimeness - are shown to be undecidable in $ 3 (we have recently proved that Th(Z, S, 1 ) is also undecidable [OR31 ) .
.
D.Richard
288
Secondly, we prove certain deep properties of saturation which may be generalized to "biordered sets"; for example we prove that the non-standard models of PA are w -saturated if and only if their reducts to < , or to I are 1
w
1
-saturated (true also for w -saturation [JFP] )
. Thirdly certain interesting
conjectures originally motivated by logic and order are equivalent to purely [JRI is number-theoretical ones; e.g. "Is Th(W) = T n ( H , S, 1) ?" equivalent to "Is-it possible to { S , 1) - define an < - increasing sequence of primes ?I' ( § 3) and also equivalent to "Is a universal k such that x = y whenever x + i and y + i have the same prime divisors for all 0 6 i < n ? " [AW] The last conjecture is implied by a conjecture due to ERDOS [ER]
.
.
First in section 0, we give all the necessary definitions and facts on nonstandard models, the two countable non-isomorphic order structures of nonstandard models of arithmetic and the classical theorems of Mac DOWELL and SPECKER, GAIFMAN, FRIEDMAN and WILKIE which are all very closely related to the natural order. In section 1, before treating problems of decidability, we concisely give the famous theorem of incompleteness of PARIS and HARRINGTON which is both combinatorial and order-theoretical. We recall some known theorems on the decidability of Th(R, S) , T h ( I N , < ) , Th(N, 1 ) . We conclude by proving the undecidability of Th(TN, S , I ) and announce the same for Th(Z, S, I )
.
Section 2 treats the saturation of Peano arithmetic and hence the possibility of defining elements by types ([CK] p.97) which are a kind of infinite existential definition. Surprisingly, everything that we can insert into a model without destroying it, can be added just using types in the languages I<) or [ I ) . We prove it for o -saturation (already published in [DRZ] and [JFP]) and we 1
.
The other results also note that these results are not extendable to Th(2L) (the extension to w -saturation, the preservation of saturation by end and cofinal extension) arelistedwithout proofs, which may be found e.g. in [JFP], [HKI,
[KSI)
.
Section 3 presents a survey of definability based on the orders < and 1 or their "parts", namely S and 1 All the proofs are given with the exception of three results which are to appear in [DR3] and [DR4]. The first proof of theorem 3 . 3 . (the definability of + and 0 from S and I ) is the original proof of Julia ROBINSON [JR] and the theorem 3.6. and its proof (the definability of + and from < and I ) are due to A . WOODS [AW] All the other proofs are new and situated in the frame-work of a coding method based on the ZSIGMONDY - BIRKHOFF VANDIVER theorem ( I 0). Theorem 3.5. gives our I + , 11 - definition of multiplication. We answer positively a question posed by Julia ROBINSON [JR] : are there I S , I ) - definitions of + , 0 and < for integers (including negative ones) ? We note that { + , 1 1 is a weaker language than { <, 11 and { S , I } which are both synonymous with I t , , < I over integers (including negative ones). The last theorem of the section is a step in the direction of the arithmetical definability of + and 0 in terms of S and 1 (which, in its turn, is equivalent so the { S , 1 1 - definability of <) since it asserts that both natural order and divisibility orders are I S , L}- definable over the powers of primes.
.
.
.
.
Sections 0-1-2 presents an overview of the domain whereas section 3 contains both a survey and new results. We must apologize to logicians for section 0 which presents, besides very known classical theorems, such an elementary result as the proposition 01 giving the order type of the countable non-standard models of Peano, but a good understanding of such very simple facts in Peano arithmetic is necessary for what follows.
Arithmetics as theories of two orders 5 0-
289
DEFINITIONS, NOTATIONS, RESULTS OF THEORY OF NUMBERS; CLASSICAL THEOREMS ON NON-STANDARD MODELS OF PEANO. (§
0.1.) -
The aritbmetics -______________
-
In a reasonable theory of sets, say ZF, the set IN of the non negative integers can be defined as a set, with a distinguished element 0 and an injective selfmap S called the sucLvssur funct7'on such that S ( W ) = IN \{O} , verifying the second order induction V A GIN
(0 € A
A V
x ( x € A + S(x)€
A))
+ A
=
.
IN
All the first order statements (i.e. those expressed by formulae in terms of two quantifiers, all logical connectives, + and 0 and variables ranging only over elements) satisfied in IN, constitute the t r u e a r i t h m e t i c Th(IN) Similarly for the set 22 of negative and non negative integers, we have the true arithmetic of arbitrary integers Th(lL). The Peano arithmetic PA is the L = {=, 0 , 1 , S, <, +, . I given by the first order theory in the language axioms 1 = S(0) ; V x ( x t 0 = x) ; V xy (x + S(x) = S(x+y)) ; V x (x.0 = 0) ; V xy (x.S(y) = x.y t x) ; x < y f-t 3 z ( z # 0 A x + z = y) ; and the infinite list of axioms
.
v
XI
...
[(
x
XI,
V(0,
..., x
) A V X(V(X, X I ,
V(S(X),
+
XI'
...,
Xn)
..., xn)))+
+
v
x V(X,
XI'
**.,
xn)]
where 7 ranges over a11 formulae in L. As usual, from the primitive symbols of L we define x = u x' ( q (x', x l , .., x,)) by and 9 (x, x l , xn) A (y < x + lv(y, x l , ..., x,))
.
x
=
max
XI
..., ( xi, x l , ...,
For an integer
I
x y and
bv
mpr.iie'
a
, we
put
,
x,)) x
Z
y(a)
whenever it exists, in an analogous way. iff
3
k Ix-yI
=
ka
; next we define
k (y = kx) ; the usual symbol P(x) stands for "x is a p r i m e " means "x and y have no common prime divisor", i.e. "x and y are iff we put 3 ! x q(x, X I , ..., xn)
3
i
q (x, x I ,
. .., xn)
(y # x
A
+
1 q
(y, x l ,
.
. . ., x,))
; obviously x
is
The symbol Supp(x) stands for the s e t of equivalent to x < y V x = y t h c 2 p r i m e dii>isors of x Now the f u Z l Inngucqe L(P) will be precisely the language we can obtain by adjoining to the basic language L all the symbols definable by formulae of L with or without induction in PA. A sublanguage of Peuno is a subset of L ( P ) The full language L(Z) is obtained by adjoining to the language L' = { + , 0 , < I all symbols definable by formulae of L' in Th(Z) A subZanguage of Th(Z) is a subset of L ( Z ) . We call a sublan(resp. L(Z)) if in guage K of PA (resp. of the Th(Z) synmyrnous with L(P) Th(IN) (resp. in Th(Z) ) + and 0 are definable by formulae of K. Actually, the synonymity of {t , 0 ) with L(P) is provable in PA (hence in Th(IN)) using the representation theorem of exponential function via the GODEL function f? (see [HE] p.245).
.
.
.
(§
0.2.)
- ------________________________________ Some key-results in theory of numbers
-
First of all is the BERTRAND'S postulate, namely the
TCHEBYCHEV's theorern such that
(see [HW] p.343). For every n >, 1, there is a prime p n < p 6 2n ; that is, if pr is the r-th prime
prtl < 2 pr
for every positive integer r
A s is customary in this fiels we u s e the
.
D. Richard
290 ('li2'ni ; L w r m i ~ n ' k t the=. ~ bl,
...,
For a r b i t r a r y i n t e g e r s
bn
0 < i
with
at,
...,
and f o r
a
p a i r w i s e c o p r i m e t h e s y s t e m of c o n g r u e n c e s
6
n
tins 'In i n f i n i t y o f s o l u t i o n s i n
x = ai (bi) iz \ Td 1 .
IN (a nd i n
The n e x t i m p o r t a n t t h e o r e m i s t h e
DTN~C'l!I,iT's t h ~ ~ r ~ rFno .r a and b c o p r i m e t h e r e a r e i n f i n i t e l y many p r i m e s o f ( c f . [HW] p . 1 3 o r [EMF] p . 2 4 2 ) . t h e form a n + b Combining t h e two l a s t t h e o r e m s i t f o l l o w s t h a t a s y s t e m of c o n g r u e n c e s s a t i s f y i n g t h e h y p o t h e s i s o f t h e C h i n e s e r e m a i n d e r t h e o r e m h a s an i n f i n i t y of p r i m e s o l u t i o n s when f o r e v e r y i (0 < i < n ) a i and b . a r e c o p r i m e . F i n a l l y t h e most i m p o r t a n t t o o l i n 8 3 o f t h i s p a p e r ( u s e d f o r c o d i n g ) i s t h e b e a u t i f u l ZSICMONDY's t h e o r e m r e d i s c o v e r e d by B I R K H O I ~ F a nd V A N D I V E R ( c a l l e d b elo w t h e "ZRV-theorem" [BV] ) . %I:lI;MOi)RDY - F~IRKlIOF1~'-VANDlVi?/f thcnrJt,iri. L e t a, b be c o p r i m e i n t e g e r s ( p o s i t i v e F o r a ny 1 ' 0 , t h e r e i s a t of n e g a t i v e ) s u c h t h a t 7 a l > Ibl
.
l e a s t a prime d i v i s o r of
m
for
<
an - bn
dividing n o n h f the
( s u c h d i v i s o r s a r e c a l l e d p~'~;rr'/,'ti~)c for
n
,
except i n the following cases
n = I
a+b=21X (azl);n - 3 ,
a = t 2 , b = ; I
t 2 a = -
,
b = t 1
a-b = I ; ;
am - bm
an - b n )
,
n = 2
n = 6 ,
( f o r m u l a t i o n d u e t o SCHINZEL i n [ S C l ) .
Note t h a t i t i s e a s y t o e x p r e s s them w i t h S . The f o l l o w i n g c o r o l l a r y i s e f f i c i e n t f o r c o d i n g t h e powers of t h e p r i m e s ( s e e t h e A ppe ndix) : COROLLARY o f t h e ZRV-Theorem.
i Proof. Supp(Z6
pa
of
Note t h a t -
> 2
F o r e v e r y p r i m e p and e v e r y p o s i t i v e i n t e g e r
I ) = {3,71
i s c h a r a c t e r i z e d by
p
26 - I
.
is the only
Supp(p" - I )
2a - I
(a
> I)
such t h a t
General s i t u a t i o n i n t h e ZBV-theorem.
n = 1
T h e r e i s n o t h i n g t o s a y f o r t h e e x c e p t i o n w he re
.-.-
.-
t h e power
.
because Supp(2 - I ) = 8 c h a r a c t e r i z e s Zn - 1 i n t h i s c a s e . t h e r e i s no e x c e p t i o n s i n c e p and b h a v e t h e same s i g n .
FIGURE 1
,
I
I
For
n
=
3
Remdrk. We illustrate the situation described in the corollary by some diar,ramms where a , h means "a divides b" in the lattice of divisibility and a ' h means "a i13" in the 1aLtice of natural order (see figures I and 2).
5 means I ; means < ; italic niimbers 'irk- primitive divisors; Supp(x) is in the corresponding circle t o x . +
f The order
:
< o f a non-standard model of
PA
is a chain of order type
where w , w:: and n are the usual order types of the non negative integers, of the negative integers and of the rationals.
(*I
+ (u" + u ) n
D. Richard
292
therefore eachnon-standard integer is greaterthan everyn ~ W . H e n c efora fixed nonstandard a and for every n e R , the integer a + n and a - n are also non-standard. Denote the set a + 21. in M . Note that a = if for a E M\W and
a
a
5
E M \W
z t 2L
there is an integer
equivalence relation. Now define in x
=
2y v x
a -+ z
the element
y
=
P
, which
is an
=
x] by With this, note that there are countably many classes
.
2y + 1
=
such that
PA
in M , each of which is of order type
iii"
+
id
.
CI
We prove that classes themselves
a dense countable order without a least or a greatest element which is therefore of <-order type q Put a << B iff a t M \ W , 5 t M\TN , form
.
a
< B and
.
a#
This relation <<
< by the equivalence that << 3 implies
I:] <<
a
<<
:
.
-a
is an order which is the quotient of
<< is dense, it is enough to check << 7 (a + 5 ) ] << which is easy to do. Similarly To prove that
#
I'
#
prevents the existence of least and greatest elements.
Remark. Roughly speaking, a countable non-standard model of IN followed by countably many copies of z! ordered a s Q : FIGLIRE 3 . -
21.
21.
~...+...+....+ O
W
21.
PA is a copy of
..........
I
n is the <-order typeyf the copies of So all countable non-standard models of
PA
0
21.
I
.
have the same order type
w + (to" + w ) q
but they need not be isomorphic; in fact, one can show the existence of continously many pairwise not isomorphic such models.
Now we quickly review (without proof) some deep results closely connected to the natural order. First we must define an end e x tension of a model M of PA V b E M (b < a) as a model M ' of the same theory such that V a E M ' \ M In the usual way we define that a model M is < - c o f i n a l in a model M ' . An elementary extension (see [CK] p.82) of a model M of PA is a model M' 3 M of n PA such that for every formula ~p EL(P) and for every (al, an) E M ,
.
...,
the formula
~p
is satisfied in M
iff it is satisfied in M ' .
The following results are classical: I ) . Theorem of Mac DOGJELL and SPECKER
[MS]. proper elementary end extension.
Every countable model of PA has a
2 1 , G A P M A N ' S theorem
[HG]. Every model M of Peano is an elementary extension of every model M ' o f Peano which is cofinal in M . Every extension of M is obtained by an end extension followed by a cofinal one.
31.
FRIEDMAN'S theorsern [ H F ] .
If M is a countable non-standard model of PA, then M is isomorphic to an initial segment of itself.
4 ) . CJLLKIE's theorem
[ALW]. For every countable non-standard model M of Peano, there is a model M ' of Peano Which is an end isomorphic extension of M and such that M' solves a diophantine equation with coefficients in M , that is not solvable in M.
Arithnietics as theories of two orders
293
Concerning the final segments of models of Peano, we answered [DRII the following question of JENSEN and EHRENFEUCHT (Problem 3 in [ J E l p.242):
"If M I and M2
(models of Peano) have final segments that are isomorphic, what properties of M M
Actually all properties are preserved since M I and 1 are preserved in M2 ?". are isomorphic as Peano models. To see this, let q be an isomorphism from a
2
final segment I1 of M
1
onto a final segment I2 of M2
E M I and for 1 This defines a mapping from M
m
€
MI , put
into M 2
.
It can easily be shown that
any
$(m)
.
Fix m
- q(m,). I $ is a bijection and an additive isomor-
= q(m+ml)
phism. The proof that it is multiplicative follows from a calculus for which we recall that ~ ( x ) is a prioz'i o n l y defined for x & m l ~ ( x - y )= q ( x ) - q(y)
that a-b
=
=
a
c means
@ ( m '+ m l + m l
§ 1
-
2
.
=
b+c
We have
-
ml ) - q ( m l )
2
=
. Moreover, note
x 2 m l , y 2 m l , x - y 2 m l and
whenever
$(m)
$(mm')
$(m')
,
so
2
= q ( m m ' + m (m+m') + m l ) 1
we are done.
LNCOYPLETENESS AND UNDECIDABILITY IN ARITHMETIC. (§
1.1.)
-
hsmpletenesp
-
In 1931, GODEL showed that if a theory T is strong enough (for instance if it contains the Peano arithmetic) and is recursively axiomatisable, then there is a sentence which is both t r u e and not provable from the axioms of T. In 1977, PARIS and HARRINGTON provided a combinatorial theorem, closely connected to the order structure of the positive integers, which is not provable in PA For this they used the notion of dense set: X is dense iff Card(X) >,min(X) Mac ALOON notes (IKMI p.04) : "la notion d'ensemble dense donne une mesure de la taille d'un ensemble fini d'entiers qui ne dspend pas uniquement de la cardinalit6 mais de "l'ordinalit6" de l'ensemble, c'est-%-dire de sa distribution dans la suite des entiers finis
.
.
.
A basic tool of the argument is the concept of KIRBY-PARIS i n d i c a t o r function which we have to define. Take a L(P)-definable function Y of two variables. Take a model M of Peano, and the set IN(M) of all the initial segments of M which are models of Peano. Suppose that for every I E IN(M) the interpretation of Y in I is exactly the restriction of the interpretation of Y in M, (this happens when Y is a b s o l u t e ) . Moreover, suppose that we can prove in PA that y' Y(x', Y) d Y(x, Y') x' d x 6 y
<
-f
Then every function Y satisfying these conditions is an i n d i c a t o r if and only i f for every model M, YM(x, y) > IN tt 3 I E IN(M) (x < I < y)
.
THEOREM ( K I R B Y - P A R I S - H A R R I N G T O N )
i
a) There is an indicator function such that N satisfies V x V y 3 z [Y(x, z ) > y] ; such an indicator is called admissible.
D.Richard
294
I
b) If Y is an admissible indicator function then the sentence V x V y 3 z [Y(x, y ) > 21 is not provable in PA although true in 7 N .
The proof uses the RAMSEY's theorem to build up an admissible indicator by coding as usual the finite sets of positive integers (see [PHI and [ K M ] )
.
-
( I 1.2.) - Undecidability
A classical result of PRESBURGER MP asserts that Th(lN, +) is complete and decidable. The proof is an example of the method of the elimination of the quantifiers. SKOLEM began the work for Th(N, .)and CEGIELSKI in [PC] gives the proof of the decidability of Th(IN, a ) and axiomatizes this theory. Then
,
Th(lN, S)
Th(N, <)
and < are obviously
,
Th(N,
1)
(+) -definable
and and
Th(N,
I
and
1
1)
are decidable since S
are of course
(a)
-definable.
For theories with two predicates we first prove the LEMMA 1.2.
Proof.
i
If K is a synonymous sublanguage of undecidable.
To every formula 9 of L ( P )
,
then Th(N, K)
is
with n variables, we can recursively
of K also with n variables such that for every
associate a formula
..., xn)
L(P)
...,
xn) is satisfied in IN the formula p(xl, x2, iff N realises q(xl, x2, In particular, if we only take sentences xn) is recursively reducible to Th(N, K) , as formulae 9 , we see that Th(W) (xl, x2,
E
INn
...,
.
Proof. -
The sublanguages concerned are shown to be synonymous with section 3 .
L(P)
in 0
THEOREM 1 . 4 .
1
The theory Th(IN, S,
1)
is undecidable.
Proof. With theorem 3.9. below, it is easy to build up in IN an inner model isomorphic to IN on the powers of 2. Then an adaptation of lemma 1.2. leads t o undecidability of Th(N, S, 1). Remark. I . )
In fact, part a) of the corollary 1 . 3 . also follows from theorem 1.4. Moreover, we do not know whether IS, 11 is synonymous with L(P) (see § 3 ) .
.
2 . ) Our first result was that Th(N, = , S, 1) is undecidable when simultaneously and independently A. WOODS proved and told us that Th(IN, S, 1) i s undecidable. Actually our proof was independent of the identity in virtue of the corollary of the ZBV-theorem.
Arithmetics as theories of two orders 8 2 -
295
SATURATION OF PEANO MODELS.
Let IY. be a cardinal. A model M of Peano is a-saturated iff for every subset X C M with fewer than n elements, the model M realizes every type E(v) of the language L(P) U Icx I x E X i which is consistent with Th(M). A model is 8 / i t r m i t r f liff i t is
(see [CKI p . 2 1 4 ) . Let Lo be a sublanguage
181-saturated
is a-snturnting for a cardinal a (or that PA is
of Peano. We say that
Lo iff for every model M of Peano, the u-saturation of the
a - ~ t 7 t z i ~ ~ ~ zover i,~d L ) 0
(M) of the structure M to Lo implies the a-saturation of M LO (see [CK] p . 4 2 5 ) . In other words, L is a a-saturating sublanguage iff whenever
r'ciizdct
Red
0
the only types expressed in Lo are realized in M , then all the positive types expressed in the full language are also realized in M.
Now we show that the
Peano arithmetic is w -saturated over order and divisibility. 1
LEMMA 2.1.
(w -saturation criterion). 1
1 A model M of PA of cardinality w-sequence
(ai)
ul
is w -saturated iff every 1
of M is an initial segment of a coded
ic w sequence of M. Proof. Let M be a non-standard model of P and I a proper initial segment of M with no last element. We show that there is no formula in L(P) for expressing x € I . Suppose we have a formula 7(x, a l , ..., an) with parameters in M such that for all b E M it is true that b E I iff M satisfies q
..., an):
(b, a l ,
Consider
X
=
{x E M
IM
satisfies
X has a first element, say xo
By the axioms of Peano,
As I is closed under successor, xo t I :
l q ( x , a],
, but
contradiction.
..., an)) .
then xo
-
1 E I ,
With this we know
that if a formula*$ is satisfied by all the x t IN in M, then there is at least an element y € M \ IN such that M satisfies $(y) This property is called oVerspiZT,. Now take a coded sequence which extends the sequence
.
(r(i))it
such that
r(i)
Let
u +qk(r(y)))
.
iew V y
(k
6y<
realizes the formulae
.
E(x) = (qi)
type
$,(u)
7. for
j
4 i of a given
be the formula
Since for every
n
t
IN
the model M satisfies
, there is an element ak E M\IN such that M satisfies $,(a,) . Since M satisfies v 6 u -+ ($,(u) + $ ,(v)) , the existence of the a; s allows us to lLk(n)
define a decreasing sequence s such that k t IN.
O(v)
V ij (i
by
every n
t
<
j
-t
v < s(j)
<
s(i)).
is true in M for every
Hence
IN and, therefore, there is an element
is true in M for every every
Jlk(s(k))
With a code of a sequence of M which extends s we can define a formula
r(y)
where
k
€
IN. Consequently the type
IN < y < a
.
e(n)
a E M\IN
E(x)
is true in M for such that
$,(a)
is realized by 0
D.Richard
296
THEOREM 2.2.
1 The Peano arithmetic is
Proof.
Take a sequence
substructure c
E
M
Red
M
w -saturated over
1
a = (a(i))ito is w -saturated.
(1) 1 realize the following (I)-type
Then the sequence of the exponents
.
Suppose that for a model M of P A , the Let
p
of the
a.
(1) .
be the n-th prime.
pi
Let
in the usual decomposition
of c form a coded sequence s (by the code c) in M.
Since a is an initial
segment of s, the previous lemma achieves the proof.
U
The same strong result is also true for < : THEOREM 2.3.
1 The Peano arithmetic is
w
I
-saturated over order.
Proof. We associate to every sequence (ai) of a model M of Peano another i is w (bi) given by bi = Z (a, + I ) . By assumption, the order structure of M j=o
iE u
J
c t M
is w -saturated so that there is an element I (ci)
c +I izCO ,< x < 2 O
n+l ;
c
c. 2 l < X <
n
c
c
.
so
-1
c 2i+2"+1
i=O
(bi)
ie u is strictly decreasing and
where c. = c - b i ir w we can code it by realizing the following type: The new sequence
such that
I
.
i=O
Then the theorem 2.3. follows from the lemma
.
2.1.
,
0
The classic results on order types, proposition 0 . 1 . and theorem 2 . 3 . lead to the characterization of the w -saturated models of Peano by their order 1 structure: COROLLARY
Proof.
1
A model M of Peano of cardinality its order type is
w + (w"
+ w)
wI
nl
Left to the reader familiar with orders.
.
is saturated if and only if
0
When we replace in the definition of the a-saturation the condition of cardinality of the type C(v) by the condition of being recursive then we get recursive saturation. The previous results are not extendable t o recursive saturation. THEOREM 2.4. [DR2].
I
In every non-standard model M of Peano, the order structures of M for < and I are recursively saturated. Consequently there is no recursive saturation of Peano arithmetic over [<) and over I 1 1 .
Arithnzetics as theories of two orders
297
Another impossibility of extension of the theorems 2.2. and 2.3. is asserted by the 1HliOREM 2.5.
I DR2 1 .
The arithmetics of arbitrary integers (and Th(iZ) in particular) are not o -saturated over the sublanguages {<) and [ I ) J I This last theorem combined with theorem 3 . 8 . b) shows that divisibility gives weaker structures when negative integers appear. More positively, the wl-saturating character of the natural order and of divisibility is preserved
.
by increasing the cardinal
wl
*
THEOREM 2 . 6 . I JPP].
i
Let a he a cardinal greatest than a-saturated with respect to
{<}
w,
.
Every model of PA which is
(or to
)
is
a-saturated.
To conclude this section, we list some results connecting cofinality and satura-
tion for the models of Peano. First SMOKYNSKI and STAVI proved
I SS I .
THEOREM 2 . 7 .
1
Every cofinal extension of a recursively saturated (resp. o-saturated) model of Peario i s itself recursively saturated (resp. o-saturated).
sirvp7c ,'.rfrn.sion of a model M of PA is the SKOLEM closure (see [CK] p . 1 4 2 ) of M U id} where d @ M, namely the smallest model of PA containing both M and d. In [HK], KOTLARSKI showed the following
A
THEOREM 2 . 8 .
i
[HK]. I f a model M of Peano is
w
extension of M, then M' is
1
-saturated and M' is a simple cofinal w
1
-saturated.
Also in [HK], KOTLARSKI asked whether the last theorem can be extended t o larger
cardinals. The answer was recently given by KAUFMANN and SCHMERL who show THEOREM 2 . 9 .
1 5
[KS].
Suppose that a model M of Peano is X+-saturated. Then M has a .. proper, proper, simple simplecofinal cofinalextension extension which which is is 1'-saturated A+-saturated hut but not not
. tt ++ -saturated. X X -saturated.
3 - DEFINABILITY FROM THE NATURAL AND DIVISIBILITY ORDERS IN ARITHMETIC.
It is well known that the full language L ( P ) of Peano arithmetic consisting of all relations and functions arithmetically definable from 0 , I , S , < , + , 0 (either with induction or without induction) is actually definable by formulae of the sublanguage { + , S ) (see [HE] p. 246) using the Godel function fi In this section, we try to find out what is definable from the natural order < and the divisibility order I . This study of the strength of expression of the sublanguages based only on two orders leads to the investigation of the on the other side. links between < , S , I , 1 on the one side and + and This point of view allows us to consider the true arithmetic Th(N) o r the Peano arithmetic PA of the first order as a theory of two distinct orders, one
.
.
I).Richard
298
b e i n g a l i n e a r e x t e n s i o n o f t h e o t t i e r . F i r s t we h a v e t o compare p a i r w i s e .:, S , ~, and 1 t o f i n d o u t w h e t h e r t h e y d i f f e r r e l a t i v e t o d e f i n a b i l i t y . S e c o n d l y we s l ~ o w 1 , ' S , [ t , ' , + , l1 and t h e synonymity ( a s d e f i n e d § O ) of t h e sublanftuagcs T<,l 1 w i t t i t h e f u l l l a n g u a g e I , ( P ) , In t h e c o n c l u s i o n o f s e c t i o n I we g i v e p a r t i a l r e s u l t s c o n c e r n i n g t h e o p e n p r o b l e m of w h e t h e r i S , l 1 i s a w e a k e r s u b l a n g u a g e of L(1'). Tn p r o p o s i t i o n I we compare t h e d e f i n i n g powers o f .:, S , and 1 u s i n g m o s t l y the PADOA's method: i f t h e r e i s a model M i n t e r p r e t i n g a r e l a t i o n K a nd a n R-autornorphism o f M n o t p r e s e r v i n g a n o t h e r r e l a t i o n K ' , t h e n K' i s c e r t a i n l y n o t R'-def i n a b l e .
'.:,I
~
PROPOSITION 3 . 1 . ( C o m p a r i s o n lemma) 1 i).
i t
P + y = S ( X ) 4 - r ( x < yA v z ( x < z P k- vx v y ( x l y *--' v z ( z I x A z I y
+
y
6 z));
->
v
t (z
I
t))).
I
i i ) . L e t R and R ' s y m b o l s t a k e n i n !<, S , , 1 1 ; w i t h t h e two e x c e p t i o n s above and i n t h e t r u e a r i t h m e t i c T h ( N ) , no symbol R is R'-definable i f R # R ' .
P r o o f . We d i s t i n g u i s h n i n e c a s e s ( s e e t h e t a b l e w h e r e I means R i s R ' - d e f i n a b l e and 0 means K i s n o t R ' - d e E i n a b l e ) . i ) . Obvious. are n o t i i ) . To show t h a t < a n d S - d e f i n a b l e c o n s i d e r a non-standard model M of 'I'h(N) and l e t a and b be d i s t i n c t elements of M \ N such that a b .
I
I
Now d e f i n e a map f f r o m M o n t o M by s e t t i n g f ( x ) = x f o r any X E M \ ( ~ + Z ) U ( ~ + Z a)n d f o r a n y ~ F Z f, ( a + y ) = b + y and f ( b + y ) = a + y : t h e o r d e r s < and a r e n o t S - d e f i n a b l e because f i s an S-automorphism p r e s e r v i n g n e i t h e r < n o r . Next t a k e N and d e f i n e b b a g : N u N by s e t t i n g g(Za 3 x) = 2 3 x when we h a v e a Z 0 , b 2 0 and x 1 6 A s g i s an (a)-automorphism, g is also a ( 1 ) - a u t o m o r p h i s m and a 1 - a u t o m o r p h i s m b u t I: p r e s e r v e s n e i t h e r S n o r < , so t h a t < and S a r e n e i t h e r ( I ) - d e f i n a b l e n o r 1 - d e f i n a b l e . Take a g a i n A n o n - s t a n d a r d model M o f T h ( N ) a n d map e v e r y n o n - s t a n d a r d e l e m e n t x o f M o n t o x + I by a f u n c t i o n h C l e a r l y ti i s a <-automorphisrc and a S-automorphism b u t p r e s e r v e s n e i t h e r [ n o r 1 F i n a l l y c o n s i d e r t h e mapping
I
I
.
.
k : N--, I
@
N
{ I , 21
,
.
such t h a t f o r every
.
k ( Z i i x ) = 2" x
x 1 2,
k ( 2 x ) = 4x
,
k ( 4 x ) = 2x
and f o r
Obviously, k respects 1 without preserving
I .
Now we show t h a t a r i t h m e t i c c a n b e c o n s i d e r e d as a t h e o r y o f two o r d e r s < a n d .'ll1e.urem 3.2 i s a c t u a l l y a c o r o l l a r y o f t h e f o l l o w i n g t ~ i e o r e m 3 . 3 . , ttte most c l a s s i c r e s u l t i n t h e a r e a of a r i t h m e t i c a l d e f i n a b i l i t y f r o m o r d e r s . However we p r e s e n t a n i n d e p e n d e n t p r o o f b e c a u s e i t i s t h e s i m p l e s t i n s t a n c e o f t h e c o d i n g method f o r t h e powers o f p r i m e u s e d l a t e r b a s e d o n t h e ZBV-theorem.
1
THEOREM 3.2.
1
The s u b l a n g u a g e
[
<,
I
1 i s synonymous w i t h L ( P ) .
I t i s c l e a r t h a t t h e g r e a t e s t common d i v i s o r ( g . c . d . ) a n d t h e l e a s t a r e t h e i n f and s u p i n t h e ( I ) - l a t t i c e of i n t e g e r s and s o ( I ) - d e f i n a b l e . The i d e n t i t y r e l a t i o n x = y i s ( 1 ) - d e f i n a b l e by
Proof.
G
n m u l t i p l e (1.c.m.)
Aritlirnelics as theories of two orders
1
299
1
V z (z x z y ) . The s u c c e s s o r f u n c t i o n i s c l e a r l y < - d e f i n a b l e . I f we c a n d e f i n e t h e m u l t i p l i c a t i o n i n {i, 11 , t h e n wc a r e done b e c a u s e i n PA t h e sum o f a a n d b i s t h e e l e m e n t c s a t i s f y i n g S ( a c ) S ( b c ) = S ( c c S(ab)) ( s e e l J R l t h e o r e m 1 . I . ) . 'To d e f i n e the p r o d u c c i n { c , 11 , f i r s t o b s e r v e t h a t t h e p r e d i c a t e s P ( x ) and P ( p , y ) meani ng " x i s a rime " a n d "p i s a - d e f i n a b l e as atoms prim e an d x a power of p d i f f e r e n t f r o m I " are We show t h a t t h e p r e d i c a t e a n d e l e m e n t s c o v e r i n g a s i n g l e atom i n t h e o r d e r 2tx E P(p , y ) me a n in g "p i s p r i m e and y = p f o r some a , 0" i s a l s o i < , 11 - d e f i n a b l e by t h e i < ,11 - f o r m u l a : I'(p, y ) ,\ 3 z P ( p , 2 ) A v q ( p ( q ) (q Y-1 -(q 2-1 v q z+l))) I
(7)
I .
-
1
I
.
1
Now we p r o v e t l i a t tl ie l o g i c a l e q u i v a l e n c e a b o v e i s a d i r e c t c o n s e q u e n c e o f t h e ZHV-thcorem ( I 0 ) . I n d e e d i t i s enough t o o b s e r v e t h a t , i f S u p p ( a ) d e n o t e s t h e s e t o f p r i m e d i v i s o r s o f a , we h a v e f o r a p r i m e p a nd p o s i t i v e i n t e g e r s 8 1
and
12
:
I)
Supp(p'-
Supp(p'{- I ) U Supp(p'+ I )
=
, 8
I
except f o r the case
a = 2
left to the reader.
Uith the p r p d i c a t e
product of
x
a nd
pa
=
y = pp
x
l e a s t primes g r e a t e r than g r e a t e s t power o f if
q = r ,
a s s e r t s tiiat x+p z = p or
p
clearly pcx < q
z
<
where t h e EP
{<,
,
y qr
.
.
p
<
.
(<,
If
q
respectively, then q r = 1.c.m.
As
ii, 1 1 - d e f i n a b l e . Rut
is
~ p ( ' pCi+'
11 - d e f i n i t i o n
we s h a l l
f o r any prime
and
less than
,a+[:+ 1
z = p
=
i f a nd o n l y i f
pB
and
I}
r
2
.
is
the
d e n o t e tlie
denotes t h e
z
(qr)
of p
-define
a nd
,
ru = 28
if
q # r
a nd
q
2
TCHEBYCHEV's t h e o r e m ( § 0 )
< r < pB+'
a nd h e n c e
z
A c t u a l l y w e know how t o e x p r e s s t l i e f a c t t h a t {< ,
a power o f e v e n e x p o n e n t o r a power of odd e x p o n e n t i n s i d e o f
.
1)
,
is
using
o n l y EP and a c c o r d i n g t o t h e s t a t u s of x and y C o n s e q u e n t l y , p a+% i s t h e g r e a t e s t power of e v e n e x p o n e n t o f p smaller t h a n q r i f E P ( p , x ) i s l o g i c a l l y e q u i v a l e n t to E P(p, y ) and t h e g r e a t e s t power o f odd e x p o n e n t o f p s m a l l e r t h a n q r i f I:P(p, x) i s n o t l o g i c a l l y e q u i v a l e n t t o E P ( p , y ) It can e a s i l y be seen t h a t t h e d e f i n i t i o n is a {<, -formula. I n s i d e I < , 11 w e c a n now e x p r e s s t h e p r o d u c t o f two a r b i t r a r y p o s i t i v e i n t e g e r s a a nd b , as t h e l e a s t p o s i t i v e i n t e g e r s u c h t h a t f o r a l l p , x a n d y s u c h t h a t P ( p , x ) and P ( p , y ) , w e h a v e x a A y b xy c C l e a r l y x = ah , concluding the proof. n
1
I
-
.
I}
I
.
Remark. T h i s p r o o f i s n a t u r a l b e c a u s e t h e p r o d u c t o f a a nd b i s b u i l t up fro m i t s s i m p l e s t f a c t o r s . A l s o t h e t o o l - t h e ZBV-theorem - a n d t h e c o d i n g t y p e p r o o f method w i l l l a t e r y i e l d s t r o n g e r r e s u l t s .
.
1)
We p r e s e n t two We weaken t h e s u b l a n g u a g e { < , by t a k i n g S i n s t e a d of < proofs; t h e f i r s t one is an a d a p t a t i o n of t h e o r i g i n a l proof while t h e o t h e r ( i n d e p e n d e n t o f t h e p r e v i o u s ) i s i n t h e m a i n s t r e a m of t h e ZBV-theorem. THEOREM 3.3. A d d i t i o n a nd m u l t i p l i c a t i o n o f p o s i t i v e i n t e g e r s a r e a r i t h m e t i c a l l y d e f i n e d i n t e r m s o f S and ( t h i s t h e o r e m i s dtir i n t h e s e t e r m s t o J u l i a ROBINSON ( s e e L J R l t h e o r e m 1 . 2 . ) ) .
I
Proof I . - L e t a , b , c be a r b i t r a r y p o s i t i v e i n t e g e r s and l e t m a n i n t e g e r coprime t o ah U s i n g t h e C h i n e s e r e m a i n d e r t h e o r e m , t h e r e a d e r may v e r i f y t h e e x i s t e n c e o f x a n d y s u c h t h a t x , y and a h a r e p a i r w i s e c o p r i m e a nd a x : by z - l ( m ) . A s i n t h e p r o o f o f t h e o r e m 2 , i t i s e nough t o I S , -define t h e m u l t i p l i c a t i o n . N o t i c i n g t h a t z = 1 + + V t (z t) , we c l a i m t h a t a b = c
.
I
1)
D. Richard
300
is logically equivalent to:
( a = I ~ b = --tc=l)v l mlS(I.c.m.(a,
x))A
V x y m [ ( x l y ~ x l a ~ x l c~ y l b~
mlS(I.c.m.(b,
y
l
c
~
I .
y)) - - , 3 u ( m ~ u - - , S ( u ) = I . ~ . m . ( c , x , y ) )
To verify the logical equivalence, we note that if ah = c then, for every convenient pair (x, y), the integer u = abxy-1 will do. Conversely, the Further from existence of the postulated u leads u s to cx 5 I ( m ) abxy : I ? c(m) for every m coprime to ab ; ax ? bx : -I(m) , we get hence c = ab
.
.
U
Proof 2. - This relies on the ZBV-theorem and on the following lemma which is a device implicit in J. ROBINSON and appears in A . WOODS "W], lemma 2.1 (11) 1 for the language I S , 1 1
.
LEMMA 3 . 4 .
Proof. -
(COMPARISON OF CARDINALITIES OF SUPPORTS)
I
There is an { S , 1 ) - formula INJ(x, y) defining tlie relation there is an injection from supp(x) into Supp(y) "
.
"
We shall first analyse the case where
I)
x
and
y
are odd and coprime.
Let A be the binary relation l'x and y are odd and coprime and there exists an injection from Supp(x) into Supp(y) ' I . Suppose (x, y) A and let F be an injection from Supp(x) into Supp(y). We shall associate to F a set of integers "coding" it. To this purpose, let us consider, for every p in Supp(x) and q in Supp(y) the system of congruences X(p, q , x, y) :
\ ==I -- 1I 2
z
z 5
I
(p) (q) (r)
for a11
r
E
has an infinite set
Observe that if p and p' are in Supp(x) and q and q ' in Sup!(y) if (P, 9) f (P', 4 ' ) then X(p, q, x, y) and X(P', q , x, Y) are disjoint: this is so because for all p SUPP(X) u SUPP(Y ) we have - 1 : 1 (p) since p # 2 We shall consider integers f "coding F" i.e. such that for every p in Supp(x) and every q in Supp(y) the set X(p, q , x, y) intersects Supp(f) if and o n l y if F ( p ) = q codes obviously exist.
are in
.
2)
.
~upp(x) u ~upp(y) \ tp, q~
By corollary 1.3, the system C(p, q, x, y) X(p, q , x, y) of prime solutions.
We are now in a position to introduce a set "3, 1 1 -definable and equal to A .
B
.
Such
which is both
We let B be the set of couples (x, y) t IN' such that: 'lx and y are odd and coprime and there exists an intezer f such that ( * ) " where ( * ) is "for every p in Supp(x), there is an unique q in Supp(y) such that the set Supp(f) intersects X(p, q, x, y) ; and also for every q in Supp(y) , there is a most one p in Supp(x) such that the s e t Supp(f) intersects X(p, q, x, y ) "
.
.
The previous analysis shows the inclusion A C B Conversely, if (x, y) E B then every f such ( x , y , f) satisfies ( * ) defines This shows a natural injection from Supp(x) into Supp(y)
.
A = B .
Arithmetics as theories of two crders
30 1
At last, let u s give an { S , I 1 -definition of B. To this effect, note that the relation z t X(p, q, x, y) is { S , I 1 -defined by the following formula ~ ( p ,q , x, y, z) which is: p
supp(s(z))
v r [(r
q E supp(s(z))
A
supp(x) u supp(y)
E
A
A
p(Z)
p 1r
A
A
q 1r)
-
r
.
~upp(z - I ) 1
E
A convenient { S , l 1 -formula INJ'(x, y) meaning (x, y) t B is 2 C sUPP(X) A 2 t SUpp(y) A x 1 y A 3 f [MAP(f, x, y) A OTO(f, x, y)] where MAP(f, x, y) is: P
suPP(x)
3q
sUPP(Y)
3
71
t
-
[ P(P, q, X, y,
Supp(f)
?T)
A
[~(p, r, x, y, a) q 1 r I1 and means that one can define a functional correspondence from the set Supp(x) into the set Supp(y) ; V r E SUPP(Y)
V q E Supp(f)
- which asserts that the above correspondence one-to-one- is:
While OTO by f ) is
v q
t
SUPP(Y)
{[ 3
i7
e sUPP(f)
v p'
'd p E SUPP(X) 3 71'
Supp(x)
-
e Supp(f) ( P(p, q, X, y,
PPP'. q', x, Y, a))] 3) For the general case we shall replace x constructed from x integer z which is A convenient formula 3X'
3y'
. )
(SUPP(X)
(2 E sUPP(Y) sUPP(Y') INJ'(x',
p I p' 1
. .
INJ(x, y)
(2 E sUPp(X) SUPP(X')
7) A
where x or y can be even or not coprime, and y by odd integers x' and y' simply and y , and compare x' and y' to another odd and coprime to x' and y' is the following:
{[21X-X'=X]
3 2
(coded
A[21yYyy'=y]A 3
u
--C
P
[P(p)
A
pl
AP # 2
X
SUPP(P))
\
SUPP(2)l)
3q [p(q)
A
ql Y
A
INJ'(y',
z) AINJ'(z,
A
9 # 2A
A
(sUPP(Y) u sUPp(q) \ SUpp(2)l) z)
A
X'l
A
.
y')}
2 A
y'l
Z A
Remark. Actually, the lemma holds with I S , 1 1 instead of I S , I 1 because P is { S , 1) -definable (see [DR3] or [ D R 4 ] ) and since for every prime p and every integer x, p 1 x is logically equivalent to 1 (pl x)
.
Now return to the promised Proof 2 of the theorem 3.3.
-
As
in the theorem 3.2., it is enough to
11 -define the multiplication. For this (and always, as in the theorem 3.2.) it is sufficient to { S , 11 -define the product pa. pB for all powers pa and p8 of a given prime p. We treat the general case p # 2 (there is a {S,
slight difference in the { S , I } -definitions of 2n.2B = 2a+8 due to the exceptional cases in the ZBV-theorem. We leave the straightfoward adaptation of the proof to the reader). The idea of the proof is illustrated in figure 4 : to every
Ea
and
Fa
such that
c1
# 8
implies that
Ea
, Fa ,
E8
pa 9
we associate
F
8 are
D.Richard
302
T
pairwise disjoined. Then we define two transversals figure. It is clear that a + B
=
T
and
fi,a
as in the
Y
if and only i f there is a bijection from
y
Y E
1 f
c1
y
=
x E E
meaning
r
+p-l+p
Y
f
+ B = y iff there is a bijection from
Now we have to express the situation in given power
pa
of
: P(p,y)
PWA
v
z
where
z
< y stands for
p
I1.
{S,
{S,
I
(p(q)
[(qlz-1 --DxZ~(~)) A((q1y-1
1y
1(y 6
A
T Y E
-define
. for a
Codl(x, y)
A q ~ 2 ) A
1 z) .
2
pY
f
onto
--
A
r
py-I
5
Tcx,
we use the following predicate
(~(p, z ) ~ Z < y ) - v q z
To
p
<-
2
E2
E
p-l
l(q1z-1))-
~~-1(q))ll]
I
IS,
Next we want to
.
1 -define
.
the height of a given power z = p of p For this consider the predicate HI(c, z ) so defined that c has exactly 6 prime divisors. Formally we put H1(c, z ) if and only if : P ( p , z ) A v q [ [ q E SUPP(C) 3 t (P(p, t) A t +
v Cod-l(x, y) 1.-
t)
A
t
I
2)
-(3
Codl(q, t))] q (Codl(q, t)
and
of
6
0)
q E
-1
z =
p6
.
;
I
SUPP(C)) I]
.
1 -define IS,
I
}-define
We note:
By the Chinese remainder theorem, for every power y at least one x such that Cod (x, y), proving that I Fa #
A
in Cod (x, y) and H1(c, z ) we is, I to get our previous Fa and H - I ( ~ , z ) which also 1
By permuting the height
[(P(P,
t
I ZA
=
pa
of
Ecl # 0
p there is (similarly
Arithmetics as theories of two orders 2.-
3.-
By t h e ZBV-theorem E,l
H ( c , z) o r 1 Supp(c) and
H - l ( ~ , z)
Ei
4.-
y = p
a'
,
Codl(x, y ' ) )
i
Fa n Fa, # 0 ) ;
(similarly
i m p l i e s t h a t t h e r e i s a b i j e c t i o n between
...,
i x l , x2,
( lCodl(x, y ) v
xSl
where e v e r y
x.
for
<
1
i ,< S
is in
; H ( c , z) A H - I ( ~ ' , z ' ) 1
Also z'
Vx
Ea, # 0 i f
proving t h a t
f?
--
y # y'
303
=
p6'
,
then
implies
c 1 c'
i s r e p r e s e n t e d by
TSS,
,
and i f
Supp(cc').
z = p6
and
The p r o d u c t of t h e
powers y and z of a f i x e d prime p i s x i f and o n l y i f : P ( p , x) v H - I ( ~ , z)
"for a l l
(a, b, c )
such t h a t
t h e r e i s a b i j e c t i o n between
Following lemma 3 . 4 . ,
H l ( a , x ) , Hl(b, y) Supp(a)
and
yz
is,
t h e l a s t formula b e l o n g s t o t h e sublanguage
From t h e f a c t s 1 , 2 , 3 and 4 we s e e t h a t t h e only i n t e g e r t h i s formula i s
and
Supp(bc)".
x
1
I
.
satisfying
.
0
..
Remark. Proof 2 i s l o n g e r t h a n proof 1, b u t on t h e o t h e r hand t h i s proof 1 i s e x t e n d a b l e t o an { S , } - d e f i n i t i o n de + , and f o r n e g a t i v e i n t e g e r s as asked i n [ J R ] p. 102. This i s done i n [DR3]
I
<
The n e x t theorem answers a n o t h e r q u e s t i o n of J u l i a ROBINSON'S ( i n [ J R ] p. 102) who asked whether i n weakening i n 1 and s t r e n g t h e n i n g < by + , we s t i l l g e t a synonymous sublanguage of L(P). J u l i a ROBINSON ( a c c o r d i n g t o A . WOODS) and A. WOODS h i m s e l f gave t h e r e s u l t and t h e i r p r o o f s a r e based on t h e SCHN1RELMA"'s theorem. ( T h i s one a s s e r t s t h a t t h e r e i s a n a t u r a l number n such t h a t e v e r y number x 2 2 i s t h e sum of fewer than n p r i m e s ) . Our proof i s d i f f e r e n t and works i n coding i n t e g e r s , r e l a t i o n s and f u n c t i o n s by means of primes.
I
THEOREM 3 . 5 .
1
Proof. -
The sublanguage
{ + , 11
The i d e n t i t y r e l a t i o n i s
i s synonymous w i t h {+, 1 1 - d e f i n a b l e .
L(P).
A convenient
{+, l ) - f o r m u l a meaning x = y i s f o r i n s t a n c e t h e f o l l o w i n g : v z v t [ ( x + z 1 t ) tf (y + z 1 t ) l
s i n c e t h i s formula i m p l i e s t h a t , f o r e v e r y prime p, w e have x y (p) . Of course the r e l a t i o n s x = 0 , x < y , y = S(x) , x = 1 , x = 2 , x = 3 , where S i s t h e s u c c e s s o r f u n c t i o n a r e r e s p e c t i v e l y d e f i n a b l e by t h e f o l l o w i n g x + x = x i+, 11-formulas: 3 2 [ Z # O A X + Z = y ] ; x < y A v z {x < 2 [ ( y < 2 ) v (y = z ) ] ) ; = I tf = s ( o ) i ; [:< = 2 f-f = s(1) i ; iX = 3 tf = s ( 2 ) l -+
rx
The unary p r e d i c a t e (X
# 0)
.
P ( x ) of p r i m a l i t y i s c l e a r l y { + , 1 ) - d e f i n e d by t h e formula: (X # 1) vy [ ( ( y # 0 ) A (y < X)) * y 1 X I
.
I
For a g i v e n prime p , t h e prime Suc(p) , t h e r e l a t i o n p x , x :y (p) a r e r e s p e c t i v e l y { + , 1 ) - d e f i n a b l e by t h e formulas: ; sUC(P) = U P ' ( p ( P ' ) A P < P ' ) ; P ( p ) A 1 (P 1 x ) v z ~ ( x + z = y v x = y + z ) + p ~ z l
the relation
.
The Chinese remainder theorem and t h e D I R I C H L E T ' s theorem a l l o w u s t o {+, 1 ) - d e f i n e , f o r e v e r y prime p > 2 and f o r e v e r y i n t e g e r x such t h a t
D.Richard
304 26x
, the prime
q ( p , x)
by the following I + , 1)-formula
A (2 p) A (2 < X ) (X < P) y = u p i ~ ( p ' ) ~ ( <x p') A b'q [(P(q) A q < We note that, if q(p, x) = q ( p ' , x') with p < and q(p',xf) : 1 ( p ) so that x E 1 ( p ) with contradiction: x = 1 , Similarly if q(p, x! then p divides lx- x'1 < p so that x = x one-to-one.
:
P(p)
.
p' l(q)lA p ' : x ( p l l . then q ( p , x) :x (1,) 2 < x < p , leading to a (p', x') with p = p ' , Therefore the function is
p)
-t
p'
Now we shall code the map x -p(x) where p(0) = 2 and p(x) is the x-th prime using the integers c(x) which we { + , 1)-define by the following formula: C(X) = UC{( 9(3,2) 1 C) A VP vy [((2,(Y)A (Y<X)A p(P) A ( V(P, Y) 1 C)) (Y(SUC(P), Y + l ) Ic)l Now it is provable by induction that p = p(x) is equivalent to [(x=o) A(p=2)]V[(x= 1) A ( p = 3 ) ] V [ 2 < x x ( q ( p 1 X+l)lc(x+l)] Similarly a code d(x, y) for multiplication is I + , 1 }-defined by d(x, Y) = v d I ( q(p(x), 2) 1 d) A (t
.
+
. .
-f
With such a code d(x, y) we can {+,I }-define the usual multiplication because it is provable by induction that xy = z is logically equivalent to [ y = o ++z=O]A[y=l-z=x]A ( 2 < y A t = m a x t ' [3q(9(q, t')Id(x,y))l A ( V(P, t) I d(x, y)) p = p(z) 1 0 + +
.
Immediately we reach the {<,l}-definability of L(P). We can only use incomparable elements for divisibility, but we have the full linear extension of the divisibility provided by the natural order. The theorem 3.6. was at first proved by A . WOODS who gives two different proofs of it in his thesis. This result could also be shown using our theorem 3.10. below. THEOREM 3.6.
i
(A. WOODS)
The sublanguage {<,I 1 is synonymous with L(P). Consequently {S,l} is synonymous with L(P) if and only if < is { S , 11-definable.
Proof. - Denote by Q(a, b) and R(a, b) for b # 0 the unique quotient and remainder under the usual euclidian division. It is enough to { S , 1 }-define x + y for given x and y We will show that z = x + y if and only if "x < y and y 4 z , x + y :z (6) , and for every prime p < z , p # 7 , Q(z, p) 5 Q(x, p) + Q(y, p) + i , with i = 0 or i = 1 I' This condition is obviously necessary. To prove that it is also sufficient, suppose that z < x + y and take p # 7 such that 2p \< x + y - z 6 4p : this is possible by TCHEBYCHEF's theorem from the fact that x + y - z = 6k # 0 The calculus proves that p has the forbidden property Q(z, p ) E Q(x, p)+ Q(y, p ) + j with j # 0 and j # 1 Now it remains to see that this condition is I<, 1 }-definable.
.
.
.
.
This follows from the logical equivalence in true arithmetic: z = p Q(x, p) [ P ( p ) A z = max z ' ( z ' < x A i(p 12')) 1 Moreover for O C i 6 6 , t 5 i ( 7 ) and t I i (6) (the last congruence is equivalent to t i ( 2 ) A t 5 i(3) ) are clearly { S , 1 }-definable. From t 3 i (7) and u 3 j ( 7 ) for some i and j (0 6 i \ < 6 , 0 \< j < 6) , one can {S, 1 ]-define without using + and 0 the congruences t + u I k (7) and ++
.
Arilhnietics as theorips of two orders
305
.
tu C ( 7 ) with 0 < k < 7 , 0 6 9. 6 7 Hence Q ( z , p ) E Q(x, p ) + Q(y, p) + i with i = 0 or i completing the proof.
= 1
is
{<, 1
}-definable, 0
Remark. This proof can be presented in such a way that we need only a {<,l )-definition o f binary predicates meaning Q(x, p ) E 1 (3) , Q(x, p ) 1 - I ( 1 ) , for every integer x and every prime p . This leads to the equivalence of the synonymity of { S , l 1 and L(P) (studied with more details below) with the I S , l 1-definability o f the remainder modulo 3 of Q(x, p) for every integer x and every prime p
.
The previous theorem asserts that the successor function S coming from < and the entire relation of divisibility are together sufficient for the description of arithmetic. Now are the "successors" defined from divisibility and the entire relation < also sufficient to the complete description o f arithmetic ? the answer is contained in the
KE:MARK __
3 . 7.
Let S ( p , x) the binary predicate meaning that " p is a prime and there is an y such that x = p y " , i.e. that x is a successor in the (I)-lattice for some y ; then the sublanguage I S ( , ) , < } is synonymous with the full language L(P)
.
-
Proof. In virtue of the theorem 6 , it is sufficient to I S ( , 1, < I -define the predicate 1 . This results from the following { S ( , )}-definition: x 1 y V p [ S ( p , x) -+ lS(P, Y) 1 0
.
At this step, one can ask what are the previous results on positive integers which are extendable to arbitrary integers. Julia ROBINSON posed the question (in [JR] p.102) noting that t h e diffic2nZty i n tkz's casp reduces to tiitrt of d e f i n i n g the n o t i o n of a p o s i t i v e integer) i n terms o f I anJ S
.
: a) The full language L(1) of Th(l) I < , 1 }-definable but is neither I + , I }-definable. 1: b) The language
is
L(Z)
IS,
1
is {+
,
I
{<,I
l-definable, l-definable nor
}-definable.
Remark. One can observe that, curiously and for all theories with negative and positive integers, I+,1 (and consequently { + , 1 } too) is a weaker 1 , since { S , 1 is synonymous with L(Z) sublanguage than [ S ,
I
I
Proof of a ) . - Recall that L(I) is synonymous with show the negative part of the theorem. Let f : Z -+ this map is an { t , I }-automorphism (and hence an the model 2 of Th(l) , but does not respect < s o I t , l } are synonymous with L ( 2 )
.
For {<, 11 and Indeed x I-+ 1x1
.
I
.
{<, +, 0 ) Firstly we 2 be such that f(z) = - z
only complete the proofs of theorem 3 . 2 . and 3 . 6 . l-definable by
{ 1 i 1. we
is
I<,
I
y = 1x1 * 'dt (t 1 y * t I X ) A (y > O V y = 0 ) V t (t I y) For non negative integers the formula x I y y = 0 ++ {<,I ]-definable (theorem 3 . 6 . ) , by a formula 7 (x, y). Replacing every
with
.
;
{+,l}-automorphisme) of that neither {+, 1 1 nor
is
.
D. Richard
306
variable v and every constant c by v 3 0 and c >/ 0 , we define easily i p ' ( x , y) meaning 1x1 1 IyI for x and y arbitrary integers. So x + 1x1 is I S , 1 }-definable by a formula analogous t o one of the case I S , 1 Similarly, as with the definitions o f x + 1x1 , we can e x p r e s s 1x1 IyI = ( z ( either in I<, 1 1 or {<, 1 ) , s o that :
.
1
The synonymity of I S , 1 1 with L ( P ) is an open and certainly very difficult question. We give below some equivalent problems and we favour those which are in the area of order theory. However some of the equivalent questions in other fields of mathematics are actually beautiful. For instance, A . WOODS shows in his thesis (see [AW]) that this question is another formulation of the following open problem in the theory of numbers: is there a universal positive k such that for every pair (x, y) of positive integers, the identity x = y is equivalent to the condition 'I X + i and y + i have the same set of prime divisors for 0 < i < k ?
.
Our investigation begins with the
I
A successor function S and a divisibility relation C$ are { S , 11-definable over Fowers of 2 so that: = S
2s(x)
(2x)
and
'2
$
'2 * x I y
Consequently an addition H and a multiplication : { S , l }-definable over powers of 2 , so that:
are
.,
2x H '2
ZX+'
=
and
2x
.
'2 '=' 2
Proof. - First we note that being a power of a s t u n r h d prime (as 2 instance) is a { S , 1)-definable property : P(2, If
X)
t--f
vyz [ y 1
z
(x 1 y V X 1
-+
f
is a mapping we write sometimes i i+l Now we define 5 We put 2 2 = 2
fx
.
v
put
t t'
[ SSt' = ?(St)
]
2)
A
instead of
for 0 6 i 6 6
[P(2, St)
i-+
P(2, SSt')
A
f(x)
.
for
convenience. 6 t > 2 - 1 we
; for A
v u ( u l S S 0 + (u 1 t
Actually this definition is equivalent in Th(lN)
l ( x 1 SS(0))
is, for
u 1 t'))
f-t
to V x
(ZSx
] =
1
,
.
S(2'))
SSt' = 2" , then for every prime x'-1 t' - 1 =q the relation q I 2 x - 1 is equivalent to q 2 2 . Conversely, assume that for every prime q this equivalence holds. By the
To prove this, note that if
St
=
2x , and
I
ZBV-theorem there is a primitive prime q 2x'-l = 1 (q) it follows that x I x ' - 1 x' - 1
1x
Next we
. Therefore
{ S , l ]-define
V t t ' [ St $ St'
3
1 (4)
and
=
2x $ 2x' x
I x'
.
.
Hence from
Similarly, it can be shown that
SSt'
and
1
=
2x' =''2
= S(St)
=
.
S(2')
by
@
* [St'
and we show that 2x
x' = x + 1
dividing -'2
1
V
iff
( P ( 2 , St)
x llx'
implies '2
.
A
P(2, St')
A
V U (U
1 t' +
Clearly, for every prime
: 1 (4)
and by definition
U
1 t))]
]
q
2x $ 2x'
for
.iiinietics as theories of t w o orders
.:,
S t = 2' then
and
St
St'
St'
theorem 3.3.,
=
we c a n
we h av e a l s o But
are
5
and
2"'
an d
and 2'
is, $1
t i o n s of
ZXy
and
+
2
by
S
Conversely i f a p r i m i t i v e prime
zX' = 1 ( 4 ) and c e r t a i n l y (5, @ ) - d e f i n e two o p e r a t i o n s
implies
formulae d e f i n i n g symbol
.
2"
I
from
by
2'
Q
@ =
.
2"'
from
and
2sx
and
2'
,
1 =
x
q
I x' . a nd
[9
ZX- I
divides
,
I f w e u s e now by t h e "same"
replacing respectively every
S(2x)
a nd
ZXy = 2x
2'
9 2'
x
iff
.
2'
1
y
,
( i n s p e c t t h e formulae used f o r t h e d e f i n i -
Obviously,
'2
and
S
Since
IS, 1)-definable j, ) .
307
13 and
{ S , 1 1 - d e f i n a b l e as a r e
are
.
Remark. T h i s t h e o r e m '3.9. i s u s e d t o p r o v e t h a t T h ( l N , S , 1) i s u n d e c i d a b l e ( s e e t h e o r e m l . 4 . ) , w hi ch i s a n e c e s s a r y c o n d i t i o n f o r h a v i n g a c h a n c e t o g e t t h e d e s i r e d synonymy b e t w e e n { S , 13 a n d L ( P )
.
To c o n c l u d e t h i s s e c t i o n , we show t h a t t h e o p e n q u e s t i o n o f t h e synonymy o f { S , 1 1 w i t h L(P) h e a v i l y depends o n t h e p o s s i b i l i t y of {S, 1 ) - d e f i n i n g e x p o n e n t i a l f u n c t i o n s o r one-t o-one ( f o r i n s t a n c e i n c r e a s i n g ) s e q u e n c e s o f p r i m e s o r o f p o we r s o f p r i m e s . To show t h i s w e need a t h e o r e m t h e p r o o f o f wh ich i s o m i t t e d b e c a u s e o f l a c k o f s p a c e .
THEOREM 3.10.
i
The r e s t r i c t i o n o f x 2x divisor is {S, l)-definable.
t o i n t e g e r s having e x a c t l y a prime
The p r o o f wo r k s by c o d i n g d i f f e r e n t k i n d s o f c r d e r b a s e d o n t h e ZBV-theorem a n d o t h e r r e s u l t s o f t h e o r y o f numbers. The i n t e r e s t e d r e a d e r w i l l f i n d i t i n IDR41. From t h e t h e o r e m s 3 . 9 . and 3.10 we g e t
,
{ S , 1 ) - d e f i n a b l e i f and o n l y i f o n e o f t h e f o l l o w i n g cond t i o n s holds:
:
a) t h e r e l a t i o n
< (or the relation
I
is
)
IS, 11-definab e,
1
::
c ) t h e r e i s a I S , 1 ) - d e f i n a b l e one-to-one ( f o r i n s t a n c e i n c r e a s i n g ) linearly s e q u e n c e u o f p r i m e s (a s u b c h a i n o f t h e atoms o f o r d e r e d by < s h o u l d b e p e r f e c t a n d t h e map x p ( x ) also),
.. c ' )
Proof.
I
Same as c ) w i t h "powers o f p r i m e s "
The condition a)
follows from theorems
3 . 6 . and 3.3.
The condition b2 follows easily theorem 3.9. p > 2
,
an obvious analogous theorem with
is necessary and sufficient. Clearly
c)
p
and
i n s t e a d of p r i m e s .
.
.
For a standard prime
2 holds so that b ) P are necessary conditions.
instead of c')
Firstly we show the sufficiency of c) : for every integer n , n +-t {S,
1)-definable by theorem 3.10. above and the prime character o f
2'(n) n(n)
is
. Rut
D.Richard
308 there is an @ which is {!3
I+,
, a )-definable
)-definable in replacing
+
and
by a formula by which
by
and
Q
. We
2'(n) = @ (2") , leading to a I S , 1)-definition of nproving that c') is sufficient, an analogous argument holds.
is
have Zn
. For CI
We have just seen that I S , 1) is synonymous with L ( P ) if and only if < is I S , 1)-definable. The corollary 3 . 1 2 . gives a necessary condition for that, since it proves that < is I S , 1)-definable over integers having no more than one prime divisor. COROLLARY 3 . 1 2 .
i
Over the powers of 2, it is natural to
Proof. 3' 2
(ZX
Now, let then
The restriction of the natural order to all the powers of all the primes is I S , 1 1-def inable.
pa
2' pa
<
= )'2
and
q8 -[(r
and it is clear that
2x
IS,
l)-define
@ '2
2x
@ '2
if and only if
by x
qB be two powers of the primes p and q respectively; a R = '2 ) A ( s = 2q ) A r @ s ] provides the desired
I S , l)-definition since the right side of this logical equivalence is, by theorem 3.10. expressed in terms of S and 1 0
.
Remark. To try to extend corollary 3.12. to all the integers is a posssible way for proving a positive answer to the Julia ROBINSON'S question. On the other hand, any eventual construction of a I S , l)-automorphism of a model M of Peano which is not an autornorphisrn of M will have to respect the natural order on the powers of primes.
.
Arithmetics as theories of two orders
309
APPENDIX.
I t a l i c n u mb e r s d e n o t e the pinitivc divi-
sors. I n s t a n c e s of an- I f o r a=2, a - 3 , n < 2 2 a n d f o r a=5, b = 7 , nd12 f o r t h e ZBVt h e o r e m . Ex a mp l e s of t h e t h e o r e m o f EULERFERMAT c a n b e c h e c k e d on t h i s t a b l e . A l s o e v i d e n t i s t h e prop e r t y of t h e l a t t i c e ( p r o v e d by CARMICHAEL) a n d b y wh ich
(1)-
1 ::1
2423.3851 2 -5.7.13.73
i s an (/)-hornorno mrphism.
1
2.3
2
2 .3 2 2.3 .19 5 2 2 .3.5
3
4 5 6 7
a
4
2.3.2801 4 2 2 . 3 .19.43 2.3.29.4733 6 2 2 .3.5 .izoi I)
2.3”. 1 9.37.1063
2 2 2 2 . 3 . 5 .13.19.43.181
1
D.Richard
310
REFERENCES:
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Annals of Discrete Mathematics 23 (1984) 313-342 0 Elsevier Science PublishersB.V. (North-Holland)
313
L' INTERVALLK EN THEORIE DES RELATIONS ; SES GENERALISATIONS FILTRE INTERVALLAIRE ET CLOTURE D'UNE RELATION Roland FraissC - UniversitG de Provence B Marseille A Ernest Corominas Abstract
<
Given a chain, or total nrdrring
, an interval can be defined in two I of the base, such
ways : absolute and relative. An ;iAsolutr intervdl is a subset x, y E I
that if
{a,bl where
and
x
, then
.
A relative interval with bound
, is the set of elemrnts x such thdt a < x < b
a'b
a relative interval with bound
F
, or all satisfy
. More
generally
(subset of the base) is a set, maximal with res-
pect to inclusion, among those sets D x
I
z E
t E F
such that, given
, all
x E D satisfy
.
x > t
A key for generalization to an arbitrary relation A , is the notion of local automorphism. A bijective function
f
is a
10Cdl
tion A/Dom f
onto
A/Rng f
. Then a
subset
A-interval, if any local automorphism f
f
map is still a local automorphism of
A
(A1
and
is an absolute
of the restriction A/I
. Any
f
is an isomorphism of the restric-
I of the base
IAl-I ; i.e. the union of
by the identity map on
A , if Dom
automorphism of
(range) are subsets of the base, and if
Rng f
f
is extensible
and of the latter identity
intersection of
A-intervals is an
A-interval. A rharacterization is given for the exterval (complement of a n interval). A subset if
D
of the base is a relative A-interval with bound
D
is disjoint from F
sets D
.
(subset of the base),
and is maximal, with respect to inclusion, among those
such that any local automorphism of
F
map o n
F
A/D
is extenseible by the identity
Every absolute A-interval i s a relative A-interval. Both notions are
identical for a chain A , but already differ for a partial ordering A. Two related notions are introduced : those of finite-val and subval. The finite-val is a boolean notion : union, intersection, complement of finite-vals are finite-vals. For a circular ordering
x
then
I
(mod A) is a
C
defined from a chain A
by
C
(x,y,z)
= +
iff
or a n y condition obtained from this by a circular permutation ;
C-subval iff
I is an A-absolute interval or exterval ; intuitively
the subval is the circular segment, or cake-portion. Every interval or exterval is a subval ; every subval is a finite-Val, but not conversely. An
A-filter (A-ultrafilter) is defined as usually, by replacing sets by
A-absolute
intervals. A P3rrpdCtdble relation is defined in two equivalent ways : ( 1 )
I , the complement of
A-interval
each A-ultrafilter tely many Finally
F
F
, associate
I is
a
finite union of
an element
for every
A-intervals ; (2) to
h ( F ) E F ; then there exist fini-
such that the union of corresponding
h (F)
covers the base.
A-ultrafilters and usual ultrafilters are used to extend to arbitrary
relations : ( I )
the closure of rationals by real numbers ; (2) the Stone closure
of a boolean lattice.
314
R. Fraisse
INTRODUCTION
P a r t o n s tle d e u x d 6 f i n i t i C i n s u s u t ~ l l t ~dse 1 ' i n t e r v a l l r d , l n s ou o r d r c t o t a l . PremiPremcnt,
I'intcrvalic~ .ibsuiii c s t un c n s e r n h l ~ c l o s p a r ii1tt. r-
mi.diarit6 : si
x r t y > x appnrtienncnt % I ' i n t c r v a l l e ,
termsdiaire
< z < y)
(x
011
F
appartient % I'intervnlle.
line p a r t i e a r h i t r a i r r dtz
de hornr
a1~11-st o u t i.li.rnt,nt
z in-
~ e u x i G m c m e n t 1 ' i i i t c ? r v a j l c >r c ' -
E l a b a s e d r 1'1 r,haint. ( r ~ i i s i ~ m h ldr r s e l6rntJ nts o r d o n n i . p a r I n c l i a i i i C ' ) ,
latif : soit et
u n c d ~.hnint7,
.
F
iin
r s t un t ~ n s r m h l e , maximdl p a r i n c l u s i o n ,
ayant I n proprigti. suivante : si
tout x de D rst
a p p e l 6 b n r n i ~ .A l o r s
F
> t.
intervslle relatif
s t a
,
parmi 1t.s 1 m r t i e s I) de L:-I'
t E F , uu b i e n t o u t x d r D e s t
P a r exrmple l o r s q u r F
F
ou h i e n
r P d u i t a u s i n g I e t i , n dc, 1'616mt-nt t ,
i l e x i s t e d r u x i n t r r v a l l e s r r l a i f s , e n s e m b l e d e s x < t e t enscmhlc, d e s x > t . I.orsque F = { t . u ) avec t
< 11,
iI
rxistr
t r o i h i n t e r v a l Jes r c l a t i f s ( p o s s i b l c m e n t v i -
dcs). T o u t i n t e r v a l l r a b s o l u D e s t u n i n t e r v a l l r r e l a t i f , l o r s q u ' o n p r e n d K-11 po u r b o r n e . I n v e r s e m r n t d a n s u n e c h a P n e , t o u t i n t e r v a l l e r e l a t i f e s t c l o s par inte rmi. d i a r i t g , donc r s t un i n t r r v a l l e a h s o l u . Ccpendant c ' r t t t l e q u i v a l e n c e nc s u h s i s t e r a p a s l o r s q u e noiis 6 t e n d r o n s l a n o t i o n d ' i n t e r v a l l e 2 u n e r e l a t i o n a r b i t r a i r e : p o u r l e c a s d ' u n o r d r e non t o t a l ,
l ' i n t e r v a l l e r r l a t i f s e r a d P j l une notion p l u s v a s t e
que l ' i n t e r v a l l e a b s o l u : v o i r $
3 ci-dessous.
La g g n g r a l i s a t i o n d e l ' i n t e r v a l l e a u x r e l a t i o n s , c o n d u i t a u x n o t i o n s d e f i n i v a l l e e t d e s u b v a l l e ( v o i r 5 4 e t 5 ) . Au § 6 , n o u s d g f i n i s s o n s l c f i l t r e e t l ' u l t r a f i l t r c i n t e s v a J l a i r e s , c e q u i permet d ' g t e n d r e a u x r e l a t i o n s l a n o t i o n d e c l o t u r e , c a l q u s e s u r l e p a s s a g e d e l a c h a i n e d e s r a t i o n n e l s 5 c e l l e d e s rGels. E n f i n a u § 7 , l ' u l t r a f i l t r e localerncnt i n t e r v a l l a i r e p e r m e t d ' 6 t e n d r e a u x r e l a t i o n s I n c l o t u r e dc'
St-onc.
ou r e p r g s e n t a t i o n p a r i n c l u s i o n d e s l a t t i s h u o l g e n s .
La n o t i o n d ' i n t e r v a l l e d ' u n e r e l a t i o n r e m o n t e aux a n n 6 e s 1950. E l l e e s t e x p o s e e chez Frai'ssE,
1971 ( v o l 1 ) p . 1 1 1 - 1 1 2 ,
r t 1977. Le p r g s e n t a r t i c l e r e p r e n d l e
c o n t e n u d e 1 9 7 7 , a v e c un c o m p l e t remani ement d e l a r G d a c t i o n , d e nouve a ux r E s u l t a t s e t de n o u v e l l e s n o t i o n s ; p a r exemple, l a g g n g r a l i s a t i o n de l a c l o t u r e de S t o n e rg p o n d 2 une c r i t i q u e c o n s t r u c t i v e d e M . Y a s u h a r a (Math. Re vie w s j u i n 1979, n o 16082, a n a l y s a n t I ' a r t i c l e dtl 1 9 7 7 ) . Parmi l e s a u t e u r s q u i o n t u t i l i s E de m a n i P r e i n t g r e s s a n t e l ' i n t e r v a l l e d e r e l a t i o n , c i t o n s F o l d P s 1973, G i l l a m 1 9 7 8 e t 1979, Lopez 1 9 7 8 , P o u z e t 1979 e t 1981. Le p r g p r i n t du p r 6 s e n t a r t i c l r a 6 t E examin6 p a r un r e f e r e e , q u i a propo-
s6 d e n o t a b l e s a m s l i o r a t i o n s ; j e l ' e n r e m e r c i r v i v e r n e n t .
I- INTERVALLE ABSOLU ; EXTERVALLE
R a p p e l o n s q u ' 6 t a n t donn6 d e u x r e l a t i o n s t i o n f d'une p a r t i e F de l a base
I
A1
A,A'
d e m@me a r i t g , u n e b i j e c -
s u r une p a r t i e F ' d e l a b a s e
IA'I,
est d i t e
3 15
L'interualle en thborie des relations
un i s v r i l o r p h i s m e l o c a l de A vers A ' , lorsque f est un isomorphisme de la restriction A / F sur A ' / F ' . A'
. . . ,f(xn)
(f(xl),
l'arit6 de A et A').
Plus pr6cis6ment lorsque =
A
(x i,...,x ) pour tous x i ,..., x
de F (l'entier n Ptant
Pour que f soit un tel isomorphisme local, i l suffit q u e f,
restreintc 5 chaque ensemble d'au p l u s n 616ments de F, soit un isomorphisme local d e A vers A ' . Lorsque A '
=
A,
nous disons que f est un a u t o m o r p h i s m e l o c a l
de A . Kappelons encore qu'une r n ~ l t i r e l ~ i t i oAn de base E est une suite finie de relations A . de base E , dites c o i a p o s a n t e s de A . Un isomorphisme local de la multirelation A vers A' est dEfini c O m e une bijection q u i , pour chaque composante A i d e A et la composante correspondante A' de A ' , est un isomorphisme local de A .
i
vers A ! ; mZme gSn6ralisation pour l'automorphisme. 1
Interva 1l e Etant donnP une relation A de base E, une partie D de E sera dite un A-intervalle
a b s o l u , ou simplement un A-intervalle, lorsque tout automorphisme
l o c a l d e la restriction A/D, 6tendu par l'identit6 sur E-D,
donne un automorphisme
local de A ; disons alors que tout automorphisme local de A/D est e x t e n s i b l e par l'identit6 sur E-D. Soit n l'aritb de A (ou le maximum
des arit6s des compo-
santes de A , lorsque A est une multirelation). Par c e qui prGcSde, pour que D soit un A-intervalle, il suffit que tout automorphisme local de A / D dEfini sur un domaine de p <--I
616ments de D, soit extensible par l'identit6 sur toute partie
1 n-p ElSments dans E-D. Si A est une cha?ne, nous retrouvons l'intervalle usuel, ou ensemble clos par intermgdiarit6. Si A
est
un ordre non total de base E, alors D est un
intervalle lorsque, pour chaque PlSment t de E-D, ou bien tous l e s 616ments de D sont
< t,
ou tous sont 1.1.
> t,
ou enfin tous sont
I
t (incomparables 1 t, modulo A ) .
Pour chaque relation, l'ensernble v i d e , l a b a s e e n t i e r e e t l e s i n -
g l e t o n d e c h a q u e g l e r n e n t d e l a b a s e , sont d e s i n t e r v a l l e s a b s o l u s . I1 exi.ste des
relations dont ce sont les seuls intervalles ; par exemple la consEcutivit6 C sur les entiers naturels : C (x,y) = + lorsque y
=
x + I , valeur - dans les autres cas.
Disons qu'une multirelation est u n a i r e lorsque s e s composantes sont des relations unaires.Si A e s t u n a i r e , t o u t e p a r t i e d e l a b a s e e s t u n A - i n t e r v a l l e . 1.2.
E t a n t d o n n e u n e r e l a t i o n A, t o u t e i n t e r s e c t i o n d e A - i n t e r v a l l e s e s t
un A - i n t e r v a l l e . 0
Partons d'un ensemble de A-intervalles D. et notons C leur intersec-
tion. Soit f un automorphisme local de A/C : i l suffit de voir que f est extensible par l'identitg sur une partie finie quelconque H de I A1 -C que Card (H) <arit6 de A ) .
Parmi les Di, soit D,,
une intersection disjointe de H. Pour chaque i
=
...,Dh ...,h
I,
(il suffit m2me
(en nombre fini h) donnant appelons u . les 616ments
de H qui n'appartiennent pas 1 D i : de sorte que H est l'ensemble des u.
R. Fra'isse'
316
...,
(i = I , h). Puisque D 1 est un intervalle, f est extensible par 1'identitG sur les u l . Puisque D2 est un intervalle, f dGj2 Gtendu par 1'identitG sur les u , de D2, est extensible par 1'identitG s u r les autres u , et sur les u 2 ; donc par l'identit6 sur tous les u 1 et u 2 ; etc. 1.3. E t a n t d o n n e u n e n s e m b l e d e A - i n t e r v a l l e s ,
f i l t r a n t par inclusion,
l e u r r e u n i o n est un A - i n t e r v a l l e . 0 Soit D les intervalles considGrEs, C leur rgunion. Soit f un autoi morphisme local de A / C , dont nous pouvons toujours supposer qu'il est de domaine
et codomaine finis. I1 existe un D. incluant 1 la fois le domaine et le codomaine ; donc f est extensible par l'identit6 sur I A1
- D i'
donc par l'identit6 sur I A1 - C .
0
1 . 4 . Soit A une multirelation, n le maximum de son aritG. S o i t F , G d e u x A-intervalles
s u p p o s o n s q u e t o u t e restriction d e A / ( F U G ) , d e c a r d i n a l
;
< n-I,
a d m e t t e u n e i s o m o r p h e p a r m i l e s r e s t r i c t i o n s d e A / ( F 17G ) . A l o r s l a r e u n i o n F U G
est un A - i n t e r v a l l e .
Dans le cas d'une chalne A , ou plus g6nsralement d'un ordre, ou mgme d'une relation binaire rGflexive, nous voyons que la r6union de deux intervalles ayant au moins un 616ment commun, est un intervalle (n'oublions pas qu'il s'agit d'intervalles absolus, notamment dans le cas d'un ordre non total). Soit f un automorphisme local de A / ( F U G ) . I1 suffit de supposer que les domaine et codomaine sont de cardinal m
f
n-I, et de prouver que f est
extensible par l'identit6 sur un ensemble de n-m 616ments arbitraires pris hors
...,
de F U G. Notons u l , . . . , u m les Gl6ments du domaine de f, et v l , vm leurs transformss par f. Par hypothlse il existe w l ,...,w appartenant 5 l'intersection m F n G , chacune des transformations des u
en wi (i = I ,
...,m)
et des w. en v i
( o u inversement) Stant un automorphisme local de A . I1 suffit donc de prouver que
la premisre transformation est extensible par l'identit6 sur n-m 616ments arbitraires hors de F U G. Nous pouvons toujours supposer qu'il existe un p < m avec les u l , 616ments de F, et les u . en w.
(i
=
U~+~,...,U
m
...,u P
Qlsments de G et non de F . La transformation des
I , ...,p) est de domaine et codomaine inclus dans F . Elle est donc
...,
extensible par l'identit6 sur les u (j = p+l, m) et sur un ensemble de n-m 616j ments arbitraires hors de F U G . Par ailleurs la transformation de w I ,...,w P' en les w. (i = I,...,m) est un automorphisme local : il suffit pour ~p+l'...,u m w ~ , u ~ ..., + ~ u, en les le voir, de la dgcomposer en la transformation des w I , m u . (i = I , m), suivie de la transformation des u . en w. La transformation
...,
...,
.
consid6r6e est de domaine et codomaine inclus dans G : elle est donc extensible par l'identit6 hors de F U G. Finalement,par composition, la transformation des u . en wi
(i
= I,
1.5.
...,m)
est extensible de mZme.0
Intervalles disjoints ; repr6sentant
S o i t A u n e r e l a t i o n , F,G d e u x A - i n t e r v a l l e s d i s j o i n t s , f u n a u t o m o r p h i s m e
l o c a l d e l a r e s t r i c t i o n A / F , e t g u n a u t o m o r p h i s m e local d e A / G .
317
L'interualle en theorie des relations A l o r s 12 r e u n i o n f U g 0
e s t u n a u t o m o r p h i s m e local de A .
Etendre les automorphismes par identiti.. p u i s composer.
0
S o i t A une multirrlatic>n,E sa b a s e , n I f maximum de I'aritP. Partageons
E en intervalles disjoints
L).
de rSlinion E . A chaque D i associons une partie finie
< n-I
d . telle que chaque restriction de A/Di 2
5lGments admette une isomorphe
restriction de A/di. Unc telle partie d i sera dite un r e p r G s e n t a n t de D i'
1.6. Considgrons une relation A de base E, une partition de E en intervalles disjoints D . et pour chaque D. un reprgsentant d.. Alors t o u t e r e l a t i o n d e 1
1
b a s e E, s d m c t t a n t l e s D . comme i r i t e r v d l l e s a v e c l e s m6mes r e s t r i c t i o n s A/Di, e t a y a n t l a m&e
r e s t r i c t i o n q u c A 9 l a r e u n i o n d c s r e p d s e n t a n t s di,
est i d e n t i q u c
B A. O
Soit A ' une relation vsrifiant les hypothPses de l'Snonc6. I1 suffit
de prouver que, pour chaque ensemble F de
616ments de E, nous avons A ' / F
=
A/F.
ou dans la rgunion des di. Supposons i F fl D i de cardinal < n-I. Par hypothPse i l existe un isomorphisme dc A/F. sur une restriction de A/di. Les isomorphismcs ainsi d6finis sont des automorphismes locaux p o u r A ' comme pour A , et leur rgunion est un auto-
Cela est 6vident s i F est inclus dans un D donc chaque intersection Fi
=
morphisme local pour A' comme pour A : l'ggalit6 A ' / F I . 7 . E t a n t d o n n e u n e r e l a t i o n A, A-intervalle
=
A/F en rgsulte.
0
un s o u s - e n s e m b l e D d e sa b a s e et un
F, 1 ' i n t e r s e c t i o n D n F e s t u n ( A I D ) - i n t e r v a l l c .
Notons qu'gtant donng un (A/D)-intervalle G, il n'existe p a s forcgment un A-intervalle F tel que G = D
f l
F. Par exemple, prenons pour A un ordre partiel,
lattis bool6en reprEsent6 par un cube usuel tridimensionnel dont le minimum u et le maximum v sont deux sommets o p p o s s s . Prenons pour D l'ensemhle d c s trois sommets x,y,z immgdiatement antsrieurs 2 v, et soit G
=
{x.y}.
Alors un A-intervalle
incluant G comprend forc6ment 1'6lGment immGdiatement antgrieur 2 x et z , donc incomparable 1 y ; donc i l comprend encore z : contradiction. E t a n t d o n n e A e t u n A - i n t e r v a l l e D, l e s ( A I D ) - i n t e r v a l l e s sont e x a c t e m e n t
l e s A - i n t e r v a l l e s i n c l u s d a n s D, o u e q u i v a l e m m e n t l e s i n t e r s e c t i o n s a v e c D d e t o u s les A - i n t e r v a l l e s .
Cons6quence immgdiate de l'dnoncE pr6cEdent.
1.8. LEMME. S o i t A u n e r e l a t i o n a d m e t t a n t comme i n t e r v a l l e s t o u t e s l e s p a i r e s d ' e l e m e n t s d e l a b a s e . C o n s i d e r o n s p e l e m e n t s x,,
h l a permutation cyclique
:
h(x.)
=
...,xP
d e l a b a s e , e t soit
(i
=
x l . S i h est
u n a u t o m o r p h i s m e l o c a l d e A , i l en e s t d e m&ne d e c h a q u e t r a n s p o s i t i o n (xi,xi)
(i,j 0
< p).
C o m m u n i q u e , a i n s i q u e 1 'enonce s u i v a n t , p a r DELEGLISE,
1983.
I1 suffit d e prouver par exemple que la transposition (x,,x2) est un automorphisme
local de A. La transformation de x2 en x,, celle de x 1 en x
x
donc celle de x 2 en P' sont des automorphismes locaux extensibles par l'identit6. Composons avec la
P' transformation simultan6e de x ,xl en xI,x2, qui est aussi un automorphisme local : P nous obtenons la transposition (xl,x2). 0
R. Fra'isst
318
Etant don&
une relation A, si toute paire d'elPments de la base est un A-lnterval-
If, alors toute partie de l a base est un A-intervalle. oSoit E la base, n l'arite de A et h un automorphisme local de A. Notons Dom h
le domaine, et Cod h le codomaine de h (transform6 par h du domaine). I1 suffit de montrer que h est extensible en un automorphisme local de A, par l'identite s u r
E
-
(Dom h U Cod h). Par les remarques du d6but du pr6sent paragraphe I , i l suf-
fit de consid6rer le cas oS Dom h et Cod h sont de cardinal G n - I . Ainsi nous pouvons supposer finie la r6unjon Dorn h U Cod h ; rGpartissons donc ses 616ments d'une
...,
x avec x = h(xi) (i < p) et X , = h(Xp) ; P i +1 d'autre part en suites maximales finies d'iteration : memes conditions mais avec
part en cycles finis d'it6ration x l ,
x 1 & Cod h et x 4 Dom h. Par le lemme prgcsdent, dans chaque cycle d'itEration, P la transposition entre deux elements quelconques d u cycle est un automorphisme local
de A, par hypothPse extensible par l'identite. Par composition, h elle-msme, restreinte au cycle, est un automorphisme local extensible. Dans une suite maximale d'iteration x I ,*..'XP' la transformation de x
en x est un automorphisme local P-1 P extensible ; composons-la avec la transformation de xp-2 en xp-I' elle-msme automorphisme local extensible, et ainsi de suite en remontant jusqu'8 la transformation de x1 en x2 : nous voyons que h, restreinte aux 616ments de la suite d'it6ration sauf x est extensible par l'identite. Finalement h toute entiere est extensible P' par l'identit6 sur E - (Dom h U Cod h).o 1.9.
Extervalle
Etant donne une relation A, appelons A-extervalle le complGment, par rapport 1 la base, d'un A-intervalle. Une partie D de la base I A I est un A-extervalle ssi, pour tout automor-
*
phisme local f de AID et toute partie G de D
=
I A I - D, ou bien f n'est pas ex-
tensible par l'identite sur G, ou f est extensible par tout automorphisme local de
*
A I D .de domaine G .
OSupposons q w D soit un extervalle, donc D*
=
IA
I - D est un intervalle. Soit
g
un automorphisme local de A/D*. de domaine G . Supposons que l'automorphisme local f de AID soit extensible par l'identit6 Ie sur G : donc f U IG est un automorphisme
local de A. Puisque D* est un intervalle, g est extensible par l'identit6 sur le codomaine F' de f, soit IF'
. Donc
la r6union f U g est un automorphisme local,
compos6 de f U IG et IF' U g . Inversement supposons que D ne soit pas un extervalle, donc D*
*
ne soit pas un in-
tervalle. Alors il existe un automorphisme local g de A/D et une partie F de D, tels que I U g ne soit pas un automorphisme local. Alors l'identit6 I est 6videmF F ment extensible par l'identit6 sur G, sans stre extensible par g, contre la condition de 1'6nonce.
0
3 19
L'interualle en t h b r i e des relations
2- LNTERVALLE ET LIBRE-INTERPRETABILITE
Rappelons qu'une relation B de mzme base que A est dite l i b r e - i n t e r p r e table en A lorsque tout automorphisme l o c a l de A est un automorphisme local de B ; s i n e s t l'arit6 d e B, i l suffit que tout automorphisme local de A , sur < n SlEments,
soit un automorphisme local de B. Equivalemment, il existe une formule logique libre (sans quantifieurs) representant B lorsque son pr6dicat est remplace par A. 2 . 1 Soit A , B deux m u l t i r e l a t i o n s de base commune, chacune l i b r e - i n t e r -
pr.c
B-intervalles sont l e s m&nes.
En consGquence, s ' i l r x i s t e uric m u l t i r e l a t i o n unaire B t e l l e que A e t B soient chdcune libre-intc?rpr&tablc en l ' d u t r e , alors toutes l e s p a r t i e s de la base sont des A-intervalles.
Exemple, la concatenee d'une relation unaire U et de 1'Equi-
valence (binaire) d6finie par les deux classes des x tels que U (x)
=
+ et U (x) =
-.
L'inverse est faux : l o chaine sur deux 6lGments admet c o m e intervalles toutes les parties de sa base, sans vErifier l a condition donnGe. De msme l'extension de cette chaync 2 une infinit6 d'6lements nouveaux x, avec valeur - pour les couples diagonaux (x,x).
2.2. Equivalence
transposition
Etant donn6 une multirelation A , disons que deux dlements u,v de sont Pguivalents par transposition (mod A),
1A I
lorsque la transposition (u,v) Gtendue
par l'identit6, est un automorphisme de A . Pour q u ' i l e x i s t e une m u l t i r e l a t i o n unaire en l a q u e l l e A s o i t l i b r e i n t e r p r e t a b l e , i l f a u t e t s u f f i t que l e s c l a s s e s d'equivalence p a r transposition
(mod A) soient en nombre f i n i . Plus prEcis6ment si ces classes sont en nombre fini, et si 5 chacune nous associons la relation unaire valant + sur la classe et
-
ailleurs, l a s u i t e
de ces r e l a t i o n s forme une m u l t i r e l a t i o n unaire en laquelle A e s t l i b r e - i n t e r p r e tab1 e .
Supposons A libre-interprgtable en une multirelation unaire U. Alors deux Glements quelconques u,v qui donnent la mEme valeur B chaque relation unaire composante de U, sont equivalents par transposition (mod A ) : les classes d'equivalence sont donc en nombre fini. Inversement supposons ces classes en nombre fini, et associons 1 chacune une relation unaire valant + s u r cette classe et
-
ailleurs.
Soit n le maximum de l'arit6 de A ; tout automorphisme local f de U, s u r un domaine F de cardinal
< n,
est la restriction au domaine F d'une permutation de F U f (F),
composee d'un nombre fini de transpositions qui conservent U, donc A : f est donc un automorphisme local de A. Done A est libre-interprstable en U.0 Notons que, si une relation B est libre-interpretable en A , i l est possible soit que t o u t B-intervalle soit un A-intervalle, soit l'inverse, soit qu'aucun des deux ensembles, des A-intervalles et des B-intervalles, ne soit inclus dans l'autre.
R. Fra'issi
320
Exemple du premier valeur A (x)
t =
A
cas :
A est une relation unaire prenant au moins deux fois la
et une fois - , et B est In relation d'gquivalence correspondante. Si (XI)
= t
et A ( y )
=
-
, la paire {x,y] est un A-intervalle, ainsi
que
tout sous-ensemble de la base ; mais cette paire n'est pas un B-intervalle, puisque la transformation de x en y est un autumorphismr local de B, non extensible par l'identitg sur
{XI).
Exemple du druxisme cas : soit A l a chaine des entiers naturels, et B l a relation unaire toujours + sur ces entiers. Exemple du troisisme cas : soit A
= (N.0)
oii
N est la chaine des entiers naturels
et 0 la relation unaire singleton de zero. Soit B la relation d'gquivalence avec deux classes, qui sont le singleton de zero
et
l'ensemble des entiers positifs.
Alors B cst libre-interprstable e n 0, d o n c e n A. La paire { O , l } est un A-intervalle mais non un B-intervalle. La paire { 1 , 3 }
est un B-intervalle mais non un A-interval -
le.
2.3. Soit A u n e multirelation de base E. Disons que deux 616ments x,y de E sont 6quivalents par isornorphie des sinyletons lorsque les restrictions A/Ix}
et A/{y) sont isomorphes. Autrement dit lorsque la transformation de x en y est un automorphisme local d c A . Si A est une relation, il ne peut exister que 2 classes pour cette gquivalence : valeur + ou valeur
-
pour A(x,
...,x).
Si A est une multi-
relation avec h composantes, il existe au plus 2h classes. 1,'Equivalence par transposition d6finie en 2.2., est identique ou plus fine que l a pri5sente gquivalence par isornorphie des singletons. Par exemple si A est une cha?ne, tous l e s 61Ements d e la base donnent d e s singletons isomorphes. Cependant aucune transposition entre d e u x 616ments distincts n'est un automorphisme local de la chaine A : donc il existe autant de classes p a r transposition que d'glgments dans la b a s e . (1)
Soit A d e b a s e E ; s i toute p a i r e , donc toute partie de E e s t un A-intervalle
e t s i x,y,z sont t r o i s dlkments d i s t i n c t s , equivalents par isornorphie des s i n y l c -
t o n s , a l o r s x,y,z sont encore equivalents p a r transposition (mod A ) . ( 2 ) En consSquence, si A e s t une multirelation form6e de h composantes, et s i toute
partie de la base est un intervalle, alors i l e x i s t e au plus Zh+l classes d'equivalence par transposition. Plus prscisbment chaque classe d'isomorphie des singlet o n s , ou bien donne une unique classe par transposition, o u donne deux de ces c l d s s e s , reduite chacune a un singleton. 0
(DELEGLISE, 1983).
Par hypothese, l a transformation d e y en z est un automorphisme local de A . La
paire { y , z > Stant un intervalle, la transformation de x,y e n
x,z
est encore un auto-
morphisme local. De mEme la transformation d e x,z en y.z puis celle de y,z en y,x. Par composition d e c e s trois transformations, la transposition (x,y) est un automorphisme local de A, extensible p a r l'identit6. 2.4.
0
Soit A une multirelation de base E. Notons UA l a rnultirelation unaire
d'isomorphie des sinqletons : chaque composante d e U
A
prenant la valeur + sur une
et
32 1
L'interualle en th6orie des relations une seule des classes d'gquivalenc? par isomorphie des singletons. Notons
Ui l a
r n u l t i r e l a t i o n u n a i r e a s s o c i e e d e m&me d 1 ' e q u i v a l e n c e p a r t r a n s p o s i t i o n (mod A). Pour q u ' i l existe une m u l t i r e l a t i o n u n a i r e U telle q u e A et
U soient c h a c u n e l i b r e -
- i n t e r p r e t a b l e en l ' a u t r e , i l f a u t e t s u f f i t q u e l e s d e u x e q u i v a l e n c e s , p a r i s o m o r p h i e d e s s i n g l e t o n s e t p a r t r a n s p o s i t i o n , se c o n f o n d e n t
;
nous pouvons alors prendre
U = U 0
= Uf; (DELEGLISE, 1983). A Supposons qu'il existe une multirelation unaire U avec A et U chacune libre-inter-
pr6table en l'autre. Alors U et A ont l r s rnzmes automorphismes locaux ; donc U se confond ci la fois avec UA et avec U o
A '
Inversement supposons U A
=
Ui. Alors les classes d'gquivalence par transposition
sont en nombre fini. Par 2.2, la multirelation A est libre-interprgtable en Uf;
.
Par ailleurs i l est 6vident que UA est libre-interprgtable en A : i l suffit donc de poser U
=
UA - -U ; ; . "
2.5. Soit A de base E ; si t o u t e p a i r e , donc toute partie de E est un A-intervalle,
(I)
alors :
i l e x i s t e u n e m u l t i r e l a t i o n u n a i r e en l a q u e l l e A est l i b r e - i n t e r p r e t a b l e
;
(2) si de plus i l existe une seule, o u aucune classe d'6quivalence par transpoun singleton, alors il e x i s t e u n e m u l t i r e l a t i o n u n a i r e U t e l l e q u e sition r6duite .i A e t U s o i e n t c h a c u n e l i b r e - i n t e r p r e t a b l e en l ' a u t r e (POUZET, 1981). Le ( I ) rgsulte de 2.2. et 2.3. (2). Le (2) r6sulte de 2.3. (2) et 2.4. 3- INTERVALLE RELATIF ; BORNE Soit A une relation de base E, et F une partie de E. Alors une partie D de E-F sera dite un ( A , F ) - i n t e r v a l l e , ou A - i n t e r v a l l e relatifadmettant F comme borne, lorsque D est maximal, par inclusion, parmi les parties D'
de E-F telles
que tout automorphisme local de A / D ' soit extensible par l'identitg sur F (donnant ainsi un automorphisme local de A ) . Si A est une chaine, les ensembles D' possibles ont, pour chaque 61Ement t de F, tous leurs 616ments
ou tous
>t
; prenant pour D un D ' maximal par in-
clusion, nous obtenons un ensemble clos par intermgdiaritg, donc un intervalle usuel. Tout intervalle absolu D est un intervalle relatif, admettant entre autres la borne E-D. L'inverse est faux, d6j2 pour un ordre non total A . Par exemple prenons a et b > a
(mod A) ; I'ensemble des x tels que a < x < b est un intervalle re-
latif admettant la borne {a,b}, entre autres. Si maintenant i l existe x,y de l'intervalle D consid6r6, et t de E - D ,
avec a < x
et t 1 y et t I b, alors l'auto-
morphisme local d u singleton {x} sur Iy} est extensible par l'identitg sur {a,b} mais non par l'identit6 sur {t}
:
l'intervalle D n'est pas absolu.
Puisque tout intervalle absolu rst relatif, nous avons, pour chaque relation, les intervalles relatifs vide, base entisre et singletons ; nous allons voir que certaines relations n'admettent p a s d'autres intervalles relatifs.
R. Fra'isse
322
A chaque cha'ine A associons la relation ternaire de cyclordre, o u ordre c y c l i q u c ? rnqendre par A : relation C de mGme base q u e A, dEfinie par C (x,y,z) x S y S z (mod A) o u lorsqu'on
a
= +
lorsque
l'une des permut6es circulaires de cette condition.
Par exemple, si A est la cha'ine des entiers naturels, le cyclordre engendrP par A est encore engendr6 par toute cha'ine form6e des entiers successifs 2 partir d'une valeur p donnge, soit p , p + l , p + 2 , Soit
C
>3
un cyclordre sur
...
suivis pour terminer de 0, I , 2, ..., p - I .
616ments, D une partie stricte de la base E. Prenons
u , v distincts appartenant 1 D : La transposition (u,v) est tin automorphisme l o c a l
de C, non extensible, &me
pas par l'idrntit6 sur u n singleton dans E-D. Donc un
tel cyclordre n'admct pour intervalles relatifs q u e le vide, la base et l e s singlctons. Par contre, l a consgcutivit6 des entirrs naturels, qui n'admet c o m e intervalles absolus que le vide, l a base et les singletdns (voir l . l ) , admet bien d'autres intervalles relatifs : par exemple l a paire { O , l } en prenant c o m e borne l'ensemhle
> 3.
des entiers
La base entisre est l e s e u l intervalle qui admette c o m e borne l'ensemble
vide (consgquence de l a condition de maximalitg). Soit D et D' 3 D deux A-intervalles relatifs ; si F est une borne de D , ni F ni aucun ensemble incluant F n'est une borne de D ' (autre consgquence de I s maximalit&). Si n est I'aritE de A, i l suffit, c o m e pour l'intervalle absolu, de prendre un automorphisme local arbitraire de A/D' sur un domaine de p S n-l
616ments
de D', et d'exiger qu'il s o i t extensible par I'identitE sur tout (n-p)-emble inclus dans F ; un intervalle relatif D 6tant encore un D' maximal par inclusion.
3.1. Etant donng
une
relation A et une partie F de sa base, l a d u n i o n
des (A,F)-intervalles est E-F. En effet, pour chaque 6lEment u de E-F, le singleton de u est un ensemble D' au sens du paragraphe 3 : il existe donc au moins un intervall e relatif qui inclut ce singleton.
Dans certains cas particuliers, c o m e le cas d'une relation binaire r6flexive A , les (A,F)-intervalles sont mutuellement disjoints. En effet ici n-l
= I
i l suffit de considgrer les automorphismes locaux dgfinis sur un unique ElEment, c'est 5 dire toutes l e s transformations d'un 6lGment de E-F en un autre, et d'exiger qu'une telle transformation soit extensible par I'identitE sur chaque Slsment de F. Autrement dit, si a dEsigne u n 616rnent de F, et x un Elsment quelconque de l'intervalle, ledit intervalle est dEfini par les valeurs A(a,x) et A(x,a)
qui
doivent rester les mEmes lorsque x varie dans le (A,F)-intervalle. 11 en rEsulte qu'un msme x ne peut appartenir 1 deux (A,F)-intervalles distincts. Passons au c a s u n peu plus gEnEral d'une relation ou multirelation bim i r e (arit6 maximum 2 ) A et d'un ensemble F. En ce cas r6partissons l e s 6lEment-s de l a base en classes
d'isomorphie des singletons (voir 2.3).
Cela donne au plus
2 classes pour une relation (A(x,x) = + ou - ) , un nombre fini de classes pour une
;
323
L'intervalle en theorie des relations
multirelation. Chaque classe s e subdivise en sous-classes, en mettant x et y dans une mzme sous-classe lorsque l'automnrphismr local de A q u i transforme x en y , est extensible par 1'identitP sur F : donc chaque sous-classe est dSfinie par les valeurs A(a,x) et A ( x , a ) qui pour a 616ment de F, doivent rester constantes lorsque x varie dans une soi~s-classe.Alors chaque (A,F)-intervalle est obtenu en prenant dans chaque classe non vide, une de s e s sous-classes non vides, p u i s la r6union des sous-classes ainsi choisies. En consgquence, Stant donnt? line relation binaire A et une partie F de sa base ; si deux (A,F)-intervalles ont un BlPment commun dans chaque classe d'isomorphie des sinyletons, ces intervalles sont identiques. ProblPme. Contre-exemple de l'Pnonc6 ci-dessus avec une relation ternaire. 3.2. Soit A une relation, F,G deux parties de sa base avec G C F ; alors c h a g u e (A,F)-intervalle est inclus dans un (A,G)-intervalle.
Dans le cas d'une relation binaire A avec G
c F,
chaque (A,G)-interval-
le non vide inclut un (A,F)-intervalle non vide. 0
Rgpartissons les ElPments du (A,G)-intervalle considErE, en classes d'isomorphie
d e s singletons (voir 2.3).
Cela donne au plus 2 classes si A est une relation, un
nombre fini de classes dans le cas gEn6ral d'une multirelation. Prenons un 61Ement t reprgsentant de chaque classe. Pour assurer la maximalit6 qui fait partie de la
dEfinition de l'intervalle relatif (voir paragraphe 3), joignons 2 chaque reprgsentant t les El6ments t' de la m@me classe, pour lesquels la transformation de t en t', qui est d6j2 extensible par tit6 sur
1'identitE sur G, est de plus extensible par l'iden-
F.0 Problsme. Existence d'une relation ternaire A avec un F et G
c F,
et un
(A,G)-intervalle qui n'inclut aucun (A,F)-intervalle. 3.3.
L'intersection de deux intervalles relatifs distincts mais ayant
une borne commune F, ne peut Stre un intervalle relatif de borne F, en raison de la maximalite. Soit A une relation binaire ; consid6ron.s un ensemble de A-intervalles relatifs Di, chacun admettant entre autres une borne Fi. Alors l'intersection des
D. est incluse dans un A-intervalle relatif ayant p o u r borne la reunion des F.. D6j2 pour une relation binaire, l'intersection de deux intervalles re-
latifs n'est p a s forcement un intervalle relatif. Prenons une base E de 6 El6ments a,b,c,c',d,e avec A(x,x) x
=
a,b,d,e, et
A(e,a)
=
- ,
tions (b,e),
-
pour x=c,c'. Orientons les paires suivantes : A(a,e)
= +
pour
= +
et
ce que nous nommons l'orientation (a,e) ; prenons de mgme les orienta(e,d),
(c,d), (d,c') ; les autres paires prenant par exemple la valeur
+ d a m les deux sens. L'ensemble A = {a,c,c') est un intervalle admettant la
borne F
=
{b,e} : on ne peut lui ajouter d en raison des orientation (a,e) et (e,d).
L'ensemble B
=
{b,c,c'} est un intervalle admettant la borne G
=
{a,e] : on ne peut
lui ajouter d en raison d e s orientation (b,e) et (e,d) : la maximalit6 est bien
R. Fraisse'
324 satisfaite. L'intersection A n B
=
{c,~'} n'est pas un intervalle.
En effet la borne ne peut pas comprendre d, en raison des orientations (c,d) et (d,c'). Donc cette borne Bventuelle est incluse dans {a,b,e). Alors la maximalit6 n'est pas satisfaite, puisqu'5 c et c' n o u s pourrons toujours ajouter d (de valeur A (d,d)
= +). 0
3 . 4 . Soit A une r e l a t i o n binaire d e base E , e t s o i t D un A-intervalle r e l a t i f ; a l o r s la reunion des bornes de D e s t une borne de D. 0
Soit u,v deux Qlgments de D, avec isomorphie des resteictions de A aux
singletons de u et de v. Alors pour chaque 616ment c de la rBunion des bornes, l'automorphisme local transformant u en v est extensible par l'identitg sur c. De plus D est maximal par inclusion, parmi les ensembles qui vBrifient ce qui prBcPde. ProblZmes. Contre-exemple P ce qui pr&cZde, dans le cas d'une relation ternaire. Existence d'une relation ternaire A, d'un A-intervalle D de borne F , d'un A-intervalle D' de borne F' avec un automorphisme local de AID' n D' inextensible par l'identit6 sur F U F'.
3.5. Contrairement 5 l'intervalle absolu (voir 1.7),
l'intervalle re-
latif n'est pas prCservC par restriction : si D est un sous-ensemble de la base de A, et si F est un A-intervalle relatif, l ' i n t e r s e c t i o n D n F n ' e s t p a s forcement un (AID)-intervalle r e l a t i f .
lorsque y
Prenons la cons6cutivit6
=
x+l. Alors l'ensemble des entiers > 2 est un intervalle, admettant
A
siir les entiers naturels : A(x,y)
= +
0
pour borne le singleton de 0. Supprimons l'ClCment 0 : nous obtenons la consscutivit6 sur les entiers positifs, pour laquelle l'ensemble des entiers un intervalle.
>2
n'est plus
0
3 . 6 . Etant donne une r e l a t i o n binaire A e t une p a r t i e f i n i e F de sa base, l e s (A,F)-intervalles sont en nombre f i n i . 0
Disons que deux ClCments u,v de I A I - F sont equivalents lorsque la
transformation de u en v est un automorphisme local de A, extensible par l'identit6 sur F. Puisque F est fini, les classes de cette 6quivalence sont en nombre fini,
ce qui entraPne l'bnonc6. L'Bnonc6 prCc6dent ne s'Ctend pas au cas ternaire. Par exemple prenons pour A un cyclordre infini ; pour prgciser, le cyclordre des entiers naturels, et prenons pour F le singleton de 0 . Chaque singleton autre que 0 , est un (A,F)-intervalle ; en effet la maximalit6 est vCrifibe, puisque l'addition d'un second entier v B l'entier u qui constitue notre intervalle, autoriserait 5 considsrer la transposition (u,v) qui est un automorphisme local non extensible par l'identite sur 0. ProblSme, suggerd par 1 . 3 . Etant donn6 un ensemble, filtrant par inclusion, d'intervalles relatifs, leur rdunion est-elle t o u j o u r s un intervalle relatif.
0
325
L'interualle en thiorie des relations 4- LE F I N I V A L L E , UNE N O T I O N BOOLEENNE
Soit A une relation, D une partie de sa base. Pour chaque entier posirt v i ( i = I , ...,p) dans D sont e q u i v a l e n t s i lorsque la transformation qui change chaque u . en v. est bijective
tif p, disons que deux p-uplrs u modulo ( A , D ) ,
et est un automorphisme local de A extensible par l'identit6 sur I A I -D. L'ensemble D est d i t un A - f i n i v a l l e
lorsque, pour chaque p, i l n'existe qu'un nombre
fini de classes d'6quivalence mod(A,D) entre p-uples. Si n est l'arit6 de A (le maximum de l'arit6, pour une multirelation), il suffit de limiter la longueur d e s suites aux p
< n-I,
et de limiter l'extensi
bilit6 1 n-p 616ments de I A I - D . L'6nonc6 1 . 7 reste valable pour les finivalles : e t a n t d o n n e u n e r e l a t i o n A et un sous-ensemble
D d e s a b a s e , t o u t A - f i n i v a l l e , intersecte p a r D
donne un
(AID)-finivalle. 4.1 . T o u t e r e u n i o n f i n i e d ' i n t e r v a l l e s d i s j o i n t s e s t u n f i n i v a l l e (la
condition de disjonction sera inutile aprSs le 4 . 4 ci-dessous). 0
Deux p-uples sont Cquivalents s i la transformation d e l'un en l'autre
est un automorphisme local, et si de plus les premiers termes sont dans un mzme intervalle, les deuxismes termes sont dans un mGme intervalle, et ainsi de suite. En particulier, pour toute relation, t o u t e n s e m b l e f i n i e s t u n f i n i v a l l e , puisque r6union finie de singletons, qui sont des intervalles ; cela se voit
aussi immgdiatement sur la dgfinition. P o u r u n e c h a i n e , l e s f i n i v a l l e s sont e x a c t e m e n t l e s r e u n i o n s f i n i e s d ' i n t e r v a l 1es
.
A chaque chaine A nous avons associ6 la relation ternaire de c y c l o r d r e
( o u ordre cyclique) voir paragraphe 3 . Nous avons vu alors que les seuls interval-
les (absolus ou relatifs) du cyclordre sont le vide, la base et les singletons. Soit alors A la chaine des entiers relatifs, C son cyclordre engendrG : l'ensemble D des entiers positifs n'est ni un C-intervalle ni un extervalle ni une r6union ou intersection finie d'entre eux ; mais D est un C-finivalle, deux p-uples 6tant Gquivalents mod(C,D) lorsque la transformation d'un en l'autre pr6serve l'ordre A . 4 . 2 . L e c o m p l e m e n t a i r e d ' u n f i n i v a l l e est u n f i n i v a l l e . 0
Soit A la relation, n s o n aritc, E sa base, D un finivalle.
Pour chaque entier positif p, il n'existe qu'un nombre fini d e classes d'Gquiva-
lence mod(A,D).
Pour chaque p-uple (p
< n-1)
dans D, prenons un unique reprGsen-
tant appartenant P la mzme classe, et appelons H la partie finie de D, rEunion de ces
reprcsentants. Pour chaque entier positif r. disons que deux r-uples dans E-D
sont Gquivalents, lorsque la transformation de l'un e n l'autre est un automorphisme local, extensible par l'identit6 sur H. Puisque H
est fini, i l n'existe, pour
chaque r, qu'un nombre fini de classes pour cette Equivalence.
R. Fra'isse
326
Pour voir que E-D est un finivalle, i l suffit maintenant de
prouver que, si g
est un automorphisme local de A/(E-D), extensible par 1'identitE sur H, alors g est encore extensible par l'identitE sur D. I1 suffit mcme de prouver que, s i g est extensible par l'identit6 sur H, alors g est extensible par l'identit6 sur toute partie F de D, de cardinal
< n-1.
Par hypothsse, il existe une partie F ' de H
et un automorphismc local f de domaine F et codomaine F'. extensible par l'identi-
t6 sur E-D, donc aussi par l'identit6 IG sur G
=
Dom g. Donc f U I et g G
u
IF'
(06 I F ' est l'identitE sur F') sont deux automorphismes locaux de A ; il en est de
mGme de f U g par composition. Puisque f est extensible par l'identit6 s u r E-D,
il l'est aussi par I G l qui est l'identit6 sur G '
=
-1
Cod g ; donc f
U I G l est u n
automorphisme local de A , et par composition I u g est un automorphisme local. 0 F En particulier, pour toute relation, l e complementaire d'une p a r t i s f i n i e quelconque de l a base e s t un f i n i v a l l e . 4.3. (1)
S o i t A une r e l a t i o n , B une autre de msme base, libre-interpr4-
table en A ; alors t o u t A - f i n i v a l l e e s t un B - f i n i v a l l e . 0
Prenons deux suites finies de mzme longueur et telles que la transfor-
mation de l'une en l'autre soit un automorphisme local de A , donc encore de B. Ces suites 6tant dans une partie D de la base, si la transformation est extensible, pour A, par l'identitE hors de D, elle est extensible de mzme pour B .
0
En particulier, s ' i l e x i s t e une m u l t i r e l a t i o n unaire en l a q u e l l e A s o i t l i b r e - i n t e r p r e t a b l e , a l o r s t o u t e p a r t i e de l a base e s t un A - f i n i v a l l e . ( 2 ) Etant donne une m u l t i r e l a t i o n B e t un B - f i n i v a l l e D , i l e x i s t e
une m u l t i r e l a t i o n A de m6me base, en l a q u e l l e B s o i t l i b r e - i n t e r p r e t a b l e e t t e l l e gue D s o i t un A-intervalle
(POUZET, 1975, non publi6).
Soit n l'arit6 de B . Pour chaque p
< n,
considgrons 1'6quivalence dE-
finie en mettant deux p-uples extraits de D dans une mzme classe lorsque la transformation de l'un en l'autre est un automorphisme local de B , extensible par l'identitb sur 1 B 1 - D. Pour chaque classe U, prenons la relation p-aire encore appel6e U, valant + pour les p-uples qui appartiennent 1 la classe et - pour tous autres p-uples dans la base. Nos classes Gtant en nombre fini, la suite formge de B et des U pour p = I ,
...,n, constitue une multirelation A
en laquelle B est
libre-interprgtable. De plus si deux p-uples dans D sont transformgs l'un en l'autre par un automorphisme local de A , ils appartiennent 1 une mzme classe, donc ladite transformation est extensible par l'identitg hors de D, pour A c o m e pour B.n ( 3 ) En consEquence, e t a n t donne une r e l a t i o n B e t une p a r t i e D d e
sa base, D e s t un B - f i n i v a l l e s i e t seulement s'il e x i s t e une A en l a q u e l l e B s o i t l i b r e - i n t e r p r e t a b l e , avec l a condition que D s o i t un A-intervalle.
Notons qu'en g6n6ra1, B gtant donnbe, il n'existe pas de multirelation A en laquelle B soit libre-interprgtable, avec la condition que t o u t B-finivalle soit un A-intervalle. En effet, tout ensemble fini 6tant un B-finivalle, il faudrait que tout ensemble fini, donc toute partie de la base, soit un A-intervalle (voir 1 . 8 ) .
327
L'interualle en thiorie des relations I1 faudrait donc que A , donc B, soit librr-interprdtable en une multirelation
unaire (voir 2.5).
4.4. (I)
Tout finivalle, augmente ou diminue d'un nombre fini d'elements
de l a base, donne un finivalle. 0
A chacun des El6ments ajoutEs, associons sa relation unaire singleton,
puis ajoutons ces relations singletons 5 la multirelation A de l'dnoncE ( 3 ) cidessus. On passe ensuite au cas d e 1 3 suppression d'un nomhre fini d'616ments, par passage au complEmentaire.
(2)
La
0
reunion et 1 'intersection de deux finivalles quelconques
c s t un finivalle (POUZET,
1976, non publiE ; ut. ax. choix dgnombrable ; ZF suf-
fisant lorsque la base est dEnombrahle). 0
Prouvons l'i.nonc6 pour la r6union ; on passera ensuite au compl6men-
taire. Rrtisonnons par l'absurde
en
supposant q u e U et V sont deux finivalles, mais
non leur rEunion. Soit p le plus petit entier pour lequel existe une infinit6 de classes de p-uples, deux p-uples dans U U V Etant dits Equivalents lorsque la transformation de l'un en I'autre est un automorphisme local extensible par l'identit6 hors de U U V. Prenons une o-suite de p-uples mutuellement non 6quivalents u les termes appartenant 5 U , et non
(ax. choix d 6 n . ) . Pour chaque p-uple, notons
1 V ; notons v l e s termes appartenant 5 V et non 1 U ; enfin w les termes appartenant 1 ]'intersection U
n V. Puisqu'il y a une infinit6 de p-uples en considEra-
tion, n o u s pouvons supposer que, pour chaque indice r
< p,
l e r-6me terme est
toujours un u , ou toujours un v, o u toujours un w. En restant dans le cas gEn6ra1, nuus pouvons supposer p
=
3 avec chaque suite formEe d'un u , un v et un w : pour
chaque entier i, nous avons donc la suite d e s trois termes u.,v.,w.. 1
1
1
Puisque U est un finivalle, et que chaque suite u i wi est formCe de deux ElEments de U, ces suites se r6partissent en un nombre fini de classes, pour 1 ' 6 quivalence d6finie par un automorphisme local extensible par l'identits hors de U. Nous pouvons donc supposer que toutes Les suites u . wi appartiennent i une m&ne
classe. Donc pour chaque i, la transformation d e u
vi wi en uo vi wo est un auto-
morphisme local extensible par Z'identitE hors d e U. Par ailleurs puisque V est un finivalle, i l en est d e &me
de V diminus d e 1'61Ement wo : voir le ( 1 ) pr6c6dent.
Les termes v appartenant Z ce nouveau finivalle, et les termes uo et wo 6tant en dehors, nous pouvons supposer que, pour chaque paire d'entiers i , j , la transformation de u
vi wo en uo vj wo es t un automorphisme local extensible par l'identi-
ti3 hors de V. Finalement pour tots i,j, nous avons un automorplisme local tralsformant u . v. w. en u 1
1
1
vi wo, puis en u
O
v. wo, puis en u J
vj v.,
avec 1 chaque 6tape
l'extersitilit6 par l'identitg h o r s de U U V : contradiction avec la d6finition de la suite infinie d e p -uple non Cquivalents 4.5.
.
0
Soit A une relation, E s a b e , D une partie de E . A l o r s pour que D
soit un finivalle, il faut et suffit qu'il existe une partie finie F du complementaire E-D,
telle que chaque automorphisme local de A I D , ou bien n'est pas extensible
R. Fra'isse
328
p a r l ' i d e n t i t k s u r F . o u b i e n e s t e x t e n s i b l e p a r l ' i d c n t i t d sur E-D t o u t e n t i e r (6nonc6 communiqu6 p a r POUZET, 1 9 7 8 ) . 0
S u p p o s o m d ' a b r d que F e x i s t e . Alors pour chaque e n t i e r p . r 6 p a r t i s -
s o m l e s p - u p l f f i d a m D e n un n o m k e f i n i d e c l a s s ~ s ,deux p ? ~ p I p s 6 t a n t d i t s
6 q u i v a l e n t s l o r s q u e l a t r a m f o r m a t i o n d e l'un e n l ' a u t r e , r 6 u n i e 3 l ' i d e n t i t s s u r F , donne un a u t o m a r h i s m e l o c a l d e A . La c o n d i t i o n d e l ' h o n c e c o n c e r n a n t F
e n t r a ' i n e a l o r s que D s o i t un f i n i v a l l e . I n v e r s e m e n t s o i t D un f i n i v a l l e ; n o t o m n l'arit6 d e A . Pour c h a q u e e n t i e r p o s i t i f p < n , r 6 p a r t i s s o f f i Iffi p i p l r s e n leiits c l a s s e s d ' P q u i v a l e n c e mod(A,D) : v o i r d 6 t u t d u p r k e n t p a r a g r a p h e 4 ; ces c l a s s -
s o n t e n nombre f i n i .
# C i , p r e n o r s un p j Gl&ments d e E - D , t e l que la
P o u r c l a q u e p e t c h a q u e p a i r e d e c l a s s e s d e p - u p l ~C ~. e t C
d ' a u p l u s n-I
u p l e dam c h a c u n e , e t u n e r s e m b l e Fi
c h o i s i pour C ne s o i t p"s e x i ' e n 'j j' L a r 6 u n i o n f i n i e F des F p o u r t o e les p < n , ij* ij v 6 r i f i e l ' 6 n o n c 6 . En e f f e t 6 t a n t donn6 deux p l l p l f f i u ' . d e C e t u' d e C la i j j' t r a m f o r m a t i o n d e u i e n u ' est p a r h y p o t h k e e x t e n s i b l e p a r l ' i d e n t i t 6 s u r E-D ; t r a n s f o r m a t i o n du p - u p l e u
i
chois i p o u r C
teffiihle par l ' i d e n t i t 6 s u r F
i
d e mEme l a traffi f o r m a t i o n d e
11
j
en u '
j'
Donc s i l a t r a m f o r m a t i o n d e u! e n u ! 6 t a i t J
e x t e n s i b l e p a r l ' i d e n t i t 6 s u r F , i l e n s e r a i t d e mZme d e l a t r a n s f o r m a t i o n de u i en u . : c o n t r a d i c t i o n . F i n a l e m e n t F v 6 r i f i e 1 ' 6 n o n c 6 p o u r l e e n t i e r s p < n , e t i l
J en r f f i u l t e que F v E r i f i e e n c o r e l ' 6 n o n c 6 p o u r u n automorphisme l o c a l q u e l c o n q u e d e AJD.
4 . 6 . En v u e des 6noncffi s u i v a n ' s , ciaprk
l e l e c t e u r r e t r o u v e r a les t r o i s lcmmffi
( d o n t l e d e r n i e r remonte 1 LOPEZ, 1 9 6 9 ) :
(I)
S o i t A une r e l a t i o n n a i r e , u , v deux 616ments d e l a h e ; s i l a
t r a n s p o s i t i o n ( u , v ) m o d i f i e A ( i . e . c e t t e trampos i t i o n , 6 t e n d u e p a r l ' i d e n t i t 6
s u r 1 s 616ments a u t r e s que u , v , n ' f f i t pas u n a u t o m o r p h k m e d e A ) , a l o r s i l e x i s t e u n e n s e m b l e H d ' a u p l u s n+l e l e m e n t s , p a r m i l e s g u e l s u e t v , t e l q u e ( u , v ) m o d i f i e A/H. ( 2 ) S o i t E un e m e m t l e , f une p e r m u t a t i o n d e E e t F une p a r t i e f i n i e d e
E ; i l e x i s t e u n e s u i t e s a n s r k p e t i t i o n d ' P l 6 m e n t s u I , ...,u,,
d e F, t e l l e que p o u r
c h a q u e Plernent x d e F , l e t r a n s f o r m e f ( x ) s o i t i d e n t i q u e a u t r a n s f o r m 4 d e x p a r l a composee d s s t r a n s p o s i t i o n s s u c c e s s i v e s (u
, f ( u I ) ) , . . . , (y,,f (r,) ) .
( 3 ) Soit A une r e l a t i o n d e b a s e E, et f une p e r m u t a t i o n d r E g u i m o d i f i e A ; a l o r s i l existe un e l e m e n t u d e E t e l g u e l a t r a n s p o s i t i o n ( u , f ( u ) ) m o d i f i e A .
4.7.
(I)
S o i t E l ' e m e m t l e dffi e n t i e r s n a t u r e l s , N l a c h a i n e u s u e l l e s u r
ces e n t i e r s . S i A e s t l i b r e i n t e r p r e t a b l e en N e t s i , p o u r c h a q u e p r o g r e s s i o n arithmetique D de raison
> 2,
i l e x i s t e d e u x e l e m e n t s x e t y > x d e D t e l s q u e la
t r a n s p o s i t i o n (X,y) s o i t u n a u t o m o r p h i s m e l o c a l d e A, e x t e n s i b l e p a r 1 ' i d e n t i t 8
sur E-D,
a l o r s t o u t e p e r m u t a t i o n d e E e s t u n a u t o r n o r p h i s m e d e A (6nonc6s ( I )
et
( 2 ) dus 1 POUZET e t YASUHARA). 0
S u p p o s o n s qu'il e x i s r e une p e r m u t a t i o n d e E qui m o d i f i e A . P a r
lffi
lemmes p r G c 6 -
329
L'interualle en theorie des relations
d e n t s , i l e x i s t e un e n s e m b l e f i n i F d ' e n t i e r s e t d e u x e l 6 m e n t s a , l a t r a n s p o s i t i o n ( a , t) m o d i f i e A / F .
Fixiins
b de F tels que
t > a . Soit u =
1 6 idges en posant
Max(2, b a ) ; s o i t v l e n o m l r e d e s 616ments < a e t w c e l u i d e s 616me nts Puisque A ffit l i b - e - i n t e r p r 6 t a b l e
rn N,
>
k da ns F .
t o u t e i n j e c t i o n c r o i s s a n t e d e doma ine F
> x+u,
e s t un a u t o m o r p h i s m e l o c a l d e A . Donc p o u r t o u s e n t i e r s x 2 a e t y
l a trans
-
p o s i t i o n ( x , y ) m o d i f i e A . P l us p r G c i s 6 m e n t e l l e m o d i f i e l a r e s t r i c t i o n d e A 1 t o u t e n s e m t l e q u i comprend a u moins v e l e m e n t s
<x
et
wi.16ments > y
e t u-l
616ments
e n t r e x e t y . En p a r t i c u l i e r G t a n t donne l a p r o g r e s s i o n a r i t h m 6 t i q u e D = { a , a+u, a+2u,.
. . 1,
p o u r d e u x 6l Gmrnt s q u e l c o n q u e s x e t y
>x
de D, l a trans -
p o s i t i o n ( x , y ) e t e n d u e p a r l ' i d e n t i t e s u r E-D, n ' e s t p a s un a u t o m o r p h i s m e l o c a l de A. 0
( 2 ) S o i t D un e r s t . m t l e i n f i n i d ' e n t i e r s , d o n t l e c o m p l g m e n t a i r e E - D e s t i n f i n i . S i t o u t e p a r t i e d e D est un A - f i n i v a l l e , a l o r s il e x i s t e d e u x elements x et y
>x
d e D,
t e l s q u e l a t r a n s p o s i t i o n (x,y) soit un automorphisme
local d e A, e x t e n s i b l e p a r l ' i d e n t i t e s u r E - D . Co rsCq u e n c e de ( I )
e t (2) : s i A e s t l i b r e - i n t e r p r e t a b l e
en N e t s i t o u t e n s e m b l e
d ' e n t i e r s est u n A - f i n i v a l l e , a l o r s t o u t e p e r m u t a t i o n d e E est u n a u t o m o r p h i s m e d e A. 0
P r e u v e du ( 2 ) . A c h a q u e p a i r e d ' 6 1 6 m e n t s x , y d e D a s s o c i o n s l e c o u p l e o r d o n n d
(x,y
> x),
c e q u i nous p e r m e t d e r E p a r t i r l e s p a i r e s e n un n o m I r e f i n i d e c l a s s e s
d ' g q u i v a l e n c e mod (A,D). P a r l e t h e o r s m e d e RAMSEY, i l e x k t e u n e p a r t i e i n f i n i e D* d e D d a n s l a q u e l l e t o u t e les p a i r e s a p p a r t i e n n e n t 1 une m t m e c l a s s e . Donc p o u r tous x < y
e t x' < y '
616ments do D*, l a t r a n s f o r m a t i o n d e x e n x ' e t y e n y ' e s t
un a u t o m o r p h i s m e l o c a l d e A , e x t e r s i t l e p a r l ' i d e n t i t g s u r E - D . P r e n o n s d a n s D* les 6lGments d e r a n g s p a i r s : s o i t D*. P u k q u e ce D* e s t un A - f i n i v a l l e , p r e n o n s P P u n e p a r t i e i n f i n i e D** d e D* d a m l a q u r l l e l e s s i n g l e t o n s s o i e n t t o u s g q u i v a l e n t s
P
avec x < y < z , l a mod ( A , D * ) . Donc d t a n t d o n n b x , z d a n s D** e t y d a m D*-D* P P t r a n s f o r m a t i o n d e x e n z e t d e y e n l u i m z m e , e s t un a u t o m o r p h i s m e l o c a l d e A ,
e x t e n s i b l e p a r l ' i d e n t i t 6 s u r E-D.
P u i s q u e x , y , z a p p a r t i e n n e n t 1 D*,
p a r composi-
t i o n l a t r a n s p o s i t i o n (x,y) e s t un a u t o m o r p h i s m e l o c a l d e A , e x t e n s i k l e p a r l ' i d e n t i t 6 s u r E-D. 4.8.
0
S o i t A u n e r e l a t i o n , E s a t a s r . Supposom q u e t o u t e p a r t i e d e E
s o i t un A i i n i v a l l e ; a l o r s ( c o m e en 2 . 3 ( 2 ) e t e n 2 . 5 ( I )
(I) fini
) :
L e s c l a s s e s d ' e q u i v a l e n c e p a r t r a n s p o s i t i o n (mod A ) sont en n o m b r e
;
(2) I 1 e x i s t e u n e m u l t i r e l a t i o n u n a i r e en l a q u e l l e A e s t l i b r e - i n t e r p r e t a b l e ( u t i l i s e l ' a x i o m e d e c h o i x d G n o m l r a t l e , ZF s u f f k a n t l o r s q u e l a h e E e s t
d6 n o m b ra b l e ; GnoncCs dus 5 POUZET). P a r c o n t r e i l n ' e x i s t e pas f o r c E m e n t u n e m u l t i r e l a t i o n u n a i r e
u
telle
que A e t U s o i e n t chacune l i t r e i n t e r p r 6 t a U e e n l ' a u t r e . Prenons e n e f f e t pour A une r e l a t i o n
t i n a i r e d ' g q u i v a l e n c e B un n o m t r e f i n i d e c l a s s e s . P o u r c h a q u e e n t i e r
R. Fra'issB
330
pour n ' i m p o r t e q u e l l e p a r t i e D d e l a
p, d e u x p - u p l e s s o n t 6 q u i v a l e n t s mod(A.D) bse qui
lffi
c o n t i e n t , p o u r v u q u ' i l s s o i r n t t r a n s f o r m & p a r une E j e c t i o n e t quc
d e u x termes de &me
r a n g d a m les d e u x p - u p l e s a p p a r t i e n n e n t 2 u n e m6me c l a s s e .
A i n s i t o u t e p a r t i e de l a Lase e s t un f i n i v a l l r ; mais une m u t t i r e l a t i o n U e n l a q u e l l e A s e r a i t l i l r e i n t e r p r g t a t l e , c o m p r e n d r a i t p a r exemple une r e l a t i o n u n a i r e valant
+ s u r une c l a s s e d e
A et
-ailleurs,
e t c e l a p o u r c h a q u e c l a s s e . Nous voyons
q u ' a l o r s U ne s e r a i t pas l i h - e - i n t r r p r 6 t d b l e 0
en A .
( I ) R e p a r t i s s o n s l e s 616ments d e E e n c l a s s e s d ' f q u i v a l e n c e p a r t r a n s p o s i t i o n
(mod A) : v o i r 2 . 2 .
Supposons q u ' i l e x i s t e une i n f i n i t 6 d e c e s c l a s s e s . P a r
l ' a x i o m e d e c h o i x d 6 n o m k r a U e , p r e n o n s un e n s e m t l e d 6 n o m l r a ' d e D d'6lGments d e l a Lase, m u t u e l l e m e n t non g q u i v a l e n t s . C o n s t r u i s o n s s u r l a Lase D u n e isomorphe N d e l a cha'ine des e n t i e r s n a t u r e l s . P a r l e thilorsme d e RAMSEY, now pouvons t o u j o u r s s u p p o s e r que l a r f f i t r i c t i o n A/D e s t l i h - e - i n t e r p r E t a b l e
e n N . Plus p r S c i s g m e n t ,
m e t t o n s dans une msme c l a s s e d e u x p a i r e s d ' E l 6 m e n t s x , y e t x ' x
e t x' < y
I ) ,
, y ' de ' es t
lorsque l a transformation de x en x ' e t y en y
D (avec
un a u t o m o r
-
phisme l o c a l d e A . A l o r s nous pouvons e x i g e r q u e t o u t e s les p a i r e s a p p a r t i e n n e n t 2 une &me
classe.
Disons q u e d e u x s i n g l e t o n s dans D s o n t G q u i v a l e n t s , o u q u e d e u x p a i r e s d ' 6 l P m e n t s d e D s o n t g q u i v a l e n t e s , l o r s q u e l ' u n i q u e automorphisme l o c a l d e N q u i t r a n s f o r m e l e premier s i n g l e t o n e n l e s e c o n d , ou l a p r e m i c r e p a i r e e n l a s e c o n d e , e t q u i e s t
e n c o r e un automorphisme l o c a l d e A , e s t de p l u s e x t e n s i b l e p a r l ' i d e n t i t 6 s u r E - D . P u k q u e D f f i t un A - f i n i v a l l e , l e s c l a s s e s d e c e t t e g q u i v a l e n c e s o n t e n n o m l r e f i n i . E n c o r e p a r RAMSEY, o t t e n o n s une p a r t i e d 6 n o m l r a U e D , d e Do = D , dans l a q u e l l e t o u s les s i n g l e t o m i o n t e q u i v a l e n t s , e t t o u t e s les p a i r e s g q u i v a l e n t e s .
It6rons en
r e m p l a p n t Do p a r D I , a v e c l ' e x t e n s i k i i l i t 6 s u r E - D l : e t a i n s i d e s u i t e n f o i s , o b t e n a n t l e s ensemUes e m b ? t & s u c c e s s i f s D Soit
3 D
0 -
2
1 -
... _> D n .
u , v d e u x glements d i s t i n c t s q u e l c o n q u e s d e D
.
Par c e q u i prscsde,
5 n+l
l a t r a n s p o s i t i o n ( u , v ) m o d i f i e A . P a r 4 . 6 , i l e x i s t e une p a r t i e H d e
d e E , parmi l e s q u e l s u e t v , t e l l e q u e ( u , v ) m o d i f i e A / H . un e n t i e r h ( I 5 h a u c u n des
5
n-l
5 n)
t e l que l a d i f f e r e n c e e n s e m b l i s t e
Glements d e H
- Iu,v}.
6lEnents
Donc i l e x i s t e a u moins
4,
- 4,
n e comprenne
Donc c e t t e d i f f B r e n c e s t d k j o i n t e d e H .
P a r l ' G n o n c 6 4 . 7 , l a t r a n s p o s i t i o n ( u , v ) e s t un a u t o m o r p h i s m e d e A / D ,
puisque toute
p a r t i e d e D e t un (A/D) X i n i v a l l e
:
donc ( u , v ) automorphisme d e A / D h . Mais ( u , v )
n ' e s t p a s un automorphisme de A / ( E
-
(Dh-I-Dh)),
Autrement d i t , B t a n t donnE u , v d e D n , phisme l o c a l d e A / D h ,
non e x t e n s i b l e p a r 1 ' i d e n t i t B s u r E
D ' a u t r e p a r t B t a n t donnB u , v , u ' , v ' la t r a m formation de u en u' e t v en v '
phisme l o c a l d e N / 9 , donc d e A / \ r a i s o n d e l a d g f i n i t i o n msme d e u,v,u',v'
de D
d u f a i t que H C E
avec u < v e t u '
,
-
(Dh-,-Dh).
i l e x i s t e u n h t e l q u e ( u , v ) s o i t un a u t o m o r -
ffi
-?I - 1 '
d e Dn a v e c u < v e t u ' < v '
t pour chaque h
(1
< h 5 n)
(mod N ) ,
un a u t o m o r -
e x t e n s i b l e p a r l ' i d e n t i t s s u r E - Dh -1 ' en p a r c o m p o s i t i o n , q u ' E t a n t donne
4.11 en r b u l t e , > v'
(mod N ) ,
i l e x i s t e un h t e l que l a t r a n s f o r -
33 1
L'intetvalle en theorie des relations
, v ' s o i t un automorphisme l o c a l d e A/Dh, non
mation de u,v respectivement en u ' extens g
H e par l'identiti. s u r E -
t un a u t o m o r p h i s m e l o c a l d e A / D
4,
n'
A pl16
f o r t e raison, c e t t e transformation
lion e x t r n s i
F i n a l e m e n t , 6 t a n t donn6 u , v , w d e D
Ue par l ' i d e n t i t e s u r E - D
avec u
< w
.
l a trans f o r -
(mod N ) ,
m a t i o n du s i n g l e t o n d e u e n r e l u i d e v e s t u n a u t o m o r p h i s m e l o c a l d e A / D n ,
non
e x t e m i hl p p a r l ' i d e n t i t i . s u r ( E - D ) U iwl. I1 e n r f f i u l t e q u e , p a r e x e m p l e , 1 ' e r s e m U e d e s GlGments d e D contradiction.
d e r o n g s p a i r s (mod N/Dn).
n ' s t p a s un A - f i n i v a l l e :
0
L ' C n o n c 6 (2) r P s u l t e du ( I ) e t d e 2 . 2 .
5- LE SUBVALLE, N O T I O N I N T E R M E D I A I R E ENTRE L'INTERVALLE ET LE FINIVALLE ; CAS DU CYCLORIIRE AVEC LE SECTEUR-PAKT-DE-GATEAU
E t d n t donne u n e r e l a t i o n A d e b
E-D,
en
ou
e E , une p a r t i e D d e E s e r a d i t e un
l o r s q u e , pour t o u t automorphisme l o c a l f d e A/D e t t o u t e p a r t i e G d e
A-suhvalle
f n ' e s t pas e x t e n s i b l e p a r l ' i d e n t i t 6 s u r G , ou f e s t e x t e n s i b l e p a r
l ' i d e n t i t e s u r c h d c u n e d e s p a r t i e s d e E-D o b t e n u e s d e G p a r un a u t o m o r p h i s m e l o c a l de A.
A i n s i q u ' i l a d 6 j h Gt6 d i t p o u r l ' i n t e r v a l l e e t l e f i n i v a l l e , i l s u f f i t q u e f s o i t d e f i n i s u r un d o m a i n e d e p s o i t de c a r d i n a l
5
n-l
616ments
(n
=
a r i t E d e A) e t que G
n-p.
Si A est u n e c h a i n e , no16
v q r o n s q u e les A - s u b v a l l e s
s o n t exactement les
i n t e r v a l l e s et les e x t e r v a l l e s . R a p p e l o l s q u e l e c y c l o r d r e engendrE p a r une chaTne e s t d 6 f i n i a u p a r a graphe 3. La n o t i o n d e s u b r a l l e d ' u n c y c l o r d r e est t r b
i n t u i t i v e : reprEsentant
l e c y c l o r d r e p a r un c e r c l e , nous r e t r o u v o n s p o u r l e s u l u a l l e l e s e c t e u r c i r c u l a i r e ,
ou part-de-g8teau.
Rigoureusement
C : les A-subvalles
s o i t A u n c y c l o r d r e e n g e n d r 6 p a r une c f i a i n e
s o n t exactement les C - i n t e r v a l l e s et les C - e x t e r v a l l e s ,
l e c a s d ' u n e b e d e 4 616ments 0
:
except6
( p o u r l e q u e l t o u t e p a r t i e est un s u b r a l l e ) .
I1 s u f f i t d ' 6 t u d i e r d e u x cas : ou l i e n f d E f i n i s u r u n d o m a i n e s i n g l e -
t o n e t G p a i r e d ' s l G m e n t s , ou f d P f i n i s u r u n e p a i r e e t G s i n g l e t o n . Dans l e p r e m i e r cas d i s o r s q u e f t r a n s f o r m e 1 ' 6 1 6 m e n t u d e D e n u ' d e D , e t s o i t G = { v , w j inclls
d a m E-D.
S u p p o s o r s D i n t e r v a l l e ou e x t e r v a l l e d e l a c h a i n e : a l o r s f e s t
evidemment e x t e n s i b l e p a r l ' i d e n t i t i . s u r C . Dans l e s e c o n d cas, f t r a n s f o r m a n t u,v de D en u ' , v '
d e D e t G = s i n g l e t o n d e 1 ' G l E m e n t w d e E-D.
c e t t e v a l e u r s u b i s t e p o u r t o u t w d e E-D A ( u ' , v ' , w) = A ( u , v , w) p o u r t o m
S i A(u,v,w)
; d e mzme a v e c la v a l e u r
-.
= +,
Donc o u h i e n
w d o n c f e x t e n s i H e S t o w les s i n g l e t o n s . Ou
A ( u ' , v ' , w) # A ( u , v , w) e t f n ' f f i t e x t e n s i H e d a u c u n s i n g l e t o n d a m E-D.
11 res t e h v o i r q u e s euls les C - i n t e r v a l l e s
ou e x t e r v a l l e s conviennent.
E x c e p t a n t l e c y c l o r d r e s u r 4 E l g m e n t s , s u p p o s o n s q u ' i l e x i s t e a u moins 5 ElEments
R. Frdisse
332
d i s t i n c t s a , b , c , d , e dans l ' o r d r e d ' u n e des c h a i n e s q u i l ' e n g e n d r e n t . A l o r s nous v q o m q u e l a p a i r e { a , c } n ' e s t pas un s u b u a l l e , l a t r a n s f o r m a t i o n d e a e n c E t a n t e x t e n s i b l e p a r l ' i d e n t i t e s u r { d , e ) mais n o n s u r { b,d}.
Par a i l l e u r s { a , c , d l
n ' e s t p a s non p l u s un s u l v a l l ~ , l a t r a f f i f i i r m a t i o n d e a e n c e t d e d e n lui-mzme e t a n t i n e x t e n s i b l e p a r l ' i d e n t i t 6 s u r l e s i n g l e t o n { b } , mais e x t e n s i k l e p a r l ' i dentit6 s u r
{el.
0
L ' e x e m p l e du c y c l o r d r e p r o u v e clue l'intersection ou la reunion d c deux subvalles n'est pas forckment un subvalle. 5.1.
T o u t intervalle ou extervalle est un subvalle.
E v i d e n t d a m l e c a s d ' u n i n t e r v a l l e . S o i t D un e x t e r v a l l e d e l a r e l a t i o n A d e b e E : donc E-D e s t un i n t e r v a l l e . S o i t f un a u t o m o r p h i s m e l o c a l d e d e domaine F e t codomaine F ' , e t s o i t g u n a u t o m o r p h k m e l o c a l d e A/(E-D),
A/D,
d e domaine G e t codomaine G ' . Supposons f e x t e n s i b l e p a r 1 ' i d e n t i t E IG s u r G :
il s u f f i t d e v o i r que f e t e n c o r e e x t e n s i b l e p a r l ' i d e n t i t e I G l f U I
G
g U I F , e s t e n c o r e un a u t o m o r p h i s m e l o c a l , donc f De p l u s g
avec f
.
La r e u n i o n
E t a n t un a u t o m o r p h k m e l o c a l d e A , e t E-D e t a n t un A i n t e r v a l l e , l a r g u n i o n
u
u g,
u g en est un, par composition. -1
IF e s t un automorphisme l o c a l , donc a u s s i g
nous o l t e n o n s f
u
IGl
U IF : p a r c o m p o s i t i o n
.
I1 e x i s t e des s u h r a l l e s q u i n e s o n t n i un i n t e r v a l l e ni un e x t e r v a l l e .
P a r exemple prenons l e q c l o r d r e d 6 j 5 6 t u d i 6 p r E c S d e m e n t , e t r a p p e l o n s q u e les s e u l s i n t e r v a l l e s ( a b o 1 u s ou r e l a t i f s ) s o n t l e v i d e , l a b e e n t i s r e e t les s i n gletons : v o i r paragraphe 3 .
5.2. Le complPmentalre d'un subvalle n'est pas forcement un subvalle. P a r t o n s d e d e u x c h a f n e s A , B e t cons t r u i s o m s u r l a r e u n i o n E d e l e u r s b f f i ,
s u p p o s 6 e s d i s j o i n t e s , l ' o r d r e daffi
incomparable B chaque elgment d e
1 Bl
l e q u e l c h a q u e BlEment d e l a h
e 1.4
I ffit
.
Notons B' un i n t e r v a l l e m e d i a n d e B ; p u i s c o n s i d e r o n s l ' e n s e m l l e D ,
reunion des lases I A
I
et
I B'I
. D'une
p a r t D e s t u n s u b a l l e ; e n e f f e t un a u t o -
morphisme l o c a l dans D , ou l i e n t r a n s f o r m e a u moins un e l e m e n t d e
I B'I ou i n v e r s e m e n t
: alors il n'est
I A I en
un d e
pas e x t a n s i kle p a r a u c u n e i d e n t i t 6 h o r s d e
D . Ou n o t r e automorphisme l o c a l e s t La r e u n i o n d ' u n automorphisme l o c a l d e A e t
d ' u n d e B ' , e t a l o r s i l e s t e x t e n s i U e p a r l ' i d e n t i t e s u r E-D.
D ' a u t r e p a r t E-D
n ' s t pas un s u b a l l e ; e n e f f e t l a t r d n s f o r m a t i o n d ' u n e l e m e n t d e B, a n t e r i e u r l ' i n t e r v a l l e B',
e n un e l e m e n t p o s t B r i e u r h B ' ,
s u r l e s i n g l e t o n d ' u n GlBment d e I A
I B'l
.
I mais
e s t extensible par l'identite
non s u r l e s i n g l e t o n d ' u n e l e m e n t d e
0
Dam l e c a s d u c y c l o r d r e ,
l e c o m p l 6 m e n t a i r e d ' u n s u h r a l l e e s t un s u t -
v a l l e . De m t m e dans l e c a s d ' u n e c h a f n e , oh les s u h r a l l e s s o n t e x a c t e m e n t les i n t e r v a l l e s e t l e s e x t e r v a l l e s . Cela now c o n d u i t a u problBme s u i v a n t . ProUBme. S i A est l i h - e - i n t e r p r e t a b l e d ' u n A s u h r a l l e e s t - i l t o u j o u r s un A - s u b a l l e .
e n une c h a i n e , l e c o m p l e m e n t a i r e
L'interualle en thiorie des relations
333
5.3. Tout subvalle est un finivalle. 0
S n i t A une r e l a t i o n n - a i r e
i l s u f f i t q u e pour chaque p < n
de
h s e E . P o u r q u e D s o i t un A - f i n i v a l l e , dans l a l a s e d o n t l a l o n g u e u r e s t p ,
Ics s u i t e s
s e r e p a r t i s s e n t e n un n o m t r e f i n i dts c l a s s e s d ' e q u i v a l e n c e mod(A,D) p a r a g r a p h e 4.
: v o i r d e b u t du
R a p p e l o ns q u e deux s u i t e s d e l o n g u e u r p d o i v e n t d e f i n i r une trans -
f o r m a t i o n d e l ' u n e e n l ' a u t r r q u i s o i t un a u t o m o r p h i s m e l o c a l d e A , e x t e n s i t l e p a r l ' i d e n t i t e s u r t o u t rnsemtle de
_<
n-p e l e m e n t s d e E-D.
5 n,
S u p p o so n s m a i n t r n a o t qcte I) s o i t u n A s u l v a l l e . P o u r q deux ensembles G , G '
d e q 61Ements d e 13-D,
disons que
s o n t Gquivalents, l o r s q u e A/G e t A/G'
i s o m o r p h e s . P r e n o n s un r r p r e s e n t a n t d a n s s h a q u e c l a s s e , e t s o i t H l ' e n s e m l - l e f i n i , reunion de ces representants
pour tous les q
< n.
-
D i s o n s q u e d e u x s u i t e s da ns D ,
n, s o n t P q u i v a l e n t e s l o r s q u e l a t r a n s f o r m a t i o n
s o i t u e t v , d ' u n e mEme l o n g u e u r p
d e u en v e s t un a u t o m o r p h i s m e l o c a l d e A , e x t e n s i b l e p a r l ' i d e n t i t i . s u r H. P u i s q u e H e s t f i n i , les c l a s s e s d e c e t t e G q u i v a l e n c e s o n t e n nomh-e f i n i . I1 s u f f i t d e
n o t e r q u e , D G t a n t un s u h r a l l e , I n t r a n s f o r m a t i o n d e u en une s u i t e G q u i v a l e n t e v de longueur p , s i e l l e est e x t e n s i b l e p a r l ' i d e n t i t e s u r H, est encore e x t e n -
s i t l e par l ' i d e n t i t i . s u r t o u t ensemtle de
< n-p
GlGments d e E-D.
L e f i n i v a l l e P t a n t une n o t i o n l o o l e e n n e , d ' a p r 2 s
0
l'enonce precedent il
e x i s t e d e s f i n i v a l l e s non s u h r a l l e s . Un f i n i v a l l e n ' e s t pas f o r c e m e n t o b t e n u 2 p a r t i r de s i n t e r v a l l e s e t e x t e r v a l l e s p a r c o m p l e t i o n t ool Genne ( u n i o n e t i n t e r s e c t i o n f i n i e s , e t c o m p l g m e n t a t i o n ) : i l s u f f i t d e p r e n d r e l e c a s des c y c l o r d r e s , 05 c e t t e c o m p l g t i o n EoolEenne n e d o n n e r a i t q u e les e n s e m t l e s f i n i s e t l e u r s c o m p l e m e n t a i r e s ,
donc n e d o n n e r a i t
pas t o u s l e s s u h r a l l e s , e t 2 p l u s f o r t e r a i s o n les f i n i v a l l e s .
P r o ' d S me . E x i s t e n c e d ' u n f i n i v a l l e non o k e n u p a r c o m p l e t i o n b o l G e n n e 2 p a r t i r d e l ' e n s e m b l e des s u l x a l l e s .
5.4. La libre-interpr6tabilite ne conserve p a s les subvalles ( a l o r s q u ' e l l e c o n s e r v e les f i n i v a l l e s : v o i r 4.3 ( I ) ) . 0
S o i t C une cha?ne, U,V deux i n t e r v a l l e s d e C , d i s j o i n t s , l a i s s a n t e n t r e
e u x un i n t e r v a l l e non v i d e , e t a v e c u n i n t e r v a l l e non v i d e p o s t e r i e u r a u s e c o n d . La r e u n i o n D un C - s u h r a l l e .
=
U U V n ' e s t n i un C - i n t e r v a l l e n i un C - e x t e r v a l l e ,
d e mOme V l a r e l a t i o n u n a i r e v a l a n t + s u r V e t (C,U,V),
d o n c n ' e s t pas
Appelons e n c o r e U l a r e l a t i o n u n a i r e v a l a n t + s u r l ' i n t e r v a l l e U ; -ailleurs.
Pour la m u l t i r e l a t i o n
l ' e m e m k l e D e s t un s u h r a l l e , e t mOme u n i n t e r v a l l e . En e f f e t s o i t f u n
a u t o m o r p h i s m e l o c a l d e (C,U,V)
d e domai ne e t d e codoma ine i n c l u s da ns D . A l o r s f
t r a n s f o r m e f o r c e m e n t u n E l ement d e U e n un e l e m e n t d e U ; d e mOme p o u r V ; donc f
est e x t e n s i h l e p a r l ' i d e n t i t s s u r
I
C
I
- D.
0
5.5. R a p p e l o n s , p a r l e g r a p h e s u i v a n t , l e s r a p p o r t s e n t r e les d i f f e r e n t s "valles".
R. Fra'isse
334 Intervalle a k o l u
>-
In~ervallerelatif
d ' i n terva I 1 es
(bolEen et conserv6 par la l i h-e-inter prEtakilit6)
6- CLOTURE D'UNE RELATION ; RE1,ATION COMPACTABLE 6.1. Filtre intervallaire Ainsi qu'il a EtE d E j P dit. i l s'agit d'gtendre aux rela ions la c 1 8 ure classique de la chaIne des rationnels par celle des rEels. Soit A une relation de lase E ; dEfinissons un A-filtre intervallaire, o u simplement A-filtre, c o m e un ensemtle non vide
F de A-intervalles alsolus non
vides, v6rifiant les deux conditions : (1)
tout A-intervalle qui inclut un ElEment de F est un ElEment de F ;
(2) l'intersection de deux 616ments quelconques de
F est un ElEment de
F (c'est un A-intervalle, par 1.2). C o m e pour les filtres usuels, un A-filtre sera dit p l u s fin qu'un autre lorsqu'il l'inclut. Un A-ultrafiltre sera u n A-filtre maximal par inclusion. Un A-ultrafiltre sera dit trivial lorsqu'il est l'ensemble de tous les A-intervalles qui comprennent un ElGment donnE de la b s e E. (Rappelom que tout singleton est un A-intervalle). Si un A-ultrafiltre est non trivial, l'intersection de ses 616ments est vide, et chacun de ses 6lEments est une partie infinie de E. Etant donnE un A-intervalle D et un A-ultrafiltre chaque 616ment de
F , et alors D appartient P F
; nu
F,
ou E e n D intersecte
i l existe un El6ment de F , dis -
joint de D. En consgquence, Etant donnE deux A-ultrafiltres distincts F et existe un El6ment D de F et un D ' de
F', il
F' avec D no' vide.
6.1.1. Si une r6union finie de A-intervalles D. est 6lEment du A-ultra1
filtre
F, alors l'un au moins des
Di ts t 616ment de
F.
6.1.2. Tout A-filtre admet un A-ultrafiltre plus fin (i.e. surensemlile du A-filtre donn6) ; &once
ottenu P l'aide de l'axiome usuel de l'ultrafiltre.
Lorsque A est la multirelation r6duite P sa b s e E , ou encore une relation valant toujours +,
nu
plus g6nEralement une relation admettant toute permutation de E
c o m e automorphisme, alors les A-filtres et ultrafiltres s e rgduisent aux filtres et ultrafiltres ensemllistes usuels, sur E. 6.2. C18ture d'une relation selon un ensemble d'ultrafiltres interval laires. Partors d'une relation A de lase E ; identifions chaque A-ultrafiltre
335
L'intervalle en theorie des relations t r i v i a l avec l ' e l e m e n t de E qiii l ' e n g r n d r e : 1 ' e r s e m E l e de t o m
les A-ultrafiltres
d e v i e n t un s u r e n s e m b l e d e E . A c h a q u e R-
u l t r a f i l t r e non t r i v i a l F , a s s a c i o n s u n e n s e m b l e a r t i t r a i r e E ( F ) ,
f i n i ou i n f i n i ,
G v e n t u e l l e m e n t v i d e ; p u i s u n e r r l d t i u n A ( F ) d e base E ( F ) e t d e mGme a r i t 6 que A,
a v e c l e s e x i g e n c e s s u i v a n t e s , q u i p e u v r n t t o u i o u m G t r e s a t i s f a i t e s . Nous s u p -
F d i s t i n c t s c o r r e s p o n d e n t deux E ( F ) d i s j o i n t s
posors qu'2 deux u l t r a f i l t r r s p l a chaque E
: de
( F ) e s t d i s j o i n t d e E . Pour c h a q u e p a r t i e f i n i e X d e E ( F ) e t c h a q u e
e l e m e n t D d e F, e x i g e o r s q u ' e x i s t e un i s o m o r p h i s m e a u moins d e l a r e s t r i c t i o n A ( F ) / X SUK une r e s t r i c t i o n d e A / D .
Dans l e c a s g e n g r a l a v e c E
(F)
infini, cette
e x i g e n c e e s t s a t i s f a i t e e n u t i l i s a n t It. lemme d e c o h e r e n c e , g q u i v a l e n t 1 l ' a x i o m e d e l ' u l t r a f i l t r e ( v o i r Note I e n f i n d ' a r t i c l e ) . Suppcsors c h o i s i e l a r e l a t i o n A ( F ) pour chaque A u l t r a f i l t r e F , e t n o t o n s E+ l a r g u n i o n d e E e t des E ( F ) . A l o r s l a c l o t u r e A+ d e A s e r a d 6 f i n i e s a n s a m b i y u i t e comme s u i t , s u r l a Lase E + . S o i t n l ' a r i t 6 d e A , e t x ] ,
de
.'E
...x
des 61Ements
S i x . a p p a r t i e n t 1 E , c o n s e r v o n s l e ( i = 1 ou...ou n ) . Groupom e n s e m b l e
l e s x . q u i a p p a r t i e n n e n t 5 un &me
E
isomorphisme l o c a l de A ( F ) v e r s A / D ,
F donc l e s e n s e m t l e s
( F ) , e t prenons l e u r s t r a n s f o r m e s x! p a r un oil D e s t un e l e m e n t d e F. Les u l t r a f i l t r e s
par n o t r e s u i t e x , ,
D concern&
...,x
s o n t en nomke f i n i :
nous pouvors donc e x i g e r q u e c e s D , q u i a p p a r t i e n n e n t 1 des F d i s t i n c t s , s o i e n t m u t u e l l e m e n t d i s j o i n t s , e t s o i e n t d i s j o i n t s des s i n g l e t o n s des x . q u i a p p a r t i e n -
,..., x
n e n t 2 E . A l o r s posons A+ ( x l
) = A(x;
,...,x:),
en posant x! = xi l o r s q u e
x . a p p a r t i e n t 1 E . La v a l e u r a i n s i d e f i n i e e s t i n d k p e n d a n t e d u choix d e s i n t e r v a l l e s d i s j o i n t s D,
du f a i t que des automorphismes l o c a u x d e A , d o n t les domaine e t
codornaine s o n t i n c l u s , p o u r c h a c u n , d a m un i n t e r v a l l e D , d o n n e n t une r e u n i o n q u i at elle-mEme un a u t o m o r p h i s m e l o c a l d r A : v o i r 1 . 5 . Notons q u e c h a q u e
A
(F)
P s t
un i n t e r v a l l e d e l a c l d t u r e A+.
R e t r o u v o r s c o m e s u i t l a c l G t u r e de l a c h a i n e Q des r a t i o n n e l s , ou p l u t s t l ' u n e des c l z t u r e s , q u i es t l a c h a i n e des r e e l s . Les u l t r a f i l t r e s i n t e r v a l l a i r e s d e Q s o n t d e c i n q s o r t e s : p o u r d7aque r a t i o n n e l u , l ' u l t r a f i l t r e t r i v i a l e n g e n d r e p a r u ; l ' u l t r a f i l t r e des i n t e r v a l l e s B d r o i t e d e u , d e l a f o r m e l u , x ] 02 x > u
;
l ' u l t r a f i l t r e dffi i n t e r v a l l e s 1 g a u c h e d e u ; e n f i n 1es u l t r a f i l t r e s 1 d r o i t e e t 2 g a u c h e d e c h a q u e i r r a t i o n n e l . I1 s u f f i t d e p r e n d r e E
(F)
v i d e pour l ' u l t r a f i l t r e
1 d r o i t e ou 2 gauche d e chaque r a t i o n n e l , a i m i q u e , p a r exemple, pour l ' u l t r a f i l t r e 1 gauche d e chaque i r r a t i o n n e l , e t E l ' u l t r a f i l t r e B d r o i t e de
(F) r g d u i t B u n u n i q u e e l g m e n t p o u r
chaque i r r a t i o n n e l .
C e t t e n o t i o n de c l s t u r e , a i m i que l a n o t i o n de r e l a t i o n compactable exposee c i - a p r k ,
r e m o n t e 5 FRAYSSl!,
deux p r o blPmes l o g i q u e s s u i v a n t s ProblSme I .
F,
une A
(F)
; e l l e donne l i e u
aux
E t a n t d o n n e une r e l a t i o n A e t un A - u l t r a f i l t r e non t r i v i a l
associons l a r e l a t i o n de b
existe-t-il
1972 ( 2 ) p . 164-165
.
e v i d e aux autrffi A - u l t r a f i l t r e s
non t r i v i a u x . Alors
d e b e n o n v i d e d o n n a n t une e x t e n s i o n l o g i q u e ( e n c o r e d i t e
R. Fraksi
336 extens i o n 616mentaire) dc A .
ProUPme 2 . S o i t F e t G deux A - u l t r d f i l t r r s
non t r i v i a u x , A ( F ) e t
A (G) deux r e l a t i o n s donnant chacune une c l 6 t u r e q u i s o i t une e x t e n s i o n l o g i q u e de A (les a u t r e s
u l t r a f i l t r e s 6 t a n t a f f e c t & d'une
o k e n u e en prenant h l a f o i s A ( F j e t A
h s e v i d e ) . Alors l ' e x t e n s i o n
(G) e s t - e l l e une e x t e n s i o n l o g i q u e .
6 . 3 . Ensemble compact de f i l t r e s , r e l a t i o n compactable Partons d'un ensemble E , e t d ' u n r n s e m l l e de f i l t r e s F s u r E . Disons qu'un t e l ensemtle de f i l t r r s e s t compact l u r s q u e , pour chaque f o n c t i o n de c h o i x h s u r les F , t e l l e que h ( F ) s o i t un 6lEment de F pour chaque F , i l e x i s t e un nomh-e f i n i de
F t e l s que l a r6union de l e u r s t r a n s f o r m &
h
( F ) recouvre E .
E t a n t donn6 un ensemble E , l'ensemble de tous les ultrafiltres sur E, est compact ( u t . ax. u l t r a f i l t r e ) . 0
Supposons l e c o n t r a i r e : i l e x i s t e une f o n c t i o n de c h o i x h t e l l e que,
pour chaque ensemUe f i n i U d ' u l t r a f i l t r e s s u r E , 1 ' e n s e m U e E U Egal B E diminu6 de l a r6union des h ( F ) pour tous les F de U, s o i t non v i d e . Alors l e s s u r e n s e m t l e s des E
U
forment un f i l t r e s u r E . Prenons un u l t r a f i l t r e plus f i n : i l e s t d i s t i n c t
de t o w l e s
F
; contradiction.
0
Revenons aux r e l a t i o n s , e t d i s o n s qu'une r e l a t i o n A es t compactable l o r s
-
que l ' e n s e m t l e des A - u l t r a f i l t r e s e s t compact. Disons qu'une r e l a t i o n A de l a s e E e s t intervallement compactable l o r s q u e , pour chaque A - i n t e r v a l l e D , l e compl6ment a i r e E-D e s t une r6union f i n i e de A - i n t e r v a l l e s .
Par exemple, t o u t e r e l a t i o n o u
m u l t i r e l a t i o n u n a i r e , o u e n c o r e t o u t e c h a i n e e s t i n t e r v a l l e m e n t compactable. Par c o n t r e l a c o n s 6 c u t i v i t Q s u r l e s e n t i e r s n a t u r e l s n ' e s t pas i n t e r v a l l e m e n t compac
-
table (voir 1 . 1 ) . Dire que A e s t i n t e r v a l l e m e n t compactakle Gquivaut 2 I ' u n e ou 2 I ' a u t r e des c o n d i t i o n s s u i v a n t e s :
( I ) pour chaque ensemble fini de L-intervalles Di, E
-
l e complementaire
(U Di) est une rgunion finie d e A-intervalles ; ( 2 ) pour chaque couple de A-intervalles C , D avec D
C-D est une reunion finie de A-intervalles
2
C, l a diffgrence
: mgme 6nonc6 en r e m p l a p n t
D p a r une
r6union f i n i e d ' i n t e r v a l l e s D . C C . 1 -
0
o l a c u n e des c o n d i t i o n s ( I ) ,
( 2 ) e n t r a i n e Qvidemment que A s o i t i n t e r -
v a l l e m e n t compactable. L ' i n v e r s e e s t v r a i pour l a c o n d i t i o n ( I ) ,
du f a i t que t o u t e
i n t e r s e c t i o n d ' i n t e r v a l l e s e s t un i n t e r v a l l e . Par a i l l e u r s pour l a ( Z ) , une r d u n i o n f i n i e d ' i n t e r v a l l e s , il en e s t de mgme pour C-D = C aux i n t e r v a l l e s i n t e r s e c t i o n s .
(E-D),
s i E-D e s t
en p a s s a n t
0
6.4. Soit A une relation intervallement compactable filtre sur la base E de A,
n
;
alors tout ultra-
une fois reduit a ses e1Qments qui sont des A-interval-
les, donne un A-ultrafiltre.
Par c o n t r e , pour l a c o n s Q c u t i v i t 6 C s u r les e n t i e r s n a t u r e l s , 1es s e u l s C - i n t e r v a l l e s Q t a n t l e v i d e , l a lase e t les s i n g l e t o n s , t o u t u l t r a f i l t r e non t r i -
L'interualle en thiorie des relations
337
vial sur la b e I C I, une fois r6duit aux C-intervalles, donne le singleton de la lase : ce n'est pas un C-ultrafiltrr. Soit F un ultrafiltre sur E, et A l'ensemkle des A-intervalles 616ments
0
de F . Supposons que A ne soit pas un A-ultrdfiltre. Alors i l existe un A-intervalle D, non elt5ment de A mais q u i intersec-te C h d q U e 6lCment de A . Ce D n'est pas element
de l'ultrafiltre F , donc E-D est el6ment de
F. Par hypothke E-D est une
reunion finie de A intervalles : l'un d'eux, soit D*, est 616ment de F, donc de A . Finalement D et D* sont disjoints, mdis D intersecte chaque 616ment de A , donc D intersecte D
:
contradiction.
0
6 . 5 . Les notions de r e l a t i o n compactable e t intervallement compactable, se confondent, moyennant l'axiome de choix. O
Soit A une relation intervallement compactable, E sa h e . Supposons A
non compactable : il existe une fonction de choix h telle que, pour chaque ensemble fini U de A-ultrafiltres, l'ensemkle E 6gal B E diminu6 de la r6union des h ( F ) U pour tous les F de U, soit non vide. En ce cas les surensemkles des EU constituent un filtre sur E. Prenom un ultrafiltre plus fin, puis r6duisons le & ses A-intervalles. Cela nous donne un A-ultrafiltre, par l'6nonc6 pr6c6dent : appelons-le H et notons que h ( H ) est 616ment de H. Pour chaque ensemble fini U de A-ultrafiltres, EU est par hypothsse une r6union finie de A intervalles (voir 6 . 3 ( 1 ) ) : l'un d'eux est 616ment de H (voir 6.1. I ) . Prenons U = s ingleton de H ; nous o kenons E = U E - h (H) ; il existe un blsment de H qui soit inclus dans E - h (H), donc disjoint de h (H)
:
contradiction prouvant que A est compactahle. Inversement supposons A compactable, et supposons qu'il existe un A-in-
tervalle D, tel qu'aucune r6union finie de A-intervalles ne donne E-D. Alors 1 chaque A-ultrafiltre F, associons h (F)
=
D lorsque D est 616ment de F ; sinon
choisissons h ( F ) 616ment de F et disjoint de D. Avec cette fonction de choix h , aucune r6union finie de h ( F ) ne recouvrr E : contradiction.
0
ProNSme. Existence d'une reldtion intervallement compactahle de lase E, et d'un intervalle D tel que E-D ne
S U ~ Lpas
une r6union finie d'intervalles m u t u e l -
lement d i s j o i n t s cela exclut les chafnes et les multirelations unaires.
7 - ULTRAFILTRE ET FAMILLE LOC-INTERVALLAIRE ; CLOTURE D'UNE RELATION PAR CES
ULTRAFILTRES ; CAS DE LA REPRESENTATION DE STONE.
7.1. Ultrafiltre loc-intervallaire Soit A une relation de b e E ; un ultrafiltre
F
sur E sera dit A-loc-
i n t e r v a l l a i r e (ahrgviation de "localement intervallaire"), soit lorsqu'il es t tri -
vial, soit lorsque, pour chaque partie finie H de E, il existe un Pigment D C E-H Hde F, tel que tout automorphisme local de la restriction A/D soit extensible par H l'identitb sur H. Par exemple, Ptant donnb un A-ultrafiltre intervallaire, au sens d u
§
6 prbcedent,
R. Fra'isse
338
tout ultrafiltre plus fin est A-loc-intervallaire. Cependant il existe bien d'autres exemples, en raison de l'6noncQ suivant. Si A est une relation binaire, tout ultrafiltre sur la base I A I est A-loc-interVal1aire. 0
Pour l'arite 2, i l suffit, dans la definition ci-dessus, que tout automorphisme
local transformant un seul QlQment de DR en un autre, soit extensible par l'identit6 sur H. Mettons dans une msme classe d'gquivalence deux GlQments quelconques u,v de
E-H, lorsque la transformation du seul u en v est un automorphisme local extensible par l'identitg sur H. Les classes ainsi dEfinies Etant en nombre fini, l'une d'elles et une seule est QlQment de l'ultrafiltre. 0 Par contre, etant donne la relation ternaire de cyclordre A de base infinie E (paragraphe 3 ) , seuls les ultrafiltres triviaux sur E sont A-loc-intervallaires. En effet, prenons pour H un singleton, et reprenons la notation DH prQcQdente. La transposition qui Qchange deux QlEments quelconques de DH est un automorphisme local du cyclordre A, inextensible par l'identitg sur H. 7.2.
Famille loc-intervallaire d'ultrafiltres
Soit A une relation de base E ; un ensemble, ou famille d'ultrafiltres dont chacun est A-loc-intervallaire, sera dit famille A-loc-intervallaire, lorsque, pour chaque partie finie H de E, et chaque suite finie d'ultrafiltres F , ,
. .. , F
de la famille (il suffit de prendre n au plus Bgal P l'aritE), on peut associer 1 chacun l'un de ses SlQments, soit D, E F , ,
.. ., Dn E Fn,
les D Stant mutuellement
disjoints, de sorte que toute reunion d'automorphismes locaux de A/D ,,..., de A/Dn, soit un automorphisme local de A, extensible par 1'identitQ sur H. Dans le cas oii certains ultrafiltres sont triviaux, on prendra Qvidemment 1'QlQment correspondant D qui se rQduit 1 un singleton (Eventuellement singleton d'un SlQment de H) ; et pour les
non triviaux, on prendra un D disjoint de H.
Exemple de la representation de Stone. Soit A un lattis boolden. Plus prQcisEment A est une relation binaire d'ordre (rgflexive, transitive, antisymgtrique) ; pour deux QlQments quelconques u,v de la base, il existe un infimum u n v tel que les x 1 l a fois Q u et Q v soient exactement les x Q u n v ; de mgme il existe un supremum u
u v,
avec distributivitg entre supremum et infimum ; de plus il existe
un minimum 0 et un maximum I et, pour chaque Cl&ent u
nlu
=
u , un complEment
7 u tel que
0 et u U l u = I . Supprimons le minimum 0, et disons qu'un ultrafiltre
ensembliste F sur la nouvelle base (aprSs suppression du 0) est stonien lorsque, pour chaque Q16ment u initial < 7 u ,
f
0, ou bien l'intervalle initial Q u , ou l'intervalle
est un Gldment de
F.
En particulier, un ultrafiltre trivial est
stonien ssi il se rQduit au singleton d'un QlEment minimal (irmngdiatement supdrieur au 0 qui a QtQ supprim6). Disons que deux ultrafiltres stoniens F,G
sont stone-
distincts, lorsqu'il existe un u pour lequel l'intervalle initial Q u est QlQment de
F, et l'intervalle initial < l u est QlQment de G.
Tout ensemble d'ultrafiltres triviaux ou stoniens, l e s stoniens etant mutuellement
339
L'interualle en the'orie des relations s t o n e - d i s t i n c t s , es t u n e f a m i 1l e l o c - i n t e r v a l l a i r e . u
L'aritg du lattis boolcien A 6tant 2, tous nos ultrafiltres sont A-loc-interval-
laires. I1 suffit donc de considGrer le cas de deux ultrafiltres non triviaux F,
G. Par hypothsse il existe un u avec l'intervalle initial F des G u , 6lGment de F , et l'intervalle initial comparable (mod
A)
G
des S l u , GlEmrnt de G. Tout ElEment de F est donc in-
2 tout Gl6ment de G. Donc la r6union d'un automorphisme local
quelconque de A / F et d'un de A / G est un automorphisme local de A. Finalement on tient compte d c l'ensemble fini H de 7.2 en remplaqant F par F pour G .
-
(F
fl H)
; de insme
0
Bien que, p o u r L ' a r i t k 2, tout ultrafiltre soit loc-intervallaire, il est faux que t o u t e n s e m b l c d ' u l t r a f i l t r e s soit l o c - i n t e r v a l l a i r e .
Par exemple, partons de la
cha'ine Q des rationnels, qui est dense, sans minimum ni maximum ; prenons deux ultrafiltres F,G
ainsi dsfinis. Pour chaque partition de l'ensemble des ration-
nels e n un intervalle initial non vide et un intervalle final, prenons, dans F c o m e dans G , l'intervalle initial. Plus ggnEralement, pour chaque partition, l'une des parties compl6mentaires au mains est coinitiale B Q, et nous la prenons si elle est la seuls des deux qui soit coinitiale. Nsanmoins nous assurons la diff6rence entre F et G en prenant par exemple l'ensemble des rationnels binaires dans F et son complgment dans G. I1 est alors Gvident que deux 616ments F de F et G de G sont coinitiaux de Q , donc qu'il existe un automorphisme local de Q/F transformant un Glement u de F en v < u ,
et un w de G intermsdiaire : la transformation de u,w
en v,w n'est pas un automorphisme local de Q. 7.3. Cl6ture d'une relation selo n une famille loc-intervallaire
Partons d'une relation A de base E et d'une famille A-lac-intervallaire d'ultrafiltres. Nous pouvons alors reprendre l'argumentation du 6.2 ; autrement dit, associer B
chacun des ultrafiltres non triviaux de la famille, soit F , un
ensemble E ( F ) et une relation A ( F ) de base E ( F ) ; les E ( F ) stant mutuellement disjoints, et disjoints de E. Comme en 6.2 il y a ambiguit6 pour le choix du cardinal de chaque E ( F ) (qui est arbitraire, nul, fini ou infini). I1 y a encore une ambiguits pour le choix de chaque A
(F), bien que toute restriction finie de A ( F )
doive n6cessairement figurer parmi l e s restrictions communes aux A / F pour tous les 616ment F de F . Mais, c o m e en 6.2, u n e f o i s l e s A ( F ) c h o i s i e s , i l ne s u b s i s t e p l u s aucune a r n b i g u i t b p o u r a c h e v e r d e c o n s t r u i r e l a c l d t u r e A
+
sur la r6union de E et
des E ( F ) . Tout c o m e en 6.2, les A (F) deviennent des intervalles de la clature A'. Bien que cela ne semble pas essentiel, notons qu'il est toujours possible, par le th6orSme de RAMSEY, d'exiger que chaque A ( F ) soit enchalnable, i.e. libre-interpr6table e n une chaTne arbitrairement donn6e sur la base E ( F ) . 7 . 4 . Pour achever la jonction avec le th6orSme de reprgsentation de Stone,
associons, 1 un lattis bool6en donn6 A , sa cl6ture A+ dEfinie c o m e suit. Nous prenons tous les ultrafiltres triviaux, p l u s les ultrafiltres non triviaux stoniens
R. Fra'isse
3 40
F.
Chaque
F est d6fini en dQcidant, pour chaque 616ment
u # 0 , si l'intervalle
initial ,< u , o u l'intervalle initial < l u , rst 616ment de
F.
Un unique
F 6tant
choisi c o m e reprssentant pour chaque fonction possible de dgcision, les F sont mutuellement Stone-distincts. A chaque F stonirn non trivial, associons un E ( F ) , donc un A ( F ) singleton, les A (F) 6tant mutuellement incomparables, c o m e il r6sulte de 7.2.
Par contre, entre un A ( F ) non trivial rt un 616ment u de E, nous
avons soit A ( F ) < u (si, au dgpart, l'intervallr initial
6tait un 616ment de
F ) , soit A ( F ) incomparable 5 u (si, au dgpart, l'intervalle initial < l u 6tait un 616ment de F ) . Cela achsve de d6finir la cl8ture At de A. Evidemment, dans la repr6sentation classique de Stone, on supprime de A+ tous les 616ments de E, except6 les 616ments minimaux (immsdiatement sup6rieurs au 0), s'il en existe. Chaque G16ment u ainsi supprim6 est remplac6 par l'ensemble des A ( F ) < u,et des 616ments minimaux < u avec la relation d'inclusion c pour remplafer
< (mod ,)'A
la rQunion
pour remplaGer le supremum, l'intersection pour remplafer l'infimum. Note 1 . Enonce du lemme de coherence (voir par exemple FRA'ISS6 1972 ( 1 ) Soit I un ensemble d'ensembles F sur chacun desquels nous avons un ensemble p. 52). fini non vide UF de relations de base commune F , et d'arit6 commune, avec les deux hypothsses suivantes : (I)
I filtrant : si F , F '
(2) si F,G E I et F
2
E
I
il existe un G E I avec G
_>
F U F' ;
G , toute relation 6lQment de UG, une fois restreinte B F ,
donne un Qldment de UF ; alors il existe une relation R de l'arit6 commune donnQe, bas6e sur la reunion des F , et telle que pour chaque F , la restriction R/F EUF.
Ce lenime 6quivaut 2 l'axiome de l'ultrafiltre. dans le cas gQn6ral ; mais ZF suffit lorsque les F sont finis et leur rsunion dsnombrable. Note 2. La notion usuelle d'intervalles contigus
se
g6n6ralise de manisre
naturelle : deux A-intervalles sont dits c o n t i g u s lorsque leur r6union est un A-intervalle. Le lecteur d6finira ainsi 1'616ment adhPrent cl6ture topologique d'un A-intervalle.
D un A-intervalle, donc la
34 1
Istvan FOLDES, 1973
-
R e l a t i o n s denses et d i s p e r s e e s
;
extension d ' m t h b o r e m e d e H a u s d o r f f .
C o m p t e s r e n d u s A c a d . S c i . P a r i s , vol 2 7 7 A , p . 2 6 9 - 2 7 1 .
Roland FRAYSSA, 1971-72Cours d e l o q i q u e mathematique, ( 1 ) R e l a t i o n et f o r m u l e l o g i q u e , 197 p . : ( 2 ) T h e o r i e d e s m o d e l e s , 1 7 7 p . : P a r i s ( G a u t h i e r - V i l l a r s ) : trad.
1973-74-
C o u r s e of m a t h e m a t i c a l l o g i c , D o r d r e c h t ( R e i d e l ) .
1977- P r e s e n t p r o b l e m s a b o u t i n t e r v a l s i n r e l a t i o n - t h e o r y
and l o g i c .
Non c l a s s i c a l l o g i c s , m o d e l t h e o r y a n d c o m p u t a b i l i t y , vol 8 9 d e S t u d i e s
i n L o g i c , Amsterdam (North-llolland).
David GILLIAM, 1978 A
-
concrete r e p r e s e n t a t i o n t h e o r e m f o r i n t e r v a l s o f m u l t i r e l a t i o n s .
Z e i t s c h r i f t M a t h . L o g i k vol 2 4 p . 4 6 3 - 4 6 6 . 1 9 7 9 - I n t e r v a l s o f b i n a r y r e l a t i o n s . I b i d e m vol 2 5 p . 5 7 - 6 0 .
Gerard L O P E Z , 1978 L ' i n d e f o r m a b i l i t e d e s r e l a t i o n s et m u l t i r e l a t i o n s b i n a i r e s . Z e i t s c h r i f t M a t h . L o g i k vol 2 4 p . 3 0 3 - 3 1 7 .
Maurice POUZET, 1979
-
R e l a t i o n m i n i m a l e p o u r son d g e . Z e i t s c h r i f t M a t h . L o g i k vol 2 5 p . 3 1 5 - 3 4 4 . 1981- R e l a t i o n s i n p a r t i b l e s . D i s s e r t a t i o n e s Mathematica
, vol 1 9 3 , p . 1 - 4 3 .
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-
PART VI ORDER AND CHAIN CONDITION ON CLASSES OF STRUCTURES
- PARTIE VI ORDRE ET CONDITIONS DE CHAINE SUR DES CLASSES DE STRUCTURES
P.D. SEYMOUR
-
Neil ROBERTSON
Some new results on the well-quasi ordering of graphs.
.... . ... .
p . 343
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Annals of Discrete Mathematics 23 (1984) 343-354 0 Elsevier Science PublishersB.V. (North-Holland)
343
SOME NEW RESULTS ON THE WELL-QUASI-ORDERING OF GRAPHS
P.D. SEYMOUR* and Neil ROBERTSON Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, Ohio 43210 U.S.A.
D e d i c a t e d t o Professor E . COROMINAS.
RESUME.-
Quelques resultats nouveaux sur la relation de be1 ordre des graphes finis.
Designons par 9 l'ensemble des graphes finis, et, lorsque G, H € 9 , Bcrivons H E G si H est isomorphe a un mineur de G, c'est-8-dire a une contraction d'un sous-graphe partiel de G . Dans les annees 1960, K. WAGNER a fait la conjecture (non publiee) que 9 est belordonne par cette relation d'inclusion. Pour des graphes finis, cela signifie que dans tout sousensemble infini un certain graphe sera isomorphe a un mineur donne d'un autre graphe. Recemment, les auteurs de cet article ont demontre des theorsmes qui laissent fortement presager que la conjecture de WAGNER est correcte. Notre methode consiste ?I montrer que l'ensemble des graphes G qui ne contiennent pas un certain graphe fixe H comme mineur a une structure definie, et de montrer alors, qu'etant donnee cette structure, l'ensemble des graphes qui la possedent est belordonnd. Si on savait le faire pour tout graphe H, il en decoulerait que ( 3 5 ) est lui-meme belordonnd, donc que la conjecture de WAGNER est vraie. La structure de graphes mentionnde p l u s haut est construite en trois Btapes: (1) plonger un graphe dans une surface S ; ( 2 ) adjoindre au plus n sommets, qui peuvent etre adjacents a n'importe quels des autres sommets, et (3) joindre les unes aux autres des copies disjointes de tels graphes en une structure d'arbre. Ici, S et n sont determines par le graphe H exclu, et les graphes adjacents dans la structure d'arbre formant G sont joints en identifiant des sousensembles de leurs sommets d'une maniere univoque. De tels sous-ensembles ne peuvent intersecter la surface de plongement S q u e trivialement, c'estA-dire au p l u s en trois sommets contenus dans la frontisre d'une face du plongement. Ce programme a maintenant BtB execute pour des classes de graphes G qui ne contiennent pas c o m e mineur un graphe planaire fixe H; c'est un cas oh il
n'est pas necessaire d'utiliser la structure de la surface. Pour tout graphe planaire H, il existe un entier w tel que tout graphe G qui ne
*
Partially supported by National Science Foundation grant number MCS 8103440
.
P.D. Seymour, N. Robertson
344
contient pas H comme mineur peut &tre construit en joignant en une structure d'arbre des graphes avec au plus w sommets. De plus, on montre que les classes de graphes G qui peuvent etre construites de cette mani6re sont belordonnees; dans cette demonstration on utilise une generalisation du th6orhe de KRUSKAL sur la structure d'arbre, que l'on prouve pour des structures d'arbre de certains hypergraphes structures (familles d'ensembles) de dimension bornee. Les consequences de ce travail sont:
(1) que toutes les anti-chaines (ensembles d'elements non relies deux B deux) dans ( 9 , C_ ) qui contiennent au moins un graphe planaire H sont finies, et ( 2 ) qu'il existe un algorithme pour determiner la relation
H
5G
(H est isomorphe B un mineur de GI lorsque H est un graphe planaire fixe et G E 9 , avec temps de calcul born6 par des polyndmes de la dimension de G . Nous pensons que nos methodes nous permettront de demontrer la conjecture generale structurelle ainsi qu'un theoreme de be1 ordre, avec les deux memes consequences qu'auparavant sans la condition que H soit planaire, c'est-5dire de demontrer la conjecture de WAGNER et d'obtenir un bon algorithm pour determiner la relation H 5 G pour tout G E 9 . ABSTRACT.It has been conjectured by K. WAGNER that finite graphs are wellquasi-ordered by minor inclusion, i.e. being isomorphic to a contraction of a subgraph. A method is reported on here that shows promise of settling this conjecture. We have proved (1) that all graphs G not including a fixed planar graph H as a minor can be constructed by piecing together graphs on a bounded number of vertices in a tree-structure, and (2) by elaborating the KRUSKAL tree theorem that the class of graphs formed by piecing together graphs of bounded size in treestructures is well-quasi-ordered. It follows from this that no infinite antichain of finite graphs can include even one planar graph and that there is a "good" algorithm for testing the presence of a fixed planar graph as a minor.
Background. Let Q be a quasi-ordered set, i.e. a set on which a reflexive and transitive relation < is defined. If x , y E Q and x < y , we say that y includes x . We write x < y if x < y % x . A subset S of Q is (i) an antichain if there do not exist distinct elements, x, y E S such that x < y , (ii) an upper order ideal if y E S whenever x < y for some x E S , (iii) a lower order ideal or lower ideal if y E S whenever y < x f o r some x E S . An upper order ideal S is finitely generated if it has a finite subset F such that S = {x E Q : f < x for some f E F} . It is an easy exercise to prove that the following conditions on a quasi-ordered set Q are equivalent: (a) all upper order ideals of I) are finitely generated, (b) in Q there is neither an infinite antichain nor an infinite descending chain (i.e. an infinite ), sequence x2, such that x > x2 > x3 > (c) for every infinite 1.
'
sequence and
XI'
xi < x.
. ..
x*,
.
...
...
1
of elements of
Q
there exist
i, j
such that
i< j
A quasi-ordered set satisfying any of the equivalent conditions
1 (a), (b), (c) is said to be well-quasi-ordered.
Well-quasi ordering of graphs
345
2.
Graph I n c l u s i o n R e l a t i o n s . Examples o f w e l l - q u a s i - o r d e r e d s e t s a r i s e n a t r i r a l l y i n a l g e b r a , g r a p h t h e o r y and l o g i c . S i n c e t h e 1940's, a b r a n c h of o r d e r e d s e t t h e o r y h a s developed from t h e s t u d y of w e l l - q u a s i - o r d e r i n t h e s e f i e l d s . The a u t h o r s of t h i s p a p e r have r e c e n t l y made p r o g r e s s on t h e open g r a p h t h e o r e t i c a l problems and r e p o r t h e r e on t h e r e s u l t s t h e y have o b t a i n e d . Denote by 9 t h e c l a s s o f f i n i t e g r a p h s . Under subgraph c o n t a i n m e n t 9 i s n o t w e l l - q u a s i - o r d e r e d , a s i s shown by t h e i n f i n i t e a n t i c h a i n o f f i n i t e c i r c u i t s . Most o f t h e t h e o r y o f g r a p h w e l l - q u a s i - o r d e r i n g c o n c e r n s t h r e e c l o s e l y r e l a t e d q u a s i - o r d e r s on 9 o f a t o p o l o g i c a l n a t u r e . To d e f i n e t h e s e r e l a t i o n s w e i n t r o d u c e some n o t a t i o n . Suppose G E 3 and X 5 V ( G ) A simple path i n G i s a p a t h w i t h a t l e a s t one edge and no r e p e a t e d v e r t i c e s , e x c e p t p o s s i b l y i t s e n d v e r t i c e s . The g r a p h of a s i m p l e p a t h ( i . e . t h e subgraph o f G d e t e r m i n e d by i t s v e r t i c e s and e d g e s ) w i t h i t s e n d v e r t i c e s i n X and no o t h e r v e r t i c e s i n X i s c a l l e d a n X-join, and i s a n x - l i n k o r an X-loop i f t i s endvertices a r e , r e s p e c t i v e l y , d i s t i n c t or e q u a l . Then P(G, X ) d e n o t e s t h e g r a p h w i t h vertex-set X , e d g e - s e t t h e s e t o f X - j o i n s o f G , and i n which e a c h X-join i s i n c i d e n t with i t s e n d v e r t i c e s i n X
.
.
X-joins i n G t h e n a c h o i c e f u n c t i o n f o r P i s a mapping s u c h t h a t f ( P ) E E ( P ) f o r a l l P E !J' Given such a c h o i c e f : f' + E ( G ) w i t h e n d v e r t e x x E X , d e n o t e by P f ( x ) t h e segment o f f u n c t i o n and P E 9) If
P
from
9' i s a s e t o f
x
.
t o t h e f i r s t vertex of
may g e n e r a t e t o t h e s i n g l e v e r t e x
i n c i d e n t with
P
f(P)
.
Note t h a t
Pf(x)
.
x
The g r a p h P ( G , X) o f X-joins can be used t o def i ne t h e i ncl usi on r e l a t i o n s f o r w e l l - q u a s i - o r d e r i n g i n a u n i f i e d manner. H ' 5 P(G, X ) f o r any X 5 V ( G ) i s embedded i n G when t h e X-joins o f E ( H ' ) meet p a i r w i s e o n l y a t common e n d v e r t i c e s . Then a g r a p h H i s c o n t a i n e d i n G by embedding i n c l u s i o n , w r i t t e n H <e G
A graph
,
when H i s i s o m o r p h i c t o a g r a p h H' embedded i n G. A graph
H'
5 P(G,
X)
f o r any
X
5 V(G)
i s immersed i n G when t h e
X-joins of E ( H ' ) have p a i r w i s e no common e d g e s . Then a g r a p h H i s c o n t a i n e d i n G by immersion i n c l u s i o n , w r i t t e n H
i s isomorphic t o a g r a p h A graph
H'
5 P(G,
X)
H'
immersed i n
f o r any
X
5 V(G)
G.
i s a minor of G when t h e r e
i s an i n j e c t i v e choice f u n c t i o n f : E ( H ' ) + E ( G ) such t h a t i f are i n c i d e n t w i t h P, Q E E ( H ' ) , r e s p e c t i v e l y , t h e n p, q E V(H') p # q i m p l i e s P f ( p ) , Q (9) are d i s j o i n t . Then a g r a p h H i s f c o n t a i n e d i n G by minor i n c l u s i o n , w r i t t e n H <m G , when H i s isomorphic t o a minor
H'
of
G
.
I n a l l t h r e e o f t h e above d e f i n i t i o n s t h e i n j e c t i v e mapping onto V ( H ' ) d e t e r m i n e d by t h e isomorphism between H and H' i s c a l l e d vertex-mapping of t h e i n c l u s i o n r e l a t i o n . I t is r o u t i n e , b u t n o t a l t o g e t h e r t r i v i a l , t o v e r i f y t h a t t h e s e t h r e e i n c l u s i o n r e l a t i o n s are q u a s i o r d e r s . An e q u i v a l e n t , and more s t a n d a r d way t o d e f i n e t h e minor i n c l u s i o n r e l a i s as f o l l o w s . F o r two g r a p h s , G , H w e s a y t h a t H 4 G i f f o r t i o n <m
a
: V(H) + V(G)
m
each x E V ( H ) w e c a n s e l e c t a c o n n e c t e d subgraph f ( x ) of G and f o r e a c h e E E ( H ) w e c a n select a n edge f ( e ) o f G i n s u c h a way t h a t ( i )t h e s u b g r a p h s f ( x ) ( x E V ( H ) ) o f G are p a i r w i s e d i s j o i n t , ( i i )t h e e d g e s f ( e ) ( e E E ( H ) ) of G are d i s t i n c t and ( i i i ) i f x and y a r e t h e v e r t i c e s j o i n e d by a n
P.D. Seymour, N Robertson
346
edge e in H , then f(e) joins a vertex of f(x) to a vertex of f(y). The equivalence of these two definitions of minor inclusion is also easy to check. It is worth pointing out that the most usual way of producing a minor of G is by a succession of three elementary operations (a) single-edge deletion, (b) single-edge contraction, and (c) removal of an isolated vertex. Indeed, to verify a property is inherited by the minors of a graph it is only necessary to check that it is preserved by these simple operations. We make a final observation that embedding inclusion is the special case of both immersion and minor inclusion where the X-joins P E E ( I 1 ' ) meet only at common endvertices. In practice our three inclusion relations are often extended to apply to graphs with vertices labelled from some well-quasi-ordered set ( Q , < ) . Let 9 5 and ( Q ) = {(G, !L) : G E 9 and !L : V(G) Q is a function). -f
If <x
denotes one of our graph inclusion relations then
means that G <x G' v -+ v'
such that
behaved under
<,
(G, !L) <x
(GI, ! L ' )
and an inclusion isomorphism exists with vertex-mapping k?(v)
<
!L'(v') for all
v E V(G)
.
We say that 3 is well-
when for every well-quasi-ordered set
(Q,<)
F ( Q ) is well-quasi-ordered under the extensed inclusion <x
the family
. Note
that
because the graphs involved are finite and their vertex-labels are taken from a well-quasi-ordered set , all three extended inclusion relations are without infinite descending chains. Thus graph well-quasi-ordered follows from the finiteness condition on antichains alone.
3.
The Context. The standard theorems about graph well-quasi-ordering concern embedding inclusion. It seems (see Krustal [31) that the subject of wellquasi-ordering began with two conjectures of A. Vazsonyi, that under embedding inclusion: (A) finite trees are well-quasi-ordered, and (B) graphs with vertex-valencies < 3 are well-quasi-ordered. The major theorem in this context, which settles Vazsonyi's conjecture (A), is Kruskal's theorem [21 that finite trees are well-behaved under embedding. A similar deep theorem for infinite trees, under the stronger better-quasiordering property, was proved by C. St. J. A. Nash-Williams [7] and Richard Laver [ 4 1
.
W. Mader [ 5 1 generalized Kruskal's theorem to sets of finite graphs with bound k on the number of pairwise edge-disjoint circuits that the graphs can contain. Erdds and Posa [l] had earlier shown that for some integer k' , any such graph can be reduced to a forest by deleting at most k' vertices. Mader proved that any family of finite graphs whose members differ by a bounded number of vertices in this way from the graphs in a well-behaved family must also be well-behaved. By Kruskal's theorem forests are well-behaved, and so the result of Mader follows. However, Mader's theorem cannot be greatly extended, at least for families of graphs containing graphs with arbitrarily many vertices of valency > 4 because of a counterexample. The set of graphs obtained from circuits by doubling each edge in parallel is an infinite antichain in ( 3 , %e'
-
The two main conjectures to avoid this difficulty work by weakening the inclusion relation to allow more inclusions so that the above counterexample does not apply. Nash-Williams conjectured in 171 that: (C) ( '$2,
Wellquasi ordering of graphs
(D)
(
9 ,4 m )
347
is well-quasi-ordered.
I n c l u s i o n s a r e t h u s c r e a t e d between doubled c i r c u i t s of d i f f e r e n t s i z e by a l l o w i n g , i n two d i s t i n c t ways, t h e p a t h s c o r r e s p o n d i n g t o t h e embedded edges t o i n t e r t w i n e . I t i s n a t u r a l t o s t r e n g t h e n c o n j e c t u r e s ( C ) and (D) t o a s s e r t t h a t 9 i s well-behaved under and <m , r e s p e c t i v e l y . Embedding
and immersion i n c l u s i o n do n o t d i f f e r on t h e c l a s s of f o r e s t s , w h i l e minor i n c l u s i o n i s a somewhat s t r o n g e r r e l a t i o n . For example, t h e t r e e shaped l i k e t h e l e t t e r X i s t h e minor o f t h e t r e e shaped l i k e t h e l e t t e r H formed by c o n t r a c t i n g t h e c e n t r a l e d g e . Thus K r u s k a l ' s theorem v e r i f i e s t h a t t h e c l a s s of f o r e s t s i s well-behaved under a l l t h r e e i n c l u s i o n r e l a t i o n s . The i n c l u s i o n r e l a t i o n a l l a g r e e on t h e c l a s s of g r a p h s w i t h v e r t e x - v a l e n c i e s < 3 i n v o l v e d i n V a z s o n y i ' s open c o n j e c t u r e (B), which t h u s forms a common s p e c i a l c a s e of c o n j e c t u r e s ( C ) and (D).
4.
Embedding S t r u c t u r e s . I n o u r o p i n i o n , t h e s e somewhat d i f f e r e n t c o n j e c t u r e s o f Nash-Williams and Wagner a r e e q u a l l y l i k e l y v a l i d and l i k e l y b o t h c o r r e c t . One p o s s i b l e approach t o t h e i r p r o o f i s t o show t h a t a l l f i n i t e g r a p h s G which do n o t i n c l u d e a f i x e d g r a p h H admit a p a r t i c u l a r t y p e o f s t r u c t u r e , and t h a t g r a p h s h a v i n g t h i s s t r u c t u r e do n o t form i n f i n i t e a n t i c h a i n s . Of c o u r s e , t h e s t r u c t u r e f o r c e d on G by e x c l u d i n g H would v a r y w i t h t h e i n c l u sion r e l a t i o n involved. a graph
An example o f s u c h a s t r u c t u r a l t y p e r e s u l t i s K u r a t o w s k i ' s theorem t h a t G embeds i n t h e p l a n e i f and o n l y i f n e i t h e r K n o r K5 embed i n
3,3
G. W e p o i n t o u t t h a t t h e c l a s s of p l a n a r g r a p h s i s c l o s e d under t a k i n g m i n o r s , from embedding i n G i s e q u i v a l e n t and t h a t t h e dominant e x c l u s i o n o f K
3,3 t o e x c l u d i n g i t a s a minor. I n f a c t , a c o n n e c t e d g r a p h
which d o e s n o t have
G
a s a minor e i t h e r i s p l a n a r or i t c a n be s t r u c t u r a l l y e x p r e s s e d a s 3,3 Given H _C G , d e f i n e and c o n s e q u e n t l y o b v i o u s l y h a s an embedded K5
K
W ( G , H) = {x E V ( H )
:
x i s i n c i d e n t w i t h some edge o f
t h a t G can b e structurally expressed a s
x2
, x3 ,
H. , 1.1
x4
,
x5
of
K5
n o t i n HI
G
.
'
Kg
.
W e say
i f there e x i s t five vertices
x
1 '
and t e n p a i r w i s e e d g e - d i s j o i n t c o n n e c t e d s u b g r a p h s
G
such t h a t
( a ) W ( G , Hi (b)
,) I3
= {xi,
U H. l < i < j < 5
for
1
<
i
<
j
<
,
5
and
. = G
The e f f e c t o f e x c l u d i n g
K
a s an embedded g r a p h i s more c o m p l i c a t e d
K5
< 5
since i t l e a v e s a l l g r a p h s w i t h some n o n p l a n a r g r a p h s l i k e
,
x.) 7
4,4
v e r t i c e s of valency
>
4
,
as w e l l
which have enough v e r t i c e s of v a l e n c y
accommodate t h e embedding. Moreover, e x c l u d i n g
K5
as
>4
a s a n immersed g r a p h l e a v e s
t h e c l a s s o f g r a p h s w i t h n o t enough v e r t i c e s o f v a l e n c y
>
4
,
b u t it d o e s n o t
l e a v e a l l p l a n a r g r a p h s . F o r example, any p l a n a r g r a p h w i t h t h e o c t a h e d r o n K a s an immersion. I n b o t h t h e s e cases t h e s t r u c t u r e o f t h e 5 g r a p h s n o t i n c l u d i n g K5 is w e l l hidden.
embedded h a s
On t h e o t h e r hand, i f
K5
to
i s e x c l u d e d as a m i n o r , t h e n a t h e o r e m - o f
P.D. Seymour, N. Robertson
348
Wagner 1161 s t a t e s t h a t t h e graphs which remain c a n be b u i l t up from d i s j o i n t (formed be adding t o an p l a n a r g r a p h s and c o p i e s o f a s p e c i f i c graph V8
octagon t h e f o u r edges j o i n i n g o p p o s i t e v e r t i c e s ) by r e p e a t e d clique-summing. I n clique-summing, t w o graphs a r e j o i n e d by choosing complete subgraphs of t h e same s i z e i n each, i d e n t i f y i n g t h e i r v e r t i c e s i n a one-to-one f a s h i o n , and d e l e t i n g t h e i r edges. T h i s produces a " t r e e - s t r u c t u r e ' ' of t h e corresponding p l a n a r graphs and V ' s , w i t h each component graph s e p a r a t i n g t h e component 8 graphs l y i n g i n d i f f e r e n t branches o f t h e " t r e e - s t r u c t u r e " . We w i l l r e t u r n t o t h i s s t r u c t u r e l a t e r i n a more p r e c i s e way, b u t remark now t h a t t h i s theorem i s a p r o t o t y p e of t h e s t r u c t u r a l r e s u l t s we o b t a i n and are c o n j e c t u r i n g . I t i s t h u s n a t u r a l t o postpone work on t h e s t r u c t u r e r e s u l t i n g from embedding o r immersion e x c l u s i o n of a f i x e d graph u n t i l t h e s i m p l e r s t r u c t u r e a r i s i n g from minor e x c l u s i o n i s w e l l understood. Wagner's c o n j e c t u r e (D) s t a t e s t h a t any upper o r d e r i d e a l i n
( 9 , <m)
i s g e n e r a t e d by a f i n i t e l i s t of minimal graphs. These graphs are a l s o c a l l e d t h e minimal excluded minors f o r t h e complementary lower o r d e r i d e a l . From t h e a v a i l a b l e evidence it seems r e a s o n a b l e t o f u r t h e r c o n j e c t u r e t h a t :
(E) any lower o r d e r i d e a l o f
(
9 ,< m )
i s c h a r a c t e r i z e d by a f i n i t e
list o f "embedding s t r u c t u r e s " which e x a c t l y g e n e r a t e i t i n t h e way p l a n a r graphs a r e g e n e r a t e d by t h e 2-sphere. A s a t i s f a c t o r y s o l u t i o n of c o n j e c t u r e s (D) and ( E ) should i n c l u d e e f f e c t i v e
a l g o r i t h m s f o r t e s t i n g minor i n c l u s i o n and s t r u c t u r e embedding. That i s : (F) given a graph H € 9 ( o r r e s p e c t i v e l y , a n embedding s t r u c t u r e .Z) t h e r e s h o u l d b e c o n s t a n t s c , r depending on H (or X ) a l o n e , such t h a t f o r any G E 9 i t can b e determined i n
Ir
IV(G) elementary s t e p s whether o r n o t G embeds i n X I
.
H
is a minor o f
G
Although we can v e r i f y many s p e c i a l c a s e s o f t h e above "exact embedding s t r u c t u r e c o n j e c t u r e " (E), and t h u s have a v a r i e t y of examples and an a p p r e c i a t i o n o f t h e i r g e n e r a l f e a t u r e s , w e have n o t y e t been a b l e t o c o n s t r u c t o r d e f i n e t h e embedding s t r u c t u r e s a p p r o p r i a t e t o c h a r a c t e r i z e a r b i t r a r y p r o p e r lower ideals i n ( 9 , <m) W e imagine an "embedding structure" 7. t o consist of a
.
s p e c i f i c a t i o n of c e r t a i n s t r u c t u r a l p r o p e r t i e s and t h e embedding r u l e s under which a graph h a s t h e s e p r o p e r t i e s . S e v e r a l e s s e n t i a l f e a t u r e s are c l e a r : ( i ) t h e class o f graphs
be a lower i d e a l
9, which embed i n a given s t r u c t u r e
1
should
( i . e . t h e lower i d e a l g e n e r a t e d by t h e s t r u c t u r e ) ;
(ii) a s t r u c t u r e s h o u l d be s u f f i c i e n t l y homogeneous t h a t t h e r e are o n l y
f i n i t e l y many e s s e n t i a l l y d i f f e r e n t ways t o embed a given g r a p h , and (iii) a s t r u c t u r e s h o u l d n o t simply embed a l l of
9
;
some graph s h o u l d
n o t embed. These t h r e e c o n d i t i o n s would allow us t o s t u d y t h e p r o p e r lower i d e a l s i n a c o n s t r u c t i v e f a s h i o n . Regarding c o n d i t i o n (iii) w e note t h e c u r i o u s f a c t ( s e e page 52 of [ 1 7 ] ) t h a t a l l G E 9 embed t o p o l o g i c a l l y [as w i t h p l a n a r embedd i n g s ) i n t h e s t r u c t u r e formed by i d e n t i f y i n g t h r e e 2-dimensional d i s k s a l o n g a common arc of t h e i r b o u n d a r i e s . An important development i n t h i s s t u d y w a s t h e r e a l i z a t i o n t h a t many o f t h e a l g o r i t h m i c and t h e well-quasi-ordering consequences e x p e c t e d from a
349
Well-quasi ordering of graphs
( 3 ,< m )
s t r u c t u r a l c h a r a c t e r i z a t i o n of t h e p r o p e r lower i d e a l s i n
would
s t i l l f o l l o w p r o v i d e d enough embedding s t r u c t u r e s c o u l d b e c o n s t r u c t e d t o c o n t a i n ( p e r h a p s p r o p e r l y ) a l l t h e p r o p e r lower i d e a l s . Suppose t h a t s u c h a f a m i l y S = ( E : i E I } of embedding s t r u c t u r e s h a s been g i v e n ; and f o r e a c h i
,
i E I
that:
a r e shown t o be w e l l - q u a s i - o r d e r e d
by
<m
’Zi
the
(’)
(2) t h e r e i s an e f f e c t i v e a l g o r i t h m t o t e s t whether
G
1
€9
over a l l
‘i G
(3)
,
E 9
g i v i n g an embedding when one e x i s t s , and
f o r each whether
E 9
H
t h e r e i s an e f f e c t i v e a l g o r i t h m f o r t e s t i n g ‘i G over a l l G f 9
.
<
H
‘i
I t i s t h e n s t r a i g h t f o r w a r d t o v e r i f y Wagner’s c o n j e c t u r e (D), and c o n j e c t u r e (F) t h a t t h e r e a r e e f f e c t i v e a l g o r i t h m s t o t e s t ( a ) f o r any f i x e d H E 9 whether H <m G , o v e r a l l G € 9 , and ( b ) f o r any f i x e d lower i d e a l
9 5 9 whether
9=
G
€7 ,
over a l l
E
G
.
9
.
To o b t a i n (D) suppose 0 5 9 i s a n a n t i c h a i n and choose H E 0 Let Now 0 - { H I 5 9 by t h e d e f i n i t i o n o f an a n t i c h a i n , € 9 : H 1;, G )
.
{G
5 5 9
and
s
on
f o r some i E I b e c a u s e 9 i s a lower i d e a l . Then c o n d i t i o n (1) ‘i i m p l i e s t h a t (3 - {H} i s f i n i t e and it f o l l o w s t h a t (3 i s f i n i t e .
3
.
E 9 : H +m G I
= {G
i E I
A s before,
9
H E
To o b t a i n ( F ) c o n s i d e r f i r s t p a r ( a ) . L e t
e x i s t s with
and
. Then
95 9 =i
G C % we t e s t whether
9
G E
then
G E
,
H <m G
,
and H e 9 ‘i ‘i by d e f i n i t i o n o f
9
u s i n g c o n d i t i o n ( 2 ) on
5
.
If
G
E 9
‘i
then
m
,
G E
If
obviously.
G
9 ‘i
.
s
u s i n g c o n d i t i o n ( 3 ) on
and
p a r t (b). For any p r o p e r lower i d e a l
9
.
5 9
G
EY
f 5
.
Suppose
H
9
E
G E
. If
H
E 9
and
‘i This gives p a r t ( a ) .
3
t h e r e i s an G €
and t e s t whether
9
then
‘i a finite list
HI
G
,
H2
, ...,
When
G
E
9,
,
‘i
w e can t e s t whether
‘i
If
s
‘i
H 4
H <m G
given
i
E
9, L .
I
Consider then such t h a t using condition ( 2 ) .
condition (1) t e l l s u s there is
i Hh
of graphs i n
9
g e n e r a t i n g t h e upper
‘i order i d e a l of
, <m)
(g
complementary t o 3
.
But t h e n c o n d i t i o n ( 3 )
‘i
a l l o w s u s t o t e s t e f f e c t i v e l y whether d e c i d e whether
G E
9
.
Hi
<m G
for
i = 1, 2 ,
..., h
and t h u s
These r e d u c t i o n s make i t p o s s i b l e t o work w i t h w e l l - d e f i n e d s t r u c t u r e s w i t h o u t f o r m u l a t i n g a g e n e r a l d e f i n i t i o n o f embedding s t r u c t u r e . I n o u r i n v e s t i g a t i o n s w e add s t r u c t u r e t o a g r a p h G € 9 i n f o u r b a s i c ways:
(a)
by l a b e l l i n g t h e v e r t i c e s of
G
w i t h e l e m e n t s from som w e l l - q u a s i -
P.D. Seymour, N . Robertson
350
(el<),
o r d e r e d set
t o a g r a p h t o p o l o g i c a l l y enbedded o n a
(b)
by an isomorphism from surface S ,
(c)
by s p e c i f y i n g a s u b s e t X 5 V ( G ) t h e s u b g r a p h G - X , and
(d)
by decomposing (T, . F ) .
G
of bounded s i z e , and hence f i x i n g
i n t o subgraphs according t o a t r e e - s t r u c t u r e
G
For t h e most p a r t it i s q u i t e o b v i o u s how t h e s e s t r u c t u r e s behave under t h e t a k i n g o f minors so t h a t t h e c l a s s o f s t r u c t u r e s i s p r e s e r v e d .
tree-structure ( T , 9 ) on
A
E V(T)} o f subgraphs
9 = {G.
: i
(2) i f
i, j , k
rates
E V(T)
from
and
G c o n s i s t s o f a t r e e T and a f a m i l y G such t h a t ( 1 ) G = U G . ( i E V ( T ) ) and
of
l i e s on t h e
k
(i, j)-path i n
.
i E V(T)
in G Normally, t h e Gi for 3 t r i c t e d and a r e g i v e n s t r u c t u r e o f t h e i r own and t h e G.
G.
T
Gi
n
G
k sepaa r e further res-
for
G.
1 satisfy conditions relative t o t h i s structure.
{ i , j j E E(T)
then
The s t r u c t u r e theorem o f Wagner d i s c u s s e d e a r l i e r i s a r e l a t i v e o f K u r a t o w s k i ' s theorem and makes a g o o d i l l u s t r a t i o n o f t h e u s e o f g r a p h s t r u c t u r e . When (T, 9 )i s a t r e e - s t r u c t u r e on G w i t h F = {G, : i E V ( T ) } , d e n o t e by
t h e g r a p h formed from
G#,
of v e r t i c e s i n
# fl
that
Gi
G!
E V(T)
i, j
3
n
Gi
become a d j a c e n t f o r a l l
G ,
1
and copies o f
E
i
V(T)
K5
$,
# 17 G!
for 3 ensures t h a t
{m
G
K
5
i4
G,
B
$m G
K
when
G
.
This ensures
G
is planar.
n
Gj)
I
it f o l l o w s t h a t
, or
(2)
included i n
<
3
,
G I K3,3
i E V(T)} )
G
E 3
and
K5
bm G
,
such t h a t , f o r e a c h
#
embeds i n t h e p l a n e with e a c h Gi a face-boundary. The l a s t c o n d i t i o n
and so when
,
Wagner's
i s a t r e e - s t r u c t u r e of planar graphs
(T, {Gi :
V8
{ i , j ) E E;T) /V(Gi
5
then
and G
i n a strong sense. S p e c i f i c a l l y , if
e i t h e r (1)
Gi
K5
V8
, fim
V8
h a s a tree-structure
,
E E(T)
.
Clearly,
G
jl
{i,
i s a l w a y s ( a t l e a s t ) a complete g r a p h f o r a l l d i s t i n c t
theorem s t a t e s t h a t i f
then
by a d d i n g e d g e s so t h a t any n o n a d j a c e n t p a i r
G.
G
G
or
V8
is internally-4-connected G
and
is planar.
Here, i n t e r n a l l y - 4 - c o n n e c t e d means t h a t G i s a s i m p l e 3-connected g r a p h h a v i n g no e d g e - d i s j o i n t d e c o m p o s i t i o n s G = H U Ha i n t o s u b g r a p h s 1 H I , H2 w i t h /V(H1 n H = 3 and IE(H1) > 3 < IE(H2)I Thus G i s
2
)I
I
.
4-connected e x c e p t f o r p o s s i b l e t r i v a l e n t v e r t i c e s , and h a s no c i r c u i t s s m a l l e r than a t r i a n g l e .
Our s t r u c t u r a l c o n j e c t u r e , which we c a l l t h e " a p p r o x i m a t e embedding s t r u c t u r e c o n j e c t u r e " h a s a s t a t e m e n t s i m i l a r t o Wagner's theorem: (G)
Suppose H € 9 is g i v e n . Then a s u r f a c e S and a n i n t e g e r n e x i s t s u c h t h a t t h e f o l l o w i n g h o l d s t r u e . For any G € 3 w i t h H %m G t h e r e e x i s t s a tree-structure (T, {Gi : i E V(T)) ) such t h a t f o r each
i
E V(T)
there exist
(a)
Xi
5 V(Gi)
with
Well-quasi ordering of graphs
lXil
#
(G. 1
< 11
and ( b ) an embedding o f
- X.) n
G.
3
1
for
# -
G.
{ i , j } E E(T)
35 1 Xi
into
S
with each
i n c l u d e d i n a face-boundary.
We can e n s u r e t h a t H d o e s n o t embed on S , and so an i n t e r e s t i n g s p e c i a l case o c c u r s when H i s p l a n a r . The s u r f a c e d i s a p p e a r s and t h e c o n j e c t u r e r e d u c e s t o t h e f o l l o w i n g : If H i s a p l a n a r graph t h e r e e x i s t s a n i n t e g e r n such t h a t f o r any G €9 with H < G t h e r e e x i s t s a t r e e - s t r u c t u r e
m
(T,
{Gi
:
i E V(T)I )
a tree-structure
G
with
IV(Gi)
I
for all
i E V(T)
.
Thus
G
is just
of g r a p h s w i t h bounded s i z e .
We d e f i n e t h e t r e e - w i d t h of admits a t r e e - s t r u c t u r e (T, (G,
G : i
t o be t h e s m a l l e s t i n t e g e r n so t h a t with jV(Gi) < n + 1 for
I
E V(T)} )
.
a l l i E V(T) T h i s convention e n s u r e s t h a t f o r e s t s a r e t h e graphs w i t h treew i d t h < 1 , which seems t o be t h e n a t u r a l number t o c h o o s e . A t r e e - s t r u c t u r e i n which t h e t r e e i s a p a t h i s c a l l e d a p a t h - s t r u c t u r e . The p a t h - w i d t h of G i s defined s i m i l a r l y t o tree-width b u t using p a t h - s t r u c t u r e s i n p l a c e of t r e e structures.
5.
N e w theorems.
I n t h e p r e s e n t s t a t e o f t h i s program ( A p r i l , 1983) t h e work on embedding s t r u c t u r e s n o t i n v o l v i n g s u r f a c e s i s c o m p l e t e d . We p r o v e d i n t h e autumn of 1981 t h a t ( 1 ) ( s e e [ 9 1 ) i f a f i x e d f o r e s t is e x c l u d e d a s a minor then t h e r e m a i n i n g g r a p h s h a v e bounded p a t h - w i d t h , and ( 2 ) g r a p h s o f bounded path-width a r e w e l l - q u a s i - o r d e r e d . I n s p r i n g of 1982 w e gave ( 3 ) e f f e c t i v e a l g o r i t h m s f o r f i n d i n g a p a t h - s t r u c t u r e of bounded width and f o r t e s t i n g minor i n c l u s i o n o v e r c l a s s e s of g r a p h s with such s t r u c t u r e . T h i s showed t h a t it w a s r e a l i s t i c t o hope t o prove o u r b a s i c c o n j e c t u r e s . The c o r r e s p o n d i n g r e s u l t s f o r p l a n a r g r a p h s were by and l a r g e a l s o o b t a i n e d e a r l y i n 1982. Of c o u r s e , c h a r a c t e r i z a t i o n s o f p l a n a r i t y and p l a n a r embedding a l g o r i t h m s a r e well-known. W e showed ( 4 ) ( s e e [ l l ] ) t h a t e x c l u d i n g a p l a n a r g r a p h H a s a minor o f a p l a n a r g r a p h G implies t h a t G h a s treewidth bounded by a f u n c t i o n o f H , and i n t h e s p r i n g o f 1982 t h a t ( 5 ) t h e r e i s an e f f e c t i v e a l g o r i t h m f o r t e s t i n g embedding i n c l u s i o n (and h e n c e minor i n c l u s i o n ) w i t h i n any c l a s s of g r a p h s h a v i n g bounded t r e e - w i d t h . F i n a l l y , i n autumn o f 1982 w e gave (6) a n o n c o n s t r u c t i v e p r o o f t h a t a n e f f e c t i v e a l g o r i t h m e x i s t s f o r t e s t i n g g r a p h s G € 3 f o r a f i x e d bound on t h e i r t r e e - w i d t h s . These During autumn o f 1982 and w i n t e r o f algorithmic r e s u l t s a r e collected i n [ l o ] 1983 w e o b t a i n e d ( 7 ) well-behavedness r e s u l t s f o r p l a n a r g r a p h s under minor i n c l u s i o n , and ( 8 ) good a l g o r i t h m s f o r t e s t i n g embedding i n c l u s i o n H < G f o r
.
a f i x e d graph
H
, and
with
G
i n t h e class o f p l a n a r g r a p h s .
During t h e w i n t e r o f 1983 w e showed t h a t ( 9 ) (see [1211 sets of g r a p h s with bounded t r e e - w i d t h a r e w e l l - q u a s i - o r d e r e d . I n March o f 1983, t h e r e s u l t w a s o b t a i n e d t h a t ( 1 0 ) (see [ 1 3 ] ) i f H i s a f i x e d p l a n a r graph, t h e n any G € 3 with H < G h a s t r e e - w i d t h bounded by a f u n c t i o n o f H Thus w e had b o t h t h e
.
w e l l - q u a s i - o r d e r i n g r e s u l t t h a t no i n f i n i t e a n t i c h a i n o f f i n i t e g r a p h s c a n i n c l u d e a p l a n a r g r a p h , and t h e a l g o r i t h m i c r e s u l t t h a t any f i x e d p l a n a r g r a p h c o u l d b e found a s a minor i n a p o l y n o m i a l l y bounded number o f s t e p s . The main d i f f i c u l t y i n p r o v i n g t h e s t r u c t u r a l theorem ( 9 ) i n v o l v e d a lemma g i v i n g c o n d i t i o n s for t h e p r e s e n c e o f l a r g e g r i d s a s m i n o r s . A 0 - g r i d i s a simple graph with v e r t e x - s e t { C i , j) : i , j E 2:,i.., €I} } such t h a t ( i , j ) i s a d j a c e n t t o ( i ' , j ' ) i f and o n l y i f 1- 1 + Ij- j'I = 1 . W e were a b l e t o show t h a t ( 1 1 ) f o r a n y p o s i t i v e i n t e g e r 0 t h e r e e x i s t s an
{i:
P.D. Seymour, N . Robertson
352 integer
8' such t h a t i f
, G2
G1
a r e subgraphs o f
G E 9
c o n n e c t e d components a p i e c e , and a r e such t h a t each c o n n e c t e d component o f
meets each c o n n e c t e d component o f
G
2
then
G
8'
with a t l e a s t G
1
c o n t a i n s a & g r i d a s a minor.
Once t h i s theorem was e s t a b l i s h e d w e showed t h a t g r a p h s of s u f f i c i e n t l y l a r g e t r e e - w i d t h c o n t a i n subgraphs G 1, G2 each w i t h 8 ' connected components which p a i r w i s e i n t e r s e c t and t h u s produce a 8 - g r i d minor. Taking l a r g e enough any p l a n a r g r a p h H c a n be produced a s a minor, t h u s p r o v i n g ( 1 0 ) t h a t e x c l u d i n g H as a minor l e a v e s o n l y g r a p h s w i t h a f i x e d bound on t h e i r t r e e - w i d t h .
An i n t e r e s t i n g c o r o l l a r y t o t h e s e r e s u l t s was a g e n e r a l i z a t i o n of t h e Erdos-Posa theorem t h a t f o r any p o s i t i v e i n t e g e r k t h e r e i s a n i n t e g e r k ' such t h a t any graph w i t h o u t k d i s j o i n t l o o p s a s a minor (i.e. k disjoint polygons as s u b g r a p h s ) d i f f e r s by a t most k ' v e r t i c e s from a g r a p h w i t h o u t a s i n g l e l o o p a s a minor ( i . e . from a f o r e s t ) . W e can show, u s i n g t h e s t r u c t u r a l theorem (10) and p r o p e r t i e s of f a m i l i e s of s u b t r e e s of t r e e s , t h a t (12) f o r such t h a t any graph w i t h o u t any p o s i t i v e i n t e g e r k t h e r e is an i n t e g e r k ' k p a i r w i s e d i s j o i n t c o p i e s o f a f i x e d p l a n a r graph H a s a minor d i f f e r s by a t m o s t k' v e r t i c e s f r o m a graph w i t h o u t H as a minor. W e can e a s i l y embed 2 k ' t 1 c o p i e s of any n o n p l a n a r g r a p h on i t s s u r f a c e of minimum genus so t h a t t h e embedded g r a p h s meet p a i r w i s e o n l y a t ( f i n i t e l y many) i n t e r n a l p o i n t s o f t h e embedded e d g e s and no such i n t e r n a l p o i n t i s i n more t h a n t w o o f t h e c o p i e s . Then t h e union o f t h e embedded g r a p h s forms a g r a p h G on S , w i t h t h e common i n t e r n a l p o i n t s v e r t i c e s , which d o e s n o t embed t w o d i s j o i n t c o p i e s o f H ( b e c a u s e H i s n o n p l a n a r and S i s i t s genus s u r f a c e ) and r e q u i r e s removal v e r t i c e s t o e l i m i n a t e a l l embeddings o f H I t f o l l w s t h a t theorem of > k ' (12) f a i l s when H i s n o n p l a n a r .
.
6.
P o s s i b l e Consequences. W e a r e o p t i m i s t i c a b o u t s e t t l i n g t h e c o n j e c t u r e s a b o u t s t r u c t u r e (E), w e l l - q u a s i - o r d e r i n g (G) and a l g o r i t h m s (F) s t a t e d above. These would imply Wagner's c o n j e c t u r e ( D ) and V a z s o n y i ' s c o n j e c t u r e ( B ) . Another c o r o l l a r y would be t h e Lovasz. and Milgram [ 1 5 ] i n t e r t w i n i n g c o n j e c t u r e , a t l e a s t i n t h e form i n v o l v i n g minor i n c l u s i o n . T h i s c o n j e c t u r e s t a t e s t h a t :
(H) If
G1
and
many minimal
G2
a r e f i x e d graphs then t h e r e a r e o n l y f i n i t e l y
G E
3
which i n c l u d e b o t h
GI
and
G2
*
Another c o n j e c t u r e , o r t r i p l e o f c o n j e c t u r e s , which f o l l o w s from Wagner's c o n j e c t u r e r e g a r d s g r a p h c o n n e c t i v i t y . W e u s e t h e term vertex-kc o n n e c t e d t o mean k-connected i n t h e u s u a l s e n s e , c y c l i c - k - c o n n e c t e d t o mean t h e m a t r o i d d u a l t o k-connected (see [ E l ) and d u a l l y - k - c o n n e c t e d t o mean b o t h v e r t e x - k - c o n n e c t e d and c y c l i c - k - c o n n e c t e d (see [14], c h a p t e r 10). D e f i n e c o n t r a c t i o n i n c l u s i o n t o mean H < G when H i s isomorphic t o a graph o b t a i n e d from G by c o n t r a c t i n g e d g e s and a l l o w i n g e d g e - d e l e t i o n s o n l y i f a p a r a l l e l edge e x i s t s . Then:
(I) t h e r e a r e o n l y f i n i t e l y many minimal (a) (b) (c) Thus maximal c h a i n s
vertex-k-connected g r a p h s under c o n t r a c t i o n i n c l u s i o n , c y c l i c - k - c o n n e c t e d g r a p h s under embedding i n c l u s i o n , and d u a l l y - k - c o n n e c t e d g r a p h s under minor i n c l u s i o n . H < H < < Hh of "k-connected" g r a p h s under t h e O Z # above t h r e e v e r s i o n s of "k-connected" and i n c l u s i o n r e l a t i o n s < , can be s a i d t o be f i n i t e l y b a s e d ( i . e ; have o n l y f i n i t e l y many p o s s i b l e i n i t i a l t e r m s HO), and i n t h i s s e n s a l l "k-connected" g r a p h s c a n be c o n s t r u c t e d from t h e
...
f i n i t e set of minimal "k-connected" g r a p h s .
Well-quasi ordering of graphs The problem of showing ( 9
, <m)
353
i s well-behaved,
which i s t h e v e r s i o n
of Wagner's c o n j e c t u r e t h a t o u r methods s h o u l d s e t t l e , i n v o l v e s d e t e r m i n i n g t h e e x c l u d e d minor s t r u c t u r e f o r ( G J ( Q ) , Gm) , where (Q,<) is an a r b i t r a r y w e l l - q u a s i - o r d e r e d s e t . Our hope i s t h a t overcoming t h e r e m a i n i n g o b s t a c l e s i n t h e program w i l l l e a d toward a d e e p e r s t r u c t u r e t h e o r y t h a t w i l l s e t t l e t h e open immersion q u e s t i o n s and b e t t e r e x p l a i n t h e f a i l u r e o f w e l l - q u a s i - o r d e r i n g on embeddings. W e c o n j e c t u r e t h a t :
(J)
t h e r e a r e e f f e c t i v e a l g o r i t h m s f o r t e s t i n g embedding and immersion inclusion i n y ( Q ) , where ( Q , -4 is any w e l l - q u a s i - o r d e r e d set which i t s e l f c a n b e r e c o g n i z e d by an e f f e c t i v e a l g o r i t h m .
This is an extremely s t r o n g c o n j e c t u r e , e s p e c i a l l y r e l a t i v e t o t h e evidence a p a r t from t h e i n v e s t i g a t i o n w e r e p o r t on h e r e . I t i s d e b a t a b l e even whether or h a s an e f f e c t i v e a l g o n o t o r d i n a r y embedding o r immersion i n c l u s i o n of K5. r i t h m , f o r example. I n k e e p i n g w i t h t h e c o n j e c t u r e s i n ( J ) we s u p p o r t NashW i l l i a m ' s immersion c o n j e c t u r e (C) and a l s o v e r s i o n s of t h e Erdds-Posa-type theorem 4 , where embedding or immersions r e p l a c e minor i n c l u s i o n .
[l]
ERDOS, P. and POSA, L . , On t h e i n d e p e n d e n t c i r c u i t s c o n t a i n e d i n a graph, Canad. J. Math. 17 ( 1 9 6 5 ) , 347-352.
[2]
KRUSKAL, J. B . , W e l l - q u a s i - o r d e r i n g , t h e tree theorem, and V a z s o n y i ' s c o n j e c t u r e , T r a n s . A m e r . Math. SOC. 95 (19601, 210-225.
131
A f r e q u e n t l y discovered KRUSKAL, J . B . , The t h e o r y of w e l l - q u a s i - o r d e r i n g : c o n c e p t , J . C o m b i n a t o r i a l Theory, S e r . A 13 (19721, 297-305.
141
LAVER, R . ,
On F r a i s s e ' s o r d e r t y p e c o n j e c t u r e , Ann. of Math. 93 ( 1 9 7 1 ) ,
89-111. 151
MADER, W . , Wohlquasigeordnete K l a s s e n e n d l i c h e Graphen, J . C o m b i n a t o r i a l Theory, S e r . B 12 (19721, 105-122.
[6]
NASH-WILLIAMS, C . S t . J . A . , On w e l l - q u a s i - o r d e r i n g Camb. P h i l . S O C . , 59 ( 1 9 6 3 ) , 833-835.
f i n i t e trees, P r o c .
[71
NASH-WILLIAMS, C . S t . J . A . , On w e l l - q u a s i - o r d e r i n g Camb. P h i l . SOC., 61 ( 1 9 6 5 ) , 697-720.
i n f i n i t e trees, P r o c .
[8]
ROBERTSON, N., t o appear.
[9]
ROBERTSON, N . and SEYMOUR, P . D . , Graph m i n o r s . I . E x c l u d i n g a f o r e s t , J. C o m b i n a t o r i a l Theory, S e r . B . 35 ( 1 9 8 3 ) , 39-61.
[lo]
ROBERTSON, N . and SEYMOUR, P . D . , tree-width, submitted.
[I11
ROBERTSON, N . and SEYMOUR, P . D . , Graph minors. J. C o m b i n a t o r i a l Theory, S e r . B , t o a p p e a r .
Minimal c y c l i c - 4 - c o n n e c t e d
g r a p h s , T r a n s . A m e r . Math. S O C . ,
Graph m i n o r s . 11. A l g o r i t h m i c a s p e c t s o f
111. P l a n a r t r e e - w i d t h ,
P.D. Seymour, A? Robertson
354 [I21
ROBERTSON, N . and SEYMOUR, P . D . , quasi-ordering, submitted.
Graph minors.
I V . Tree-width and w e l l -
[13]
ROBERTSON, N . and SEYMOUR, P.D., graph, submitted.
Graph minors. V . E x c l u d i n g a p l a n a r
1141
TUTTE, W.T., C o n n e c t i v i t y i n Graphs, Mathematical E x p o s i t i o n s , No 15 (Univ. of T o r o n t o P r e s s , T o r o n t o , O n t a r i o ; Oxford Univ. P r e s s , London, 1966).
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UNGAR, P . , A d i s s e c t i o n problem and a n i n t e r t w i n i n g problem, i n : Bondy, J . A . and Murty, U.S.R., ( e d s . ) , Graph Theory and R e l a t e d T o p i c s . (Academic P r e s s , N e w York, 1 9 7 9 ) , 370-371.
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[171
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B e w e i s e i n e r Abschwachung d e r Hadwiger-Vermutunq, 153 ( 1 9 6 4 ) , 139-141.
Math. Ann.
.
- PART VII ORDER AND COMBINATORICS -
PARTIEVII ORDRE ET COMBINATOIRE
Ivan RIVAL Linear extensions of finite ordered sets. . . . . . . . . . . . . . . . . . . . . .
p . 355
Michel HABIB Comparability invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p . 371
Andrk BOUCHET Codages e t dimensions de relations binaires. . . . . . . . . . . . . . . . . . . p . 387 Jerrold R. GRIGGS The sperner property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p . 397
GBrard VIENNOT Chain and antichain families grids and young tableaux. . . . . . . . . . p . 409
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Annals of Discrete Mathematics 23 (1984) 355-370 0 Elsevier Science PublishersB.V. (North-Holland)
355
LINEAR EXTENSIONS OF F I N I T E ORDERED SETS
Ivan RIVAL
Department of Mathematics and Statistics The University o f Calgary Calgary, Canada T 2 N 1 N 4 Dedicated to E.Corominas R e s urn&
La theorie des extensions lineaires des ensembles ordonnes est fkconde. Dans le cas infini les rdsultats principaux sont souvent ensemblistes tandis que les resultats et les methodes dans le cas fini s o n t typiquement algorithmiques. Les rdsultats les plus connus d'extensions lindaires traitent du cas infini - les m@mes rgsultats dans le cas fini sont souvent triviaux. Malgr6 tout, il y a eu recement des r6sultats concernant surtout les ensembles ordonngs finis. En informatique la raison d e l'importance d e s extensions lindaires d'ensembles ordonnes finis est Claire.
11 y a correspondance
evidente entre l'id6e d'extension lineaire et l'idhe de tri topologique. Dans cet article nous allons considerer trois themes qu'on rencontre dans l'6tude des extensions lin6aires des ensembles ordonnks finis :
(i)
Q u e l l e e s t l a t a i l l e de 1 'ensemble des extensions l i n e a i r e s d'un ensemble ordonnP f i n i ?
(ii)
Duel e s t l e p l u s p e t i t nombre d'extensions l i n 6 a i r e s d'un ensemble ordonnt? f i n i dont 1 ' i n t e r s e c t i o n redonne c e t ensemble ordonne ?
(iii) Comment construit-on une extension l i n e d i r e optimale
The work reported here was supported in part by N . S . E . R . C .
A4077.
?
I. Rival
356
CONTENTS How many l i n e a r e x t e n s i o n s a r e t h e r e ? The o r d e r dimension Optimal l i n e a r e x t e n s i o n s
INTRODUCTION The t h e o r y of l i n e a r e x t e n s i o n s of o r d e r e d sets i s r i c h and v a r i e d : i n t h e i n f i n i t e c a s e t h e main r e s u l t s and methods a r e t y p i c a l l y s e t - t h e o r e t i c and modelt h e o r e t i c i n c h a r a c t e r ; i n t h e f i n i t e c a s e t h e r e s u l t s and methods a r e c h a r a c t e r i s t i c a l l y i n s p i r e d by a l g o r i t h m i c c o n s i d e r a t i o n s .
A linear extension L of an o r d e r e d s e t P i s a t o t a l l y o r d e r e d set on t h e same s e t a s P such t h a t z < y i n L whenever z < y i n P. For i n s t a n c e , t h e t o t a l l y o r d e r e d s e t of n a t u r a l numbers 1 < 2 < 3 .; 4 < 5 is a linear e x t e n s i o n of t h e o r d e r e d s e t of n a t u r a l numbers w i t h r e s p e c t t o d i v i s i b i l i t y 1.; 2 < 4 1< 3 < 6 e t c . The p o i n t about a l i n e a r e x t e n s i o n i s t h a t i t "extends" t h e o r d e r r e l a t i o n t o a t o t a l o r d e r r e l a t i o n , w i t h o u t " e n l a r g i n g " t h e u n d e r l y i n g s e t . L e t L ( P ) s t a n d f o r t h e s e t of a l l l i n e a r e x t e n s i o n s of t h e ordered s e t P .
...
...,
...,
The b e s t known r e s u l t s about l i n e a r e x t e n s i o n s of o r d e r e d s e t s a r e encountered i n the i n f i n i t e case - the so-called "general" case. I n the f i n i t e case t h e s e same r e s u l t s a r e o f t e n a b a r e t r i v i a l i t y . Even t h e s t a r t i n g p o i n t of t h e s u b j e c t b e a r s o u t t h a t s e n t i m e n t : L ( P ) i s nonempty f o r every o r d e r e d s e t P (E. SZPILRAJN [ 1 9 3 0 ] ) . A more r e c e n t q u e s t i o n , whether o r n o t an o r d e r e d s e t h a s a complete l i n e a r e x t e n s i o n , i s w i t h o u t any t h e o r e t i c a l i n t e r e s t f o r f i n i t e o r d e r e d s e t s ; t h e " g e n e r a l " c a s e , on t h e o t h e r hand, i s q u i t e engaging (M. POUZET and I . RIVAL [ 1 9 8 l ] ) . There i s much t h e same dichotomy f o r t h e problem of c h a r a c t e r i z i n g t h e o r d e r e d sets f o r which t h e r e i s , up t o isomorphism, j u s t one l i n e a r e x t e n s i o n (R. ASSOUS [ 1 9 8 2 ] ) . Many of t h e s e and r e l a t e d q u e s t i o n s a r e t a k e n up i n t h e survey paper of R. BONNET and M. POUZET [1982]. There i s a l s o t h e r e c e n t book of J . ROSENSTEIN [1982] which s y s t e m a t i c a l l y t r e a t s of t o t a l l y o r d e r e d sets - almost always i n t h e " g e n e r a l " c a s e . A l l of t h i s evLience n o t w i t h s t a n d i n g , t h e r e i s a s u b s t a n t i a l body of r e s u l t s r e c e n t l y emerging j u s t about l i n e a r e x t e n s i o n s of f i n i t e o r d e r e d s e t s . A t a f i r s t g l a n c e t h i s i s an u n l i k e l y c l a i m e s p e c i a l l y as t h e r e a p p e a r t o be s o few approaches t o t h e very c o n s t r u c t i o n of a l i n e a r e x t e n s i o n . T y p i c a l l y , a l i n e a r < z 1 of a n n-element o r d e r e d s e t P i s c o n s t r u c t e d e x t e n s i o n Izl < r 2 <
...
a c c o r d i n g t o an a l g o r i t h m which s t a r t s w i t h a minimal element zlof P , p r o c e e d s
z2< ... < zn' z2,..., I of t h e elements of P d e t e r m i n e s a l i n e a r e x t e n s i o n of P, i f each xi+, i s chosen from one element a t a t i m e , t o d e l i n e a t e t h e t o t a l o r d e r i n g z1 <
ending w i t h a maximal element
z of P.
A p e r m u t a t i o n zl,
among t h e minimal elements of t h e "up s e t "
i
So x1 E min(P) = min(Ul) and, f o r
i
=
l,Z,...,n-l,
C min(Ui+l).
Figure 1
357
Linear extensions o f finite ordered sets
N
i l l u s t r a t e s a four-element o r d e r e d s e t t o g e t h e r w i t h a l l ( f i v e ) of i t s linear extensions. I n t h e o r e t i c a l computer s c i e n c e c i r c l e s t h e s e i d e a s have come t o be known as " t o p o l o g i c a l s o r t i n g " ( c f . A.B. KAHN [ 1 9 6 2 ] , D . E . KNUTH and J . L . SZWARCFTTER [19741).
a
One a l g o r i t h m f o r c o n s t r u c t i n g a l i n e a r e x t e n s i o n of a n o r d e r e d s e t P can b e d e s c r i b e d u s i n g t h e depth f u n c t i o n 6 of P :
6(x)
=
b
P
max{lCI/C chain i n P and z = i n f C}.
The d e p t h - f i r s t l i n e a r e x t e n s i o n chooses, a f t e r e a c h s t e p i, an element of " g r e a t e s t depth" i n Ui+l, t h a t i s , z E min(Ui+l)
i+ 1
Ll
L2
such t h a t
L3
L4
L5
Figure 1 =max{&(x)12 ~ m i n ( U ~ + ~ ) l .
For i n s t a n c e , each of t h e l i n e a r e x t e n s i o n s L , , L 2 , L 3 , L , i s d e p t h - f i r s t .
Only
L s i s n o t s i n c e , a f t e r x1 = b 6 min(U1) = min P , min(Uz) = { a , d } and 6 ( a ) = 2 ,
6(d)
=
1.
Another a l g o r i t h m c o n s t r u c t s a greedy l i n e a r e x t e n s i o n : a f t e r each s t e p i, ziflC min(Ui+l) which, i n P, i s g r e a t e r
i t i s p r e d i s p o s e d t o s e l e c t i n g an element
E min(Ui+l)Ix xi+l E min(Ui+l) i f {x C min(Ui+l) l z > 2.1 than z..
L1, L2, L
x1
=
b
That i s , xi+l C
5
i s greedy.
-+ x2
=
f:free lattices
L
3
13:
> 2.) i f t h e r e i s one, o r , any =
0.
Each of t h e l i n e a r e x t e n s i o n s
i s n o t greedy s i n c e min(U ) = { a , d } and x1 = b 2
i
d , but
a.
g:partition lattices
h h t t i c e s of preorders
Of c o u r s e , t h e s e a r e n o t t h e only algorithms f o r t h e construction of a l i n e a r e x t e n s i o n of a f i n i t e o r d e r e d s e t . They a r e r a t h e r common though. Imagine t h e p r o p o s a l t o w r i t e a monograph on o r d e r e d s e t s and suppose t h a t t h e c h a p t e r themes and t i t l e s a r e a l r e a d y s e l e c t e d . I n which ( t o t a l ) o r d e r s h o u l d t h e chapt e r s b e a r r a n g e d ? Some c o n s t r a i n t s o r i g i n a t e from t h e need t o p r e c e d e c e r t a i n c h a p t e r s by o t h e r s which contain the prerequisite s k i l l s . These precedence c o n s t r a i n t s cons t i t u t e an o r d e r on t h e u n d e r l y i n g s e t of c h a p t e r s . An i n t e r e s t i n g example i s t h e i n t r o d u c t o r y t e x t on o r d e r e d s e t s by M. BARBUT and B. MONJARDET [ 1 9 7 0 ] . I t c o n s i s t s of e i g h t c h a p t e r s which ( f o l l o w i n g the authors' preface) a r e linked
by t h e precedence c o n s t r a i n t s i n d i c a t e d i n t h e F i g u r e 2 . There may w e l l be d i f f e r e n t b e n e f i t s from d i f f e r e n t l i n e a r e x t e n s i o n s . I f t h e r e a d e r i s meant t o a c q u i r e a working knowledge of a v a r i e t y of t o p i c s , a t t h e expense of e a r l y s p e c i a l i z a t i o n , then a d e p t h - f i r s t l i n e a r e x t e n s i o n such as L 1 may be a p p r o p r i a t e . I f , i n s t e a d , i t i s p r e f e r a b l e t o r e a c h some s p e c i a l i z e d t o p i c s e a r l y , a t t h e expense of d i v e r s i t y , t h e n a greedy l i n e a r e x t e n s i o n L , may b e b e t t e r . And of course, there a r e other l i n e a r extensions too, such a s L 3 . c r i t e r i a , too. text?)
There a r e o t h e r
(What do you suppose was t h e a c t u a l t o t a l o r d e r i n g used i n t h a t
The aim o f t h i s paper i s t o s u r v e y b r i e f l y t h e t h r e e p r i n c i p a l themes encountered i n t h e s t u d y of l i n e a r e x t e n s i o n s of f i n i t e o r d e r e d s e t s . (i) (ii)
BOW
IL(PI I ?
big i s
What i s the l e a s t number k of linear extensions k L1, L,, Lk C L ( P I such t h a t fl Li = P?
...,
(iii)
i= 1
How i s an optimal member o f L(PI constructed?
HCW MANY LINEAR
ARE THERE?
EXTENSIONS
Counting i s h a r d ! And c o u n t i n g t h e l i n e a r e x t e n s i o n s of a n o r d e r e d s e t i s no e x c e p t i o n . T h i s c o u n t i n g problem i s t r a c t a b l e o n l y f o r some s p e c i a l and v e r y simple c l a s s e s o f o r d e r e d s e t s . An n-element a n t i c h a i n h a s n ! l i n e a r
L e t L ( P ) = Il(P)I f o r an o r d e r e d s e t P. e x t e n s i o n s . I n symbols,
n
+
L(
L) = n !
,
i=1 The c a r d i n a l sum of an n-element
chain
2 has
n+m l i n e a r extensions.
The d i r e c t p r o d u c t of t h e c h a i n s L(2
*
x
n) IV
2 and m, h a s
=
n+l
l i n e a r e x t e n s i o n s . The n-element fence ("zig-zag") F = { a l c a,, a 2 > a 3 , a 3 < a4,. 1 i s s a i d t o s a t i s f y
..
i=O and from t h i s , t h e 2n-element c?om C2n = { a l < b,, b , > a 2 , a , < b ,
pi:]
,...,
L ( F * i ) L(Fn'n-1-2i )
an < bn, bn
>
a 11 s a t i s f i e s
There i s l i t t l e hope f o r a u n i f y i n g formula e x p r e s s i n g L ( P ) i n terms of some s t r u c t u r a l i n v a r i a n t s o f an o r d e r e d s e t P. N e v e r t h e l e s s , in r e c e n t y e a r s , t h e r e have appeared s e v e r a l a r t i c l e s which d a r e t o c o n f r o n t t h i s c o u n t i n g problem fairly directly.
359
Linear extensions o f finite ordered sets
Let P he an ordered set. For elements 2 , y of P such that y $ z let P ( z < y ) stand for the transitive closure of P with the additional comparability 3: < y , that is, u < u in P ( z < 9) if u < u in P or if u < z in P and y < v in P. How likely is it that a "typical" extension L of P contains z < y? One way to find out is by a "simple majority" criterion. Write zBy if
L(P(z < Y)) > L(P(y < z)). Observe that L(P(x < y ) ) = L ( P ) if 3: < y in P and in this case just put L ( P ( y < 2 ) ) = 0. For instance, suppose the elements of P represent candidates for some elective office and the order u < U in P stands for v is unanimously preferred to u. Now suppose each elector ranks all of the candidates in a total order. If there are L ( P ) electors each selecting a different linear extension then the binary relation ii summarizes the outcome according to simple majority rule. Notice that R need not impose a total ordering on P: if P is an antichain then L ( P ( z < y ) ) = L ( P ( y < 2)) for each z # y s o D leaves P totally unordered. What is surprising though is that R need not even he a transitive order on P. A n example to illustrate this (see Figure 3) was first') put forward by FISHBURN [ 1 9 7 4 ] . This 31-element ordered set P has a three-element "6-cycle": L ( P ) = 6270; L ( P ( z iy ) ) = 3218 so zBy; L ( P ( y < z ) ) = 3192 so yBz: L ( P ( z < x)) = 3150 so Z E X . (Recently both M. AIGNER [1982] and P.M. WINKLER [1982] have remarked that 6 is a transitive ordering of P if and only if li decomposes P into a linear sum of antichains.) P.C.
A more encouraging direction is proposed in the series of articles R.L. GRAHAM, A . C . YAO and F.F. YAO [1980], L.A. SHEPP [1980], [1982], P.M. WINKLER [1982a], [1982b]. At this writing the most important result is this so-called "transitivity inequality": f o r elements a, b, c of G f i n i t e ordered set P
X
Figure
L P ( G < b ) ) . L ( P ( a < c)) 5 L ( P ( a < b , a < c)).
L(P)
(L.A. SHEPP [1982]). Why is this "transitivity inequality" interesting? One motivation is an algorithmic one. It is concerned with the problem of sorting numbers a , b , c,... by binary comparisons " a : b" to construct successively "finer" orders P on { a , b , c, ) until a total order is finally established. It is said that a quantity fundamental in deciding the expected efficiency of such algorithms is
....
the "probability" that the result of a : b is a c b , assuming that all linear extensions of P are equally likely. (In this model P is thought to be some intermediate state of order (disorder) of { a , b , c, . . . I and a , b are not yet + as in Figure 4 , then sorted.) For instance, if P
22 2
1) The well known "voting paradox" example of CONDORCET [1785] will not do. Suppose there are three candidates x, y, z and three voters ranking the z < x. Evidently y is "preferred" candidates by z < y < z, z < z < y and y to x, z is "preferred" to y , but x is "preferred" to z . Nevertheless, if a l l of the linear extensions of a three-element antichain are considered then, by symmetry, B still leaves all pairs noncomparable.
360
I. Rival
'1 1:
Q
P
while
Pr(d < b(P)
=
0.
Moreover, "conditional probabilities" can also be considered. For instance
C
Pr(a c blP(a < d ) )
9(P) Figure 4
=
L(P(a < b , a < d)) L(P(a < d ) )
=
2 5
and
Pr(a
<
d / P ( b < c))
=
7. 4
Probably a more profound motivation of this "transitivity inequality" is concerned with the very character of transitivity among all of the binary relations on a set { a , b , c , . . . ) . We can express the "transitivity property'' in this way: for elements a; b , c
a < c and c < b impZy a < b. In particular, every linear extension of an ordered set P = {a, b , c , which a < c and c < b , must include the inequality a < b , that is,
. . . I in
P r ( a < b l P ( a < c , c < b ) ) = 1. And, unless a < c and c < b are both satisfied in P itself, we will, in general, ju s t have
o
5 P r ( a < b l ~ 5)
1.
However, what if we have "partial" information, say a < c? This reasoning in part led I. RIVAL and B. SANDS [1982] to conjecture the transitivity inequality. The proof of the transitivity inequality is itself interesting (L.A. SHEPP [1982]). There are three highlights and the last is ingenious! The first step is an item of background, the "down set inequality" for distributive lattices ( D . J . KLEITMAN [1966], cf. C.M. FORTUIN, P.W. KASTELEYN, and J . GINIBRE [19711). For down s e t s A , B of a f i n i t e distributive Zattice D IAl
- 1BI
5
\A fl BI
*
(DI
In Figure 5 , the shaded elements enumerate an example of the corresponding subsets
A
An 0
0 Figure 5
D
36 1
Linear extensions o f finite ordered sets The second s t e p i s on t h e s u r f a c e a m a t t e r of formalism
Pr(a < b l ~ =) l i m
I,PcQ < b , 1 If1
N-
3 is
t h e d i s t r i b u t i v e l a t t i c e of a l l o r d e r - p r e s e r v i n g maps of t h e o r d e r e d s e t P i n t o the chain F = { O < 1 < 2 < < N - 1 1 o r d e r e d (componentwise) by f 5 g i f f(z) C g(x) f o r each x C P. The p o i n t about t h i s a s y m p t o t i c e s t i m a t e i s t h a t t h e t r a n s i t i v i t y i n e q u a l i t y can now b e c a s t i n t h e language of d i s t r i b u t i v e l a t t i c e s . I t i s enough t o p r o v e , f o r any i n t e g e r N ,
...
w
I t seems we a r e done by merely o b s e r v i n g t h a t D = that
f
is a d i s t r i b u t i v e l a t t i c e ,
i y ( - b , a < b L A " , w
where
A = $(a
< b,
N
and
and a p p l y i n g t h e down s e t i n e q u a l i t y .
There i s a h i t c h .
Neither
$ ( a .= b ) N
nor
$ ( a c c) need be down s e t s i n
$.
w
This i s i l l u s t r a t e d i n Figure 6, using three-element a n t i c h a i n .
=
{0 < 1) and P = {a, b , e} a
Figure 6 The f i n a l s t e p i n t h e proof of t h e t r a n s i t i v i t y i n e q u a l i t y i s t h e n o v e l i d e a t o change t h e o r d e r on t h e u n d e r l y i n g s e t of
and
s o t h a t both
I. Rival
362
a r e down s e t s and y e t distributive!
$ remains
I n f a c t , t h i s can
be accomplished as f o l l o w s .
f, g E
(1,0,0)
$, d e f i n e f a
g
For
i f , (as
integers with the usual ordering)
f(a) 5 g(a) and, f o r
z# a,
f ( a ) - f(z)5 g ( a )
-
lDIC
g(d.
'L
(componentwise order)
An example i s i l l u s t r a t e d i n
(O,I,?)
datbtc ' ( aorder)
'L
F i g u r e 7.
Figure 7
And F i g u r e 6 i s then transformed i n t o F i g u r e 8.
2{o
,-u
2b
2b
,-L
,-L
da,b,cl ,-L
Figure 8
The down set i n e q u a l i t y i s t h e f o r e r u n n e r of s e v e r a l g e n e r a l i z a t i o n s commonly denoted by t h e acronym "FKG i n e q u a l i t y " ( c f . C.M. FORTUIN, P.W. KASTELEYN, and J . G I N I B R E [1971], P.D. SEYMOUR and D . J . A . WELSH [1975], D . E . DAYKIN [19791, R. AHLSWEDE and D . E . DAYKIN [ 1 9 7 9 ] , D.B. WEST [ 1 9 8 2 ] . The most i m p o r t a n t o u t s t a n d i n g q u e s t i o n i s whether t h e t r a n s i t i v i t y i n e q u a l i t y i s a c t u a l l y a s t r i c t i n e q u a l i t y i n a l l b u t t h e t r i v i a l c a s e s , t h a t i s , i f a, b , c are pairwise %on-
comparable i n a f i n i t e ordered s e t P then i s L(P(a < b ) )
- L(P(a
< c)) < L ( P ( a < b , a < c))
- L(P)?
THE ORDER DIMENSION Suppose a and b a r e noncomparable elements of an o r d e r e d s e t P. Then t h e r e a r e l i n e a r e x t e n s i o n s L1, L, of P such t h a t a < b ( L 1 ) and b < a ( L p ) , s o , i n
L1
n L,, a
and b a r e noncomparable and y e t
z < y ( L 1 f' L2) i f z < y ( P ) .
How many
of a l l of t h e l i n e a r e x t e n s i o n s of P are "needed" t o determine P? The (order) dimension dim P of P i s t h e l e a s t number Ll,L2,...,Lk of members of L(P) s u c h t h a t
k
n
i=l that is,
3: iy ( P )
i f and only i f
3:
L.
'
=
P,
< y ( L i ) f o r each 5 = 1, 2 ,
..., k .
How a r e
Linear extensions o f finite ordered sets
363
such "minimal" s e t s of l i n e a r e x t e n s i o n s c o n s t r u c t e d ? These are q u i t e o l d q u e s t i o n s and t h e r e i s a l r e a d y a n e x t e n s i v e l i t e r a t u r e and many deep r e s u l t s about them. Perhaps t h e d e e p e s t r e s u l t i n t h e dimension t h e o r y of o r d e r e d s e t s i s t h e c o n s t r u c t i o n of a l l of t h e 3 - i r r e d u c i b l e o r d e r e d s e t s . An o r d e r e d s e t P i s n-irreducible i f dim P = n and i f , f o r each
5
E P,
dim(P -
{z})< n
.
The p o i n t a b o u t n - i r r e d u c i b l e o r d e r e d s e t s i s t h a t a n o r d e r e d s e t P h a s dimension a t l e a s t n i f and o n l y i f i t c o n t a i n s a n n - i r r e d u c i b l e s u b s e t . For i n s t a n c e , dim P 3 2 i f and o n l y i f P c o n t a i n s a two-element a n t i c h a i n , (which i s t h e only 2 - i r r e d u c i b l e o r d e r e d s e t ) . The d e s c r i p t i o n of a l l of t h e 3 - i r r e d u c i b l e o r d e r e d s e t s h a s t u r n e d o u t t o be a much more d i f f i c u l t t a s k (D. KELLY [ 1 9 7 7 ] , W.T. TROTTER, J r . and J.I. MOORE [19761). The 3 - i r r e d u c i b l e o r d e r e d s e t s w i t h a t most seven elements a r e , up t o d u a l i t y , i l l u s t r a t e d i n F i g u r e 9. But t h e r e a r e 3 - i r r e d u c i b l e o r d e r e d s e t s of a l l s i z e s , b i g and t a l l .
??$dk&k& Figure 9
There a r e two p a r a l l e l and q u i t e d i f f e r e n t approaches t o t h e c o n s t r u c t i o n of t h e complete l i s t . ( " I t i s t h e r e f o r e q u i t e u n l i k e l y t h a t any 3 - i r r e d u c i b l e h a s been missed!") The one approach of D. KELLY [1977] i s based on a c h a r a c t e r i s a t i o n of p l a n a r l a t t i c e s . The l i n k t o l a t t i c e t h e o r y i s f o r g e d by t h e normal completion (completion by c u t s ) L(P) of an o r d e r e d s e t P. The p o i n t i s t h a t dim P = dim k ( P ) and dim L ( P ) 5 2 i f azd only i f L ( P ) i s a p l a n a r l a t t i c e . Now i f P i s 3 - i r r e d u c i b l e t h e n L(P7 i s a n o n p l a n a r l a t t i c e - a n d s o L ( P ) must c o n t a i n one of t h e "minimal" nonpl%ar l a t t i c e s ( c f . D. KELLY and I . %IVAL [ 1 9 7 5 ] ) . Moreover, t h e u n d e r l y i n g s u b s e t of j o i n i r r e d u c i b l e o r meet i r r e d u c i b l e elements of a "minimal" n o n p l a n a r l a t t i c e t u r n s out i t s e l f t o be 3-irreducible. The o t h e r approach of W.T. TROTTER, J r . and J . I . MOORE [1976] i s based on a c h a r a c t e r i s a t i o n of c o m p a r a b i l i t y g r a p h s . For a n o r d e r e d s e t P, dim P 5 2 i f and only i f t h e n o n c o m p a r a b i l i t y graph of P i s i t s e l f a c o m p a r a b i l i t y graph. I f P i s 3 - i r r e d u c i b l e t h e n i t s n o n c o m p a r a b i l i t y graph i s not a c o m p a r a b i l i t y g r a p h and s o i t must c o n t a i n one of a l i s t of "minimal" f o r b i d d e n subgraphs ( c f . T . GALLAI [ 1 9 6 7 ] ) . Moreover, i f t h e complement of a "minimal" f o r b i d d e n graph i s i t s e l f a c o m p a r a b i l i t y g r a p h t h e n , up t o d u a l i t y , i t h a s a unique o r i e n t a t i o n and t h i s o r i e n t a t i o n y i e l d s a 3-irreducible ordered s e t . The c o m p a r a b i l i t y graph h a s p r o v i d e d t h e s e t t i n g f o r a n o t h e r q u i t e s t r i k i n g
I. Rival
364
r e s u l t about dimension. I t i s t h i s . I f o r d e r e d s e t s P and Q have graph isomorphic c o m p a r a b i l i t y graphs t h e n dim P = dim Q . T h i s r e s u l t i s e s t a b l i s h e d by many a u t h o r s ( c f . J . C . ARDITTI [ 1 9 7 8 ] , J . C . ARDITTI and H.A. JUNG [ 1 9 7 9 ] , R. G Y S I N [ 1 9 7 7 ] , W.T. TROTTER, J r . , J . I . MOORE and D . P . SUMNER [ 1 9 7 6 ] ) . A r e l a t e d r e s u l t , c r e d i t e d t o R.P. STANLEY, i s t h i s . I f P and Q have isomorphic c o m p a r a b i l i t y graphs t h e n L ( P ) = L(Q). I n f a c t , t h e v a l u e of t h e s e r e s u l t s i s t h a t they highl i g h t t h e b a s i c r e s u l t which t i e s t h e enumeration of a l l of t h e o r i e n t a t i o n s of a c o m p a r a b i l i t y graph t o t h e i d e a of l e x i c o g r a p h i c decomposition ( c f . L.N. SHEVRIN and N.D. F I L I P P O V [ 1 9 7 0 ] , M. GOLUMBIC [198O]).
Q b e a n o r d e r e d s e t and l e t (P Iq E Q) b e a f a m i l y of o r d e r e d s e t s 4 Q. The lex
indexed by
is
1exicographicaZly nondecornposable. In F i g u r e 1 0 w e i l l u s t r a t e t h r e e p a i r s of o r d e r e d s e t s . E a c h i n t h e f i r s t p a i r
Figure 10 is l e x i c o g r a p h i c a l l y nondecomposable and t h e one i s t h e d u a l of t h e o t h e r . Each i n t h e second p a i r h a s a l e x i c o g r a p h i c decomposition w i t h i n d e x s e t t h e f i v e element c h a i n and b l o c k s which are e i t h e r two-element a n t i c h a i n s o r s i n g l e t o n s ; t h e one i s o b t a i n e d from t h e o t h e r b y "permuting" t h e b l o c k s o v e r t h e i n d e x s e t . Each i n t h e t h i r d p a i r h a s a l e x i c o g r a p h i c decomposition w i t h i n d e x s e t t h e 4-fence and b l o c k s which a r e s i n g l e t o n s e x c e p t f o r t h e "lower endpoint" of t h e 4-fence where t h e b l o c k i s a 3-fence; t h e one i s i n t h i s c a s e o b t a i n e d from t h e o t h e r by d u a l i z i n g a n o n t r i v i a l b l o c k . Each member of a g i v e n p a i r h a s i d e n t i c a l dimension and t h e same number of l i n e a r e x t e n s i o n s . No wonder t h e i r c o m p a r a b i l i t y g r a p h s ( s e e F i g u r e 11) are graph isomorphic. And t h i s i s t h e h e a r t of t h e m a t t e r . I f a n o r d e r e d s e t P i s l e x i c o g r a p h i c a l l y nondecomposable t h e n s o i s e v e r y o r i e n t a t i o n of i t s c o m p a r a b i l i t y graph: up t o d u a l i t y , i t h a s p r e c i s e l y one orientation.
If P =
1
4 FQ graph of P ( q € Q) and Q ' , 4
Pq, and P ' i s an o r i e n t a t i o n of t h e c o m p a r a b i l i t y 4 t o o , i s a n o r i e n t a t i o n of t h e c o m p a r a b i l i t y graph of Q
Linear extensions of finite ordered sets
(a 1
365
(b) Figure 11
(C)
1 P' has the same comparability graph as P. All orientations of q€Q the comparability graph of P are constructed in this way. I proposed this as a conjecture to R. Wille recently. He has since shown that it does follow from a careful analysis of lexicographic decompositions (see R. WILLE [1982]).
then P'
=
Recently, M. HABIB [1981] (cf. M. HABIB [1983]) reports that a real-valued function defined for ordered sets is "invariant" over all orientations of a given comparability graph provided that it is invariant with respect to its dual and it is invariant over all orientations in which the index set is a chain. The dimension of an ordered set is such an invariant. A definitive account of the dimension theory of ordered sets is found in D. KELLY and W.T. TROTTER, Jr. [1982].
OPTIMAL LINEAR EXTENSIONS Among all of the linear extensions of an ordered set how is an "optimal" one constructed? Such questions arise in the design of a "schedule" to process a set of jobs; technological or other constraints on the jobs may impose precedence constraints on the jobs. The "schedule", too, must comply with them. Whether a schedule is "optimal" depends, of course, on the objectives o f the schedule: earliest completion time, minimum maximum lateness, etc. The literature on this topic is graving rapidly; an important introduction to it is i n E.L. LAWLER and J . K . LENSTRA [19821. We shall consider here just one such problem. Suppose a single machine is to perform a set of jobs, one at a time; a set of precedence constraints prohibits the start of certain jobs until someaother jobs are already completed. Any job which is performed immediately after a job which is not constrained to precede it, however, requires a "setup" or "jump" entailing some fixed additional cost. Our problem is this: scheduZe the
jobs t o minimize the nwnber of j w s . The problem seems to be posed first in J. KUNTZMA" and A . VERDILLON [1971] (cf. M. CHEIN and P. MARTIN
N P
b
I. Rival
366
[1972] and a l s o , G. CHATY, M. C H E I N , P . MARTIN and G . PETOLLA [19741). I n t h e language of o r d e r e d s e t s t h e problem i s commonly r e n d e r e d a s f o l l o w s . Let P b e a f i n i t e o r d e r e d set and f o r L C L ( P ) s e t
1 { (a,b)
s(P,L) =
C PxP(
a c o v e r s b i n L and a
%
b i n PI
I
and S ( P ) = min { s ( P , L ) ( L
c L(P)~
t h e jump number of P. The problem t o c h a r a c t e r i z e s ( P ) i s c o n s i d e r e d i n s e v e r a l r e c e n t a r t i c l e s ( e . g . G. CHATY and M. CHLIN [1979], M. C H E I N and M. HABIB [1980], D. DUFFUS, I . RIVAL and P.M. WINKLER [1982], G . G I E R Z and W . POGUNTKE [ 1 9 8 2 ] , M . HABIB [19811, W.K. PULLEYBLANK [ 1 9 8 2 ] ) . But t h i s "jump number problem" h a s a b a s i c a p p e a l even a p a r t from any supposed c o n n e c t i o n t o s c h e d u l e s . T h i s i s because of i t s a p p a r e n t l i n k t o t h e "chain decomposition" theorem of R . P . DILWORTH [1950]: i n a f i n i t e o r d e r e d s e t P t h e minimum number of d i s j o i n t c h a i n s whose s e t union is a l l of P e q u a l s t h e maximum number of p a i r w i s e noncomparable elements of P . Now, if C,, C2, C i s any minimum f a m i l y of d i s j o i n t c h a i n s of a f i n i t e o r d e r e d s e t
...,
m P whose set union i s P then t h e l i n e a r sum C, eR C2 @ ... CH Cm of t h e s e c h a i n s need not b e a l i n e a r e x t e n s i o n of P . On t h e o t h e r hand I any L E L(P) can b e expressed as t h e l i n e a r sum L = C, N C, @ @ C of c h a i n s C. i n P ( s o k chosen t h a t s u p Ci $ i n f p Ci+, f o r each i) and while i=1 Cz. = P , t h i s f a m i l y of c h a i n s need not
...
b
be t h e minimum number whose s e t union i s P ( s e e F i g u r e 1 3 ) . The "jump number problem'' i s p e r h a p s t h e f i r s t one t o s e e k a r e c o n c i l i a t i o n between t h e " c h a i n decomposition" theorem and t h e " l i n e a r e x t e n s i o n " theorem ( s e e I . RIVAL [ 1 9 8 2 ] ) .
)
Figure 13
The jump number of a n o r d e r e d s e t satisfies
s ( P ) 2 w(P)
-
1
where w ( P ) s t a n d s f o r the w i d t h of P . At l e a s t i n t h e case t h a t P contains no 2n-crown C2n(n 3 2)
s(P)
=
w(P)
-
1
(D. DUFFUS, I . R I V A L , and P.M.
WINKLER [19821 and c f . M. C H E I N and P . MARTIN 119721, M. HABIB [1975], M. C H E I N and M. M. HABIB HABIB [1982]). [1982]). Another Another wide wide ccllaassss of of oorrddeerreedd sets sets ffoorr which which tthhee jump jump number number is is known known iiss tthhee ccllaassss of of series-parallel o r d e r e d s e t s . An ordered ordered sseett PP iiss series-parallel, s e r i e s - p a r a l l e l , iiff iitt can can bbee decomposed decomposed iinnttoo ssiinngglleettoonnss u s i n g o n l y t h e c o n s t r u c t i o n s of l i n e a r sum and c a r d i n a l sum ( s e e F i g u r e 15). I n f a c t , t h e jump number of a seriesp a r a l l e l o r d e r e d s e t can b e computed by c o u n t i n g t h e number of + ' s i n i t s unique s e r i e s - p a r a l l e l decomposition ( c f . 0. COGIS and M. HABIB [ 1 9 7 9 ] ) .
b2" 03''
bih a1
a2
a3
02"
Linear extensions o f finite ordered sets
367
More useful is the fact that s(P)
=
s(P,L)
for any greedy linear extension of a series-parallel ordered set.
Actually
this follows from this more general result.
Call any ordered set isomorphic
to the ordered set P of Figure 1 or Figure 12 an N ("en").
A f i n i t e ordered s e t P whose diagram contains no N s a t i s f i e s s(P)
=
s(P,L)
for any greedy linear extension L of P (I. RIVAL [19821). This result generalizes the series-parallel class rather substantially. First of all the ordered set illustrated in Figure 16(a) is not series-parallel, yet it contains (in its diagram) no N . (A finite ordered set P is series-parallel just if its directed comparability graph (a) contains no N . ) Second, any ordered set can be embedded in one which contains, in its diagram, n o N (see Figure 16(b)). The theme of N ' s in the diagram of P figures in a result of P. GRILLET [1969] (cf. B. LECLERC No N's and B. MONJARDET [19731). A c6 finite ordered set contains no N (b) in its diagram if and only if Figure 16 each maximal antichain meets each maximal chain.
N
M-M
REFERENCES M. Aigner (1982) Private communication. J.C. Arditti (1978) Partially ordered sets and their comparability graphs, their dimension and their adjacency, in ProblDmes combinatoires e t t k b o r i e des grupkes, Orsay 1976, Co1oq.int.C.N.R.S. No. 260, pp. 5-8. J.C. Arditti and H.A. Jung (1980) The dimension of finite and infinite comparability graphs, J . London Math. Soc. (2) 21, 31-38. R. Assous (1982) Private communication. M. Barbut and B. Monjardet (1970) Ordre e t CZassification, AZgDbre e t Combinatoire. Paris, Hachette.
368
I. Rival
R. Bonnet and M. Pouzet (1982) Linear extensions of ordered sets in Ordered S e t s (I. Rival, ed.) D. Reidel Dordrecht, pp. 125-170. C. Chaty and M. Chein (1979) Ordered matchings and matchings without alternating cycles in bipartite graphs, U t i Z i t a s h'atlz. 16, 183-187. (;.
Chaty, M. Chein, P. Martin, C. Petolla (1974) Some results about the number of jumps in acircuit digraphs, U t i l i t a s !lath., Cong. Nwn. X , 267-279.
M. Chein and M. Habib (1980) T h e jump number of dags and posets: an introduction, Annals of Discrete Math. 9, 189-194. M. Chein and M. Habib (1982) Jump number of dags having Dilworth number 2, preprint. M. Chein and P. Martin (1972) Sur le nomhre de sauts d'une forst, C.R. Acuii. Sci. Paris 275 A . 159-161. 0.
Cogis and M. Habib (1979) Nombre de sauts et graphes sgrie-parallsles, R.A.I.R.O. I n f . Th. 13, 1 , 3-18.
M. De Condorcet (1785) I:ssai sur l'application de l'analyse h la probabilit6 des decisions rendues B la pluralit6 des voix, Paris (Chelsea), New York (1974). R.P. Dilworth (1950) A decomposition theorem for partially ordered sets, Ann. oj' Math. 51, 161-166. D. Duffus, I. Rival and P. Winkler (1982) Minimizing setups for cycle-free ordered sets, Proc. h e r . Math. Soc. P.C. Fishburn (1971) On the family of linear extensions of a partial order, J . Combin. Th. ( B ) 17, 240-243. C.M. Fortuin, P.W. Kasteleyn and J. Ginibre (1971) Correlation inequalities on some partially ordered sets, Corn. Math. Phys. 22, 89-103. T. Callai (1967) Transitiv orientierbare Graphen, Acta Math. Acad. Sci. Hungar. 18, 25-66. C. Gierz and W, Poguntke (1982) Minimizing setups for ordered sets: a linear algebraic approach, SIAM J . Algebraic Discrete Methods. M. Golumbic (1980) Algorithmic Graph Theory and P e r f e c t Graphs, Academic Press, New York. R.L. Graham (1982) Linear extensions of partial orders and the FKG inequality, in Ordered S e t s (I. Rival, ed.) D. Reidel, Dordrecht, pp. 213-236. R.I..
Graham, A . C . Yao and F.F. Yao (1980) Some monotonicity properties of partial orders, S I N J . Algebraic Discrete Methods 1, 251-258.
P.A. Grillet (1969) Maximal chains and antichains, Fund. Math. 65, 157-167. R. Gysin (1977) Dimension transitiv orientierbarer Graphen, Acta Math. Acad. Sc. Hungar. 29, 313-316. M. Habib (1975) Partitions en chemins des sommets et sauts dons les graphes sans circuits, Thsse, Universite P. et M. Curie, Paris. M. Habib (1981) Substitution des structures combinatoires, th6orie et algorithmes, ThPse, Univ. Pierre et Marie Curie, Paris.
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M. Habib (1983) Comparability i n v a r i a n t s , p r e p r i n t . B . J 6 n 5 s o n (1982) A r i t h m e t i c of o r d c r c d s e t s i n Ordered S e t s ( I . R i v a l , e d . ) I ) . R e i d e l , D o r d r e c h t , p p . 3-41.
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D. K e l l y and I . R i v a l (1975) P l a n a r L a t t i c e s , Canad. J . !lath. 27, 636-665. D . Kelly and W.T.
T r o t t e r , J r . (1982) Dimension t h e o r y f o r o r d e r e d s e t s , i n
Orderad S e t s ( I . R i v a l , e d . ) D. R e i d e l , D o r d r e c h t , pp. 1 7 1 - 2 1 1 . D . I . Kleitman (1966) F a m i l i e s of n o n - d i s j o i n t
s e t s , J . Combin. T k . 1, 153-155.
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Knuth (1973) T h e A r t of Computer Programming, Vol. 1 , V o l . 3, Addison-Wesley.
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P u l l e y b l a n k (1982) On minimizing s e t u p s i n precedence c o n s t r a i n e d s c h e d u l i n g ,
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L.A.
Shepp (1982) The xyz c o n j e c t u r e and t h e FKG i n e q u a l i t y , Annals of
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S h e v r i n and N . D . F i l i p p o v (1970) P a r t i a l l y o r d e r e d s e t s and t h e i r comparab i l i t y g r a p h s , Siberian Math. J . 11, 497-509.
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T r o t t e r , J r . , J.I. Moore and D . P . Sumner (1976) The dimension of a c o m p a r a b i l i t y g r a p h , Proc. Amer. Math. Soc. 60, 35-38.
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Annals of Discrete Mathematics 23 (1964) 371-386
37 1
0 Elsevier Science PublishersB.V. (North-Holland)
COMPARABILITY
INVARIANTS
Michel HABIB Ecole des Mines de SAINT ETIENNE 158, Cours Fauriel 42023 SAINT ETIENNE CEDEX
C e t a r t i c l e e s t d e d i e au p r o f e s s e u r E .
RESUME.-
COROMINAS.
Nous presentons ici une caracterisation des invariants de comparabilit6 (i.e. des fonctions qui donnent la meme valeur aux graphes transitifs ayant, a l'isomorphie prPs, le meme graphe de comparabilit6 sous- jacent)
.
Cette caracterisation a l'air tres interessante car elle permet l'unification de plusieurs resultats connus: - un theoreme de DILWORTH 161 sur l'indice de partition des sommets en chemins, - un theoreme dfi 2 ARDITTI [ I 1 et 2 TROTTER, MOORE, SUMNER [ I 9 1 sur la dimension d'un ordre partiel, notion introduite par DUSHNIK et MILLER 171, - un theoreme de STANLEY [lo] sur le nombre d'ordres totaux superieurs 2 un ordre partiel donne. En outre, nous obtenons un resultat analogue pour le nombre de sauts d'un ordre partiel (nous avons present6 ce dernier invariant dans [31 1 .
TIETRACT -_ _ - - - - - : we present here a characterization of comparability invariants. Let us recall that a comparability invariant is a directed graph invariant which keeps the same value over the whole class of transitive directed graphs that have the same underlying comparability undirected graph. This characterization seems to be a very practical tool since it allows the unification of several results: - a theorem of DILWORTH [61 on the path-partition number, - a theorem of ARDITTI [ l ] , and TROTTER, MOORE, SUMilER [191 on the dimension of partial orders, as defined by DUSHNIK and MILLER [ 7 ! , - a theorem of STANLEY [ l o ] on the number of linear extensions of a given partial order. Eesides, we obtain a similar result for the jump number of a partial order (we have introduced this invariant in [ 3 1 ) .
I -
INTRODUCTION
We deal in this paper with finite simple graphs (i.e. without loops and multiple edges). We u s e standard graph terminology ( s e e for example BERGE [ 2 ] 1 .
M . Habib
312 Therefore f o r a given graph
G = (X,
t h e induced subgraph a s f o l l o w s : d e n o t e by
IGI =
1x1
U) , and f o r e v e r y
G ( A ) = (A, V)
,
with
A
5X ,
V = U
n
we define
, and w e
A2
t h e number o f v e r t i c e s of t h e g r a p h G .
A d i r e c t e d graph G = ( X , U) i s c a l l e d t r a n s i t i v e l y o r i e n t e d i f U i s a t r a n s i V x, y , z 6 X , ( x , y ) E U and ( y , z) E U t i v e b i n a r y r e l a t i o n on X : i . e . implies ( x , z) E U ( S i n c e l o o p s are n o t a l l o w e d , w e j u s t o n l y c a r e w i t h t r i p l e s made up w i t h distinct vertices).
.
S i m i l a r l y a t r a n s i t i v e l y o r i e n t e d g r a p h which h a s no c i r c u i t i s c a l l e d a p a r t i a l order o r a poset. L e t u s d e n o t e by (resp. (resp. p a r t i a l orders),.
9
)
t h e c l a s s of a l l t r a n s i t i v e l y o r i e n t e d g r a p h s
For a g i v e n u n d i r e c t e d g r a p h G = (X, E) , w e d e f i n e an o r i e n t a t i o n $ of G , a s a d i r e c t e d g r a p h H = ( X , U) , d e n o t e d by $ ( G ) , such t h a t t o any edge $ ( [ a , b l ) 5 { ( a , b ) , (b, a ) } and U = $(e) [ a , bl E E , we a s s o c i a t e eeE
u
A c o m p a r a b i l i t y graph i s an u n d i r e c t e d g r a p h which h a s a t r a n s i t i v e o r i e n t a t i o n
(which i s n o t n e c e s s a r i l y a p a r t i a l o r d e r ) . For any graph.
G Ez
, we
d e n o t e by
Comp(G)
t h e a s s o c i a t e d undirected comparability
-
Two g r a p h s G and G ' , e l e m e n t s of t , are e q u i v a l e n t ( i n symbol G G ' ) i f and o n l y i f t h e i r c o m p a r a b i l i t y g r a p h s are isomorphic ( i . e . Comp(G) w Comp(G') ) . Note t h a t i f t h e r e i s several isomorphisms from Comp(G) t o Comp(G') , w e s h a l l choose one o f them (which we c a l l " t h e " isomorphism). O u r r e s u l t s do n o t depend on t h e c h o i c e d isomorphism. F o r u s g r a p h - i n v a r i a n t i s a mapping f from 7; t o R E 7; , G w H implies f(G) = f(H)
.
G, H
such t h a t f o r every
F u r t h e r m o r e , w e now r e a c h t h e main d e f i n i t i o n . DEFINITION:
The g r a p h - i n v a r i a n t f : 7; -B R ( r e s p . f : 9 -t R ) * i s a c o m p a r a b i l i t y 7; - i n v a r i a n t (resp. 3 - i n v a r i a n t ) i f and only i f : (resp. G, G' E ) t h e condition For e v e r y G , G ' E 6 G G' implies f (G) = f ( G I )
-
For any d i r e c t e d g r a p h
.
w e d e f i n e i t s i n v e r s e , G-
G = ( X , U)
by r e v e r s i n g a l l t h e a r c s , i . e . w i t h t h e above d e f i n i t i o n .
U- = {(x, y )
I
( y , x)
E
U]
=
(X, U-) o b t a i n e d
. Therefore
G
-
G-,
A t l e a s t , w e d e f i n e t h e t w o f o l l o w i n g t r i v i a l g r a p h s on p v e r t i c e s :
*
S = ( [ I , p ] , 0) P p-independent;
*
K
P '
t h e e m p t y graph,
t h e complement o f S
P
,
a l s o c a l l e d p - s t a b l e or
a l s o c a l l e d t h e symmetric complete g r a p h .
L e t u s r e c a l l two famous examples o f c o m p a r a b i l i t y i n v a r i a n t s t!iat in literature.
can b e found
-
The p a t h - p a r t i t i o n number o f a g r a p h G ( i . e . t h e minimum c a r d i n a l i t y of a s e t of e l e m e n t a r y p a t h s t h a t p a r t i t i o n s t h e v e r t i c e s of G ) , d e n o t e d by p ( G ) .
*
T h i s h y p o t h e s i s i s n o t c r u c i a l , we c o u l d e a s i l y g e n e r a l i z e t h e f o l l o w i n g r e s u l t s t o i n v a r i a n t s d e f i n e d from z ( r e s p . ? .' ) t o o t h e r s t r u c t u r e s t h a n R .
Comparability invariants
373
The well known theorem of DILWORTH [61 which states the equality between p(G) and a(G1 (the stability number) of any transitive graph G enables us to class p as a comparability %-invariant.
-
The dimension of partial orders defined by DUSHNIK and MILLER [ 7 1 , has also been classed as a comparability ?-invariant, nearly at the same time by ARDITTI [11, and by TROTTER, MOORE, SUMNER [191.
In the following, we shall prove a general result (theorem 3 ) that allows the classification of many comparability invariants and in particular of the two previous ones. Our approach is derived from the study of substitution decompositions of directed graphs, HABIB [ll]. For other studies on closely related decompositions see also: CUNNINGHAM, EDMONDS [5], MAURER [ I s ] , [161, and MOHRING, RADERMACHER [17]. Other authors have found similar results when studying comparability graphs, for a complete bibliography see GOLUMBIC [lo], chapter 5. We end this paper by using this characterization to prove that the jump number is a comparability .Ip -invariant, we have introduced this invariant in 131.
I1 -
A CHARACTERIZATION OF COMPARABILITY INVARIANTS
Let us first recall the definition of the substitution operation on directed graphs. DEFINITIONS:
Let
H = (Y, V ) , K = (Z, W)
be two directed graphs with
~ n z = 0 . We define
x u
G = (X,
=
(Y - {a])
=
wu
=
U)
H
for any
a E Y
,
as follows:
uz
t(x,y) E
VI
x
=
a, y = a1
u
u
i(x,y)
1 XEZ, ~
E Y ,(a,y) E
I(X,Y) 1 X E Y , Y E Z , (x,a) E
vl
VI
.
This operation is called an elementary substitution as we can consider the graph G obtained from the graph H by replacing the vertex a by the graph K. Furthermore Z is called an externally related subset of G. For a complete study and a bibliography on the substitution operation see HABIB [ l l ] , or more recently MOHRING, RADERMACHER [17]. We just notice that 9 is closed under this operation, but not
z .
Obviously there exist trivial elementary substitutions which are: For any graph G = ( X , U) and any a E X
,
Thus we can now define: G is irreductible when it admits only the above trivial decompositions by elementary substitution, otherwise G is called decomposable. L e t us notice that this definition implies that all the graphs with strictly less than 3 vertices are irreducible. W e define now a general substitution operation.
G
=
(..
.
(H :l) 1
...)
Gm m
where xi for i = 1,
-..,m
are distinct vertices of H.
M . Habib
314
Using t h e a s s o c i a t i v i t y of t h i s o p e r a t i o n , t h e p a r e n t h e s e s can be o m i t t e d and
When f o r
P
=
{XI,
...,
X
m
)
graph H whose v e r t i c e s a r e
t h a t p a r t i t i o n s t h e v e r t i c e s of G , t h e r e e x i s t s a
al,
...,
a
. ..
G = H G(X1) ('m) then al m ( T h e r e f o r e P is made up w i t h e x t e r n a l l y
such t h a t
m
P i s c a l l e d a c o h e r e n t p a r t i t i o n o f G. r e l a t e d s u b s e t s of G ) .
DECOMPOSITION RESULTS: L e t u s r e c a l l a w e l l known decomposition r e s u l t ( d i s c o v e r e d by many a u t h o r s ) . "Every d i r e c t e d g r a p h h a s a unique s u b s t i t u t i o n decomposition i n t o s u b g r a p h s , e a c h of whose members i s an i r r e d u c i b l e g r a p h , a complete graph, a s t a b l e o r a t o t a l o r d e r . " (For a p r o o f of t h i s r e s u l t , s e e f o r example CUNNINGHAM, EDMONDS [ 5 1 , MAURER [ 1 5 ] , [ 1 6 1 or HABIB [ I l l ) . I n o t h e r words, f o r e v e r y d i r e c t e d g r a p h G o n a v e r t e x s e t X , unique c o h e r e n t p a r t i t i o n d e n o t e d by P ( G ) = {X1, Xk}
...,
0
every i ,
1
<
i
G(Xi)
there e x i s t s a such t h a t f o r
i s a n i r r e d u c i b l e g r a p h , a complete g r a p h , a
s t a b l e or a t o t a l o r d e r . Therefore i, i
<
i
PO(G) i s t h e minimal non t r i v i a l c o h e r e n t p a r t i t i o n o f G ; f o r e v e r y
< k,
t h e subgraph
G(XI)
w i l l be c a l l e d a p r i m e component o f G . W e c a n
a p p l y t h i s p r o c e s s r e c u r s i v e l y and d e f i n e t h e f o l l o w i n g f i n i t e sequence of g r a p h s : Go = G , G1 = G / P O ( G ) , , G i + l = Gi/Po(Gi) ,
...
... .
Then we have c a n o n i c a l sequence of prime components and t h e i r p r o j e c t i o n on X y i e l d what we c a l l g e n e r a l i z e d prime components o f G.
W e must n o t i c e t h a t i n t h e above d e f i n i t i o n such a prime component K i s whether minimal and non t r i v i a l w i t h t h i s p r o p e r t y when K i s i r r e d u c i b l e o r maximal i n t h e t h r e e o t h e r cases. Of c o u r s e , a l l t h e s e n o t i o n s ( s u b s t i t u t i o n o p e r a t i o n , e x t e r n a l l y r e l a t e d s u b s e t , c o h e r e n t p a r t i t i o n , i r r e d u c i b l e , prime component, ) can b e d e f i n e d f o r u n d i r e c t e d g r a p h s , and e x a c t l y si m i l a r r e s u l t s h o l d . F u r t h e r m o r e , i n t h i s p a r t i c u l a r c a s e , prime component c o r r e s p o n d s t o m u l t i p l e x d e f i n e d by GOLUMBIC i n [ 9 1 .
...
RELATIONSHIP BETWEEN DECOMPOSITION OF G AND COMP(G) Of c o u r s e t o e v e r y s u b s t i t u t i o n d e c o m p o s i t i o n of a d i r e c t e d g r a p h , w e c a n a s s o c i a t e t h e same s u b s t i t u t i o n d e c o m p o s i t i o n o f t h e u n d e r l y i n g u n d i r e c t e d graph. The c o n v e r s e i s i n g e n e r a l f a l s e , b u t f o r t r a n s i t i v e g r a p h s , w e have some p a r t i a l r e s u l t s r e l a t e d w i t h prime components a s d e f i n e d p r e v i o u s l y . B u t b e f o r e g o i n g f u r t h e r , w e must mention a q u i t e s t r a n g e r e l a t i o n s h i p between s u b s t i t u t i o n decomposition and f o r c i n g c l a s s e s o f e d g e s . L e t u s r e c a l l t h e d e f i n i t i o n of t h e f o r c i n g classes o f e d g e s . L e t G = ( X , E ) b e a n u n d i r e c t e d g r a p h , we s a y t h a t [ a , b ] E E forces [b, c ] E E i f and o n l y i f [a, c ] E E or a = c
.
The t r a n s i t i v e c l o s u r e of t h i s b i n a r y r e l a t i o n d e f i n e d on t h e e d g e s o f G , p a r t i Ek t i o n s E i n t o f o r c i n g classes of edges: El ,
...,
.
When l o o k i n g a t t h e t r a n s i t i v e o r i e n t a t i o n s of G , we see t h a t t h e o r i e n t a t i o n of t h e whole f o r c i n g c l a s s i s c o m p l e t e l y d e t e r m i n e d ( f o r c e d ) by t h e o r i e n t a t i o n of one o f i t s e l e m e n t s .
375
Comparability invariants
,
i E [l, k l
Furthermore for every
an externally related subset of G
X. =
{x E
X
1
3 e E Ei
, e n x
=
01
is
,
This relationship between forcing classes of edges and substitution decomposition was first noticed by GALLAI [ 8 1 , and rediscovered by GOLUMBIC 191. This relationship can be strengthened, for details see GOLUMBIC [ l o ] , and this yields 3
an O(n ) algorithm for the substitution decomposition of undirected graphs, see HABIB, MAURER [12]. In the following we only need a partial result, although generalized in the directed case.
-
THEOREM 1
If G E , having strictly more than 2 vertices, is is reduced to G, irreducible then the class of G modulo
-
and G-
.
Hints for t h i s p r o o f : In 1111, I have proposed a generalization of the .................... forcing relation, let us say the "directed f o r c i n g relation". This new relation keeps the same property about externally related subsets. Besides an irreducible transitive graph G with n > 2 vertices is necessarily a partial order. At least for the class of partial orders the forcing relation and its generalization are identical and thus we can apply the similar result for undirected graphs, see for example GOLUMBIC [ l o ] . THEOREM 2 -
Let
G E
then
G
=
, and K be a prime component of H G (K)
Comp(G),
(Where G(K) is the subgraph of G associated with K and where H is the simple graph obtained by contracting the subgraph G(K) of G in a single vertex a E GI.
x C H- a
Proof: ----- Let us suppose there exists
edge of
Comp(H)
.
Since
such that
[x, a]
, necessarily all the vertices
Comp(G) = Comp(H)E
of G(K) are connected with x. This yields a partition P = {XI, X2, X 3 j that : X1 = {y E K I (x, y) E G , (y, x) 4 GI
X2
=
{y E K
X j = {y E K
1 I
G
(x, y)
(x, y) F G
is an
, (y, x) E , (y, x) E
such
GI
.
GI
IKI > 3 . Necessarily G ( K ) is also irreducible. Since there is no irreducible partial order having 3 vertices, we can suppose IK/ > 4 . Using the transitivity of G and G(K), the partition P yields necessarily a substitution decomposition of G ( K ) , as we have supposed G(K) irreducible IKI > 4 implies IPI = 1 this decomposition must be a trivial one. But G (K) And thus we have: G = H
1 ) When K is irreducible and
.
2) When K is a stable, therefore G(K) is also a stable. Then using the transitivity we have:
*
*
If \ X 3 \ # 0 then 1X 1 = 1 and IX I = IX 1 = 0 and so 3 1 2 reduced to a single vertex and the result is trivial. If
IX
have:
3
I
=
0
G = H
,
then whether
G (K)
IX
2
I
=
0 or
IX
1
I=
3 ) When K is a complete graph. If I P / = 1 the result is obtained, let us now suppose:
0
IPI
G(K)
and thus we
>
2
.
is
M.Habib
316
As K is a prime component of Comp(G), Ktx is not a prime component of Comp(G). Thus there exists y E H - a - x such that [x, 21 is an edge of Comp(H), and [y, 21 is not an edge of Comp(H) for some z E K . /PI = 2 and using transitivity there exists necessarily some vertex E G(K) connected in G with y , and thus using the substitution decomposition of Comp(G), we can see that K+x is a prime component of Comp(G) , which is excluded. and G = H G (K) And thus ( P I = 1
As z'
Since (KI = 2 and K irreducible implies that K is a stable or a complete graph, we have considered all the cases. From this result we can easily deduce the very useful following result. COROLLARY
-
Let G, H, K E ^ e
and
G = H
K
with
a E H
.
If Comp(K)
is a prime component of Comp(G) , then for any G' E % G'
H'
-
-
, , there exist H' and K' E % , a' E H' such that: K' (where a' corresponds H , K' K , and G' = H' a ,
G
-
to a in the isomorphism from Comp(H') to Comp(H)
)
.
In other words if K is a prime component of an undirected comparability graph G, then every transitive orientation $(GI admit a substitution decomposition "around" $(K) Of course this result is still valid for any generalized prime component of Comp(G) .
.
Now we can express the main result of this paper. THEOREM 3 -
Let f : % + R (resp. ? + R ) a graph-invariant then f is a comparability Z-invariant (resp. ? -invariant) if and' only if f satisfies the following conditions (O), (i) and (ii):
*
.
t ,
(Where T and K2 denote respectively 2 the total order and the symmetric complete graph on two vertices).
( 0 ) f (T2 = f(K2)
(i) For every G E Z (ii) For every H, H' E^G For every K, K' E t
(resp. 3'
)
(resp. 3 (resp. 9
) )
f(G) such that such that
=
H K
.
f(G-)
--
H' K'
, ,
For a E H and a ' its image . by -the isomorphism from Comp(H) to Comp(H') , the condition f(H) = f(H') and f(K) = f(K') implies K K' f(H a) = f ( H ' a,)
.
pof:
These conditions are obviously necessary. Let us consider their sufficiency, first in the case of comparability %-invariants.
The proof goes by induction on the number of vertices, and since there exists only one simple directed graph having one vertex, the result is obvious in this case. Let u s suppose for every have: f ( G ' ) = f (G")
.
W e consider now
G',
GI, G2 E ?f,
G" E %
such that
GI
-
G"
and
(GI(
< n , we
two transitive orientations of the same comparabi-
Comparability invariants
lity graph having n vertices
(i.e. Comp(G
)
1
Comp(G2)
=
Comp(G1) ,
Let us take K a prime component of
377
)
.
IKl > 1
with
,
(there exists
at least one such prime component, Comp(G1) itself). Then using theorem 2, we have: Gl(K) and
rily we have: 1") If
GI a(K)
Gl(K)
K # Comp(G
-
G2(K)
,
)
1
and
l
H1
H
N
IG1(K)I < n
then
2") If K
H2 G2 (K) where a2 associated with K. Necessa-
,
=
f(H2)
=
Comp(G)
2 '
.
By induction hypothesis we know that
f (H 1
G2 =
and 1 G2(K) are the subgraphs of resp. G1, G2 G1 = H
f (GI(K)) = f (G2(K)) and
hence we conclude with the condition
,
H1 = ({a,}, p )
then
H2
fi:
,
(ii).
I
and
/G1(K)
=
n
.
a) If K is irreducible and IKI > 3 using theorem 1 and condition (:)
,
b) If K is a stable, then G1 and
are also isomorphic stables.
Therefore G
1
G
M
and
2
G2
.
then the result is obtained by
f(G1) = f(G2)
since f is a graph-
invariant. c) If K is a complete graph. We notice that any transitive orientation of Comp(G1) can be obtained by substituting complete graphs in a total order.
cl, Thus
G1
- O1
xl,
..., c ..., xP
D1, and
is a total order on p vertices
...,
D
. . . I
Yq
G2 = 0
2 Y1'
(resp. q ) , and
,
C . D. I' I'
where
1 < i
O1 (resp. 0 2 )
1
<
j
are complete graphs. Let us choose x G2 = F
J2 x:
,
E
C1 and
where
Of course, we have
x2
F 1 , F2
F1
-
F2
E
D1
,
then we may write:
have only two vertices namely
and
J
-
J2
J1 F 1 x, , and 1 {x,, xi} and
G1
=
and hence using the induction hypo-
thesis and conditions ( O ) , (ii), we have finished. And exactly similar proof gives the corresponding result for comparability
9 -invariants.
1.-
Conditions ( 0 ) and (i) are obviously satisfied by many graphinvariants, and there is no need of condition ( 0 ) for comparability P -invariants.
2.- To establish condition (ii), we shall prove, in each particular application, a formula to compute
f(G)
when
G = G
G2 a
,
and
show that this formula yields condition (ii), this can be written as corollaries.
M . Habib
378
Indeed we have written the two corollaries that follows for the purpose of classification of three graph-invariants: path-partition number, dimension and jump-number. These results give the flavor of similar properties that have to be obtained from theorem 3 in order to classify a particular graph-invariant.
COROLLARY 1. If f : 6 -+ B (resp. 9 satisfies (01, (i) and:
+
(iii) There exists a mapping V G1, G2 € 6
G2 f (G1 a ) ^c;
Proof: _----
=
We just have to show
)
(v) There exists a mapping G1, G2 E^c; G1
,
-+
R
such that:
,
a E GI
then f is a comparability
(iii) implies (ii)
(iv) V G E ^c; (resp. 3 f(G) < \GI .
E
2
v
f (G2))
-+
V a
R
:
(resp. 9 -invariant).
COROLLARY 2. If f : Tt; N (resp. P ! and satisfies (01, (i) and:
v
$
(resp. .'p ) ,
$ (f( G 1 ) ,
-invariant
is a graph-invariant and
R )
N
-+
, which
is a graph-invariant
)
which is not a stable $ : NL
3
N
such that:
(resp. 3 ) ,
f(G1
G2 a)
f' =
$ (f(G2),f(G1
then f is a comparability invariant).
K1
-
,
K2
If we take Comp(H2)
,
IK1( + IH1( = n
al E H1
and
a2
and
,
K 1 , K2 € ' G
f(H ) = f(H2) 1
(G2)
and
such that
(resp. 9 -
that conditions H1
f(K1) = f(K2)
its image by the isomorphism from
1
a
r:-invariant
Proof: _---- We shall prove it by induction on n = \HI + IKI (iv) and (v) imply the condition (ii) of theorem 3. Let u s now consider H1, H2 E 6
is obvious
-
H
.
Comp(H1) to
= $
(f(K1), f(H1
1
'f(K1) a 1 1
$ (f(K2), f(H2
f ' (KZ) a 1
I
and
H
,
f(H2
=
2
2
a) Let K1 is not a stable, then using condition (iv), \ H I + IK1 I < n thus by induction we have finished.
K
1
,
and then apply (v) we have:
f(H1
b ) If
H2
is a stable, let us consider
component of
Comp(G1)
K1 G1 = HI al
is a stable, then
.
If every generalized
Comp(G1) admits only two different
transitive orientations. (SHEVRIN and F I L I P P O V have first noticed this fact in [ 1 8 1 ) . And then the result is given by condition (i)
.
Else Comp(G1) admits a generalized prime component T which is not a stable and
Comparability invariants
K1
H1(T)I K 1 therefore:
G1 =
HI1
'
bl
H'l a
=
al
379
, HI(")
.
Hence applying (v), we
1 ' bl
K
IH'll
Since
I
ISf(H1(T))
+
I11 -
,
< n
we can use the induction hypothesis
APPLICATIONS
A) Path-partition number This invariant p : 3 + N satisfies trivially conditions ( 0 ) and (i). In order to obtain condition (ii) we may use the following result. PROPOSITION 1 -
Let G, G1, G2 E Z
such that
G = G
l
G1 a
a E G
and
1 '
S P(G2)
then
p(G)
=
Proof: -_-- The inequality
p(G)
< p(Gl
p(G
l S
-
change each vertex of
.
h
> p(G2 )
Let us suppose these paths are
If
h
p(G2)
If
h > p(G2)
=
, ,
P(G2) is obvious as we can
)
a
Let us consider now the other inequality. Let
G2
.
by a path of a path-partition of G
S P(G2)
partition of G. There exist
1
a
we have finished we contract in
P
{Cl,
=
...,
2'
Ckj
a path-
paths of this partition P that cross cl,
. . . I
.
'1'
' * . I
.
h'
the vertices of
'p(G2)
G
2
to o n e
vertex (for example the first one of the path in G2). Thus we obtain CI1
, ...,
Sp (G2) C'
PtG2)
paths
Of
Furthermore, we delete from possible as C'
p(G2)+1 ,
Thus Since
...,
Cp(G2)+1
. , ...,
Ck
the vertices of
G
2
(thus is
and we obtain
new paths.
Cth
S
...,
IPI = IP' I
,
=
a
GI was supposed to be transitive),
iC'l,
P'
G1
Ctk}
is a path-partition of
G1
P(G-) 2 a
we have the desired result. m
Thus by proposition 1, it is not very hard to notice that we can apply the corollary 2 of theorem 3, with mapping @ equal to the second projection mapping. So the path-partition number is a comparability 3 -invariant. An
exactly argument shows that the path-covering number is a comparability
6 -invariant.
M . Habib
380
Note -
I n t h e f o l l o w i n g examples, we s h a l l now o n l y c o n s i d e r p a r t i a l o r d e r s and hence c o m p a r a b i l i t y 9 - i n v a r i a n t s .
DEFINITION:
L e t P be a p a r t i a l o r d e r , we d e n o t e by C ( P ) t h e s e t of a l l t o t a l ord ers g reat er than P ( a l s o c a l l e d t h e s e t of a l l l i n e a r extensions of P )
.
P a r t i a l o r d e r dimension
W e r e c a l l t h a t t h e dimension of a p a r t i a l o r d e r P , d e n o t e d by d e f i n e d by DUSHNIK, MILLER i n [ 7 ] , a s t h e minimum i n t e g e r h ,
n
P =
with
T~
Ti
d i m ( P ) , was such t h a t :
c(P)
E
1 6 i c h For a r e c e n t s u r v e y on p a r t i a l o r d e r dimension t h e o r y , s e e KELLY, TROTTER [141 W e j u s t r e c a l l t h e p r o o f s made by A R D I T T I [ l ] and TROTTER, MOORE, SUMMER 1191 a s t h e y i n s e r t t h e m s e l v e s v e r y w e l l i n o u r frame.
.
HIRAGUCHI f i r s t noticed i n t 1 3 1 :
,
V P1, P2 E .’P
V a
E P1
dim(P p 2 ) l a
:
.
= max { d i m ( P 1 ) , d i m ( P 2 ) )
With t h i s formula and t h e p r e v i o u s c o r o l l a r y 1 o f t h e theorem 3 , w e can t r i v i a l l y show t h a t t h e dimension i s a c o m p a r a b i l i t y 9 - i n v a r i a n t , by i d e n t i f y i n g $ w i t h t h e a p p l i c a t i o n max. Number of l i n e a r e x t e n s i o n s of a p a r t i a l o r d e r
I
1
.
L e t u s adopt t h e following n o t a t i o n L(P) = f (P) From GOLUMBIC [ 101, ?-invariant. w e know t h a t STANLEY h a s proved t h a t L ( P ) i s a c o m p a r a b i l i t y S i n c e it i s very e a s y t o be c o n v i n c e d of t h e f o l l o w i n g formula P V P1, P2 E 9 and V a E P1 , L ( P 1 a 2 ) = L ( P 2 ) . L(P1
”,2
f o r any
o2 E
c
(P2)
,.
t h e n w e c a n p r o v e , a s i n c o r o l l a r y 2 of theorem 3 , t h a t t h i s formula i m p l i e s t h e c o n d i t i o n ( i i ) o f theorem 3 . Here t h e argument g o e s by i n d u c t i o n on t h e number of uncomparable p a i r s of v e r t i c e s of t h e p a r t i a l o r d e r . Hence w e have a n o t h e r p r o o f of STANLEY’S r e s u l t .
E ~ t g-
T h i s r e s u l t c a n n o t b e g e n e r a l i z e d t o t h e number of t o t a l p r e o r d e r s g r e a t e r than a given p r e o r d e r .
D) J u m p number
For
P E9
(x., x .
and
1
<
T =
i
xl,
..., x
such t h a t
C
E xi
(P)
< xi+l
,
a “jump“ of T i s a c o u p l e
.
P
W e d e f i n e a l s o t h e j u m p number o f T , d e n o t e d by u ( 7 , P ) a s t h e number o f a ( P ) = m i n o (T, P) , t h e jump number o f J . such c o u p l e s . S i m i l a r l y we d e f i n e T E f (P) W e have f i r s t b e l i e v e d t h a t : V P 1 , P2 E 9 and f o r any x E P1 , P o ( P I x 2 ) = IJ ( P 1 ) + 0 ( P 2 )
.
And we wrote it i n
[111
.
38 1
Comparability invariants N e v e r t h e l e s s , COGIS, POGUNTKE and R I V A L [ 4 ] showed u s a counter-example ( c f . f i g u r e 1). n
r 2 ) , u ( P 1 ) and a ( P 2 ) i s a l i t t l e more 1 x s o p h i s t i c a t e d a s w e s h a l l see i n t h e f o l l o w i n g theorem 4 . T h i s theorem means r o u g h l y t h a t t h e p o s e t d e s c r i b e d i n f i g u r e 1 i s t h e " o n l y " counter-example o f t h e above formula. Indeed t h e formula between a ( P
To prove t h i s theorem, w e need two lemmas and u n t i l t h e end of t h e p r o o f w e now assume
P = P
p2 1 x
w i t h P 1 , P 2 E .'P
.
x E P1
and
For any T E C ( P ) w e use i t s c a n o n i c a l d e c o m p o s i t i o n i n t o maximal s e q u e n c e s , (resp. P 2 ) , d e n o t e d by a . ( r e s p . h i ) , made up w i t h v e r t i c e s o f P1
,
Furthermore t o e a c h o f t h e s e s e q u e n c e s a . or b .
1
w e c o n s i d e r two d i s t i n g u i s h e d
v e r t i c e s , which are r e s p e c t i v e l y t h e l e a s t and t h e g r e a t e s t e l e m e n t w i t h r e s p e c t t o the t o t a l order T W e d e n o t e t h i s e l e m e n t s by s u p ( l ) , i n f ( 1 ) for such a sequence 1 .
.
LEMMA 1
-
...
... E
a. bi+l T = b. 1 1+1 ( s u p ( b i ) , i n f ( a i + l ) ) and
If
jumps o f T
f
then
(P)
(sup(ai+l), inf(bi+l))
are
.
-----
P r o o f : By t h e d e f i n i t i o n o f e l e m e n t a r y s u b s t i t u t i o n a l l t h e v e r t i c e s of t h e s e q u e n c e s b . and b . have t h e same neighbourhood o u t s i d e o f P 2 ' 1+1 Since T E f (P) , necessarily the v e r t i c e s of a. are n o t comparable w i t h 1+1 t h o s e o f b . and b . 1+1 *
1
-
LEMMA 2
a (P ) 1
+ u
more, when
-
(P ) 2
1< a ( P ) < o ( P 1 ) +
u (P2)
>
1
, u
and o n l y i f t h e r e e x i s t s
u(.rl, P
that
)
= a(P1)
(P) =
T~
,
=
1
a x a a
1 2 3 sup(a ) < x
1
and f u r t h e r -
0(P2)
u(P ) + o(P2)
-
1
f (P1)
E
,
<
x
if
such
inf(a3)
,
l P
1 p1 and x i s n o t comparable w i t h t h e v e r t i c e s o f a2 ( s u p ( a 2 ) , i n f ( a 3 ) ) i s a jump o f T
,
and
E f
(P2)
.
Emof: Let T E 1 Thus w e make
1") The r i g h t i n e q u a l i t y i s n e a r l y o b v i o u s . W e c a n decompose
that satisfies O(P) 2 O ) Let
< U(P1) T(P)
't1
U(T,
+ U(PJ
= T 1' x T 1"
P) = U ( T l ,
. P1)
+ a ( ?2 , P 2 )
f (P1) , T = T
'
T
2
T2 TI"
and w e have :
.
be an optimal l i n e a r e x t e n s i o n
(i.e.
U(T,
P) = U ( P ) )
E
. c (P)
M. Habib
382 a)
T
= al bl
+
o(P1)
... b9
o(P2)
<
a
q+l
o(P)
.
q
If
.
O (T,
P) 4
= al b
T'
.
P)
U(T,
b
1
, when
T
a
2
2
a
T
=
a l b l a 2 b2 a 3 E
.
f(P)
a l b l b2 a 2 a j E
=~
'
C (P)
>
a
...
3 b3
q
Thus we can a p p l y t h e same p r o c e s s u n t i l T~
>
q
.
2
(sup(bl), inf ( a 2 ) ) , (sup(a2), inf ( b 2 ) ) ,
( s u p ( b 2 ) , i n f ( a 3 ) ) a r e jumps of
that
then obviously
L e t u s c o n s i d e r t h e c a s e where
From lemma 1 , we know t h a t
Then we can make:
,
1
=
>
.
3
3
C
E
aq + l
( P ) , such
and o b t a i n
I n t h i s c a s e , we can always make
.
But i f s u p ( a ) 4 i n f ( b l ) , 1
p1
s u p ( b ) 4 i n f ( a 2 ) and s u p ( a 2 ) , i n f ( a ) 2 3 p1 then O ( T [ ' ~ ,P ) = U ( T ~ ,P ) t 1 , e l s e
a r e n o t comparable i n P
P)
U(TI0,
= U ( T ~ , P)
1'
.
(The f i r s t c a s e can o c c u r , s e e f i g u r e 1 ) . Anyway, we always have: O(P)
>
O(P1) +
U P 2 )
+
1
= U(P )
+
P)
P) 4 O(To,
"(TIo,
and t h e r e f o r e
.
- 1
L e t u s c o n s i d e r t h e two o t h e r c a s e s .
@
)
~
= ba
...
-
when
q = 1
-
when
q
>
2
such t h a t y ) ~ = ab l
...
a
q
, ,
b
.
q+l
we t r i v i a l l y have
u s i n g lemma 1 , we make
U(T'
,
<
P)
o(P)
i s n e c e s s a r i l y a jump o f
...
T
i
2
...
= al bl
T'
.
a or b q q o(P1) + o(P2)
=
=
o(P )
b
9
a
9
E
f ( P
P).
O(T,
z
where
z
two s e q u e n c e s t h e n
T' = a l b l b 2
O(P)
I f T i s made w i t h o n l y
.
Else
(sup(bl), inf(all
and we c o n s t r u c t :
z E f (P)
u ( T ' , P)
with
<
O(T,
P)
.
Hence i n c a s e 6 o r y , w e b u i l d a n o t h e r l i n e a r e x t e n s i o n on which w e c a n a p p l y t h e r e s u l t o f case a Thus, we have proved t h e f i r s t i n e q u a l i t i e s o f lemma 2 .
.
L e t u s now prove t h e e q u i v a l e n c e s t a t e d i n lemma 2 . 1 ) I f t h e r e e x i s t s such a l i n e a r e x t e n s i o n
T~
E f (P1), since
o(P2)
>
1
,
t h e r e e x i s t s a n o p t i m a l l i n e a r e x t e n s i o n of P 2 ,
T = bl b2 such t h a t 2 Therefore T = a l b a b a T ( s u p ( b l ) , i n f ( b ) ) i s a jump o f 1 2 2 3 E 1 ( P ) 2 and s i n c e ( s u p ( a 2 ) , i n f ( a 3 ) ) i s a jump o f T , w e h a v e :
.
U(T,
P)
< O(P1) +
O(P2)
- 1
. o(P) = o(P )
2 ) C o n v e r s e l y , i f w e suppose
1
+
o(P2) - 1
we n o t i c e t h a t as i n
t h e p r e v i o u s p r o o f o f t h e l e f t i n e q u a l i t y o f lemma 2 , we must n e c e s s a r i l y find a linear extension: sup(al)
< P
inf(bl)
,
T~
sup(b ) 2
b a b a E C(P) 1 1 2 2 3 i n f ( a 2 ) , and s u p ( a 2
= a
< P
,
with
n o t comparable w i t h
Comparability invariants
383
i n f ( a 1.
3
Thus t h e r e e x i s t s and
sup(a2)
= al x
T~
a2 a j E
i s n o t comparable w i t h
t h e v e r t i c e s of a
Proof of theorem 4 : -----__-----_---__
=
o(P ) 1
t
o(P2)
inf(a ) 3
2 '
I t j u s t r e m a i n s t o p r o v e f o r any
o(P)
sup(a ) < x , x < i n f ( a ) , 3 I P P and x i s n o t comparable w i t h
with
C(P1)
- 1
P2 E
, with
3'
u(P2)
>
1
S,
i f and o n l y i f
L e t u s look a t t h e e q u i v a l e n c e s t a t e d i n lemma 2 , we n o t i c e t h a t t h e r i g h t c o n d i t i o n depends o n l y on P1 and t h u s f o r any P 2 E ,? , w i t h o(P2) > 1 , the
c o n d i t i o n i s t h e same and t h u s we can t a k e t h e s m a l l e s t p a r t i a l o r d e r w i t h u(P2) > 1 , i.e. s2 . From t h i s theorem 4 , w e deduce: COROLLARY 1 : L e t P
1'
P
2
E .'P
and
x € PI
,
then
o(P )
2
>
1
implies
Proof: _ _ _ _ _ T h i s r e s u l t i s o b v i o u s l y o b t a i n e d by a p p l y i n g t h e above theorem 4
so (P2)
P
t o the posets:
p1 x
COROLLARY 2 :
and
P
1
x
The jump number i s a c o m p a r a b i l i t y
3 -invariant.
Proof: _---- S i n c e V P E 9 , o ( P ) < /PI , t h e p r e v i o u s c o r o l l a r y 1 e n a b l e s u s t o u s e t h e r e s u l t c o n t a i n e d i n t h e c o r o l l a r y 2 o f t h e theorem 3 w i t h a mapping I$ e q u a l t o t h e second p r o j e c t i o n mapping minus o n e .
AS-n-coNczusLo!! _______________ : I t seems t h i s approach h e l p t o c l a s s i f y a wide c l a s s o f c o m p a r a b i l i t y i n v a r i a n t s , f u r t h e m o r e it y i e l d s some o t h e r p r o b l e m s . L e t u s go back t o t h e f i r s t example c i t e d : t h e p a t h - p a r t i t i o n number. W e know from a theorem o f DILWORTH [ 6 ] , t h a t t h i s number i s e q u a l t o t h e c a r d i n a l i t y of a maximum i n d e pendent s e t i n t h e a s s o c i a t e d c o m p a r a b i l i t y g r a p h .
T h u s t h e r e i s no need o f t h e o r i e n t a t i o n t o compute t h e v a l u e o f t h e p a t h p a r t i t i o n number. T h i s f a c t c o u l d be g e n e r a l i z e d f o r o t h e r c o m p a r a b i l i t y i n v a r i a n t s and t h e r e f o r e , w e c a n a s k t h e f o l l o w i n g q u e s t i o n . How i n t e r p r e t o r compute a c o m p a r a b i l i t y i n v a r i a n t on t h e a s s o c i a t e d u n d i r e c t e d comparability graph ? Is t h i s p r o p e r t y always r e l a t e d w i t h min-max p r o p e r t i e s ? ( A s f a r as we know, t h e r e i s no such r e s u l t f o r t h e dimension o r t h e jump number)
.
@-O".EEEMEWs_ __________-----
:
We c o u l d l i k e t o t h a n k M. C H E I N ,
U. FAIGLE, R. SCHRADER, W. POGUNTKE and an anonymous r e f e r e e f o r t h e i r c a r e f u l r e a d i n g and h e l p f u l c r i t i c i s m s o f t h e f i r s t d r a f t s of t h i s p a p e r .
M . Habib
38 4
:Hi; 1
p1
2
Figure 1
[l]
-
[2]
- C. BERGE, Graphes e t h y p e r g r a p h e s , Dunod. Paris ( 1 9 7 3 ) .
[3]
-
M.
[4]
-
0.
[5]
-
W. H. CUNNINGHAM, J. EDMONDS, A c o m b i n a t o r i a l d e c o m p o s i t i o n t h e o r y . Canad. 3. Math., Vol. 3 2 , N o 3 , (1980) 734-765.
[61
- R.P. DILWORTH, A d e c o m p o s i t i o n theorem f o r p a r t i a l l y o r d e r e d s e t s ,
J.C. ARDITTI, G r a p h e s d e c o m p a r a b i l i t . 4 e t d i m e n s i o n d e s o r d r e s , Note de Recherches, CRM 607. Centre de Rech. Math. Univ. Montreal ( 1 9 7 6 ) .
CHEIN, M. HABIB, T h e j u m p number of d i g r a p h s and p o s e t s a n i n t r o d u c t i o n . Annals of Discrete Mathematics, 9 (1980) 189-194. COGIS, W. POGUNTKE, I. RIVAL, P e r s o n a l Communication.
Ann. Math. Ser. ( 2 )
51
(19501 161-166.
[71
-
[81
- T. GALLAI, T r a n s i t i v o r i e n t e r b a r e g r a p h e n . Acta Math. Acad. Sci. Hangar.
B. DUSHNIK, E.W. MILLER, P a r t i a l l y o r d e r e d s e t s . h e r . J. Math, 6 3 , (1941) 600-610.
18
(1967)
25-66.
[91
-
M.C. GOLUMBIC, C o m p a r a b i l i t y g r a p h s and a n e w m a t r o i d . J. Combin. Theory (B), 22 (1977) 68-90.
[lo]
-
M.C. GOLUMBIC, A l g o r i t h m i c g r a p h t h e o r y and p e r f e c t g r a p h s . Academic Press, N.Y. ( 1 9 8 0 ) .
1111
- M. HABIB, S u b s t i t u t i o n d e s s t r u c t u r e s c o m b i n a t o i r e s , theorie e t a l g o r i t h m e s . These Sci. Math. Universite P. et M. Curie
1121
(1981).
- M. HABIB, M.C. MAURER, On the X - j o i n d e c o m p o s i t i o n f o r u n d i r e c t e d g r a p h s . Discrete Applied Math. 1 ( 1 9 7 9 ) .
201-207.
Comparability in variants [I31
- T. HIRAGUCHI, On t h e d i m e n s i o n of p a r t i a l l y o r d e r e d s e t s . Sci. Rep. Kanazaws Univ. 1, (1351) 77-94.
[I41 - D. KELLY, W.T. TROTTER Jr., D i m e n s i o n t h e o r y f o r o r d e r e d s e t s , i n O r d e r e d s e t s . I. RIVAL ( e d . ) D. REIDEL Publish. Comp. (1980) 171-211. [I51
- M.C. MAURER, U n i c i t e d e l a d e c o m p o s i t i o n d ' u n g r a p h e en j o i n t s u i v a n t un g r a p h e j o i n t i r r k d u c t i b l e d ' u n e f a m i l l e d e ses s o u s - g r a p h e s . C.R.A.S., Paris, t. 282, serie A (1976) 289-292.
(161 - M.C. MAURER, J o i n t s e t d 6 c o m p o s i t i o n . s p r e m i i . r e s , d a n s l e s q r a p h e s . These de troisieme cycle. Universite P. et M. Curie, (1977). [I71
-
R.H. MOHRING, F.J. RADERMACHER, S u b s t i t u t i o n d e c o m p o s i t i o n f o r d i s c r e t e s t r u c t u r e s a n d connections w i t h c o m b i n a t o r i a l o p t i m i z a t i o n . Preprint.
[I81 - L.N. SHEVRIN, N.D. FILIPPOV, P a r t i a l l y o r d e r e d s e t s a n d t h e i r c o m p a r a b i l i t y g r a p h s . Siberian Math. J. 11, (1970), 497-509.
[I91
- W.T. TROTTER, J.I. MOORE, D.P. SUMNER, T h e d i m e n s i o n o f a c o m p a r a b i l i t y g r a p h . Proc. Amer. Math. SOC. 60, (1976) 35-38.
385
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Annals of Discrete Mathematics 23 (1984) 387-396
387
0 Elsevier Science Publishers B.V. (North-Holland)
C O D A G E S E T DIMENSIONS D E RELATIONS B I N A I R E S A . IiOUCHET Dkpartement d e MathGmatiques e t I n f o r m a t i q u e U n i v e r s i t t ? d u Maine R o u t e d e L a v a 1 - 72017 L E MANS C e d r x FRANCE ABSTRACT.-
5 E 1 X E 2 be a b i n a r y r e l a t i o n . The means t h a t ( e l , e . ) v e r i f i e s t h e r e l a t i o n .
L e t E l a n d E 2 be two s e t s a n d p notation
tf
,
p e2
-
2
A c o d i n g of p i n t o t h e complete l a t t i c e El T , (p2: E 2 + T s u c h t h a t e l p e 2
(pl:
+
T i s d e f i n e d by two maps (pl(el)
< W2(e2)
.
I f E l and E 2
a r e f i n i t e , t h e d i m e n s i o n o f p i s t h e m i n i m a l v a l u e o f d s u c h t h a t p c a n be coded i n t o a p r o d u c t o f d c h a i n s . I t i s shown t h a t d i s e q u a l t o t h e d i m e n s i o n o f D u s h n i k a n d M i l l e r when and p d e f i n e s a p a r t i a l o r d e r . A s a n a p p l i c a t i o n o f t h e main r e s u l t w e El E2 d e r i v e a s h o r t p r o o f o f Mac N e i l l c ' s c o m p l e t i o n t h e o r e m . The c o d i n g s i n t o B o o l e a n l a t t i c e s a r e more e s p e c i a l l y s t u d i e d .
7
INTRODUCTION Chaque r e l a t i o n b i n a i r e c o n s i d e r k e s e r a d b f i n i e s u r un p r o d u i t c a r t b s i e n E X E d e d e u x e n s e m b l e s q u i p o u r r o n t G t r e d i s j o i n t s ou n o n , f i n i s ou i n f i n i s . 1 2 Une t e l l e r e l a t i o n s c r a n o t C e [ E , , p, E 2 1 , p b t a n t l e s y m b o l e r e l a t i o n n e l . On Ccrira alors pour s i g n i f i e r que I e couple ( e l , e ) v b r i f i e l a r e l a t i o n . 2 clpe2 A i n s i , u n e r e l a t i o n d ' o r d r e p a r t i e l s u r u n e n s e m b l e E p o u r r a Ctre n o t e e [ E , <, E l . S o i t une r e l a t i o n b i n a i r e = (pl
[T, <, TI
: E
1 + T
et
.
Un c o d a g e (p2 : E 2
+
R = [ E l , p, E21
(Q,, T
(p,)
: R
t e l l e s que
+G
e t un t r e i l l i s c o m p l e t
est c o n s t i t u b p a r deux f o n c t i o n s
e l P e2
Q
vl(el)
< v21e2) . <, E l
c'=
Considerons m a i n t e n a n t une r e l a t i o n d ' o r d r e p a r t i e l [E, i m m e r s i o n d e o d a n s e e s t u n e a p p l i c a t i o n u, : E + T t e l l e q u e e l < e2 v ( e 1 ) < w(e,). L e c o u p l e ((p, cp) e s t a l o r s u n c o d a g e d e Q
. Une fl d a n s Z .
A i n s i l ' e x i s t e n c e d ' u n e i m m e r s i o n i m p l i q u e c e l l e d ' u n c o d a g e mais l a r e c i p r o q u e n ' e s t pas bvidente. Le r e s u l t a t p r i n c i p a l d e c e t r a v a i l c o n s i s t e e n un t h b o r h m e q u i admet p o u r c o n s k q u e n c e l ' b q u i v a l e n c e e n t r e c o d a g e s e t i m m e r s i o n s . On d C f i n i t a l o r s l a dimens i o n d ' u n e r e l a t i o n b i n a i r e R c o m e l e p l u s p e t i t e n t i e r d t e l q u ' i l e x i s t e un c o d a g e d e R d a n s Nd Le t h 6 o r b m e e s t a l o r s a p p l i q u b p o u r m o n t r e r qiie c e t t e n o t i o n d e d i m e n s i o n g 4 n b r a l i s e l a d i m e n s i o n d b f i n i e p a r D u s h n i k e t Miller [51 p o u r l e s o r d r e s p a r t i e l s . Finalement on s ' i n t e r e s s e aux codages c o n t r a c t b s q u i s o n t l a g r i n b r a l i s a t i o n l a p l u s f i d 6 l e d e s i m m e r s i o n s e n c e sens q u e t o u t c o d a g e c o n t r a c t 6 (cp 1 ' cp2): 0 +l5 d ' u n o r d r e p a r t i e l d a n s un t r e i l l i s c o m p l e t e s t t e l q u e tPl = cp2
.
.
Tous c e s r b s u l t a t s f o n t p a r t i e d e n o t r e t r a v a i l d e t h h s e p r e s e n t 6 e n 1971. L ' e x p o s e e n e s t i c i s i m p l i f i b . Les a l g o r i t h m e s d e r e c h e r c h e d e l a d i m e n s i o n q u i b t a i e n t a u d e p a r t l a m o t i v a t i o n p r i n c i p a l e o n t b t e l a i s s b s d e cste. En e f f e t , s i c e s a l g o r i t h m e s e t a i e n t p l u s p e r f o r m a n t s que l e s a l g o r i t h m e s connus 2 l ' e p o q u e , notamment g r z c e a u f a i t q u ' i l s t r a v a i l l a i e n t s u r une s o u s - r e l a t i o n g C n b r a t r i c e a u l i e u d e t r a v a i l l e r s u r t o u t un o r d r e p a r t i e l , i l s r e s t e n t n o n p o l y n o m i a u x , e n a c c o r d a v e c un r e s u l t a t r e c e n t d e Y a n n a k i s . L e l e c t e u r i n t b r e s s b p o u r r a d o n c c o n s u l t e r [2] p o u r u n e 6 t u d e p l u s c o m p l k t e .
A . Bouchet
388
Cette etude traite egalement l a d4termination des codages contractes dans N d alors que le present travail se limite h Zd. L'essentiel de la methode est inchange, il s'agit de couvrir la relation binaire complGmentaire par des paves maximaux. Signalons l'etude recente de Cogis [ 3 ] sur la dimension des relations binaires. Cette etude e s t dans son esprit plus combinatoire que notre methode algebrique qui donne des resultats valides dans les cas infinis. Le lecteur trouVera des applications aux sciences humaines et l'extension de certaines proprietes au cas infini dans le travail de Doignon, Ducamp et Falmagne [ 4 1
.
2 - DEFINITIONS ET NOTATIONS
D'une facon genbrale nous a w n s conservb les notations de 121 notions de base sont exposees dans 1 1 1
. Les
.
Soit une relation d'ordre partiel O = [ E , 5 , E]
.
tine section initiale
principale sera notee 2 (ensemble des y 5 x ) et une section finale principale sera
"
notee x (ensemble des y
5
x). Un sous-ensemble U C E (resp. I c E ) sera dit V-gcne-
rateur (resp.A -g&nerateur) si tout element x de E est borne supbrieure (resp.inferieure) d'un sous-ensemble contenu dans U (resp.1). On verifie aisement que U est V-ggnerateur (resp.1 est A-gengrateur) si et seulement si x (resp. x
Inf
=
(y n I))
rateur, on dit que : g Un
p1
X J &
n U)
pour tout x de E. Si U est V-gknerateur et I est A- gene=
[U, < , I] est une sous-relation generatrice de 0 ,
p2 d'une relation binaire R
X
Sup ( 2
=
=
[E1,p , E ] est constitub par
2
un sous-ensemble p, C E 1 et un sous-ensemble pg C E2 tels que e, € p1 et e2 e p2 e l P e2. =)
3 - THEOREME DU CODAGE
- Soit [U, 5 , I] une sous-relation generatrice d'une relation d'ordre 5 , E l . Alors pour tout x dans E, (2 n U) x (y n I) est un pave maximal de [U, 5 , I1 . ./ Preuve : Posons A : 2 n U et B = x n I. Si on considere a € A et b 8 B , on a : 3 . 1 .LEMME
partiel [E,
a
L
x
2
b. Ainsi A
X
B est bien un pa&
de [ti,
5,
11
.
S i ce pavC n'est pas maxi-
mal on peut trouver u € U \ A ou i € I \ B de telle fason que (A A
X (B
u {u}) x
B ou
lJ {i}) soit encore un pave. Dans la premiere eventualit&, u estun niinorant de B. Mais d'autre part
x est borne inferieure de B puisque I est A-generateur. I1 vient alors u contredit u € U \ A .
utilisant le fait que 3.2.THEOREME
-
u
5
x qui
i 8 I\B
Dans la seconde eventualit6 on contredit de &me
en
est V-gCn&rateur..
Soit un codage
( v l , v2)
: [U,
5 , I1
+
[T,
est une sous-relation generatrice d'une relation d'ordre
5,
TI o h g
0=
[E,
=
[U,
5 , El
5 , I]
Codages et dimensions de relations binaries
=
[T, 5
,
TI est u n treillis complet. Soient 0, : E
g l ( x ) = sup ql(ii
rl
+ 0
u),
des immersions de (I dans
(XI =
Inf
u,(X' n I).
Preuve : Montrons d'abord que l'application
(u")
u2(v).
codage.Ainsi
(il
x'
(1
I
=
fi I C
<" n
T definies par
0 , et 0, sont
est necessairement injective. (i
1
( u ' ) = , q l ( u " ) , et montrons que
(i
1
Cette derniere inkgalit6 implique u"
x'
-+
fl I.
Considbrons un Cl&ment v dans
5
: E
Les applications
On a alors u ' s v avec u ' € U et v € I. Ceci implique (cl
T et 0,
.
Soient donc u ' et u" deux elkments de U tels que u ' = u".
+
389
3'fl I.
(v). D'oh : 2 est un v puisque (ql, (u')
5 q
v2)
Nous etablirions de m6me l'inclusion inverse. Donc
I. Mais on a u '
=
Borne inf
puisque I est kgBn6rateur. Ainsi u '
=
(y' n I)
et u"
=
(x" fl I)
Borne inf
u".
On montrerait de facon duale que q2 est injective. Ainsi, si on pose
Q
P = ql(U),
=
q2(I)
et R
5 , Q] ,
= [P,
( q l , q2) est un codage bijectif de g s u r R.
Nous allons maintenant montrer que pour tout x dans E, on a 1'CgalitC : fi (A) q 1 ( ? nu) = 0 , ~P
n
n U) x cc 1 ( 2 n U)
L'ensemble (P
W
( x , n I) est un pave maximal de g d'apres le lemme (3.1 .).Donc v
n
x cc ( x I) est un pave maximal de R puisque ( e l , v2) est un 2 A v codage bijectif. D'autre part, l'ensemble ( 0 , ( x ) n P ) x ( g l ( x ) Q) est aussi un l'ensemble
-n
pave de R car on a a s 0 1 (x) 5 b pour tout a dans 0 1 (x) v Q. Montrons l'egalit6 de ces deux paves.
@,(x)
n
P et tout b dans
n
D' u n e
part, on sait que 0 , ( x ) est la borne superieure de q ( 2 (1 U). I1 1
en resulte :
i)
n (x)
0,
Ti P z.q 1 ( a n u ) .
D'autre part, le fait que q l ( n
ii)
v
x q2(x
n I)
est un pav6 de R signifie
I) majorent 0 (x) qui est la borne sup de q (? 1 1
que tous les elements de q2(? I1 en rCsulte :
n U)
w 0 , (x) fl Q
v 2 q2(x
n I). A
n U).
w
n P)
x ( 0 (x) ( 1 Q) 1 (1 I). Puisque le second de ces pavCs est maximal
Les inclusions i) et ii) montrent que le pave ( 0 (x), 1
contient le pave q (? 11 U) x q (x' 1 2 les deux sont kgaux. L'egalite (A) resulte alors de l'Bgalite de ces paves. Soient x et y dans E. L'inCgalitb x s y equivaut h 2
U est V-gbnerateur pour la relation d'ordre [E,
v1 ( 2 n U)
5,
El
n U _C
. L'inclusion
9
nU
puisque
precddente
C_ ( 9 U ) puisque q1 : U P est bijective. Cette dernihre inclusion implique g1 (x) = Borne sup q ( a n U ) s Borne sup 9, (9 n U ) = 0, (y). 1 P. D'oh : 'Inversement si on a 0 , (x) O 1 ( y ) , il vient @ , ( X I n P _C 0 , ( y ) v (P II U) _C ql (?( n U ) grsce h l'egalite (A). On remonte alors h x 5 y . Ainsi x y Bquivaut h
-+
1 Bquivaut h 0 , ( x ) s g l ( y ) , et 0 , est bien une immersion. La &me
montrerait de facon duale pour
g2.
propriCt6 se dB-
A. Bouchet
390
3 . 3 . COROLLAIRE - En conservant les notations du theorkme, s'il existe un codagc
4
e
(9 dans
Ze
-
, alors il existe une immersion
.
de (5 d a n s Z
APPLICATIONS Le theorbme prCcCdent permet une dtmonstration rapide du thearkme de
Mac Neille. Soit une relation d'ordre partiel (9 = [ E ,
5,
I:].
Soit M le sous-ensemblc
d e s parties d e E defini par P 6 M si et seulement s i P = E ou il cxiste X
que P
?. Le treillis complet M ( (9)
I-
=
=
[K,
5,
5" tel
t i ] est l'extension minimale de
XBX
0 en treillis complet. L'immersion y : x
-t
? est l'immersion canonique de 0 dans
M ( 0 ) .
e , alors
THEOREME (Mac Neille) - Si 0 est immergeable dans un treillis complet M( d ) est immergeable dans 0
.
Preuve : Considerons l'immersion canonique y : 0
M ( 0 ) , Tout Glement de M
est par definition une intersection d'el4ments de y(E) ; ainsi y(E) est A - g t n b rateur pour M(
0 1. Par ailleurs, tout bl4ment
de M est une section initiale de
8 , et une telle section initiale est union de sections initiales principales. Etant donne que y(E) est constitue de l'ensembledes sections initiales principales
6 ,y(E) est V-genGrateur pour M( 0 M( f i ) sur Y(E) est generatrice. de
) . Ainsi la sous-relation g induite par
Considerons maintenant une immersion(c:fl+ G immersion de g dans G
. Puisque
.
Alors
$
o y
-1
est une
g est une sous-relation gbneratrice de M( 0 ) , il
suffit d'appliquer le corollaire.. Supnosons maintenant quc 0 est une relation d'ordre partiel sur un ensemble fini. La dimension de 0 est le plus petit entier d tel qu'il existe une immersion
(c
:
0
-f
INd. Par extension, nous appelons dimension d'une relation bi-
naire finie R le p l u s petit entier d tel qu'il existe un codage
((c
1
, V ) de R dans 2
N d , Le coroIlaire du theorbme nous permet d'affirmer que ces deux notions de dimension coincident lorsque l'on considkre un ordre partiel. 5
-
CARACTEKISATION DE LA DIMENSION D'UNE RELATION BINAIRE
Soit une relation binaire finie R =[E 1 , p , E2]. Pour tout e l 8 E 1 on pose L(ei) = {e, 6 E2 : e l p e2} et on dit que L(e ) est une ligne de R. Pour tout I e 2 8 E2, on pose C (e ) = { e € E, : e l p e21 et on dit que C ( e , ) est une colonne 2
de R.
La relation R est une relation de Ferrer si a l
p
b2, c,
p
b2, c 1 p d g im-
pliquent a l p d2.
5.1. PROPOSITION [ 6 ] - Les quatre propriCtes suivantes sont equivalentes :
i) ii)
R est une relation de Ferrer, Les lignes de R sont totalement ordonnbes par inclusion,
Codages et dimensions de relations binaries iii) iv)
39 1
Les colonnes de R sont totalement ordonnCes par inclusion,
R est de dimension 1.
En fait, seules l e s trois premitres bquivalences sont Ctablics dans [ G I . L'Gquivalence de iv) avec ii) ou iii),est B peu prPs Cvidente. Si on reprksente
?
R par une matrice de 0 et de 1 et si on ordonne les lignes et colonnes selon les ordrcs totaux dCcrits en ii) et iii), on obtient la forme d'un escalier ; d ' o h le nom quc nous donnons en [ l ]
3. c e s relations.
-
Une relation binaire finie R =[IJ1,p , E 2 1 est codable dans N si e t seulemcnt si elle est Cgale B l'intersection de d escaliers.
5.2. PROPOSITION
Preuve : Soit un codage ( U considgrons q . j : E. Soit ij
p j , E2] la
I
, q,L )
: R
Nd
.
Pour chaque i
=
1
1,2 et j 2
=
N de telle facon que q.(e.) = (V. (e.),V. (e.) 1
1
1
1
relation binaire dbfinie par e l p j e 2
1
1
1 ,2,.
d
. . ,d
,..., qid (ei)).
y , j ( e l ) s q2'(ez).
Le couple (qlj,q2j) est Cvidemment un codage de 6' dans N ; ainsi [ j est un escalicr. Par ailleurs q (e ) 9 q2(e2) e s t vCrifibe si et seulement si (ilJ(el)s q (e 1 1 2 2 est vkrifiP pour tout j = l , Z , ..., d. Ainsi R est l'intersection des escaliers *,- 1 1 r, 2 , . . . , r , d
.
Rkciproquement, si R est donnee comme intersection d e d escaliers 6 1, F 2 ,...,Ed, on construit pour chaque j = 1,2 ,..., d un codage (U,', U),' E J + N e t on en dCduit un codage ( q 1 , V2) : R +Nd.
:
5.3. COROLLAIRE - La dimension d ' u n e relation binnire est l e nombre minimal d'escaliers d o n t elle est l'intersection.
5.4. COROLLAIRE - La dimension d ' u n e relation binaire est le nombre minimal d'escaliers dont sa complbmentaire est l'union. Pour l e second corollaire, i l suffit de remarquer que l e compl6mentaire d'un escalicr est un escalier d'aprPs l a caractkrisation ii) des escaliers. Cette caractsrisation de la dimension par la complCmentaire d e R est particulihrement intCressante d a n s le cas de la 2-dimension dont nous faisons maintenant une etude directe. 6 - CODAGES BOOLEENS
L'ensemble des parties d'un ensemble D est not6 P(D). Soit unc relation binaire R
= [E,,p
, E2] qui n'est plus nkcessairement
' P (D )
ct une application V2 : E 2 - + P ( D ) . Pour chaque d dans D, on note P l d le sous-cnsemble d e s 61Cments e, de E l tels que finie, un ensemble D, une application
(4
: El
de Ez tels que d k? V2(e2).
La
A. Bouchel
392 Si
( v1'
( c ) est un codage de R dans P(D) muni de l'ordre induit par
2 l'inclusion, on dira que c'est un codage booleen de R. La dimension booleenne ( o u 2-dimension) de R est le cardinal minimal de D tel que le codage existe. 6.1.
PROPOSITION
-
Le couple de fonctions (VlLq2)
est un codage booleen si et
seulement si le domaine de definition de la relation complementaire de R est @gal B l'union des sous-ensembles de la famille ( Pd 1 ~ Pd2 ) d
.
.
d d Supposons que ( v ,(c ) est un codage et montrons U P 1 x P2 1 2 d€ D que P est le domaine de definition de la relation complementaire de R. Soit (el,e2) Preuve : Posons P =
dans P, et raisonnons par equivalences successives. d d (el,e2) € P 9 d € D : el 6 P 1 et e2 € P2 Q
,
3 d 8 D : d € ql(el) et d G I q(e2), Qt
tCl(el)
v2(e2),
ca
e l p e2 non verifie.
Reciproquement, supposons que P est le domaine de definition de la relation complementaire de R. On reprend le raisonnement par equivalences successives en partant de la dernigre assertion. On arrive alors B l'avant-dernikre assertion. Ainsi ( q ,q est bien un codage.. 1 2 6.2. COROLLAIRE
-
La dimension booleenne de R est egale au cardinal minimal d'une
famille de paves couvrant la relation complementaire de R. 7
-
CODAGES CONTRACTES Soit un treillis completT( et une sous-relation [ P , s
'P
, Q]
de t
. Notons
l'ensemble des bornes superieurcs des sous-ensembles contenus dans P et Q'
l'ensemble des bornes inferieures des sous-ensembles contenus dans Q. La relation [P,
2
, Q ] est contractee si P CQ' et Q CP'.
p , E 1 et un codage ( ( c 1 % ) : R + T ( . Le codage (V1,V2) 1' 2 1 2 est contract6 si la sous-relation image [(c (E ) , s , cc. ( E ) ] est contractee. 1 1 2 2
Soit R = [ E
-
Le codage (CC. , q ) : R +T( est contracte si et seulement s i les ega1-2 lites suivantes sont verifiees pour tout e dans E et tout e dans E2 : 11 -2 ql(el) = inf v2(L(el)),
7.1. LEMME
q2(e2)
=
sup v1(C(e2)).
Preuve : Posons P = v,(El) et Q
=
v2(E2).
Les egalites de l'bnonc6 sont evidem-
ment suffisantes pour avoir P CQ' et Q CP'.
Reciproquement, supposons P cQ.et
considerons un bl&nent e, de El. Les sous-ensembles q2(L(el)) egaux car (Vl,V2) est un codage surjectif de R s u r [P,s , Q ]
et
.
\ I (e )
c+
1
1
nQ
sont
Codages et dimensions de relations binaries
393
Pour demontrer la premiere Cgalite du lemme, il suffit donc de montrer que vl(el)
=
v
-
inf (Vl(el) nQ). L'616ment %,(el) est dans P et l'on a PCQ' ; il
(4 (e ) = Inf K. Le sous-ensemble K est contenu dans 1 1 v v V1(el) n Q ; on a donc Inf K 5 Inf (vl(e ) nQ). D'oL v (el) 2 Inf ($,(el) flQ). 1 1 L'inPgalite opposee s'obtient en remarquant que v (e ) est un minorant de
existe donc K C Q tel que
\ I
v 1 (e1 ) n Q .
1 1 On a donc bien l'egalitb desir6e. La deuxikme partie de 1'Cnonce s'ob-
tiendrait de facon duale en partant de Q C P ' . Soit une relation d'ordre partiel &ratrice
g
[U,
=
5,
o = [E, 5 , Elet
une sous-relation g6-
I1 de cet ordre partiel. Considerons toujours le treillis
( v1, ( 4 2)
complet 5 et un codage
prolonge l e codage ( q , V 1 2 % i I.
: g
+
Z
.
On dira qu'une immersion cc : 0
si q1 est la restriction de
(i
+
5
B U et (i2 est la restric-
7.2. PROPOSITION - Si l e codage (Ul,v2) est contract6 alors les immersions 0 et 1theoreme 3. 2. sont des prolongements de ce codage.
g2 du
Preuve : Posons P = vl(U) et Q
o1(x) =
Sup
vl( 2
V2(I).
=
Soit x un element de E. Rappelons que v
U) et 0,(x)
Inf v2(x
=
un pave de la sous-relation [U,
5,
n I).
L'ensemble ( 2
I]. L'ensemble ql(?
n U)
x
n U)
(z
x
v2(v" n I)
(I I) est est donc
un pave de la sous-relation [ P , i _ , Q ] puisque (v1,v2) est un codage. I1 en rksulte sup q1( 2
ri U)
Inf
2
v 2 G fi I)
; soit :
0,
0, (x)
(A)
(x)
Consid6rons maintenant un element e, dans U. L'element q (e ) appartient B
vl(^el fl U)
1
(B)
Vl(el)
=
vl(el)
5
@,(el).
Puisque ( ( 4 , v ) est contracte, on sait par le l e m e que : 1 2 v Inf v2(L(el)). Dans ce cas L(el) est egal B e l n I. L'CgalitG pr6cPdente
est donc equivalente B V. ( e ) = @,(e,).
1
Vl(el)
=
1
; d'oL :
@,(el) = @,(el).
En combinant avec (A) et (B) il vient alors
1 On montrerait de fason duale 9 (e ) 2
2
=
0 (e 1
) =
2
02 (e2)
pour tout e2 dans 1.m Definition : Soient deux relations binaires R = [El, p , E2] et R'
=
Un isomorphisme de R' sur R est un couple de bijections i, : El1
+
[El1 , p ' , E >
1.
El,
i2 : El2 + E2, telles que e' p ' el2 a il(e' 1) p i2(er2). La relation R ' peut ttre un ordre partiel - et alors E ' = El2 - alors que la relation isomorphe R peut 1 stre definie entre deux ensembles distincts - El # E2 -
.
7 . 3 . COROLLAIRE
-
Soit une relation binaire R =[El, P , E21, un treillis completz
et un codage contract6
( v1 2v2)
: R +'7?
. La relation binaire R est
une relation d'ordre partiel si et seulement si tC1&q telles que vl<E1)-q
(E ) 2-2--'
isomorphe
2 sont deux injections
,
A . Bouchet
394
-
Preuve : La condition suffisante est 6vidente puisque (q1,(r2)
.
[(c (E ) , s , V (E ) I 1 1 2 2 K&ciproquement, consid6rons un ordre partiel 61 = [E,
phisme de K sur l'ordre partiel phisme (il,i2) : 0
R. Le couple de fonctions (i o
-f
avec ce codage contract6 et (!/ =
i
o q1
i2 o
=
< , Llet un isomor-
i2 o CC2) est un codage Appliquons le thCor&me 3.2. et la proposition prtcedente
.
contract6 de c." dans 5 que 8
Ptablit un isomor-
v2.
=
(cl,
g. On trouve alors une immersion 0 : 0
+
6 tellc
Une immersion d'un ordre partiel dans un treillis est
necessairement une injection. Ainsi 0 est une injection et il en est nbcessairepour (cl et v2. Enfin on a q ( E ) = tF (E ) = 0 ( E ) . W 1 1 2 2 Ainsi les codages contractbs g6nCralisent parfaitement les immersions
ment de &me
car pour tout ordre partiel
,
6 et tout treillis complet
( V ,q ) :
1
2
0
+
G est
un codage contract; s i et seulement si Y1
= 'p et cette fonction est une immersion. 2 Revenons au cas d'un codage booleen ( ( c l , q 2 ) : R P ( D ) . Nous avons vu que la donn6e d'un tel codage equivaut B celle d'une famille (P, d x P2)deD d de -f
pav6s couvrant l a relation complementaire de I?. 7.4.
PROPOSITION
-
Le codage booleen (vl,q2)
chacun des pav6s de la famille (P:zPi)dt?.D
est contract6 si et seulement si est un pave maximal de la relation
compl6mentaire de R. Preuve : Rappelons qu'un pave P
,
x P2 d'une relation R est maximal si et seulement
si : P1
=
{el 6 E l : P2 C L (e,)]
P2
=
{ e , 6 E2 : P1
et C
C
(e,))
Nous nous intbressons ici aux pav6s de la relation compl6mentaire de R. Pour cette relation compl6mentaire de R , notons donc L'(e un 616ment e , t?. E
1
et C'(e
2
) l a colonne relative
1
ment L'(e 1 ) = E2 \L(el) et C'(e2) = El\ C(e2). Un pave Pd 1 pour l a relation compl6mentaire de R si et seulement si : P: = {el 6 E~ : p2d c ~ ' ( e , ) ~ et
d
) la ligne relative 5
B un 616ment e 2 6 E2 .On a 6videmd X
P2 sera donc maximal
d
P 2 = {e, € E g : P, cC'(e2)}
Appliquons le lemme et consid6rons l a premikre condition caractbristique d'un codage contract6 : V e l 6 E , : tcl ( e l ) = Inf p2 (L (el))
Raisonnons par equivalences successives B partir de cette condition. Nous obtenons successivement :
e 1 € E l , V d t?. D : (d 6 vl(el)CV
e2 6 L(el) : d 8 q2(e2))
Codages el dimensions de relations binaries V el 6 El, V d 6 D Ve
: P
: (el 8
G V e2 6 L(el) : e 2 !6 P2d )
€ E , , V d € D : (el € Py
1
L ( e l ) c E 2 \ P 2d
V e l € E , , V d 6 D : ( e l 8 P, -P;
V d
€
D : P;'
=
{el 6 El
395
CL'(e,))
: P2d C L ' ( e l ) }
D e facon duale, la deuxibme condition caracterisant un codage contract4
donnerait par equivalences successives la deuxibe condition caractbrisant un pavb maximal de la relation complbmentaire de K. W La proposition prbc6dente peut Stre gbnkralis4e dans le cas d'un codage
dans INd ; ceci est fait dans [ 2 1
.
La proposition suivante montre que les codages
contractCs ne sont pas moins g4nciraux que les codages.
7.5. PROPOSITION - S i une relation binaire K admet un codage dans un treillis complet Z
,
alors elle admet aussi un codage contract6 d a n s Z
Preuve : Soit [P,
5,
.
Q ] la sous-relation image par le codage de R dans
5,
allons montrer qu'il existe un codage de [ P , relation contractke de Z
. La
'e . Nous
Qldont l'image soit une sous-
composition des deux codages donnera le codage con-
tract6 d4sirC. Soit f la fermeture dCfinie s u r e ayant pour ensemble de fermes l'ensemble Q ' d e s bornes infkrieures d e s sous-ensembles de Q. Pour tout blkment x de % et pour tout Clkment y invariant par f, la relation x s y est v4rifi4e si et seulement s i f ( x ) d e [P,
5 , Ql
5
y. P.insi le couple de fonctions (f,l) permet de definir un codage
sur [P,
5,
(A)
Ql avec ? P C Q'
=
f (P). On
n
alors :
Considerons maintenant la fermeture duale g dCfinie sur 5 ayant pour ensemble d'invariants l'ensemble P+ des bornes superieures des sous-ensembles d e P. De &me que ci-dessus, le couple (1,g) permet de dkfinir un codage de [P,5 ,Q] sur
[P,5 ,q]
avec
3 (B)
=
g ( Q ) . On a alors :
Q
5 Pt
La composition des deux codages precedents donne un codage sur Compte-tenu de (B) il reste 5 montrer q u e
FQ'pour
4tablir que [ P ,
5
[P, 5 ,a]
,Ti] est
une
sous-relation contractee. D'aprks (A) tout 614ment z 8
P est
borne inferieure d'un sous-ensemble
A C Q . Soit z' la borne inferieure d e g ( A ) . L'GlCment z minore A et c'est un in-
variant de l a fermeture duale g ; ainsi z minore g(A) et z
5
z'. Chaque element
de A majore son image dans g ( A ) puisque g est une fermeture duale ; ainsi la borne inferieure de A majore la borne infkrieure de g(A) et z est donc borne inferieure de g ( A ) , et l'on a bien
CG'.
2'.
L'elCment z
.
396
A . Bouchet REFERENCES
[I
1
G.BIRKOFF, Lattice theory, A.M.S.Colloquium Publications, 1948.
[2] A.BOUCHET, Etude combinatoire des ordonnes finis, Applications, ThPse de Doctorat d'Etat, Universite Scientifique et Medicale de Grenoble, 1971. [31 O.COGIS, On the Ferrers dimension of a graph, Discrete Mathematics 38 (1982) 47-52. [41 J.P.DOIGNON, A.DUCAMP, J.C.FALMAGNE, On realizable biorders and the biorder dimension of a relation, 1 paraftre. [5] B.DUSHNIK, E.W.MILLER, Partially ordered sets, Amer.J.Math.63 (1941) 600-610. [6]
J.RIGUET, Les relations de Ferrers, C.R.Acad. Sci. Paris 232 (1951) 1729-1730
Annals of Discrete Mathematics 23 (1984) 397-408 0 Elsevier Science Publishers B.V. (North-Holland)
397
THE SPERNER PROPERTY
*
Jerrold R. Griggs
Department of Mathematics and Statistics University o f South Carolina Columbia, S.C. 29208 USA Dedicated to Professor Corominas A finite ranked poset has the Sperner property if no antichain is larger than the largest rank in P. This was first shown to hold in the lattice of subsets of an n-set by Sperner in 1928, Efforts to prove analogues of Sperner's theorem for other posets have led to the emergence of an entire theory. A big advance has been the introduction into the field of techniques from linear algebra, category theory, and even algebraic geometry. This is a survey of progress made in this area over the past few years.
R6 s um6 Soit Ai @ A
j
{Al,
A2,.. Akl
d5s que
une famille de sous-ensembles de
i # j. Pour n
,
{l,
...,n}
telle que
fix6, quelle est la taille maximum d'une telle
),;,(
famille? En 1928, Sperner [Sr] a montr6 que k
I;
.
Cette borne est la est Ai El,.. .,nl ou de tous
meilleure possible et il y a 6galite' si et seulement si la famille des la famille de tous les sous-ensembles les sous-ensembles 2
[I
61gments.
a'
E]
6lgments de
En termes d 'ensembles partiellement ordonngs (posets) le thgorgme de Sperner se traduit par: dans l'algkbre de Boole Bn de tous les sous-ensembles de {l,, ,n] (ordonne's par inclusion), aucune antichaine n'est plus grande que le rang maximum. Quelques quarante ans plus tard, Rota [ R o ] 6mit une cglsbre conjecture:le treillis n des partitions de 11, n1 (ordonngs par raffinement) 9 aurait la meme propriet; ou plus prgcisement, aucune antichaine ne contiendrait plus d'glgrnents que le maximum (pris pour h variant del'an) du nombre de partitions de {l,. ..,n1 en h parties. Cela est vrai pour des petites valeurs de n, mais, il y a presque dix ans, Canfield [CL] a montrg que cette conjecture est Cet ordre de grandeur fausse lorsque n est suffisamnent large, n ?I 6 x 5X1O6 [Sh, p. 3 1 . a e't6 abaissc depuis 'a n
..
...,
Bien que cette conjecture soit fausse, les essais pour la prouver ont entrain6 la dgcouverte de toute une thCorie s u r les "posets" qui satisfont la proprigth du thdorbme de Sperner. D e rgcentes contributions 5 ce domaine tr6s vivant ont utilisg, avec succSs, des outils empruntgs $ l'alggbre linGaire, la
* Supported
in part by the National Science Foundation and by a U.S.C. Research and Productive Scholarship.
J.R. Griggs
398
thgorie des catGgories, la programmation liniaire et mgme 5 la g8omGtrie alggbrique. Ici notre principal intsrgt portera sur les relations entre la proprigti. de Sperner et les conditions d'existence de certaines familles d'ensembles totslement ordonnis dans un "uoset" et de certaines transformations lingcires dansl'espace vectoriel associg au"poset". Une synth'ese plus complgte doit etre diff4r6 mais la bibliographie couvre tout l e domaine. Pour de larges synthhses couvrant entiirement l e domaine de la thgorie des ensembles extrgmaux, voir [ C K 3 , We21.
1,
INTRODUCTION Suppose A1 ,...,4, are subsets of
i=j.
{l,.
.. ,n)
such that
How large can such a collection of sets be, for given
[Sr] showed that only if the A.'s
k
5
(Ln;2J),
are all the
Ai a; Aj n?
unless
In 1928 Sperner
This bound is sharp, with equality holding if a n d Ln/2l-subsets of
...,
11,
111,
or all the
rn/2~1-
subsets. In the language of posets, Sperner's theorem states that in B the Boolean algebra of all subsets of {l,..,,nl (ordered by inclusion), no anpichain is larger than the largest rank. Some forty years later Rota [no] made a famous conjecture that nn! the lattice of partitions of {l,...,nl (ordered by refinement), would have a similar property: No antichain should contain more elements than the nl into k parts. This maximum, over k, of the number o f partitions {l, is true for small n , but it was almost ten years before Canfield [Cl] showed that the conjecture fails for sufficiently large n, n 6 X l o z 4 , This has since been lowered to n T, 5 x l o 6 [Sh,P3].
...,
Although the conjecture failed, attempts to prove it inspired the discovery of an entire theory about posets which satisfy analogues of Sperner's theorem. Recent contributions to this lively field have successfully employed tools from linear algebra, category theory, linear programming, and even algebraic geometry. Here we shall focus on connections between the Sperner property and conditions on the existence of certain chain collections and linear transformations. A more complete survey on the Sperner theory must be postponed, but the list of references covers the whole area. For broad surveys of the entire field of extremal set theory, see [ G K 3 , We2]. 2.
DEFINITIONS
All posets in this paper are assumed to be finite and ranked (graded). A poset P has rank n if it is the disjoint union of nonempty subsets Po,Pl,:..,P n' called ranks, where Po is the set of minimal elements of P and for all 1 Pi+l' whenever x E P. and y covers x, then Y Each rank P. is an antichain (totally unordered set). P has the Sperner property if no antichain in P is larger in size than the largest rank. Thus Bn has the Sperner property, but En does not in general. A k-family in P is a subset of P such that no (k+l) elements lie on the same chain. Equivalently, a k-family is the union of k (or fewer) antichains. For example, the union of k ranks of P is a k-family. In 1945 Erdb's [ E l generalized Sperner's theorem by showing that f o r all k the union of the k is a k-family of maximum size. Motivated by this example, largest ranks in B we say P is k-Speker if no k-family is larger than the union of the k largest ranks. P is strongly Sperner if it is k-Sperner for all k. This notion of strongly Sperner posets is important in the theory now emerging.
The Sperner property
399
Many r e s u l t s a l s o d e p e n d c r i t i c a l l y on p r o p e r t i e s o f t h e s e q u e n c e o f W h i t n e y ( r a n k s i z e s , \ P i \ ) . A p o s e t P of r a n k n i s s a i d t o b e r a n k - u n i m o d a l i f , f o r some j ,
~numbers
I i....5
[ p [ i: [ P 0 1
(P.1 J
s*..<
\Pn/.
P is rank-symmetric i f f o r a l l k \ P k / = IPn-kl. P i s Peck i f i t i s r a n k s y m m e t r i c , r a n k - u n i m o d a l , a n d s t r o n g 1 . y S p e r n e r . Many i n t e r e s t i n g e x a m p l e s of S p e r n e r p o s e t s t u r n o u t t o be P e c k . 'These i n c l u d e B ; D , t h e l a t t i c e o f th:! l a t t i c e o f s u b s p a c e s o f d i v i s o r s o f n ( o r d e r e d by d i v i s i b i l . i t y ) ; a n d I. ((1): a n n - d i m e n s i o n a l v e c t o r s p a c e u v e r (:F(q) ( o r d e r E d by c o n t a i n m e n t ) . A c h a i n i n P i s s a t u r a t e d i f f o r some i 5 j i t meets c o n s e c u t i v e r a n k s i , i + l , .. . , j a n d meets no o t h e r r a n k s . A c h a i n i s s y m m e t r i c i f i t i s s a t u r a t e d and j = n - i . m denotes a chain with m elements.
The ( d i r e c t ) p r o d u c t o f two p o s e t s , I' a n d Q , i s d e n o t e d by P x Q a n d h a s i f p 5 p' and e l e m e n t s { ( p , q ) : p c P , q t Q } o r d e r e d by ( p , q ) 5 ( p ' , q ' ) q 5 q'. The q u o t i e n t i f a r a n k e d p o s e t P u n d e r a g r o u p o f ( o r d e r - p r e s e r v i n g ) a u t o m o r p h i s m s C i s d e n o t e d by P/C. I t s e l e m e n t s are t h e o r b i t s of P u n d e r G , o r d e r e d by A S B i n P/C i f t h e r e e x i s t a E A , b E B s u c h t h a t a 5 b i n P .
3.
C H A I N CONDITIONS
For any c h a i n C and k-tamily t i t i o n C o f P i n t o c h a i n s c:,
The C r e e n e - K l e i t m a n (ranked o r not) has h o l d s above. Thus t h a t t h e sum o n t h e
I C n FI S M i n ( l C / , k ) ,
F,
Thus f o r a n y p a r -
theorem [ C K l ] a s s e r t s t h a t f o r a l l k e v e r y f i n i t e p o s e t P a k - s a t u r a t e d p a r t i t i o n , which i s a C s u c h t h a t e q u a l i t y P is k-Sperner i f and o n l y i f t h e r e i s a p a r t i t i o n C s u c h r i g h t e q u a l s t h e sum o f t h e k l a r g e s t W h i t n e y n u m b e r s o f P .
An e s p e c i a l l y n i c e t y p e of p o s e t i s a s y m m e t r i c c h a i n o r d e r . This is a poset P which c a n b e p a r t i t i o n e d i n t o symmetric c h a i n s . T h i s p a r t i t i o n s u f f i c e s t o g u a r a n t e e t h a t P i s k-Sperner f o r a l l k . Indeed, P i s Peck. Some e x a m p l e s o f s y m m e t r i c c h a i n o r d e r s i n c l u d e B n , D n a n d I,,(q). A18cther i n t e r e s t i n g p o s e t i s i n t e g e r s i n t o p a r t s 3 = (al,a2, ordering is
-f
-+
a 5 a'
if
a . 2 a! 1
1
L(m,n),
..., a m ) ,
t h e l a t t i c e of a l l p a r t i t i o n s of w h e r e n 2 al 2 a2 t...> a t 0 . The
for all
-f
i.
a
m
has rank
Cai.
Equivalently,
L(m,n) i s t h e l a t t i c e o f o r d e r i d e a l s o f 2 x 2 ( o r d e r e d by i n c l u s i o n ) . L(m,n) c a n a l s o b e v i e w e d as t h e F e r r e r s d i a g r a m s w h i c h f i t i n t o a n mxn r e c t a n g l e . -f L(m,n) is rank-symmetric s i n c e f o r any a of r a n k k t h e r e c o r r e s p o n d s t h e n-al) o f r a n k mn-k. I n t h e a b s t r a c t [ S t l ] S t a n l e y announced e l e m e n t (n-a m i s S p e r n e r f o r a l l m a n d n . T h e r e i s a c h a i n c o n d i t i o n , (;t), w h i c h t h a t L(m,n) S t a n l e y showed i s t r u e f o r L(m,n) and which i m p l i e s t h e S p e r n e r p r o p e r t y . A r a n k - s y m m e t r i c p o s e t P o f r a n k n s a t i s f i e s c o n d i t i o n (") i f f o r e a c h i=O,l, I.n/ZJ, t h e r e exist \ P i \ d i s j o i n t s a t u r a t e d c h a i n s g o i n g f r o m Pi U P to P n-1
,...,
..., ..
The c o l l e c t i o n s o f c h a i n s i n (") c a n b e u s e d t o c o n s t r u c t a p a r t i t i o n c of I c h a i n s . To do t h i s , t a k e t h e s u b c h a i n s b e t w e e n P . a n d i n t o only IpLn/2J a n d P . f r o m t h e c h a i n s g o i n g b e t w e e n Pi a n d P P. a n d b e t w e e n Pn-i-l n-i' 1+l 11-1 no t h i s f o r e a c h i < Ln/21 a n d c o m b i n e t h e s u b c h a i n s t o f o r m c. From c i t
P
400
J.R. Griggs
follows that P is Sperner and rank-unimodal. I noticed [Gg3] that in fact P must be k-Sperner for all k, for in the same spirit one may construct, for each k, a chain partition Ck such that C min(k, C. ) is the sum of the k largest Thus we may conclude that if' P is rank-symmetric and satisWhitney numbers. fies ( * ) , then P is Peck. Conversely, if P is Peck, the k-saturated partitions of P (which exist by the Greene-Kleitman theorem) can be used t o generate the chains needed for ( A ) , and P is rank-symmetric because it is Peck.
I I
is Peck. We can now see that Stanley's (") condition implies that L(m,n) It remains an open problem t o determine whether it is a symmetric chain order. This has been confirmed after considerable effort for Min(m,n) 5 4 [LnZ], Wel, Ri], but it may require some "global" technique t o prove this in general. Note that for different values of i, the chains in (A) may differ considerably, s o that a Peck poset need not be a symmetric chain order. (Figure 1 is an example of such a poset.)
Fig. 1 The equivalence above between (*) and Peck for rank-symmetric posets inspired me to probe for connections between the strongly Sperner property and chain conditions without restrictions on the Whitney numbers [Gg3]. Here is an appropriate way to extend (*) to arbitrary ranked posets. P satisfies condition T if for all k there exists a collection of disjoint chains in P which each intersect each of the k largest ranks and which cover the kth largest rank. This reduces to (*) if P is rank-symmetric and rank-unimodal. (What I originally called "condition S" is now called the strongly Sperner property. I named condition T after Stanley.) Here are the results which generalize the equivalence of the Peck and ( * ) conditions. If P is strongly Sperner, then it satisfies condition T. (This again can be proven by the existence of k-saturated partitions [Gg3]. Another proof employs network flows [GSS].) The converse is false, in general. For example, the poset in Figure 2 satisfies T but is not even Sperner.
Fig. 2 However, if P is rank-unimodal and satisfies condition T, then it is strongly Sperner. The original proof [Gg3] required the existence of simultaneously k- and (k+l)-saturated partitions, but a simpler, elementary proof has been found [GSS].
40 1
The Spernerproperty 4.
LINEAR ALGEBRA
In a landmark paper [St3], Stanley introduced linear algebra as a tool in studying the Sperner property. This has created a new wave of activity in Sperner theory. A s he announced earlier in the abstract of [Stl], the fact that L(m,n) is Sperner follows from its satisfying the chain condition ( * ) . This condition is implied in turn by a third condition involving linear transformations. Here is the actual theorem from [St3], which included the equivalence of (") and Peck, from [Gg3], discussed in the last section. denotes the complex vector space Pi with basis P..
Theorem 1. Let P be a finite rank-symmetric poset of rank following three conditions are equivalent: (i)
P
is Peck.
(ii)
P
satisfies
(iii)
I-
i
oi: Fi
+
ii+l
(a)
0i i
For
<
i
n, there exist linear transformations
For each
with the following two properties:
Ln/2], the composite transformation
-
...
$n-i-l @n-i-2 (b)
The
( 9 ~ ) ~
0
For
n.
i, 0
@i+l @i : Pi
-
+
Pn-i
i i n, and each
i
x
E
is invertible. Pi, Qi(x) =
C
cYy Ycpi+l
c = 0 unless x < y. Y Condition (iii) permits the introduction of algebraic tools to prove posets are Peck. Stanley employed techniques from algebraic geometry to prove that certain posets QX, derived from a class of algebraic varieties X, are Peck. An important instance of this is when QX is defined in terms of the Bruhat order of a Weyl group. These include, in turn, L(m,n) and M(n). M(n) is the poset with elements being the subsets of 11,. ,n], ordered as follows: where
..
,... ,a.] J
Suppose A = {a, with
al <...<
for 0 M(n)
=
2
i
2
a
j'
b
and bk.
B
1 For instance,
j-1.
where
<...<
P
=
Z1.
{bl
=
,...,bk 1
Then A
2
B
are subsets of
if
{1,3) i {1,2,4}.
j 5 k
and
{l,.. .,nl
a. 3-i
bk-i
One may also define
,
Now consider a seemingly unrelated conjecture of Erdzs and Moser: For any set S of 2L+1 distinct real numbers, the maximum number of subsets of S whose element sums are equal is attained by taking S = {-L, -L+l,. , L l and all subsets of S whose element s u m s are zero. Lindstrh [Lnl] noticed a connection between this conjecture and M(n), and he conjectured that M(n) is Sperner. Stanley came across M(n) as one of the posets QX and proved it is Sperner, indeed Peck, without any knowledge o f the work of LindstrEm or Erdgs and Moser. Harper pointed out the connection, and Stanley went on to prove the Erdgs-Moser conjecture and generalize it. Thus an elementary number theory problem was solved via algebraic geometry and Sperner theory!
..
Stanley's solution of the ErdEs-Moser problem can be reduced to using linear algebra in place of algebraic geometry but this proof is still motivated by deeper considerations [Prl, P3]. A more elementary proof that L(m,n) is Peck is described in the next section.
J.R. Griggs
402
Another n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r a p o s e t P t o b e Peck has b e e n d i s c o v e r e d by P r o c t o r [ P r 2 ] . The c o n d i t i o n i s t h a t P c a r r i e s a c e r t a i n r e p r e s e n t a t i o n o f t h e L i e a l g e b r a sR(2.C). This condition l e a d s to s h o r t e r p r o o f s , f r e e o f a l g e b r a i c g e o m e t r y , t h a t some B r u h a t o r d e r s a r e P e c k .
W e s a w i n t h e l a s t s e c t i o n t h a t t h e e q u i v a l e n c e of (") a n d P e c k c a n b e gene r a l i z e d t o t h e e q u i v a l e n c e of s t r o n g l y S p e r n e r and T f o r rank-unimodal p o s e t s . [ P S S ] o b s e r v e d how t o u s e l i n e a r a l g e b r a a n d t h i s r e s u l t t o g e n e r a l i z e t h e t h e o r e m a b o v e . A s b e f o r e , 7, i s t h e s p a c e w i t h b a s i s Pi. Let be the s p a c e w i t h b a s i s P. A r a i s i n g ( r e s p e c t i v e l y , l o w e r i n g ) o p e r a t o r o n P i s a
p:
linecir transformation x
E
covers
is
+
(resp.,
A:
is
=
Theorem 2 . L e t a r e equivalent.
P
-t
F)
such tlmt f o r a l l
implies
be a f i n i t e r a n k e d p o s e t .
c
Y
=
0
(i)
P
is rank-unimodal
and s t r o n g l y Sperner.
P
i s rank-unimodal
and s a t i s f i e s c o n d i t i o n
There e x i s t a r a i s i n g o p e r a t o r
X
operator
and
on P
(a)
lPil
5 1p.1
3
(b)
lPil
2 IPjI
=$
J
p
such t h a t f o r a l l pj-i
I Pi
xj-i
unless
y
The €allowing t h r e e c o n d i t i o i - s
(ii) (iii)
5.
is
C cyy ( r e s p . , A(x) = C c y ) Yep y€P x i n P (resp., x cuvers y i n P).
P, p ( x )
T.
and a l o w e r i n g i < j,
is i n j e c t i v e , is injective.
PRODUCTS AND QUOTIENTS
Many i n t e r e s t i n g p o s e t s c a n be formed by a p r o d u c t of p o s e t s o r by t h e (See s e c t i o n 2 f o r d e f i n i q u o t i e n t o f a l a r g e r p o s e t by a n a u t o m o r p h i s m g r o u p . tions.) F o r e x a m p l e , B a n d D a r e p r o d u c t s o f c h a i n s a n d L(m,n) is a q u o t i e n t of
Bmn
under an a p p r o p r i a t e group.
Some o f t h e m o s t p r o f o u n d S p e r n e r
t h e o r e m s c o n c e r n how S p e r n e r p r o p e r t i e s a r e p r e s e r v e d u n d e r p r o d u c t s and q u o t i e n t s . C a n f i e l d [C2] a n d P r o c t o r , Saks, a n d S t u r t e v a n t [PSS] i n d e p e n d e n t l y d i s c o v e r e d t h a t t h e p r o d u c t o f Peck p o s e t s is Peck. T h e i r p r o o f s i n v o l v e l i n e a r a l g e b r a , w h i c h seems t o b e t h e p r o p e r " b o o k k e e p i n g " t o o l t o s t o r e a l l t h e i n f o r m a t i o n n e c e s s a r y t o show t h a t PXQ i s s t r o n g l y S p e r n e r when P a n d Q a r e P e c k . (PXQ i s e a s i l y s e e n t o b e r a n k - s y m m e t r i c a n d r a n k - u n i m o d a l . ) Proctor [Pr2] has f o u n d t h e s h o r t e s t known p r o o f o f t h e p r o d u c t t h e o r e m , u s i n g t h e r e p r e s e n t a t i o n condition. T h e r e i s n o known p u r e l y c o m b i n a t o r i a l p r o o f o f t h i s c o m b i n a t o r i a l result. [PSS] g e n e r a l i z e d t h e Peck p r o d u c t theorem, a n d Saks [SaZ] f u r t h e r s t r e n g t h e n e d t h i s as f o l l o w s . P a n d Q a r e s a i d t o b e corn a t i b l e i f t h e r e e x i s t s a t s u c h t h a t f o r a l l i a n d j , l P i l < IPj o n l5 y* i IQ T h i s means, t-1 t-J e s s e n t i a l l y , t h a t f o r a l l k, the l o c a t i o n of t h e kth l a r g e s t rank i n P r e l a t i v e t o t h e l a r g e s t r a n k c o r r e s p o n d s o p p o s i t e l y t o t h e l o c a t i o n o f t h e kth l a r g e s t Saks' theorem s t a t e s t h a t t h e p r o d u c t rank i n Q r e l a t i v e t o t h e l a r g e s t rank. o f two c o m p a t i b l e , s t r o n g l y S p e r n e r p o s e t s i s S p e r n e r . E x a m p l e s show t h a t t h i s r e s u l t c a n n o t b e improved. The P e c k p r o d u c t t h e o r e m i s n o t a c t u a l l y c o n t a i n e d i n t h e t h e o r e m , b u t i t f o l l o w s from i t by an e l e g a n t s h o r t a r g u m e n t [ P S S ] .
I
. I.
The Sperner property
403
The p r o d u c t o f s y m m e t r i c c h a i n o r d e r s i s a s y m m e t r i c c h a i n o r d e r , w h i c h i s e a s y t o show [ K a Z ] . P e r h a p s t h e m o s t u s e f u l p r o d u c t t h e o r e m i s d u e t o H a r p e r [ H l ] w i t h a n o t h e r p r o o f d u e to H s i e h a n d K l e i t m a n [HK]. I t s t a t e s t h a t i f 1' a n d (1 h a v e a c e r t a i n p r o p e r t y , c a l l e d the LYM p r o p e r t y , a n d h a v e l o g a r i t h m i c a l l y conc a v e IJhitney numbers ( t h i s i s s t r o n g e r t h a n r a n k - u n i m o d a l i t y ) , t h e n PYQ h a s t h e same p r o p e r t i e s . S e e [ G H 3 J f o r t h i s a n d more on t h e LYM p r o p e r t y , w h i c h i s stronger than t h e strongly Sperner property. For Harper's "global" category t h e o r y t r e a t m e n t of t h i s , see [ H 3 ] . [Gg4] a p p l i e d t h e p r o d u c t t h e o r e m s a n d o t h e r techniques t o o b t a i n c o l l e c t i o n s of s u b s e t s w i t h t h e Sperner p r o p e r t y . The m a i n r e s u l t t h e r e i s g e n e r a l i z e d v i a c a t e g o r y t h e o r y i n [Wa]. T h r e e p a p e r s from 1 9 7 6 a n d b e f o r e [Kn, LW, P o ] i n c l u d e r e s u l t s a b o u t a permut a t i o n group a c t i n g on a f i n i t e s e t which c a n be u s e d t o s h m tli:it B /C, t h e rankB o o l e a n a l g e b r a modulo a p e r m u t a t i o n g r o u p C o n i t s g e n e r a t o r s , i s u n i m o d a l a n d S p e r n e r . T h i s i s d o n e u s i n g l i n e a r t r a n s f o r m a t i o n s on t h e s p a c e gene r a t e d by t h e e l e m e n t s o f B,/G. I n f a c t , simple l i n e a r a l g e b r a i c techniques, inc l u d i n g Theorems 1 a n d 2 o f t h e l a s t s e c t i o n o f t h i s p a p e r , c a n b e e m p l o y e d t o prove t h a t Bn/G i s P e c k [H4, PR, S t 4 ] . T h e r e are two p a r t i c u l a r l y n i c e a p p l i c a t i o n s o f t h i s r e s u l t . The f i r s t o n e i s t o o b s e r v e t h a t t h e r e i s a n e l e m e n t a r y ( l i n e a r a l g e b r a i c ) p r o o f t h a t L(m,n) is P e c k [H4, PR, S t 4 1 . T h i s f o l l o w s f r o m t h e o b s e r v a t i o n t h a t L(m,n) is t h e q u o t i e n t of Bmn, c o n s i d e r e d as t h e B o o l e a n a l g e r s o f a l l s u b s e t s o f a n mxn a r r a y o f s q u a r e s , modulo tile g r o u p C g e n e r a t e d by t h e p e r m u t a t i o n s o f t h e e l e m e n t s o f e a c h row a n d t h e p e r m u t a t i o n s o f t h e r o w s w i t h e a c h o t h e r . The s e c o n d a p p l i c a t i o n i s t h a t t h e p o s e t o f all s i m p l e u n l a b e l l e d g r a p h s on n v e r t i c e s , o r d e r e d by containment (embedding), i s Peck. T h i s f o l l o w s b e c a u s e t h i s p o s e t i s t h e q u o t i e n t of t h e p o s e t B(;) o f a l l s i m p l e l a b e l l e d g r a p h s o n n v e r t i c e s modulo t h e p e r m u t a t i o n s of t h e v e r t i c e s [H4,PR]. F o r w h a t r a n k e d p o s e t s P a r e a l l q u o t i e n t s P/G P e c k ? H a r p e r [H4] a n d S t a n l e y [ S t 4 1 f o u n d t h a t P/C i s P e c k i f P i s u n i t a r y P e c k , w h i c h m e a n s t h a t the "unitary" l i n e a r transformations $ i ( x ) : Fi+ Pi+1 d e f i n e d by bi(x)
=
C y, Y€P. 1+1
for
x
E
P.
Y>X
t o g e t h e r s a t i s f y c o n d i t i o n ( i i i ) ( a ) o f Theorem 1 i n s e c t i o n 4 . T h a t i s , P i s u n i t a r y P e c k i f i t i s e n o u g h t o t a k e all c = 1 f o r x < y i n c o n d i t i o n ( i i i ) (b) i n o r d e r t o o b t a i n t h e mappings t o p r o y e t h a t P is Peck. Not all P e c k p o s e t s are u n i t a r y P e c k , e . g . , F i g u r e 3 .
Fig. 3 T h e r e are e x a m p l e s known w h i c h show t h a t n e i t h e r t h e P e c k p r o p e r t y n o r t h e u n i t a r y Peck p r o p e r t y i s p r e s e r v e d u n d e r q u o t i e n t s . The p r o o f s of t h e P e c k p o s e t t h e o r e m c a n b e e x t e n d e d t o p r o v e t h a t t h e p r o d u c t o f u n i t a r y P e c k p o s e t s i s u n i t a r y P e c k [H4, S t 4 ] . Thus Bn a n d D,,, w h i c h a r e T h a t L n ( q ) , t h e l a t t i c e s of s u b s p a c e s , p r o d u c t s o f c h a i n s , are u n i t a r y Peck. i s u n i t a r y P e c k w a s shown i n e f f e c t by K a n t o r i n 1 9 7 2 [ K n ] . I t t h e n f o l l o w s by t h e p r o d u c t t h e o r e m t h a t m o d u l a r g e o m e t r i c l a t t i c e s are u n i t a r y P e c k [H4]. S t a n l e y ' s a l g e b r a i c g e o m e t r y a r g u m e n t s [ S t 3 1 show t h a t L(m,n) a n d M(n) are u n i t a r y P e c k , a l t h o u g h q u o t i e n t m e t h o d s a l o n e o n l y e s t a b l i s h t h a t L(m,n) i s Peck.
J.R. Griggs
404
Pouzet and Rosenberg [PR] derived the theorem that Bn/G is Peck as part of a more general study o f relation systems, permutation groups, and posets. Independently, Harper [H4] obtained the more general unitary Peck quotient theorem as part of an even more general study using category theory. Stanley's derivation [St41 of the unitary Peck quotient theorem is more compact, requiring only some linear algebra. More recently, Proctor [Pr3] noted that there is another short proof of the quotient theorem which u s e s the sR(2,C) representation characterization of Peck posets. I n conclusion, we have made great progress in recent years in studying the Sperner property and building an elegant theory, The introduction of algebraic tools has been particularly fruitful, Some nagging questions remain open, and it is hoped that efforts to answer them will lead to even deeper insights.
ACKNOWLEDGEMENT Thanks to JKrgen Stahl for editorial assistance, and to Madeleine Paoli for the French translation. M. Aigner, "Lexicographic matching in Boolean algebras, 14 (1973), 187-194.
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Annals of Discrete Mathematics 23 (1984) 409-464 0 Elsevier Science Publishers B.V. (North-Holland)
409
CHAIN AND ANTICHAIN F A M I L I E S
G R I D S AND YOUNG TABLEAUX G . Viennot U.E.R.
de Math6matiques e t Informatique UniversitC de Bordeaux I 33405 - Talence - FRAXCE
Dedicated t o Professor Corominas
A b s t r a c t . - G r e e n e ' s T h e o r e m r e l a t i n g c h a i n and a n t i c h a i n f a m i l i e s with m a x i m a l cardinalities i n a poset (partially o r d e r e d s e t ) w a s motivated by t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e . T h i s c o r r e s p o n d e n c e i s a b i j e c t i o n b e t w e e n p e r m u t a t i o n s a n d p a i r s (P,Q ) of s t a n d a r d Young t a b l e a u x having t h e a m e s h a p e . T h i s b i j e c t i o n o r i g i n a t e d i n t h e r e p r e s e n t a t i o n t h e o r y of the s y m m e t r i c group. The underlying combinatorics i s v e r y d e e p and a l s o takes r o o t s i n t h e t h e o r y of s y m m e t r i c f u n c t i o n s a n d a l g e b r a i c g e o m e t r y . T h e p u r p o s e of t h i s t a l k i s t h r e e f o l d . F i r s t we g i v e a brief s u m m a r y of t h e p r i n c i p a l c o m b i n a t o r i a l p r o p e r t i e s of the R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e , e s p e c i a l l y t h o s e having a n o r d e r - t h e o r e t i c a l f l a v o r . Second we s h e d s o m e l i g h t on i t s r e l a t i o n s h i p with p o s e t t h e o r y a n d Greene's Theorem. T h e t h i r d p u r p o s e of t h i s t a l k i s t o s o l v e t h e following o p e n p r o b l e m : give a n i n t e r p r e t a t i o n of t h e v a l u e l o c a t e d i n t h e ( i , j ) c e l l of t h e Young t a b l e a u x P a n d Q . T h i s "local" c h a r a c t e r i z a t i o n of t h e c o r r e s p o n d e n c e is c o m p l e t e l y s y m m e t r i c i n r o w s a n d c o l u m n s a n d r e q u i r e s t h e c o n c e p t of g r i d s a n d e x t e n d a b l e g r i d s . T h e s e l a s t r e s u l t s c a n be e x t e n d e d t o a r b i t r a r y p o s e t s . C o m p l e t e p r o o f s w i l l be g i v e n e l s e w h e r e . R k s u m 6 . - L e th6or'eme de G r e e n e s u r l e s c a r d i n a u x m a x i m a u x d e s f a m i l l e s de c h a r n e s e t d ' a n t i c h a r n e s e x t r a i t e s d ' u n e n s e m b l e ( p a r t i e l l e m e n t ) ordonnk a s o n o r i g i n e d a n s la c o r r e s p o n d a n c e de R o b i n s o n - S c h e n s t e d . C e t t e c o r r e s p o n d a n c e e s t une b i j e c t i o n e n t r e l e s p e r m u t a t i o n s e t l e s p a i r e s ( P , Q ) de t a b l e a u x s t a n d a r d s d e Young de meme f o r m e . C e t t e b i j e c t i o n p r o v i e n t e n f a i t de la t h k o r i e d e s r e p r C s e n t a t i o n s du g r o u p e s y m k t r i q u e . L a c o m b i n a t o i r e s o u s - j a c e n t e e s t f o r t r i c h e e t p r e n d Cgalement r a c i n e d a n s la t h 6 o r i e d e s f o n c t i o n s s y m 6 t r i q u e s e t e n geomktrie algebrique. L e but de c e t e x p o s 6 e s t t r i p l e . D ' a b o r d n o u s donnons un bref a p e r s u d e s propriCtCs c o m b i n a t o i r e s l e s p l u s c l a s s i q u e s de c e t t e c o r r e s pondance, n o t a m m e n t c e l l e s e n r e l a t i o n a v e c la m6thodologie d e s e n s e m b l e s ordonnCs. L e deuxi'eme but de cet e x p o s 6 e s t de f a i r e s e n t i r ?I u n public motivC p a r l e s e n s e m b l e s ordonne's l ' i n t k r s t de m i e u x c o n n a i t r e
G. Viennot
410
c e t t e m e r v e i l l e u s e c o r r e s pondance . Enfin une t r o i s i ' e m e p a r t i e e s t c o n s a c r k e 'a l a rCsolution d ' u n p r o bl'eme o u v e r t : donner une i n t e r p r k t a t i o n s y m k t r i q u e e n l i g n e s e t c o l o n n e s de l a v a l e u r s i t u 6 e d a n s l a c a s e ( i , j ) d e s t a b l e a u x de Young P e t Q . C e t t e dkfinition "locale" de la c o r r e s p o n d a n c e de RobinsonS c h e n s t e d u t i l i s e l e s nouveaux c o n c e p t s de g r i l l e e t de g r i l l e p r o l o n geable. C e s d e r n i ' e r e s i d k e s peuvent Etre 6 t e n d u e s a u x e n s e m b l e s o r donnks q u e l c o n q u e s , L e s p r e u v e s c o m p l e t e s de c e s nouveaux r 6 s u l t a t s s e r o n t donnkes a i l l e u r s .
4
1 - Introduction. In 1938, G . de B . Robinson i n t r o d u c e d [77] a c o r r e s p o n d e n c e between
the
n !
p e r m u t a t i o n s of the s y m m e t r i c g r o u p
6n
and p a i r s of c e r t a i n c o m b i -
n a t o r i a l o b j e c t s c a l l e d s t a n d a r d Young t a b l e a u x . T h i s c o r r e s p o n d e n c e g i v e s in f a c t a bijective ( c o n s t r u c t i v e ) proof of the following identity 2
n ! = 13 f X ,
x
w h e r e the s u m m a t i o n extends o v e r all i r r e d u c i b l e r e p r e s e n t a t i o n s ( o v e r t h e f i e l d of c o m p l e x n u m b e r s ) of t h e s y m m e t r i c g r o u p
'5
and
fh
denotes the d e g r e e
of the c o r r e s p o n d i n g r e p r e s e n t a t i o n . T h i s identity i s c l a s s i c a l i n the r e p r e s e n t a t i o n t h e o r y of finite g r o u p s , w h e r e the left h a n d - s i d e i s the o r d e r of the g r o u p .
6
, e a c h of t h e i r r e d u c i b l e r e p r e n s e n t a t i o n s i s i n b i j e c t i o n with a v e r y s i m p l e c o m b i n a t o r i a l o b j e c t : a p a r t i t i o n of In t h e c a s e of t h e s y m m e t r i c g r o u p
the i n t e g e r
n
below) with
n
.
Such a p a r t i t i o n i s v i s u a l i z e d a s a F e r r e r s d i a g r a m ( s e e f i g u r e 2 c e l l s . In a s e r i e s of p a p e r s ( s e e the c o l l e c t e d p a p e r s [118]) ,
Young i n t r o d u c e d his f a m o u s t a b l e a u x a s a c o m b i n a t o r i a l tool i n t h e study of the r e p r e s e n t a t i o n s of t h e s y m m e t r i c g r o u p . T h e s e s o - c a l l e d Young tableaux a r e l a belings of the F e r r e r s d i a g r a m s with i n t e g e r s s u c h t h a t t h e n u m b e r s a r e i n c r e a sing in r o w s and c o l u m n s ( s e e definition b e l o w ) . The dimension
fA
i s i n f a c t t h e n u m b e r of s t a n d a r d Young t a b l e a u x
a s s o c i a t e d with a g i v e n p a r t i t i o n
A
. The
r i g h t - h a n d s i d e of (1) i s i n t e r p r e t a t e d
a s t h e n u m b e r of p a i r s of s t a n d a r d Young t a b l e a u x having the s a m e shape
(i.e.
F e r r e r s diagram). T h e p r o b l e m to find
fA
b e c o m e s a p u r e l y c o m b i n a t o r i a l p r o b l e m , and
41 1
Chain and antichain families i s i n f a c t a p o s e t p r o b l e m : f i n d t h e n u m b e r of l i n e a r e x t e n s i o n s of t h e p o s e t defined by the c e l l s of the F e r r e r s d i a g r a m . C o m b i n a t o r i c s of Young t a b l e a u x a p p e a r s a l s o in many o t h e r a r e a s ,
. , .), i n v a manifolds, . . . ) . The
a s f o r e x a m p l e the t h e o r y of s y m m e t r i c f u n c t i o n s ( S c h u r f u n c t i o n s , riant theory o r a l g e b r a i c g e o m e t r y (Schubert calculus, flag i n t e r e s t e d r e a d e r w i l l s e e t h e books [72],[80], recent
t h e s u r v e y p a p e r [7] a n d s o m e
w o r k of L a s c o u x a n d S c h u t z e n b e r g e r . We s h a l l r e s t r i c t o u r s e l f to t h e
c o m b i n a t o r i a l point of v i e w . T h e c o r r e s p o n d e n c e i n t r o d u c e d by Robinson w a s r e d i s c o v e r e d by S c h e n s t e d [89] a n d defined m o r e c l e a r l y i n p u r e l y c o m b i n a t o r i a l way by a r e c u r s i v e a l g o r i t h m : the "bumping p r o c e s s " . Since t h e n , t h i s c o r r e s p o n d e n c e h a s been known a s t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e a n d a p p e a r s a l s o t o be of p u r e l y c o m b i n a t o r i a l i n t e r e s t . Following S c h u t z e n b e r g e r , m u c h c o m b i n a t o r i a l w o r k h a s b e e n done. A good s u r v e y of the s t a t e of t h e a r t i n 1972 c a n be found in K n u t h ' s book ;52],
s e c t i o n 5 . 1 . 4 , a n d m o r e r e c e n t l y i n t h e book [13] e d i t e d
by D . F o a t a a n d r e l a t e d t o the S t r a s b o u r g " t a b l e r o n d e " in A p r i l 1976. The Robinson-Schensted correspondence l i n k s with p o s e t t h e o r y . With e v e r y p e r m u t a t i o n
0
U
*
(P(O),
Q(u))
has strong
, one c a n a s s o c i a t e a p o s e t
having d i m e n s i o n G 2 , a n d c o n v e r s e l y a n y s u c h p o s e t c a n be o b t a i n e d i n t h i s way. Increasing subsequences c o r r e s p o n d to chains and d e c r e a s i n g subsequences c o r r e s p o n d t o a n t i c h a i n s . S o m e c o m b i n a t o r i a l p r o p e r t i e s of t h e c o r r e s p o n d e n c e a r e i n f a c t p r o p e r t i e s of t h i s p o s e t . F o r e x a m p l e t h e well-known S c h e n s t e d p r o p e r t y g i v e s t h e length of the l o n g e s t i n c r e a s i n g ( r e s p . d e c r e a s i n g ) s u b s e q u e n c e a s t h e n u m b e r of e l e m e n t s of t h e f i r s t r o w ( r e s p . c o l u m n ) of t h e F e r r e r s d i a g r a m common to
P(u)
and
Q(o) . A g e n e r a l i z a t i o n h a s b e e n g i v e n by G r e e n e
[35], i n t e r p r e t a t i n g t h e e n t i r e s h a p e ( F e r r e r s d i a g r a m ) of
P(0) a n d
Q(0)
i n t e r m s of a f a m i l y of c h a i n s a n d a n t i c h a i n s with m a x i m u m c a r d i n a l i t i e s . T h i s w a s t h e s t a r t i n g point f o r t h e d e e p a n d now c l a s s i c a l G r e e n e ' s T h e o r e m [37] : t h e duality between f a m i l y of c h a i n s and a n t i c h a i n s c o m i n g f r o m the R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e ( p o s e t s with d i m e n s i o n C 2 ) i s v a l i d for any poset. F r o m a p o s e t point of view, i t m a y be f r u i t f u l t o h a v e a good u n d e r standing of t h e o r d e r - t h e o r e t i c a l p r o p e r t i e s of t h e R o b i n s o n - S c h e n s t e d c o r r e s pondence ( t h i s c a n be a l r e a d y v e r y d i f f i c u l t ) a n d t h e n t o l o o k f o r p o s s i b l e e x t e n s i o n s t o a r b i t r a r y p o s e t s . In p a r t i c u l a r s o m e g e n e r a l i z a t i o n s of t h e c o r r e s p o n -
G. Viennot
412
dence h a v e been p r o p o s e d f o r a r b i t r a r y p o s e t s by F o m i n [16] a n d G a n s n e r [ 2 7 ] . G r e e n e ' s proof of h i s t h e o r e m r e l i e s on s o m e j o i n t w o r k w i t h K l e i t m a n [39] g e n e r a l i z i n g t h e well-known D i l w o r t h ' s T h e o r e m . T h i s w o r k i s p a r t of the s o - c a l l e d " e x t r e m a l p r o p e r t i e s " a n d a s u r v e y of t h i s f i e l d c a n be found i n W e s t [113]. Since the p a p e r s of G r e e n e a n d Kleitman, e x t e n s i v e r e s e a r c h h a s b e e n done a n d s e v e r a l o t h e r p r o o f s have been given ( s e e b e l o w ) . In p a r t i c u l a r , G r e e n e ' s T h e o r e m c a n a l s o be d e d u c e d f r o m i n t e g e r p r o g r a m m i n g t e c h n i q u e s , using the m i n i m a l c o s t flow a l g o r i t h m of F o r d a n d F u l k e r s o n [23], [47]
.
F o r a s u r v e y of
the u s e of l i n e a r p r o g r a m m i n g i d e a s i n poset t h e o r y , s e e Hoffman [ 4 5 ] . T h i s p a p e r is a r r a n g e d i n the following way. F i r s t in s e c t i o n 2, we r e c a l l G r e e n e ' s T h e o r e m ( f o r a r b i t r a r y p o s e t s ) . T h e n we give d i f f e r e n t v e r s i o n s of the Robinson-Schensted c o r r e s p o n d e n c e : the o r i g i n a l definition of S c h e n s t e d in s e c t i o n 3, a " p l a n a r i z a t i o n " given by Viennot [lo91 in s e c t i o n 5, t h e n the s y n t h e s i s m a d e by S c h i i t z e n b e r g e r a n d L a s c o u x with t h e 'ljeu de taquin" and c u l m i nating i n the plactic monoid [62], [97] in s e c t i o n 6 , S o m e c l a s s i c a l c o m b i n a t o r i a l p r o p e r t i e s of the c o r r e s p o n d e n c e a r e given in s e c t i o n 4 . In s e c t i o n 7 we i n t r o duce the new c o n c e p t s of g r i d s and extendable g r i d s . We a r e t h u s a b l e t o give a n i n t e r p r e t a t i o n of t h e s h a p e of t h e Young t a b l e a u x
P(0) a n d
Q(a)
that i s
c o m p l e t e l y s y m m e t r i c in r o w s a n d c o l u m n s . M o r e o v e r , we give a n e x p l i c i t m i n Q. .(u)) l o c a t e d i n t h e ( i , j ) c e l l 'J 1J ( r e s p . Q(0)). In s e c t i o n 8 we extend t h e s e i d e a s to a r b i t r a r y p o s e t s .
m a x f o r m u l a f o r the v a l u e of
P(a)
P..(u) ( r e s p .
T h i s e x t e n s i o n r e l i e s on s o m e t h e o r e m s of F r a n k [23], p r o v e d by l i n e a r p r o g r a m m i n g t e c h n i q u e s , f r o m which G r e e n e ' s T h e o r e m c a n be d e d u c e d . An o p e n p r o b l e m is t o prove d i r e c t l y the r e s u l t s of s e c t i o n 8 without
the u s e of l i n e a r
programming. In t h i s p a p e r , we do not p r e t e n d t o give a n e x h a u s t i v e s u r v e y of t h i s huge s u b j e c t a n d apologize f o r o m i s s i o n s a n d unquoted p a p e r s . A l s o we c o n f e s s t h a t we c h o s e s o m e of the m o s t s p e c t a c u l a r p r o p e r t i e s r e l a t e d t o the high u s e of t r a n s p a r e n c i e s in the t a l k . Unfortunately, t h e s i m p l i c i t y of the c o m b i n a t o r i a l c o n s t r u c t i o n s , t o g e t h e r with t h e m a g i c of t h i s v e r y beautiful c o r r e s p o n d e n c e , cannot be w r i t t e n down i n a p a p e r a s e a s i l y a s i t c a n be d e s c r i b e d in a n o r a l c o m m u n i c a t i o n with a f r i e n d o r using s u p e r p o s i t i o n of p i c t u r e s with t r a n s p a r e n c i e s . T h i s i s the r u l e i n c o m b i n a t o r i c s , N e v e r t h e l e s s , we hope t h a t the r e a d e r , not having a t t e n t e d t h e t a l k c o r r e s p o n d i n g t o t h i s p a p e r , will a g r e e with Knuth's s t a t e m e n t ([52], page 6 0 , l i n e 21) : " T h e unusual n a t u r e
Chain and antichain families
413
of t h e s e c o i n c i d e n c e s m i g h t l e a d u s t o s u s p e c t t h a t s o m e s o r t of w i t c h c r a f t i s o p e r a t i n g behind the s c e n e s !
4
2 -
".
Greene's Theorem. Let
P
be a f i n i t e p o s e t ( p a r t i a l l y o r d e r e d s e t ) . An a n t i c h a i n i s a
P
s u b s e t of p a i r w i s e i n c o m p a r a b l e e l e m e n t s of t a l l y o r d e r e d by t h e induced o r d e r of k - f a m i l y ) i s a s u b s e t of c h a i n s ) . We denote by chain f a m i l i e s Example :
P
P
.
(resp.
P
Let
k anti-
d (P)) the m a x i m u m c a r d i n a l i t y of
k
1,2, . . ., 9
.
P
.
F o r t h e r e a d e r u n f a m i l i a r with p o s e t t h e o r y , we
4
P
of
t i v e c l o s u r e of t h e following r e l a t i o n : x d y y
chains ( r e s p .
be t h e p o s e t defined by t h e d i a g r a m d i s p l a y e d i n f i g u r e 1.
r e c a l l that t h e ( p a r t i a l ) o r d e r r e l a t i o n
i s below
k
( r e s p . a n t i c h a i n f a m i l i e s ) of t h e p o s e t
The vertices a r e
x
A chain i s a s u b s e t that i s to-
A chain k-family ( r e s p . antichain
t h a t i s t h e union of
ck(P)
.
i s d e f i n e d a s t o be t h e t r a n s i -
iff a n d e d g e l i n k s
.
F i g u r e 1 : A poset
.
x
and
y
and
414
For
G. Viennot
t h e following s u b s e t s a r e c h a i n k - f a m i l i e s with m a x i m u m c a r -
k = 1,2, 3
dinality
The subset
Y2
= [1,2,5,8,93
k = l ,
Y
k = 2 ,
Y2 = { l , 2, 3 , 4 , 6 , 7 , 8 , 9
k = 3 ,
Y = P = [ l ., . . , 9 ) ,
1
3
3
is a union of two c h a i n s ,
i s a union of t h r e e c h a i n s y3 c2(P) = 8 , c3(P) = 9 .
Y2 = [ 1, 2, 3 , 4 ) U { 6 , 7 , 8 , 9
1,
while
Y3 = { l , 2 , 3 , 4 3 U { 5) U ( 6 , 7 , 8 , 9 ] . T h u s c l ( P ) = 5 ,
Note t h a t , i n t h i s p a r t i c u l a r e x a m p l e , t h e r e e x i s t s only one c h a i n k - f a m i l y with m a x i m u m c a r d i n a l i t y , f o r For
k = 1, 2 , 3 , 4, 5
k = 1, 2 , 3
,
, t h e following s u b s e t s a r e a n t i c h a i n k - f a m i l i e s
with m a x i m u m c a r d i n a l i t y . k = l ,
61 = { 3 , 5 , 7 3 ,
k = 2 ,
6 = {1,61 U { 3 , 5 , 7 1 , 2
k = 3 ,
6
k = 4 ,
6 4 = (1,6] U {2,7) U r3,8)
k = 5 ,
6
We h a v e w r i t t e n e a c h Thus
dl(P) = 3
Here, for each
bk
3
5
= (1,6) U {3,5,71 U ( 4 , 8 ) ,
= P = {1,67
a s a union of
k
u
U { 2 , 71 U f 3 , 8 7 U [ 4 , 97 U [ 5 ? .
antichains.
, d2(P) = 5 , d3(P) = 7 , d4(P) = 8 , d5(P) = 9 k , 1G k
with m a x i m u m c a r d i n a l i t y
<4
,
(4,9)
.
, t h e r e e x i s t s s e v e r a l distinct antichain k-families
dk(P)
.
In t h e l i t e r a t u r e , c h a i n k - f a m i l i e s a r e often c a l l e d k - c o f a m i l i e s , a n t i c h a i n k - f a m i l i e s a r e a l s o c a l l e d k - f a m i l i e s , while a n t i c h a i n k - f a m i l i e s with maximum cardinality dk(P) Usually
dk(P)
are c a l l e d S p e r n e r k - f a m i l i e s . T h e notation
i s in honor of Dilworth (Saks c a l l s t h i s t h e kth D i l w o r t h n u m b e r ) . ck(P)
i s denoted by
throughout t h e p a p e r ,
ck(P)
2k ( P ) ,
We have c h a n g e d t h i s notation b e c a u s e
i s a s s o c i a t e d with t h e "shape"
X
of the Young
t a b l e a u x obtained by t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e , while a s s o c i a t e d with the "conjugate s h a p e " , usually d e n o t e d by A p a r t i t i o n of a n i n t e g e r
n
^x
or
A*
dk(P)
is
.
i s a n o n - i n c r e a s i n g s e q u e n c e of non z e r o
Chain and antichain families
415
's, \ . . . 3 k > O s u c h t h a t n = X t . . . t X . Usually a p a r t i t i o n 1 2 P 1 P A i s v i s u a l i z e d by a " F e r r e r s d i a g r a m " F ( s e e f i g u r e 2 , i n " F r e n c h notation") th T h e n u m b e r of e l e m e n t s i n t h e i row i s X i . If :1 d e n o t e s t h e n u m b e r of J e l e m e n t s i n t h e jth c o l u m n , we obtain a n o t h e r p a r t i t i o n X = (X 2 . . . > X ) , 1 9 c a l l e d the con,jugate p a r t i t i o n of 1 . We have d e n o t e d by q t h e n u m b e r of e l e -
X
integers
P .
m e n t s of the f i r s t row of
With t h e s e definitions we c a n s t a t e t h e well-known G r e e n e ' s T h e o r e m
a s follows T H E O R E M 1.
(Greene)
Let
P
be a f i n i t e p o s e t with
n
elements,
q
( r e s p . p) t h e length of t h e l a r g e s t c h a i n ( r e s p . a n t i c h a i n ) . We define t h e n u m bers
k i ( P ) = ci(P) - c.
1-1
&ri(P) = di(P) - di-l(P)
(P) and
(with t h e convention
c , ( P ) = d o ( P ) = 0 ) . T h e n we have t h e following p r o p e r t i e s (i)
A1(P)3 . . . 1 1 ( P ) ,
(ii)
y1(P)I
(iii)
t h e p a r t i t i o n s d e f i n e d by ( i ) a n d ( i i ) a r e c o n j u g a t e .
P
. . . 31 u
9
(P) ,
T h e F e r r e r s d i a g r a m a s s o c i a t e d with t h e p a r t i t i o n defined by ( i ) w i l l b e c a l l e d t h e G r e e n e d i a g r a m of
Example :
P
G(P)
a n d d e n o t e d by
.
T h e G r e e n e d i a g r a m a s s o c i a t e d with t h e p o s e t defined by f i g u r e 1
i s displayed i n figure 2 .
a
= (5,3,1)
*
h = (3,2,2,1,1)
Figure 2.
T h e F e r r e r s d i a g r a m of
X
= (5, 3 , l ) .
416
G. Viennot
Note t h a t r e l a t i o n s ( i ) a n d ( i i ) of G r e e n e ' s T h e o r e m a r e not a t a l l o b v i o u s . In p a r t i c u l a r , a c h a i n k - f a m i l y ( r e s p . a n t i c h a i n k - f a m i l y ) with m a x i m u m c a r d i n a l i t y i s not n e c e s s a r i l y obtained by adding a c h a i n ( r e s p . a n t i c h a i n ) f r o m a c h a i n (k-1)-family ( r e s p . a n t i c h a i n ( k - 1 ) - f a m i l y ) with m a x i m u m c a r d i n a l i t y . T h e poset of f i g u r e 1 h a s a unique c h a i n of length 5 a n d a unique c h a i n 2 - f a m i l y of c a r d i n a l i t y 8 which i s a union of two c h a i n s of length 4. O r i g i n a l l y , G r e e n e p r o v e s h i s r e m a r k a b l e t h e o r e m [37] by using s o m e w o r k done with K l e i t m a n [39] t h a t e x t e n d s t h e c l a s s i c a l D i l w o r t h ' s T h e o r e m [ l l ] , d ( P ) of the l a r g e s t a n t i c h a i n in t h e p o s e t P i s the 1 m i n i m u m n u m b e r of c h a i n s which c o v e r P . Given a n y c h a i n p a r t i t i o n C of
which s t a t e s that the s i z e
m ( C ) = C m i n ( k , c ) . Since e a c h c h a i n c o n t r i b u t e s a t m o s t k e l e CEC m e n t s t o a k - f a m i l y , dk(P)4 m ( C ) . If equality holds then C i s c a l l e d a k k - s a t u r a t e d partition. Greene and Kleitman proved that f o r any k t h e r e e x i s t s
P , let
a chain partition
C
t h a t i s both
k
and ( k t 1 ) - s a t u r a t e d . T h i s f a c t i s known
a s t h e t - p h e n o m e n o n , a n d i m p l i e s condition ( i i ) of t h e o r e m 1. Since t h e n , m a n y o t h e r p r o o f s have been g i v e n : S a k s c o n s i d e r e d a n t i c h a i n k - f a m i l i e s i n a p r o d u c t of two p o s e t s [86], €881 a n d gave a s h o r t proof of the e x i s t e n c e of k - s a t u r a t e d p a r t i t i o n s [87]. T h e t o o l s of l i n e a r a l g e b r a c a n be u s e d : S a k s [86] and G a n s n e r [27] p r o v e d that t h e n u m b e r s
ck(P)
c a n be c o n s i -
d e r e d a s the i n v a r i a n t s of a nilpotent m a t r i x , t h a t i s the block s i z e s of i t s J o r d a n c a n o n i c a l f o r m ; thus t h e y p r o v e d condition ( i ) of t h e o r e m 1. Dual i n t e r p r e t a t i o n s d (P) . A n o t h e r kind of proof c o m e s f r o m l i n e a r p r o g r a m k m i n g using the m i n i m a l c o s t flow a l g o r i t h m of F o r d a n d F u l k e r s o n : Hoffman and
exist for the numbers
S c h w a r t z [47], F o m i n [16],
F r a n k [23],
s e e a l s o H o f f m a n ' s s u r v e y p a p e r on the
u s e of l i n e a r p r o g r a m m i n g i n p o s e t t h e o r y [ 4 5 ] ,
$ 3
-
T h e Robinson-Schensted a l g o r i t h m . In t h e c a s e of a p o s e t of d i m e n s i o n 2 , t h e r e is a v e r y convenient way of
computing the G r e e n e d i a g r a m : j u s t apply t h e R o b i n s o n - S c h e n s t e d a l g o r i t h m . We denote by group on
[n]
the s e t
[n]. F o r e v e r y p e r m u t a t i o n
t o be t h e s e t of points
{ l ,2 , 0
6 = { ( i , O ( i ) ) ,i E [n] )
o r d e r of t h e u s u a l p r o d u c t o r d e r :
., ,.n)
a n d by
En
the s y m m e t r i c
6 we a s s o c i a t e a p o s e t P o s ( ~ ) n o r d e r e d by t h e i n d u c e d c [n] x [n]
of
Chain and antichain families ( x , y ) & (XI,y ' )
(2)
E v e r y p o s e t of d i m e n s i o n G 2
iff
A
and
y
< y'
i s a s u b p o s e t of the p r o d u c t of two c h a i n s , and
c a n in f a c t be identified with a p o s e t Let
x 4 x'
417
be a p a r t i t i o n of
g r a m . A Young t a b l e a u of s h a p e
A.
-
f o r a certain permutation
0
n
and
0
.
the associated F e r r e r s dia-
Fx
i s a l a b e l i n g of t h e c e l l s of
with i n t e -
F
x
g e r s s u c h t h a t t h e s e n u m b e r s a r e s t r i c t l y i n c r e a s i n g in t h e c o l u m n s (down-up i n t h e " F r e n c h notation") a n d a r e weakly i n c r e a s i n g i n t h e r o w s ( l e f t t o r i g h t ) . If the e n t r i e s of t h e Young t a b l e a u a r e d i s t i n c t i n t e g e r s and a r e the i n t e g e r s 1,2,
. . .,n
, t h e n we h a v e a s t a n d a r d Young t a b l e a u on
[n]
( s e e f i g u r e 3 ) . In
o t h e r w o r d s , a s t a n d a r d Young t a b l e a u i s a n e m b e d d i n g in t h e c h a i n poset a s s o c i a t e d with
5
of t h e
[n]
( w h e r e the o r d e r i s the i n d u c e d o r d e r of t h e p r o d u c t
order).
m
h 1
2
2
3
m
4
A Young t a b l e a u , a n d a s t a n d a r d Young t a b l e a u o n [9] with s h a p e A. = ( 5 , 3 , l ) .
Figure 3 .
We define now r e c u r s i v e l y a p r o c e s s t h a t i n s e r t s a n i n t e g e r Young t a b l e a u tinct f r o m
x If
I(T,x)
T
.
.
We s u p p o s e t h a t t h e e n t r i e s of
x
If not, t h e n l e t
T
.
Then
x
.
Let
I(T,x)
I(T,x).
i s g r e a t e r t h a n e v e r y e l e m e n t of t h e f i r s t r o w of
z
x
in a
a r e all d i s t i n c t a n d d i s -
T h e r e s u l t i s a Young t a b l e a u d e n o t e d by
i s t h e Young t a b l e a u o b t a i n e d by adding
t e r than
T
x
T , then
a t t h e end of t h i s f i r s t r o w .
be t h e s m a l l e s t v a l u e of t h e f i r s t r o w of
T
grea-
be t h e Young t a b l e a u o b t a i n e d by d e l e t i n g the f i r s t r o w of T1 i s t h e t a b l e a u o b t a i n e d f r o m T by r e p l a c i n g z by x
418
G. Viennot
a n d by i n s e r t i n g ( r e c u r s i v e l y ) t h e v a l u e
in the tableau
z
T1
T h e a r r a n g e m e n t i s a Young t a b l e a u a t e a c h s t a g e b e c a u s e the r e p l a c e m e n t of
by
z
x
s a t i s f i e s the r e q u i r e d i n e q u a l i t i e s i n i t 5 r o w a n d c o l u m n .
Note t h a t t h e shape of shape of
T
(T,x)
. An e x a m p l e of
i s obtained by adding a c e l l on the b o r d e r of the t h i s "bumping p r o c e s s ' ' i s given i n f i g u r e 4 .
I(T,3)
F i g u r e 4.
Now if tableaux
P(O)
u and
The tableau
P = 6 , P1 = ( a ( l ) ) , . 0 pitl = 1 ( P i , ~ ( i + 1 ) ) .
T h e bumping p r o c e s s .
is a p e r m u t a t i o n of
Q(o)
6, . , . , Q n = Q(u)
6 , we define t w o s t a n d a r d Young
n by t h e following.
P(u) i s o b t a i n e d by s u c c e s s i v e i n s e r t i o n s :
. ., Pn ( 0 ) = P(o)
T h e tableau Qo=
=
Q(O)
where, for
it1
i=l,
. , . , n - 1 , Qitl
,
is obtained f r o m
on the b o r d e r of t h e t a b l e a u
s a m e p o s i t i o n as t h e unique c e l l t h a t is in the s h a p e of s h a p e of
. . ., n-1
i s o b t a i n e d b y c o n s t r u c t i n g a s e q u e n c e of t a b l e a u x
such that, f o r
by a d d i n g a new c e l l l a b e l e d
i = 1,
Qi
, in the
Pi+1 but not i n t h e
.
pi In o t h e r w o r d s , the t a b l e a u
Q(u)
s h a p e s o c c u r i n g i n t h e s e q u e n c e of i n s e r t i o n s
i s a coding of t h e s u c c e s s i v e
P1,
. . . , Pn .
i'
Chain and antichain families
419
u - 6 1 7 2 5 8 3 9 4.
Example:
~
-.
IT
I j
r6-1
3
Q
J.
I /
/
Figure 5.
T h e Robinson-Schensted a l g o r i t h m .
It i s e a s y t o s e e t h a t t h i s p r o c e s s c a n be r e v e r s e d a n d the c o r r e s pondence
u + ( P ( u ) , Q ( u ) ) is a b i j e c t i o n . T h i s is t h e R o b i n s o n - S c h e n s t e d c o r -
respondence. The tableau
P(u)
is t h e P - s y m b o l , w h i l e
Q(u)
i s the
Q- symbol. THEOREM 2 -
(Robinson-Schensted)
The correspondence
defined above i s a bijection b e t w e e n t h e n
I p e r m u t a t i o n s of
d
* (P(o),Q(u))
en
and the
G. Viennot
420 p a i r s of s t a n d a r d Young t a b l e a u x on
[n]
having the s a m e s h a p e .
T h i s c o r r e s p o n d e n c e g i v e s a bijective proof of identity (1). The number
fX
of s t a n d a r d Young t a b l e a u x of s h a p e
X
c a n be c o m -
D i f f e r e n t f o r m u l a e have been found and h a v e r e c e i v e d m u c h
puted e x p l i c i t l y . investigations.
h. = ( A , > .
Let let
X'. = X i
i-p-i
. . 5 X P)
be a p a r t i t i o n of
n
and f o r
i , 1 6i 6 p
,
Using c o m p l i c a t e d a l g e b r a i c m e t h o d s ( g r o u p c h a r a c t e r s , s y m m e t r i c p o l y n o m i a l s ) , Young [I181 and F r o b e n i u s [ 2 5 ] independently gave the f o r m u l a
A c o m b i n a t o r i a l proof of ( 3 a ) , using d i f f e r e n c e m e t h o d s , w a s found by Mac Mahon, i n t e r m s of a " n - c a n d i d a t e ballot p r o b l e m " [7 4 ], of " l a t t i c e p e r m u t a t i o n s " [73]
o r equivalently
, page 133.
Another way t o prove ( 3 a ) i s to d e r i v e t h i s f o r m u l a f r o m the d e t e r m i nantal e x p r e s s i o n (3b)
= 0
(with t h e convention
when
X.-itj
0). See f o r example
(A, -itj) Z e l e v i n s k y [120], page 9 i . A v e r y s i m p l e "bijective" proof h a s b e e n g i v e n by
G e s s e l a n d Viennot [30] a s p a r t of a m o r e g e n e r a l w o r k i n t e r p r e t a t i n g m a n y d e t e r m i n a n t s of c o m b i n a t o r i c s a s t h e n u m b e r of c e r t a i n n o n - c r o s s i n g configur a t i o n s of p a t h s ( a n d a bijection between Young t a b l e a u x and s u c h c o n f i g u r a t i o n s ) . T h e f o r m u l a ( 3 a ) c a n be r e a r r a n g e d i n t o a m o r e s i m p l e and e l e g a n t form. Let
x
be a c e l l of the F e r r e r s d i a g r a m
define t h e hook length of x , including
Fx
a s t h e n u m b e r of c e l l s of
x
a s s o c i a t e d with
X
FX l o c a t e d above
. We or a t
itself ( s e e f i g u r e 6 ) . Using t h e f a c t t h a t , f o r any th i , 1 G i G p , the p r o d u c t of the hook length of t h e i r o w is Xi ! / n (,I\h.; ), jbi we o b t a i n f r o m ( 3 a ) t h e v e r y well known a n d c l a s s i c a l f o r m u l a of F r a m e ,
the r i g h t of
Robinson a n d T h r a l l [22]
x
42 1
Chain and antichain families
(3c)
(hooklength formula)
f
X
n! = n h ' x x
w h e r e the p r o d u c t i s t a k e n o v e r a l l hook l e n g t h s
Hook length
h the ( 1 , ~ cell )
of
FX .
Hook lengths of the Ferrers diagram ( 5 , 3 , 1 )
= 5 of
Figure 6 .
hx
Hook l e n g t h s .
A p r o b a b i l i s t i c proof of ( 3 b ) h a s been g i v e n by G r e e n e , N i j e n h u i s and Wilf [41]. B i j e c t i v e p r o o f s of ( 3 c ) h a v e been g i v e n by R e m m e l [76] a n d Z e i l b e r g e r [119]. In f a c t , R e m m e l c o m b i n e d the n o n - c r o s s i n g c o n f i g u r a t i o n s of p a t h s m e n t i o n e d above a n d i n t e r p r e t a t i n g ( 3 b ) , with a bijective d e r i v a t i o n of ( 3 a ) a n d ( 3 c ) f r o m ( 3 b ) . using t h e "involution p r i n c i p l e " i n t r o d u c e d by G a r s i a a n d Milne [29] i n t h e i r bijective proof of t h e c e l e b r a t e d R o g e r s - R a m a n u j a n i d e n t i t y . N e v e r t h e l e s s , no s i m p l e proof of ( 3 c ) i s known a n d n o d i r e c t c o m b i n a t o r i a l c o r r e s p o n d e n c e e x p l a i n s the r o l e of t h e hook l e n g t h s . A s f o r t h e R o g e r s - R a m a n u j a n i d e n t i t i e s , i t i s s t i l l a n open p r o b l e m to know w h e t h e r t h e r e e x i s t s a " s i m p l e " a n d " n a t u r a l " bijective d e r i v a t i o n of ( 3 c ) . Perhaps,
the b e s t c o m b i n a t o r i a l i n t e r p r e t a t i o n of t h e hook l e n g t h i s
due t o G r a s s l a n d H i l l m a n [31]. T h e hooks a p p e a r n a t u r a l l y i n a c o m b i n a t o r i a l c o r r e s p o n d e n c e involving the s o - c a l l e d p l a n e p a r t i t i o n s . T h i s s u b j e c t i s v e r y c l o s e to o u r t o p i c , but w i l l not be t o u c h e d i n t h i s p a p e r . F r o m G r a s s l a n d H i l l m a n ' s c o r r e s p o n d e n c e , f o r m u l a ( 3 ) c a n be d e r i v e d using a n a s y m p t o t i c argument. K r e w e r a s [53] gave a n e x t e n s i o n of f o r m u l a (3b) t o "skew Young
422
G. Viennot
tableaux" ( s e e definition i n
9 6
b e l o w ) . Canfield and W i l l i a m s o n [I171 gave a n
" o p e r a t o r " f l a v o u r f o r hook l e n g t h s . T h e n u m b e r s
fX
a r e a l s o related t o the
( w e a k ) B r u h a t o r d e r ( s e e the end of t h i s p a p e r ) . T h e p r o b l e m of computing p r o b l e m i n poset t h e o r y . If
P
of
P
i s a p a r t i c u l a r c a s e of a c l a s s i c a l
is a poset with
n
( o r o r d e r - p r e s e r v i n g m a p ) i s a bijection
V(X)~.Cp(Y)
if
X 4 Y
1,2,
. . . ,n
e l e m e n t s , a l i n e a r extension
cp : P
+
[n]
such that
( i n P ). T h i s is a l s o c a l l e d topological o r d e r i n g ( o r
topological s o r t i n g ) . T h e i n t e g e r s are
fX
rp(x)
c a n be c o n s i d e r e d a s l a b e l s (the l a b e l s
a n d e a c h n u m b e r a p p e a r s once a n d only o n c e ) . S o m e t i m e s ,
s u c h l a b e l i n g with l a b e l s i n c r e a s i n g along t h e c h a i n s of
P , a r e also called
" n a t u r a l l a b e l i n g " . Such e x a m p l e i s given in f i g u r e 1. The F e r r e r s diagram
Fx c a n be c o n s i d e r e d a s a l o w e r i d e a l of
Nx lN ( o r d e r e d by t h e r e s t r i c t i o n of the p r o d u c t o r d e r ( 2 ) ) . T h e n u m b e r the n u m b e r of l i n e a r e x t e n s i o n s of t h i s p o s e t
FA
.
fX
is
The problem for general
p o s e t s i s connected with m a n y o t h e r c o n s i d e r a t i o n s ( s e e f o r e x a m p l e S t a n l e y
[ 9 9 ] ) a n d i s widely o p e n . Two o t h e r kinds of p o s e t s a r e known, giving r i s e t o a n analogous f o r m u l a f o r t h e n u m b e r of l i n e a r e x t e n s i o n s a s a r a t i o of
n 1
by a p r o d u c t of
n u m b e r s playing the r o l e of "hook l e n g t h s " . One kind i s t h e s h i f t e d F e r r e r s d i a g r a m s , that is the induced p o s e t s obtained by deleting the c e l l s above t h e diagonal f r o m F e r r e r s d i a g r a m s ( T h r a l l [108]). A n o t h e r f a m i l y of p o s e t s i s t h e f a m i l y of t r e e s ( w h e r e t h e o r d e r m e a n s "to be a d e s c e n d a n t of" ). It i s e a s y t o prove t h e analog of f o r m u l a ( 3 ) w h e r e the "hook length" i s t a k e n a s t h e s i z e of a s u b t r e e ( s e e Knuth [52], e x e r c i s e 20, s e c t i o n 5 . 1 . 4 . ) . It is a m a j o r p r o b l e m t o find a n o t h e r f a m i l y of p o s e t s having a "hook length" f o r m u l a ( s e e Sagan [82]). T h e c o r r e s p o n d e n c e of t h e o r e m 2 c a n e a s i l y be g e n e r a l i z e d t o s e q u e n c e s with r e p e t i t i o n of l e t t e r s , t h a t i s w o r d s . T h e t a b l e a u t a b l e a u while t h e t a b l e a u
Q
P
i s a Young
i s a s t a n d a r d Young t a b l e a u . Knuth gave i n [51]
a n u l t i m a t e g e n e r a l i z a t i o n w h e r e both i n d i c e s
i
and v a l u e s
o(i)
c a n be
repeated.
5
4 -
S o m e c l a s s i c a l p r o p e r t i e s of t h e Robinson-Schen,r3ted c o r r e s p o n d e n c e . The dihedral group
en
D4
( s y m m e t r i e s of t h e s q u a r e ) a c t s on t h e s e t
of p e r m u t a t i o n s ; i n o t h e r w o r d s , p e r m u t a t i o n m a t r i c e s c a n be f l i p p e d o r
42 3
Chain and antichain families rotated t o obtain other permutation m a t r i c e s . T o r e l a t e this to the RobinsonS c h e n s t e d c o r r e s p o n d e n c e , it s u f f i c e s t o view a p e r m u t a t i o n
6
embedding
= {(i,O(i)))
by the two s y m m e t r i e s :
0
(1
a s t h e obvious
in the s q u a r e [n] x [ n ] . T h e a c t i o n c a n be g e n e r a t e d -1 *u ( i n v e r s e ) a n d u + U* with U * = O(n). . , ~ ( 1 )
( m i r r o r i m a g e ) . T o study t h e a c t i o n , l e t u s e x a m i n e the e f f e c t of the o p e r a t i o n s
u*
u + U-1 and
on the P - s y m b o l a n d Q - s y m b o l .
6*
E v e n though the c o n s t r u c t i o n of the
P
and Q-symbol a r e completely
d i f f e r e n t , we have t h e s u r p r i s i n g l y s i m p l e (and not t r i v i a l ) p r o p e r t y ( S c h u t z e n b e r g e r [91]) :
PROPOSITION 3 -
Q(5) ,
P(m-') =
Q(u-') = P ( u )
F o r the e f f e c t of t h e m i r r o r i m a g e we n e e d t o define two o p e r a t i o n s
Y
on s t a n d a r d Young t a b l e a u x . T h e t r a n s p o s e of a s t a n d a r d Young t a b l e a u t h e t a b l e a u o b t a i n e d by r e v e r s i n g r o w s a n d c o l u m n s a n d i s d e n o t e d by second operation, called
dual, w a s
YT
is
.
The
i n t r o d u c e d by S c h u t z e n b e r g e r [91] a n d i s
more elaborate,
Y
Let trace
Tr(Y)
lowing : x
1 1 4 p < k , if
.
be a s t a n d a r d Young t a b l e a u o n [n]
to be the l o n g e s t s e q u e n c e
. ..
{x,
F i r s t we define t h e
defined by t h e f o l k is t h e value of the (1.1) c e l l ( t h a t i s t h e s m a l l e s t e n t r y of Y ) ; for x
P
(i, j ) , then
i s t h e value of t h e c e l l
of t h e two e n t r i e s l o c a t e d in the c e l l s
(i,j t l )
Y , let
x
and
is the s m a l l e s t P+l ( i t l , j ) , (if one of t h e s e x
be the value i n t h e c e l l that i s ) . By P " l o n g e s t s e q u e n c e " , we m e a n t h a t t h e v a l u e x i s l o c a t e d i n a c e l l (i, j ) k s u c h t h a t t h e c e l l s ( i , j t l ) a n d ( i t l , j ) a r e not i n t h e s h a p e of Y . Such a c e l l s i s not i n the s h a p e of
c e l l of
Y
x
will be c a l l e d a c o r n e r c e l l of t h e F e r r e r s d i a g r a m of We define the t a b l e a u
replacing each value t h e c e l l with v a l u e
x
x
P
d(Y)
Y
. Y
by
of
Tr(P)
2,
by the v a l u e
x
. . . ,n , and shape contained in the
.
LEMMA 4 - (Schtitzenberger) T
Y
, deleting P+l and keeping i n v a r i a n t all o t h e r v a l u e s . Obviously, d ( P ) (1 d p < k )
k i s a Young t a b l e a u with d i s t i n c t e n t r i e s s h a p e of
as the tableau obtained f r o m
Let
u
be a s e q u e n c e of d i s t i n c t i n t e g e r s , and
be t h e s e q u e n c e o b t a i n e d by deleting t h e m i n i m u m e l e m e n t . T h e n we have
P ( T )= d ( P ( 0 ) ) .
G. Viennot
424 Y
Let
YJ
be a s t a n d a r d Young t a b l e a u . We define t h e t a b l e a u
the following l a leling. T h e c e l l containing the g r e a t e s t value
a
xk
of
by
Tr(Y)
is
r e l a b e l e d 1 . F o r 1 , 1 G 4 n , the c e l l which i s in d ( Y ) , but not i n Rt1 a ( Y ) ( t h a t is the c e l l containing t h e g r e a t e s t v a l u e of T r ( d ( Y ) ) ) , i s r e l a d beled
Y
k+l
.
The tableau
with t h e i n t e g e r s
1, 2 ,
YJ
i s a labeling of t h e F e r r e r s d i a g r a m u n d e r l y i n g
. . .,n
such that rows and columns a r e s t r i c t l y d e -
c r e a s i n g , Such a t a b l e a u i s c a l l e d a r e v e r s e s t a n d a r d Young t a b l e a u a n d i s n a m e d the
dual of
.
Y
An e x a m p l e is shown i n f i g u r e 7 .
d(Y) =
b,, , m,,, 5
6
8
4
2
m
& a
7
Figure 7.
5
4
2
6
8
5
4
7
5
4
IZBn 4
5
6
8
2
2
[q,, 9
m
1-51
2
T h e dual of a s t a n d a r d Young t a b l e a u , obtained by "vidage - r e m p l i s s a g e " ,
PROPOSITION 5 - ( S c h e n s t e d , S c h i i t z e n b e r g e r ) . F o r any p e r m u t a t i o n u of T J T p(0") = p ( 0 ) , QC(o*) = (Q ) ( 0 ) , w h e r e QC denote the r e v e r s e e n , t a b l e a u o b t a i n e d f r o m Q by r e p l a c i n g e a c h v a l u e i by n t l - i .
425
Chain and antichain families
T h e r e a d e r i s u r g e d t o t e s t t h i s p r o p e r t y w i t h t h e p e r m u t a t i o n of f i g u r e 5 u s i n g t h e c o n s t r u c t i o n of f i g u r e 7 . By r e v e r s i n g t h e o r d e r , o n e c a n d e f i n e t h e d u a l of a r e v e r s e s t a n d a r d Young t a b l e a u . T h e a b o v e p r o p o s i t i o n i m p l i e s t h e r e l a t i o n
(4)
J J (Y ) = Y
f o r a n v s t a n d a r d Young t a b l e a u
.
In [ 9 4 ] , S c h u t z e n b e r g e r h a s e x t e n d e d t h e c o n s t r u c t i o n t r a r y posets. Let
P
be a p o s e t a n d
PL
s t a n d a r d Young t a b l e a u x , w e c a n d e f i n e
Y
+ YJ
a linear embedding in
for arbi[n] . As f o r
T r ( P L ) , a n i n c r e a s i n g s e q u e n c e of l a -
b e l s c a l l e d t h e t r a c e of
Using the o p e r a t o r analogous to d(Y) , it i s posPL . s i b l e t o d e f i n e a d u a l " r e v e r s e n a t u r a l " l a b e l l i n g of PL . A d e e p r e s u l t i s t h a t relation (4) i s still valid. Knuth's transpositions. W e s t u d y t h e c l a s s of p e r m u t a t i o n s h a v i n g t h e s a m e P - s y m b o l . T h e n u m b e r of s u c h p e r m u t a t i o n s i s o b v i o u s l y
fk
where
X
i s t h e s h a p e of
P
.
We m a y
a s k how to g e n e r a t e all t h e s e p e r m u t a t i o n s f r o m o n e of t h e m , u s i n g e l e m e n t a r y t r a n s f o r m a t i o n s . T h i s i s d o n e by t h e s o - c a l l e d Knuth t r a n s p o s i t i o n s ( K n u t h [51]). Let
DEEn
If t h e v a l u e or
z =
and U(i-1)
x = u(i) ,
or
z =
y = O(it1)
O(it1)
be t w o c o n s e c u t i v e e l e m e n t s of
is b e t w e e n
y < z C x ) , t h e n t h e t r a n s p o s i t i o n of
x
and
y
( t h a t is
u .
x
and
y
i s c a l l e d a Knuth t r a n s -
x < z
position. T h e P - s y m b o l i s invariant under Knuth's transpositions. Conversely, any permutation
T
such that
P(0) = P ( T ) c a n be o b t a i n e d f r o m
u
by a s e q u e n c e
of K n u t h ' s t r a n s p o s i t i o n s .
A p e r m u t a t i o n i s c a l l e d r o w c a n o n i c a l ( r e s p . c o l u m n c a n o n i c a l ) if t h e s e quence
~ ( 1 ). .. ~ ( n ) c a n b e o b t a i n e d by r e a d i n g t h e e n t r i e s of t h e P - s y m b o l i n
t h e f o l l o w i n g w a y : r e a d e a c h r o w f r o m l e f t to r i g h t w i t h r o w s o r d e r e d t o p - d o w n ( r e s p . r e a d e a c h c o l u m n top-down with the c o l u m n s o r d e r e d f r o m l e f t to r i g h t ) . In e a c h class h a v i n g t h e s a m e P - s y m b o l t h e r e i s o n e a n d o n l y o n e c a n o n i c a l p e r mutation. Example -
The permutation
5
=
6 3 581 247 9
i s row canonical. Possible
K n u t h ' s t r a n s p o s i t i o n s a r e t h e t r a n s p o s i t i o n s of c o n s e c u t i v e v a l u e s
( 6 ,3)
G. Viennot
426 and (8,l).
Of c o u r s e , w e c a n d u a l l y d e f i n e K n u t h ' s t r a n s p o s i t i o n s o n t h e v a l u e s i n s t e a d of t h e p o s i t i o n s a n d k e e p t h e Q - s y m b o l i n v a r i a n t . G r e e n e ' s i n t e r p r e t a t i o n of t h e s h a p e . We c o m e b a c k t o t h e m a i n m o t i v a t i o n of t h i s t a l k .
PROPOSITION 6 with t h e p o s e t P(o) a n d
-
For any permutation
Pos(u)
Q(u)
( d e f i n e d in
8
the Greene diagram associated
3
2 ) i s t h e s a m e a s t h e F e r r e r s d i a g r a m of
o b t a i n e d by t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e .
In t h e o t h e r w o r d s , t h e n u m b e r of e l e m e n t s of t h e f i r s t c o l u m n s ) i s the m a x i m u m c a r d i n a l i t y of s u b s e q u e n c e s of k
k
rows ( r e s p .
w h i c h a r e union of
0
increasing (resp. decreasing) subsequences. The particular case
k=1
i s known a s S c h e n s t e d ' s T h e o r e m [89] a n d w a s
t h e m o t i v a t i o n of S c h e n s t e d ' s o r i g i n a l c o n s t r u c t i o n . G r e e n e p r o v e d p r o p o s i t i o n 6 in [35] by s h o w i n g i t s i n v a r i a n c e u n d e r K n u t h ' s t r a n s p o s i t i o n s , a n d noting t h a t t h e p r o p e r t y is t r i v i a l f o r r o w c a n o n i c a l p e r m u t a t i o n s . We s h a l l s e e b e l o w a d i r e c t g e o m e t r i c c o n s t r u c t i o n of s u c h s u b s e q u e n c e s ( t h a t i s , u n i o n s of
k
increasing (decreasing) subsequences with m a x i -
m u m c a r d i n a l i t y ) , a n d a l s o a c h a r a c t e r i z a t i o n of t h e s h a p e of t h e P - s y m b o l c o m p l e t e l y s y m m e t r i c with r e s p e c t t o r o w s a n d c o l u m n s .
A n i c e c o r o l l a r y of S c h e n s t e d ' s t h e o r e m is a p r o p o s i t i o n of E r d 6 s a n d S z e k e r e s [12] : a n y p e r m u t a t i o n c o n t a i n i n g m o r e t h a n
n2
elements h a s mono-
t o n i c s u b s e q u e n c e ( i n c r e a s i n g o r d e c r e a s i n g ) of l e n g t h g r e a t e r t h a n
n
(this
p r o p o s i t i o n i s a l s o a c o r o l l a r y of D i l w o r t h ' s T h e o r e m , a s a p p l i e d to p o s e t s of d i m e n s i o n 2 ) . N o t e t h a t t h e r e e x i s t p e r m u t a t i o n s with
.
m o n o t o n i c s u b s e q u e n c e s of l e n g t h g r e a t e r t h a n
n
e x a c t l y t h o s e having a P - s y m b o l with a s q u a r e
nxn
n2
elements having no
These permutations are shape.
Many i n v e s t i g a t i o n s h a v e b e e n m a d e of t h e n u m b e r of p e r m u t a t i o n s h a v i n g a m a x i m u m - l e n g t h i n c r e a s i n g s u b s e q u e n c e . No e x a c t f o r m u l a i s known ( s e e [791
f o r e x a m p l e ) . T h i s p r o b l e m is e q u i v a l e n t t o e n u m e r a t i n g s t a n d a r d Young t a bleaux having
k
r o w s , a n d is r e l a t e d t o p r o b l e m s i n a l g e b r a (see R e g e v [ 7 5 ] ) .
*.
It h a s b e e n p r o v e d by H a i n m e r s l e y [43] t h a t t h e m e a n of t h e l e n g t h of maximal increasing subsequence is asymptotically
c n
L o g a n a n d S h e p p [71]
proved
Chain and antichain families c
2 , while K e r o v a n d V e r s h i k [5O] p r o v e d
c Q2
427
.
c = 2 , as was sug-
Thus
g e s t e d by B a e r and B r o c k [122]. A l s o F r e d m a n [24] gave a n o p t i m a l a l g o r i t h m in nlogn-nloglogn c o m p a r i s o n s t o c o m p u t e t h i s m a x i m a l l e n g t h .
"Line-of - r o u t e " and up-down s e q u e n c e An i m p o r t a n t notion i n t h e e n u m e r a t i v e t h e o r y of p e r m u t a t i o n s i s t h e up-down s e q u e n c e of a p e r m u t a t i o n with two l e t t e r s
t
,
-
.
It i s defined a s a w o r d of l e n g t h n-1
a s follows with
UD(0) = w l ' . .Wn-l
(5)
UEEn
w,1 =
+
~-
iff
o(i) < o(it1) ( r i s e )
if
o ( i ) 2 o ( i + l ) (fall)
T h e dual notion of a r i s e ( r e s p . f a l l ) i s t h a t of a d v a n c e ( r e s p . r e t r e a t ) : x = o(i)
i s a n a d v a n c e ( r e s p . r e t r e a t ) if x t l = O ( j ) with i j -1 y i e l d s t h e a d v a n c e s and r e t r e a t s ( r e s p . i b j ) . T h e up-down s e q u e n c e of 0
the value
.
a
of
S h i i t z e n b e r g e r s h o w e d t h a t the value
u
iff t h e v a l u e
tableau with
xtl
is a n a d v a n c e of t h e p e r m u t a t i o n
x
i s l o c a t e d t o the S o u t h - E a s t of the v a l u e
P(o) (that is
x
i s in the cell
(i, j )
and
xtl
x
i n t h e Young
in t h e cell
(i',jt)
it* i , j ' 4 j ) . Note t h a t if
W e s t (that is
x+l
i s not S o u t h - E a s t of
h a s b e e n c a l l e d t h e l i n e of r o u t e by F o u l k e s . -1 Dually, by taking t h e i n v e r s e u of of r o u t e of
x
, then
xtl
i s a t the North-
i ' 4 i , j ' > j ) . T h i s s u c c e s s i o n of S o u t h - E a s t o r N o r t h - W e s t s t e p s
Q(u)
a
a s m e n t i o n e d a b o v e , the l i n e
g i v e s the up-down s e q u e n c e of the p e r m u t a t i o n ( s e e e x a m p l e
on figure 8 ) . Example -
For
u = 6 1 7 2 5 8 3 9 4 , UD(u)
= -
1
+-
- + - and - -+t+ .
t t
UD(o- ) = t t t
T h e l i n e s of r o u t e of t h e P - s y m b o l a n d Q - s y m b o l a r e t h e following
G. Viennot
428
Figure 8
-
L i n e s of r o u t e of t h e P - s y m b o l a n d Q - s y m b o l
F o u l k e s gave beautiful a p p l i c a t i o n s of t h i s i n t e r p r e t a t i o n to e n u m e r a t i v e p r o b l e m s of p e r m u t a t i o n s [19], [ZO], [Zl] duality
J
Y+Y )
k-matchings
P
in the
-
. Other applications
( u s i n g a l s o the
c a n be found i n F o a t a , S c h u t z e n b e r g e r [15].
A m a j o r q u e s t i o n i s t o give a d i r e c t i n t e r p r e t a t i o n of t h e v a l u e s
a n d Q - s y m b o l . G r e e n e g a v e a n i n t e r p r e t a t i o n of t h e s e t of v a l u e s
which a r e above t h e kth r o w i n the P - s y m b o l . F o r t h a t p u r p o s e , he i n t r o d u c e d the new c o n c e p t of k - m a t c h i n g s . A k - m a t c h i n g of a p e r m u t a t i o n
o
is a n array
(aij)lG i G p , 1 4 j 4 k
i n t e g e r s s u c h t h a t e a c h r o w i s a d e c r e a s i n g s u b s e q u e n c e of of e a c h c o l u m n a r e d i s t i n c t . T h e s e t
all,
. . . , a Pl
u
of
a n d the e l e m e n t s
i s c a l l e d the s o u r c e of t h e
k - m a t c h i n g . F i r s t , t h e n u m b e r of e l e m e n t s i n t h e r o w s above a n d including t h e kth r o w of
P ( u ) is e q u a l t o t h e m a x i m u m s i z e ( a m o n g k - m a t c h i n g s , k f i x e d )
of the s o u r c e of a k - m a t c h i n g of t h e p e r m u t a t i o n [ 3 8 ] . F o r the s e c o n d r e s u l t , we o r d e r s u b s e t s of [n] A
a ppe a r s in
A
t h a t is
by l e x i c o g r a p h i c o r d e r ,
iff t h e s m a l l e s t e l e m e n t t h a t o c c u r s i n only one of
.
A
and
B
G r e e n e [38] showed t h a t t h e s e t of e l e m e n t s l o c a t e d i n the r o w s
above a n d including t h e kth r o w of
P ( u ) is the l e x i c o g r a p h i c a l l y m i n i m u m
s o u r c e a m o n g the s o u r c e s of m a x i m u m - s i z e d k - m a t c h i n g s of t h e p e r m u t a t i o n . G a n s n e r [27] g a v e p r o o f s of t h e s e two f a c t s using t o o l s f r o m l i n e a r a l g e b r a , a n d gave a n a n a l o g of t h e f i r s t p r o p e r t y f o r c o l u m n s , by introducing t h e c o n c e p t of k - s c a t t e r s of a p e r m u t a t i o n . We s h a l l d i s c u s s below the e x t e n s i o n of t h i s t o a r b i t r a r y p o s e t s .
Chain and antichain families § 5 -
429
Planarization In o r d e r t o give a s i m p l e e x p l a n a t i o n of p r o p o s i t i o n 3 ( a n d o t h e r p r o p e r t i e s
such a s G r e e n e ' s i n t e r p r e t a t i o n s ) , Viennot i n t r o d u c e d [lo91 a g e o m e t r i c v e r s i o n of the R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e , t h a t is a p i c t o r i a l d e s c r i p t i o n with "shadows" a n d " b o r d e r s of shadows" i n t h e p l a n e . We c o n s i d e r t h e p o s e t
TT = Z x
Z
o r d e r e d by t h e u s u a l p r o d u c t o r d e r ( 2 ) .
We i m a g i n e s o m e light coming f r o m t h e S o u t h - W e s t . T h e shadow of a point
.
x2 i , y B j
(i, j )
i s the s e t of points
F
T h e shadow of a f i n i t e s u b s e t
dows of e a c h point of
F
.
of
(x,y)
of
IRx
IR
with
i s the union of the s h a -
IT
T h e b o r d e r of t h i s shadow ( i n t h e t o p o l o g i c a l s e n s e )
i s a " z i g - z a g l i n e " s u p p o r t e d by the points t h a t a r e "in f u l l light", t h a t i s the
F
points which a r e not i n t h e shadow of a n o t h e r point of
A finite subset
F
of
11
.
i s s a i d t o be a q u a s i - p e r m u t a t i o n iff n o two
d i s t i n c t points a r e o n the s a m e v e r t i c a l o r h o r i z o n t a l l i n e . T h i s t e r m i n o l o g y i s not c l a s s i c a l . S e v e r a l o t h e r n a m e s a r e u s e d , i n p a r t i c u l a r independent s e t . When
TI
i s considered a s a poset ( d i r e c t product) such s e t s a r e called
s e m i a n t i c h a i n s . We i n t r o d u c e h e r e t h e t e r m " q u a s i - p e r m u t a t i o n ' ' b e c a u s e i t w i l l c o r r e s p o n d below t o a s u b s e q u e n c e of a p e r m u t a t i o n viewed a s a s e q u e n c e , i n which the i n f o r m a t i o n a b o u t t h e p o s i t i o n s h e l d by t h e e l e m e n t s i s r e t a i n e d . Let L1,
. . ., L k
shadow of
F
be a q u a s i - p e r m u t a t i o n of
.
We define a s e q u e n c e
of b r o k e n l i n e s by t h e following p r o c e s s :
F , and f o r
L1,. . . , Li
o u t s t a n d i n g of
F
i,
1 6i
k ,
Li+l
i s t h e b o r d e r of t h e L1 i s t h e b o r d e r of the shadow of t h e
Fi+l o b t a i n e d by deleting f r o m
quasi -pe rmutation the l i n e s
n
.
(see figure 9 ) . The lines
L1,
F
t h e points t h a t a r e on
. . .,Lk
w i l l be c a l l e d t h e
G. Viennot
430
-I !;:.+-I Figure 9
-
T h e outstanding l i n e s a n d t h e s k e l e t o n of t h e permutation u = 6 1 7 2 5 8 3 9 4 .
In t h i s c o n s t r u c t i o n , s o m e r e m a r k a b l e points a p p e a r . Following the
F , one c h a n g e s d i r e c t i o n f o r e a c h point of
outstanding l i n e s of
a l s o a change of d i r e c t i o n f o r o t h e r points t h a t a r e not i n points i s a q u a s i - p e r m u t a t i o n of
S(F)
Z x
0
be a P e r m u t a t i o n of
= ( ( i , o ( i ) ) , i E [n]
3
,
Z c a l l e d t h e s k e i e t o n of
. T h e y a r e m a r k e d with c r o s s e s i n f i g u r e Note t h a t IS(F)I t k = IF I . Let
F
Bn
,
F
. There
is
T h e s e t of t h e s e
F
and denoted
9.
We c o n s i d e r t h e q u a s i - p e r m u t a t i o n
a n d apply the c o n s t r u c t i o n
F + S (F) a s many t i m e s as
we need t o obtain the e m p t y q u a s i - p e r m u t a t i o n . We t h u s c o n s t r u c t a s e q u e n c e
s
( a ) = 6 , . . . , sp-l(3) w h e r e f o r
i , 1d i
ep ,
Si+l(6) = S(Si(6)) , a n d
Sp(G))= $
(see figure lo).
An outstanding l i n e of a q u a s i - p e r m u t a t i o n s e g m e n t s (joining a point of
F
with a point of
F
is f o r m e d with f i n i t e
S(F))
and with two infinite
h a l f - l i n e s , one i s v e r t i c a l a n d t h e o t h e r i s h o r i z o n t a l . T h i s v e r t i c a l ( r e s p .
Chain and antichain families horizontal) half-line h a s coordinate x ' s axis
v(L)
(resp. h(L))
43 1 with r e s p e c t with the
(resp. y's axis).
We a r e now r e a d y t o s t a t e t h e " g e o m e t r i c " v e r s i o n of t h e R o b i n s o n Schensted correspondence :
PROPOSITION 7 of the P - s y m b o l h(Lki)
(Viennot) P(0)
Let
0
be a p e r m u t a t i o n of
( r e s p . Q-symbol
Q(u))
G n . The ith row
.
h(L1), . . ,
i s the sequence
( r e s p . v(L1), . . . , v ( L k , ) ) of c o o r d i n a t e s of t h e h a l f - l i n e s c o m p o s i n g 1
t h e outstanding l i n e s of
si-l(8).
9
a 7
Outstanding lines of
6
-
5
- 0 -
C S(0)
--_- s2
(Q-)
4 3 2 1
F i g u r e 10 - P l a n a r i z a t i o n of t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e
432
G. Viennot
The permutation
0
c a n be r e c o n s t r u c t e d when knowing, f o r e v e r y
the c o o r d i n a t e s of the h a l f - l i n e s a p p e a r i n g i n the outstanding l i n e s of the i
iM, th
S.(6). T h i s c o m e s f r o m t h e following g e o m e t r i c p r o p e r t y .
skeleton
F , let
F o r a quasi-permutation
I
and
on t h e two c o o r d i n a t e a x i s , T h e d i r e c t p r o d u c t
F
s u p p o r t of
and i s denoted by
J
F
be t h e p r o j e c t i o n s of
Zx Z
Ix J
i s c a l l e d the
S u p p ( F ) . A f u n d a m e n t a l l e m m a is t h a t t h e
F + ( S u p p ( F ) , S ( F ) ) i s a n i n j e c t i o n f r o m the s e t of q u a s i - p e r m u t a t i o n s of
map
into the s e t of p a i r s
[n] x [n] the f o r m
Ix J
, and
B
(A, B )
where
A
i s a s u b s e t of
i s a q u a s i - p e r m u t a t i o n included in
it d o e s not s e e m obvious to give a c o n s t r u c t i o n of
F
A
when
[n]x [n] ,
of
At f i r s t s i g h t ,
(Supp(F), S ( F ) )
i s known. N e v e r t h e l e s s , t h e r e e x i s t s a n e a s y g e o m e t r i c c o n s t r u c t i o n ( s e e Viennot [109]).
F
F = T ( S ( F )U C ) , w h e r e
is given by
(with 2 k points if
S(F)
has
k
C
i s a c e r t a i n s u b s e t of
outstanding l i n e s ) a n d
T
Zx
Z
i s the notation
f o r a n analogous definition of the s k e l e t o n of a q u a s i - p e r m u t a t i o n , but with t h e "light" c o m i n g f r o m t h e N o r t h - E a s t . T h u s , by r e p e t i t i v e a p p l i c a t i o n of t h i s r e l a t i o n , the p e r m u t a t i o n
u
can
be r e c o n s t r u c t e d when knowing the ( d e c r e a s i n g ) s e q u e n c e of s u p p o r t s of the
. , .,S
(8). T h i s s e q u e n c e is c o m p l e t e l y defined by P-1 t h e c o o r d i n a t e s of the h a l f - l i n e s a p p e a r i n g i n the outstanding l i n e s of S o ( 6 ) , . . . , different skeletons
So(6),
T h e r e a d e r i s u r g e d t o t a k e a r e c t a n g u l a r p i e c e of p a p e r , hide t h e points of
6
f o r the p e r m u t a t i o n d i s p l a y e d o n f i g u r e 10, t h e n move t h i s p i e c e of p a p e r
f r o m l e f t to r i g h t , a n d look c a r e f u l l y what happens e a c h t i m e a new c o l u m n of the [n] x [n]
g r i d a p p e a r s , When p a s s i n g f r o m c o l u m n
i
to
it1
, the c h a n g e of
the outstanding l i n e s c r o s s i n g t h e l e f t - h a n d s i d e of the p i e c e of p a p e r i s nothing but a g e o m e t r i c coding of the "bumping p r o c e s s " i n s e r t i n g of the w o r d
it1
i n the P - s y m b o l
~ ( 1 )... u ( i ) . A new v e r t i c a l h a l f - l i n e a p p e a r s i n c o l u m n
c o r r e s p o n d i n g outstanding l i n e belongs t o a c e r t a i n s k e l e t o n
S,(U)
it1
.
.
The
T h e index
i s a coding of the " Q - p a r t " of t h e R o b i n s o n - a l g o r i t h m . T h i s i s e s s e n t i a l l y the proof of p r o p o s i t i o n 7 . It would be p o s s i b l e to m o v e t h e piece of p a p e r f r o m t h e b o t t o m t o t h e t o p and t h u s d e s c r i b e d a dual v e r s i o n of the bumping p r o c e s s . In f a c t t h i s duality i s -1 , which c o r r e s p o n d s t o exchanging r o w s nothing but t h e t r a n s f o r m a t i o n 0 + 0
Chain and antichain families
433
a n d c o l u m n s i n t h e p i c t u r e . V e r t i c a l c o o r d i n a t e s of t h e h a l f - l i n e s of t h e o u t standing l i n e s a r e e x c h a n g e d with t h e h o r i z o n t a l c o o r d i n a t e s . T h u s p r o p o s i t i o n 3 -1 is m a d e c r y s t a l c l e a r : t h e duality 0 3 0 c o r r e s p o n d s t o exchanging t h e
P - and Q -symbol s. The symmetry
U
3
u*
( m i r r o r i m a g e ) c o r r e s p o n d s t o m a k i n g the "light"
c o m e f r o m the S o u t h - E a s t . In f a c t , t h e r e a r e f o u r d i f f e r e n t p o s s i b l e d i r e c t i o n s f o r the "light" a n d t h u s f o u r d i f f e r e n t c o n s t r u c t i o n s of s e t s of o u t s t a n d i n g l i n e s a n d s k e l e t o n s . Unfornately, t h e r e do not s e e m t o b e s i m p l e g e o m e t r i c r e l a t i o n s b e t w e e n t h e s e f o u r c o n s t r u c t i o n s . In p a r t i c u l a r , p r o p o s i t i o n 5 d o e s not s e e m t o have a si mpl e explanation in t h i s g eo met r i c model. N e v e r t h e l e s s , G r e e n e ' s T h e o r e m (proposition 6 and k-matching i n t e r p r e t a t i o n ) c a n be p r o v e d in a c o n s t r u c t i v e g e o m e t r i c way, without changing t h e p e r m u t a t i o n t o a c a n o n i c a l one with Knuth's t r a n s p o s i t i o n s , T h e r e a d e r will f i n d in [lo91 a g e o m e t r i c c o n s t r u c t i o n of a m a x i m a l s i z e d a n t i c h a i n k - f a m i l y , using a n o p e r a t o r , s o m e w h a t " i n v e r s e " t o t h e o p e r a t o r
F
*
S ( F ) , sending e v e r y s u b s e t of
S(F)
t o a s u b s e t of
F
according to some
"light" c o m i n g f r o m the N o r t h - E a s t . A k - m a t c h i n g with l e x i c o g r a p h i c a l l y m i n i m u m m a x i m u m - s i z e s o u r c e c a n be o b t a i n e d in t h e following w a y . We s t a r t with t h e s e t of y - c o o r d i n a t e s of the S k - l ( ~ ). Below e a c h of t h e s e p o i n t s , t h e r e i s a unique point of
points of Sk-2(u)
.
T h e s e c o n d c o l u m n of the k - m a t c h i n g will be f o r m e d with t h e y - c o o r -
dinate of t h e s e p o i n t s , (which a r e d i s t i n c t v a l u e s ) . Going down f r o m s k e l e t o n t o
6 g i v e s v a l u e s t h a t c a n be a r r a n g e d i n a r e c -
s k e l e t o n t i l l r e a c h i n g points of
t a n g u l a r a r r a y having t h e k - m a t c h i n g p r o p e r t y .
F o a t a ' s m a t r i x c h a r a c t e r i z a t i o n a n d k e r n e l of t h e i n v e r s i o n p a i r g r a p h . When using t r a n s p a r e n c i e s , t h e d i f f e r e n t s k e l e t o n s c o l o r e d black, r e d , blue, g r e e n , points of We h a v e a
by t h e v a l u e
S.(U)
nxn
Si(U)
, iS 0 , a r e
. . . and e v e r y t h i n g is c l e a r . H e r e , w e l a b e l t h e . All o t h e r points of [n] x[n] a r e l a b e l e d 0 .
it1
m a t r i x with i n t e g e r e n t r i e s . A q u e s t i o n i s t o f i n d a c h a r a c t e -
r i z a t i o n of t h i s m a t r i x . T h i s h a s b e e n done by F o a t a i n [14]. We need t o define t h e "hook" joining two p o i n t s . L e t be two points of
Z x Z
or
and
(XI,
y')
s u c h t h a t t h e s e c o n d is at t h e S o u t h - E a s t of t h e f i r s t
o n e . T h e h o o k joining t h e s e two points i s t h e s e t of points x
(x, y )
i = x' , y 3 j
> Y'
( s e e f i g u r e 11).
(i, j )
with
j = y
,
434
G. Viennot A m a t r i x with i n t e g e r e n t r i e s h a s the f o r m defined above (coding of t h e
d i f f e r e n t s k e l e t o n s of a p e r m u t a t i o n
IJ
en )
of
iff i t s a t i s f i e s the t h r e e f o l -
lowing conditions : (i)
t h e r e e x i s t s one a n d only one 1 i n e a c h r o w and column
(ii)
for every entry
i t 1 2 2 , t h e r e e x i s t s a n e n t r y i both on the l e f t a n d
below the e n t r y
it1 ,
(iii)
f o r e v e r y hook joining two points l a b e l e d it1
i 3 1 , there exists an entry
l o c a t e d on t h e hook between t h e m .
Such m a t r i c e s h a v e been i n t r o d u c e d by F o a t a [14] u n d e r t h e n a m e of Viennot's m a t r i c e s . Example -
0 0 0 1 0 0 0 0 0
0 0 0 2 0 0 0 0 1
0 0 1 0 0 0 0 0 0
0 0 2 0 0 0 0 1 0
0 0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 0 3 2 0 1 0 0
1 0 0 0 0 0 0 0 0
0 2 0 0 0 1 0 0 0 F i F " 11
- A hook
joining two p i n t s .
T h e above c h a r a c t e r i z a t i o n c a n a l s o be w r i t t e n in a n i c e way i n t e r m s of graph theory. Let
0
t a t i o n . We denote by [n] x [n]
6 and 5 [n] x [n] i t s p l a n a r r e p r e s e n n the g r a p h whose v e r t i c e s a r e a l l t h e points of
be a p e r m u t a t i o n of
GI(o)
s u c h t h a t t h e r e e x i s t s a point of
6
o n the l e f t a n d below t h e m ( t h a t
i s t h e union of all the c o r n e r s of t h e hooks joining points of edge b e t w e e n t h e v e r t i c e s y = y'
and
pairs
(i. j )
x > x'
.
(x.y)
and
(x', y')
iff
x = x'
5
). We put a n
and
y > y'
or
T h e n u m b e r of v e r t i c e s of t h i s g r a p h i s t h e n u m b e r of
such that
i
<j
n u m b e r of t h e p e r m u t a t i o n
and 0
.
o ( i ) 3 O(j) , i . e . t h e s o - c a l l e d i n v e r s i o n s
Such d i a g r a m s have been i n t r o d u c e d by Rothe
[8l] i n o r d e r t o give a n i c e ( g e o m e t r i c ) proof of t h e f a c t t h a t t h e i n v e r s i o n
435
Chain and antichain families
n u m b e r i s invariant under the t r a n s f o r m
0
=;b
u
-1
.
In t h e i r r e c e n t w o r k r e l a t i n g c o m b i n a t o r i c s (Young t a b l e a u x , p l a c t i c m o noid, S c h u r f u n c t i o n s , . . . ) a n d a l g e b r a i c g e o m e t r y (flag m a n i f o l d s ,
. . .)
[611,
1651, r66], L a s c o u x and S c h u t z e n b e r g e r showed that t h e s e " i n v e r s i o n p a i r s " g r a p h s play t h e s a m e r o l e f o r S c h u b e r t f u n c t i o n s a s F e r r e r s d i a g r a m s f o r S c h u r f u n c t i o n s . T h e y c a l l e d t h e s e g r a p h s Riguet d i a g r a m s .
F i g u r e 12
-
T h e i n v e r s i o n p a i r s g r a p h (on Riguet d i a g r a m ) of a p e r m u t a t i o n .
A c l a s s i c a l notion i n t h e t h e o r y of d i r e c t e d g r a p h s i s t h e k e r n e l of a g r a p h , that i s a s e t of points s u c h t h a t e v e r y v e r t e x of the g r a p h i s t h e s o u r c e of a n edge ending i n the k e r n e l , a n d e v e r y edge having i t s s o u r c e i n t h e k e r n e l h a s i t s end not i n t h e k e r n e l ( s e e B e r g e [5]). Such a s e t i s unique if i t e x i s t s . It i s a l s o t h e s e t of "winning p o s i t i o n s " of a " N i m g a m e " played on t h e g r a p h . F r o m [lo91 t h e s k e l e t o n
S(0)
of t h e p e r m u t a t i o n
CJ
is e x a c t l y the
G. Viennot
436
k e r n e l of i t s i n v e r s i o n p a i r s g r a p h . Writing t h i s p r o p e r t y f o r all s k e l e t o n s S.(U) leads to Foata's characterization. F r o m F r e d m a n [24] a n d Viennot [lo91 t h e c o n s t r u c t i o n of the l o n g e s t s u b -
s e q u e n c e of a p e r m u t a t i o n
u
n e e d s the c o n s t r u c t i o n of the s k e l e t o n . T h u s ,
t h e c o n s t r u c t i o n of the s k e l e t o n c a n be done by a n o p t i m a l a l g o r i t h m ( f o r t h e w o r s t c a s e i n "on-line" r e a d i n g ) i n
nlogn-nloglogn
comparisons, For more
u n d e r s t a n d i n g about t h e s e t h e o r e t i c a l c o m p u t e r s c i e n c e c o n c e p t s , s e e f o r e x a m p l e Knuth [52] o r Aho, H o p c r o f t , Ullman [2]
6
6
-
.
Plactic monoid. We have s e e n t h a t t h e Robinson-Schensted c o r r e s p o n d e n c e i s r e l a t e d t o
t h e r e p r e s e n t a t i o n t h e o r y of t h e s y m m e t r i c g r o u p . T h e c o m b i n a t o r i c s of Young t a b l e a u x i s a l s o i n t i m a t e l y c o n n e c t e d with t h e t h e o r y of s y m m e t r i c f u n c tions. A b a s i s of s y m m e t r i c f u n c t i o n s h a s b e e n i n t r o d u c e d by J a c o b i [48] a s a quotient of a n t i s y m m e t r i c functions ( e x p r e s s e d a s d e t e r m i n a n t s ) . Following F r o b e n i u s [25), S c h u r [9O] d i s c o v e r e d t h e i r r e l e v a n c e t o t h e r e p r e s e n t a t i o n t h e o r y of t h e s y m m e t r i c g r o u p s a n d t h e g e n e r a l l i n e a r g r o u p s . T h e s e f u n c t i o n s h a v e b e e n c a l l e d S c h u r f u n c t i o n s ( o r S - f u n c t i o n s ) by Littlewood and R i c h a r d s o n [TO),
A l s o they a r e r e l a t e d t o s o m e w o r k i n i t i a t e d by P i e r i i n 1873, f o l l o w e d by
S c h u b e r t , G i a m b e l l i a n d o t h e r s , a n d which i s nowadays n a m e d a "cohomology r i n g of G r a s s m a n n v a r i e t i e s " . We s h a l l not t o u c h t h i s d e e p a n d huge s u b j e c t . We s h a l l only give t h e w e l l known c o m b i n a t o r i a l definition of t h e S c h u r f u n c t i o n s ( s e e f o r e x a m p l e Littlewood [ 6 9 ] ).
X
Let
be a p a r t i t i o n a n d
be a Young t a b l e a u of s h a p e
Y
X
(with
e n t r i e s s t r i c t l y i n c r e a s i n g i n c o l u m n s a n d w e a k l y i n c r e a s i n g i n r o w s ) . We d e i xp" , w h e r e i.!, i s t h e n u m b e r of e n note by m(Y) t h e m o n o m i a l
4'. . .
' trie s equal to
.!,
in the tableau
s u m of all m o n o m i a l s
m(Y)
and e n t r i e s taken among Example
-
s ( 2 , 1) (xl,
X2'
1,2,
Y
. T h e S c h u r function
Sx (xl,
. . . ,xn)
e x t e n d e d o v e r all Young t a b l e a u x with s h a p e
is t h e
X
. . ., n .
x 3 ) = x12 x 2 + x 2 x3+ x3 2 xl+ x12 x3+ x 2 XI+ x 2 3 x2 + 2x1 x2 x 3 *
Chain and ontichain families
437
c o r r e s p o n d i n g t o the e i g h t Young t a b l e a u x d i s p l a y e d i n t h e following f i g u r e .
F i g u r e 13 -
A Schur function.
L a s c o u x a n d S c h u t z e n b e r g e r i n t r o d u c e d the p l a c t i c monoid a s a n o n - c o m m u t a t i v e c a l c u l u s , unifying a n d extending p r e v i o u s p r o p e r t i e s of the R o b i n s o n S c h e n s t e d c o r r e s p o n d e n c e a n d of the t h e o r y of s y m m e t r i c f u n c t i o n s [62]. F o r example, the r a t h e r m y s t e r i o u s , s o - c a l l e d "Littlewood-Richardson rule" (giving t h e p r o d u c t of two S c h u r f u n c t i o n s a s s u m of S c h u r f u n c t i o n s ) i s m a d e clear. T h e c o m b i n a t o r i a l p a r t of t h i s t h e o r y is b a s e d on t h e "jeu d e taquin", [62] ,[97].
Y
+
d(Y)
T h i s r u l e f o r moving l a b e l s i n d i a g r a m s g e n e r a l i z e s t h e o p e r a t i o n m a d e i n t h e c o n s t r u c t i o n of t h e d u a l
YJ
of a t a b l e a u
Y
.
We s h a l l only g i v e h e r e a f e w h i n t s of t h i s beautiful t h e o r y . T h e i n t e r e s t e d r e a d e r w i l l s e e t h e p a p e r s of L a s c o u x a n d S c h t i t z e n b e r g e r [58] ,[59], [60], [62], [63], [95], [97]
a n d of T h o m a s [103], [104], (105], [lob], [107], t h e e x t e n s i o n to
t h e n i l p l a c t i c monoid [66] a n d t h e s u r v e y p a p e r of C a r t i e r a t t h e " S k m i n a i r e B o u r b a k i [7]. F o r t h e c l a s s i c a l t h e o r y of s y m m e t r i c f u n c t i o n s , s e e t h e book of Macdonald [72]
o r S t a n l e y 6983, F o u l k e s [l8]
,
J e u de t a q u i n . Let
X
contained i n
and
Fx
.
U
be two p a r t i t i o n s s u c h t h a t t h e F e r r e r s d i a g r a m
The diagram
g r a m s , is a n i n t e r v a l of
TI
F
cr
is
Fx\FcI , t h e d i f f e r e n c e of t h e t w o F e r r e r s d i a -
= Z x Z
A s k e w Young t a b l e a u of s h a p e
( o r d e r e d by t h e p r o d u c t o r d e r ( 2 ) ) . X\P
is a l a b e l i n g of
FA\Fcr with i n t e -
g e r s s u c h t h a t they a r e weakly i n c r e a s i n g i n t h e r o w s ( f r o m l e f t t o r i g h t ) a n d s t r i c t l y i n c r e a s i n g i n t h e c o l u m n s (down-up, i n t h e " F r e n c h notation").
G. Viennot
438
p = (5,5,2) = (8,7,5,5,2) 0
comer cell of F
r
F i g u r e 14 - A s k e w Young t a b l e a u .
T h e lljeu de taquin" i s d e s c r i b e d as f o l l o w s . F i r s t we c h o o s e one of t h e c o r n e r c e l l s (defined i n
9 4) of the F e r r e r s d i a g r a m F . We t a k e the m i n i m u m CI
of the t w o v a l u e s that a r e l o c a t e d at t h e North a n d E a s t , a n d move t h i s v a l u e into the c o r n e r c e l l . T h i s c r e a t e s a "hole" i n t h e s k e w Young t a b l e a u . In a sim i l a r way t o t h e c o n s t r u c t i o n of the d u a l i n
5 4, we
hole h a s moved t o the N o r t h - E a s t o u t s i d e of
Fx
r e p e a t t h e p r o c e s s until t h e
. In c a s e the
two v a l u e s a t t h e
N o r t h o r the E a s t of a hole a r e e q u a l , we move i n the hole the value l o c a t e d a t t h e N o r t h (thus p r e s e r v i n g the condition t o have a s k e w Young t a b l e a u ) . A l s o note t h a t if the hole i s a t t h e " b o r d e r " of
Fx , at m o s t one of the N o r t h a n d E a s t
c e l l s i s n o n - e m p t y . We move a l s o t h i s s i n g l e v a l u e i n t h e h o l e . T h e p r o c e s s t e r m i n a t e s when both t h e N o r t h a n d E a s t c e l l of t h e hole a r e e m p t y (that i s i n f a c t when the hole i s a c o r n e r c e l l of
Fx
1.
T h e t e r m i n o l o g y b n d t h e t h e o r y !) lljeu de taquin" is due t o S c h u t z e n b e r g e r . T h e lljeu de taquin" is a g a m e played with a n x n b o a r d . I n s i d e the b o a r d , 2 t h e r e a r e n -1 e l e m e n t a r y c e l l s . On e a c h c e l l i s p r i n t e d a l e t t e r . At e a c h s t e p of the g a m e , one c a n move one of t h e n e a r e s t c e l l s to t h e unique e m p t y p o s i t i o n . At e a c h s t e p t h i s e m p t y position t a k e s the p l a c e of the c e l l which h a s m o v e d . T h e r u l e i s t o obtain some w o r d s ( o r l e t t e r s i n a l p h a b e t i c o r d e r ) .
439
Chain and antichain families
F i g u r e 15
-
T h e lljeu de taquin"
We o b t a i n a n o t h e r s k e w Young t a b l e a u with s h a p e ( r e s p . FuI ) i s obtained f r o m
X'\bi',
Fx
1
( r e s p . F@) by r e m o v i n g a c o r n e r c e l l .
Fx , Now, if we r e p e a t t h e p r o c e s s going f r o m a t a b l e a u
t o a t a b l e a u with s h a p e
where
Y
with s h a p e
I\@
F is e m p t y , we o b t a i n a U Young t a b l e a u , c a l l e d t h e " r e d r e s s d " of t h e s k e w Young t a b l e a u Y u n d e r t h e A'\P'
until t h e d i a g r a m
s u c c e s s i v e c h o i c e s of t h e "jeu de taquin". A r e m a r k a b l e f a c t i s t h e i n v a r i a n c e of the " r e d r e s s C " u n d e r t h e "jeu de taquin",
i . e . the " r e d r e s s d " i s independant
of t h e s u c c e s s i v e c h o i c e s of t h e c o r n e r c e l l s of e a c h d i a g r a m
F
u
. This inva-
r i a n c e h a s b e e n p r o v e d by S c h a t z e n b e r g e r [ 9 7 ] a n d T h o m a s [lo51 ( s e e a n i d e a of t h e proof b e l o w ) . T h e " r e d r e s s C " of the s k e w Young t a b l e a u
Y
is d e n o t e d by
R(Y). We follow a g a i n S c h u t z e n b e r g e r ' s t e r m i n o l o g y ; " r e d r e s s 6 " i s b a d l y t r a n s l a t e d by s a y i n g t h a t t h e South-West b o r d e r of s k e w Young t a b l e a u
Y
is n o
m o r e a zig-zag line. F o r example, the tableau where
a
11
d(Y)
defined i n
8
4 is nothing but
R ( Y \ all)
d e n o t e s t h e ( s k e w ) Young t a b l e a u o b t a i n e d by d e l e t i n g t h e v a l u e
Y \ all (1,l) c e l l f r o m the Young t a b l e a u
i n the
Y
.
In the e x a m p l e d i s p l a y e d i n f i g u r e 16, we h a v e s h o r t e n e d t h e c o n s t r u c t i o n by giving only the t a b l e a u x o b t a i n e d o n c e t h e "hole" i s o u t s i d e of the d i a g r a m
Fx
. We h a v e
c i r c l e d t h e v a l u e s which m o v e f o r e a c h c h o i c e of t h e c o r n e r c e l l s
of t h e s u c c e s s i v e d i a g r a m s
F
U
.
,w
G. Viennot
440
&+
Y =
3 8 9
choice of the
comer cell
0
F i g u r e 16
-
The "redressk"
v a l u e w i n g in the " j e u d e taquin"
R ( Y ) of a s k e w Young t a b l e a u
Y
.
6 . We denote by Y(o) the s k e w Young n t a b l e a u obtained by w r i t t i n g t h e s u c c e s s i v e v a l u e s U(l), ,u(n) f r o m N o r t h Let
u
be a p e r m u t a t i o n of
...
West t o S o u t h - E a s t ( s e e f i g u r e 17). A f u n d a m e n t a l p r o p e r t y of t h e lljeu de taquin", a n d giving a n o t h e r definition of t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e is t h e following
PROPOSITION 8 .
(Schiitzenberger)
-
F o r any permutation
" r e d r e s s k " of t h e s k e w Young t a b l e a u P-symbol
P(u)
Y(U)
coding
u
u
of
en ,
the
i s identical to t h e
o b t a i n e d by t h e Robinson-Schensted c o r r e s p o n d e n c e .
Define a s t r i p t o be a d i a g r a m
Fcl a r e disjoint and s u c h t h a t
Fx\Fu
Fx\ Fu
F
s u c h t h a t t h e b o r d e r s of
d o e s not c o n t a i n a n y s q u a r e
x
F
(292)
and of
Chain and antichain families f o u r c e l l s . Such d i a g r a m s with
n
44 1
c e l l s a r e coded by a w o r d of l e n g t h
two l e t t e r s . T h e r e e x i s t s a t r i v i a l b i j e c t i o n between p e r m u t a t i o n s of s k e w Young t a b l e a u x with d i s t i n c t e n t r i e s
1, 2,
. . .,n
6,
n-1
on
and
a n d with a s t r i p s h a p e ,
F r o m s u c h t a b l e a u x , we a s s o c i a t e a p e r m u t a t i o n by r e a d i n g t h e e n t r i e s f r o m N o r t h - W e s t t o S o u t h - E a s t following t h e s t r i p . C o n v e r s e l y , f r o m a p e r m u t a t i o n 0
we define a s k e w Young t a b l e a u
the c e l l containing c e l l containing
iff
having a s t r i p s h a p e a n d s u c h t h a t
i s c o n s e c u t i v e a n d a t t h e E a s t ( r e s p . South) of t h e
a(it1)
O(i)
Str(0)
0(i) < o(it1)
( r e s p . 0 ( i ) 3 o ( i t 1 ) ) . In o t h e r w o r d s ,
t h e s t r i p i s nothing but a coding of t h e up-down s e q u e n c e of t h e p e r m u t a t i o n
8
4
Y h)
-
defined i n
.
k-, 3 9
F i g u r e 17
-
T h e s k e w Young t a b l e a u x Y ( 0 ) a n d Str(ff) coding a p e r m u t a t i o n 0 .
Obviously, the t a b l e a u
Str(0)
c a n be o b t a i n e d f r o m
Y(0)
by applying
t h e "jeu de taquin", a n d t h u s , by S c h f i t z e n b e r g e r ' s t h e o r e m s ( i n v a r i a n c e of t h e " r e d r e s s k " u n d e r t h e "jeu de taquin" a n d p r o p o s i t i o n 8 ) , t h e " r e d r e s s k " R(Str(0))
i s the P-sy mb o l
P(0) of t h e p e r m u t a t i o n
U
.
T h e r e a d e r will
c h e c k t h i s f u n d a m e n t a l fact with f i g u r e 16 f o r t h e g e n e r i c p e r m u t a t i o n a p p e a r i n g i n all the e x a m p l e s of t h i s p a p e r . T h e i n v a r i a n c e of t h e " r e d r e s s k " u n d e r t h e lljeu de taquin" i s p r o v e d by introducing t h e following f u n c t i o n s , b a s e d on G r e e n e ' s i n t e r p r e t a t i o n of the s h a pe d the t a b l e a u x
P(0) a n d
Q(0)
(we s h a l l a s s u m e t h a t t h e e n t r i e s a r e d i s -
t i n c t , t h e g e n e r a l c a s e c a n be d e d u c e d f r o m t h a t ) .
G. Viennot
442 Let
Y
be a s k e w Young t a b l e a u (with p o s s i b l y a "hole" i n s i d e ) . We d e -
f i n e a n i n c r e a s i n g s u b s e q u e n c e of
< xk
such that f o r e v er y
E a s t of t h e c e l l w h e r e we define
i , 1G i
x.
to be a s e q u e n c e of i n t e g e r s k , x.
1t1
. ..
x
1 i s i n a c e l l which i s a t t h e S o u t h -
i s l o c a t e d . T h e n , f o r any i n t e g e r s
k
a2
and
1,
Y
to be the m a x i m u m c a r d i n a l i t y of s u b s e t s of e n t r i e s of
Gk,,(Y)
t h a t a r e unions of
Y
k
.
i n c r e a s i n g s u b s e q u e n c e s with v a l u e s
i s i n v a r i a n t u n d e r the e l e m e n t a r y 'k, sliding of t h e 'ljeu de taquin". T h e f a c t mentioned above i s p r o v e d f r o m the r e It c a n be shown t h a t t h e function
m a r k t h a t a s t a n d a r d Young t a b l e a u is c h a r a c t e r i z e d by the function
G
k,
.
u + (R(0) , R( 0-l))
T h e n one c a n s h o w d i r e c t l y that the c o r r e s p o n d e n c e
i s a b i j e c t i o n . T h e "bumping p r o c e s s " c a n be s e e n as p a r t i c u l a r c a s e of t h e s l i d i n g s of the lljeu de taquin", a n d t h u s we h a v e p r o p o s i t i o n 8 . F r o m t h e r e , G r e e n e ' s i n t e r p r e t a t i o n and p r o p e r t i e s of Knuth's t r a n s p o s i t i o n s c a n be d e d u c e d . Other p r o p e r t i e s mentioned in
5
4 a r e a l s o d e d u c e d . F o r e x a m p l e , the p r o p e r t y
r e l a t i n g t h e up-down s e q u e n c e and t h e "line of r o u t e ' ' i s e a s i l y shown a s a p r o p e r t y i n v a r i a n t u n d e r the "jeu de taquin". A l s o , taking t h e dual of a t a b l e a u c o r r e s p o n d s to r e v e r s i n g the I'jeu de taquin", t h a t is r e v e r s e the o r d e r b e t w e e n i n t e g e r s and move t h e e n t r i e s t o t h e N o r t h - E a s t ( i n s t e a d of t h e S o u t h - W e s t ) .
P r o d u c t s of Young t a b l e a u x . T h e above c o m b i n a t o r i a l c o n s i d e r a t i o n s c a n be e x p r e s s e d in a m o r e a l g e b r a i c way by introducing the plactic m o n o i d . Let
Y
and
Z
be two Y o u n g t a b l e a u x . Denote by
Young t a b l e a u obtained by putting
Z
the skew
Y
a s shown o n
a t t h e S o u t h - E a s t of
f i g u r e 18. T h e p r o d u c t of the two Young t a b l e a u x the " r e d r e s s 6 "
Y. Z
Y
Z
and
i s defined to be
R(Y. Z ) . By t h e i n v a r i a n c e of t h e " r e d r e s s & " u n d e r t h e lljeu de
taquin", t h i s p r o d u c t i s a s s o c i a t i v e . T h e e m p t y t a b l e a u i s t h e identity e l e m e n t . A , and with the
T h e s e t of Young t a b l e a u x with e n t r i e s i n the s e t of i n t e g e r s p r o d u c t j u s t defined, i s t h e p l a c t i c monoid g e n e r a t e d by
A
.
443
Chain and antichain families
=B 1
1
2
Y.2
-@ F i g u r e 18 -
=%
n Young t a b l e a u x r e d u c e d t o one c e l l
insertion process
I(T,x)
with the s i n g l e - c e l l
x
R] 2
2
4
T h e p r o d u c t of two Young t a b l e a u x .
P r o p o s i t i o n 8 s a y s nothing but t h a t t h e P - s y m b o l p r o d u c t of the
m =
R(Y*Z)
5
defined in
P(0)
is the p l a c t i c
O(l), 0 ( 2 ) ,
. . . ,~
( n .) T h e
3 i s in f a c t t h e p l a c t i c p r o d u c t of
T
placed a t the South-East.
A n o t h e r way t o define t h e p l a c t i c monoid c o m e s f r o m K n u t h ' s t r a n s p o s i be a t o t a l l y o r d e r e d a l p h a b e t . L e t
tions . Let
A
n e r a t e d by
A , t h a t i s the s e t of w o r d s
w = wl.. .w
a n d with p r o d u c t the c o n c a t e n a t i o n p r o d u c t : f o r
be the f r e e monoid g e with l e t t e r s
n
of
x,y,z
A
x y z s y x z
(6) f o r any l e t t e r s
x,y
of
A
x y x s y x x
such that and
x c z
in
A
v = vl. . . v q ,
t o be the c o n g r u e n c e
or
y < z c x ,
z x y s z y x ,
such that and
and
= ul'.
%
f o r any letters
w. 1
. We define t h e p l a c t i c c o n g r u e n c e l', vl.. A* g e n e r a t e d by the r e l a t i o n s
uv= u of
A*
x < y ,
y x y - y y x .
In e a c h e q u i v a l e n c e c l a s s , t h e r e i s one and only one "tableau" ; t h a t i s ,
a w o r d t h a t c a n be obtained by r e a d i n g a Young t a b l e a u r o w by r o w , f r o m l e f t t o r i g h t i n e a c h row, t h e r o w s being o r d e r e d up-down ( t h i s c o r r e s p o n d s t o the r o w - c a n o n i c a l p e r m u t a t i o n s defined in monoid above.
A* / s
4). It c a n be shown t h a t t h e quotient
is i s o m o r p h i c t o the p l a c t i c monoid g e n e r a t e d by
A
defined
G. Viennot
444
4
7
-
Grids. T h i s s e c t i o n p r e s e n t s a s u m m a r y of a f o r t h c o m i n g p a p e r of t h e a u t h o r [127],
w h e r e c o m p l e t e p r o o f s of the new r e s u l t s announced will be g i v e n . A word
with
0
pq
Y
e x i s t s a Young t a b l e a u such t h a t the
pq
and c o l u m n s of
distinct letters i s called a with d i s t i n c t e n t r i e s a n d
e n t r i e s of
Y
Y
(p,q)-grid
pxq
rectangular shape
a r e exactly t h e l e t t e r s of
a r e (monotone) s u b s e q u e n c e s of
It is e a s i l y s e e n t h a t s u c h a t a b l e a u
u
iff t h e r e
0
, and the r o w s
.
is unique a n d i s i n f a c t t h e P-
Y
P ( 0 ) . A l s o we have the following l e m m a
symbol
L EMMA 9
-
Let
u
be a w o r d with d i s t i n c t l e t t e r s ( o r p e r m u t a t i o n ) . T h e f o l -
lowing conditions a r e equivalent i s a grid,
(i)
d
(ii)
the s h a p e of
(iii)
f o r any
P(u) i s r e c t a n g u l a r ,
i ' r l , the ith c o l u m n of
y - c o o r d i n a t e s of t h e points of
a^
l i n e of
6
P(0)
i s c o m p o s e d of t h e th lying o n t h e i outstanding
.
P e r m u t a t i o n s with r e c t a n g u l a r s h a p e f o r t h e a s s o c i a t e d G r e e n e d i a g r a m have m a n y o t h e r nice p r o p e r t i e s . T h e n u m b e r of s u c h p e r m u t a t i o n s , with a fixed
px q
r e c t a n g l e a s s o c i a t e d G r e e n e d i a g r a m , h a s b e e n c o n s i d e r e d by
H i l l e r [44], i n t e r m of the "Schubert c a l c u l u s " d e s c r i b i n g t h e "cohomology of the c o m p l e x g r a s s m a n manifold". In t h i s s e c t i o n , we a r e m o s t l y i n t e r e s t e d i n s u b g r i d s of p e r m u t a t i o n s , t h a t i s s u b s e q u e n c e s that a r e g r i d s (see e x a m p l e i n f i g u r e 19). When a p e r m u t a t i o n
u
is viewed a s a p o s e t
POS(U) , the concept of s u b g r i d s i s c l o s e l y r e l a t e d t o
the c o n c e p t of "orthogonal f a m i l i e s " i n t r o d u c e d by F r a n k [23] i n h i s proof of G r e e n e ' s T h e o r e m ( s e e definition
5
8 below).
F i r s t , we h a v e the following
L E M M A 10 - L e t
0
has Greene diagram contained in
be a p e r m u t a t i o n f o r which the a s s o c i a t e d p o s e t G(Pos(0))
(shape of
P(u)).F o r a n y
G ( P o s ( U ) ) , t h e r e e x i s t s a (p, q ) - s u b g r i d of
u
pxq
.
POS(U)
rectangle
Chain and antichain families T h e proof r e l i e s o n G r e e n e ' s i n t e r p r e t a t i o n of a n t i c h a i n s k - f a m i l i e s of T
of
0
diagram
p
G ( P o s ( 0 ) ) with c h a i n s a n d
( p r o p o s i t i o n 6 ) : f i r s t one takes a s u b s e q u e n c e
Pos(0)
t h a t i s a union of
445
chains (increasing subsequences). The Greene
G ( P o s ( 7 ) ) i s not n e c e s s a r i l y the s a m e a s t h e d i a g r a m
obtained f r o m
C ( P o s ( 0 ) ) by deleting all r o w s i n d e x e d by
b e r of e l e m e n t s of t h e
pth
pth
r o w of
Gp(Pos(0)),
i > p , but t h e n u m -
G ( P o s ( T ) ) i s not l e s s t h a n t h e n u m b e r of
G ( P o s ( 0 ) ) . Applying t h e same r e a s o n i n g o n P with a n t i c h a i n s ( d e c r e a s i n g s u b s e q u e n c e s ) l e a d s t o l e m m a 10. e l e m e n t s of t h e
r o w of
T
A s shown i n f i g u r e 19, t h e c o n v e r s e of l e m m a 10 i s not t r u e . A m a j o r p r o b l e m i s to give a c h a r a c t e r i z a t i o n of s u b g r i d s giving e x a c t l y t h e G r e e n e d i a g r a m G(Pos(0))
a n d t h u s o b t a i n a n i n t e r p r e t a t i o n of
G(Pos(0)) symmetric in rows
and c o l u m n s ( i . e . c h a i n s a n d a n t i c h a i n s ) .
Cxeene diagram of C = 6 1 7 2 5 8 3 9 4
T = 6 1 7 2 8 3 9 4 is a (2,3)-subgrid of r.
F i g u r e 19
-
A ( 2 , 4 ) - s u b g r i d of a p e r m u t a t i o n with n o c o r r e s p o n d i n g 2 x 4 - r e c t a n g l e .
G. Viennot
446
Let
u
t h e i n t e r i o r of the g r i d of
T
and T be a s u b g r i d of a . We define n t o be t h e i n t e r s e c t i o n of the f o u r shadows ( s e e 4 5)
be a p e r m u t a t i o n of T
6
r e l a t e d to the f o u r p o s s i b l e d i r e c t i o n s of t h e light : N o r t h - W e s t , South-
W e s t , S o u t h - E a s t a n d N o r t h - E a s t . In o t h e r w o r d s , t h i s i s a l s o the s e t of points (x,y)
of
IRxIR
7
s u c h that t h e r e e x i s t s a t l e a s t one point of
f o u r q u a d r a n t s with o r i g i n a t
in e a c h of t h e
(x, y ) .
T h e i n t e r i o r of a g r i d i s a union of e l e m e n t a r y c e l l s having the p r o p e r t y t h a t f o r e v e r y p a i r of c e l l s on the s a m e v e r t i c a l ( r e s p . h o r i z o n t a l ) , all the c e l l s between t h e m a r e i n t h i s union. Such o b j e c t s a r e known i n c o m b i n a t o r i c s a s convex polyominoes ( s e e [9] a n d the r e f e r e n c e s t h e r e i n ) .
1
2
3
F i g u r e 20
Let
P ( T )= Y
T
be a
-
5
6
7
8
9
T h e i n t e r i o r of a g r i d .
( p , q ) - s u b g r i d of the p e r m u t a t i o n
d
of
en
and
t h e unique c o r r e s p o n d i n g t a b l e a u i n t h e definition of g r i d s . W e s h a l l
s a y t h a t the s u b g r i d
quences
4
al,
. . .,aP
T
of
i s extendable (in u ) iff t h e r e e x i s t s i n c r e a s i n g s u b s e 0
and decreasing subsequences
PI,.
..#Pq
of
0
Chain and antichain families
441
s a t i s f y i n g t h e t h r e e following conditions ( s e e f i g u r e 21). (i)
j ,
for any
4j
s u b s e q u e n c e of (iii)
, the i
th
r o w of
Y
is a (increasing) subse-
a. ,
q u e n c e of (ii)
i , 14 i
f o r any
e v e r y point of
6
t h e jth
c o l u m n of
Y
i s a (decreasing)
J
t h a t i s i n the i n t e r i o r of
al. . . . , n , B,,
the s u b s e q u e n c e s
P
T
b e l o n g s t o one of
. . . ,p . 9
T h e definition of e x t e n d a b l e s u b g r i d s i s not c h a n g e d i f we r e q u i r e t h a t t h e s u b sequences
al, . . . , a
P
a r e pairwise disjoint, and sim ila rly f o r
pl,
. . , ,p . q
T h e ( 2 , 4 ) - s u b g r i d d i s p l a y e d i n f i g u r e s 19 and 20 is not e x t e n d a b l e . T h i s c o m e s f r o m the f a c t t h a t ( 2 , 5, 8 ) a n d ( 7 , 5, 3 ) a r e r e s p e c t i v e l y i n c r e a s i n g a n d
, with t h e point l a b e l e d 5 in t h e i n t e r i o r of
d e c r e a s i n g s u b s e q u e n c e s of
0
M o r e g e n e r a l l y , a point
of the p e r m u t a t i o n
f o r the s u b g r i d
T
x
0
u
.
i s s a i d t o be a c r i t i c a l point
iff t h e r e e x i s t s e n t r i e s of the Young t a b l e a u
P(0) s u c h
) is a n i n c r e a s i n g s u b s e q u e n c e a n d 1 ,x,a. * x~ a i t l , jtl 1, j t l (aitl, j i s a d e c r e a s i n g s u b s e q u e n c e of 0 . Obviously, s u c h c r i t i c a l point x i s i n t h e
that
(a. . 1, j
i n t e r i o r of t h e ( 2 , 2 ) - s u b g r i d (of
7)
is a l s o i n the i n t e r i o r of the subgid
{ a i , j , a i t l , , ai, j + l , ai+l, j t l 3 , a n d t h u s . Now one c a n s e e t h a t x c a n n o t be
T
" i n c o r p o r a t e d " i n one of t h e ( m o n o t o n e ) s u b s e q u e n c e s of r o w s a n d c o l u m n s of t h e Young t a b l e a u
0
f o r m e d with t h e
Y = P ( o ) , ( s e e f i g u r e 22).
T h u s , a s u b g r i d having a c r i t i c a l point i s not e x t e n d a b l e . Note t h a t t h e c o n v e r s e i s not t r u e ( s e e f i g u r e 2 3 ) . Note a l s o t h a t t h e s h a p e ( 4 , 2 ) of t h e s u b g r i d of f i g u r e 23 i s not c o n t a i n e d i n t h e G r e e n e d i a g r a m .
G. Viennot
448
// a" ,a
0
0
0
\ I
\
\
0
0
p i n t s of the subgrid
T
X
pints of u i n the i n t e r i o r of T but not not i n T
0
okher points of the pnmtation u
Figure 21
-
- - - I
border of the i n t e r i o r of the subgrid T increasing subsequences o^) (chains of decreasing subsequences (antichains of o^)
A n extendable (3,4)-subgrid.
449
Chain and antichain families
11 10 9
8
7 6 5 4 3 2 1
u = 10 3 2 7 1 6 5 11 9 8 4 T
=lo 3 2 111 9 8 4
Greene diaqran
G(o) =
F i g u r e 22
-
A c r i t i c a l point of a s u b g r i d .
F i g u r e 23
-
A non-extendable subgrid having no c r i t i c a l p o i n t .
T h e m a i n r e s u l t of t h i s s e c t i o n is
PROPOSITION 11
-
Let
u
be a p e r m u t a t i o n and
by t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e . T h e v a l u e (i, j )
c e l l of
i t s P - s y m b o l obtained
P(U)
P..(o)l o c a t e d i n the 'J
P(a) is e q u a l t o P..(u) = m i n ( m a x 'J 7
w h e r e t h e m i n i m u m i s t a k e n o v e r all e x t e n d a b l e
T)
,
(i, j ) - s u b g r i d s
T
of
d
.
G. Viennot
450 In o t h e r w o r d s , if 1< x
u
i s a p e r m u t a t i o n in
, t h e r e e x i s t s a unique p a i r
of a n extendable
( i ,j )
F u r t h e r m o r e , this
.
and
such that
x
( i , j ) - s u b g r i d a n d s u c h t h a t no extendable
e x t r a c t e d f r o m the s u b s e q u e n c e of
u
(i, j )
en
u
X
an integer
i s t h e m a x i m u m value ( i , j ) - s u b g r i d c a n be
obtained by deleting a l l v a l u e s
i s p r e c i s e l y the position of
x
2 x
,
i n the P - s y m b o l of
T h i s kind of "topological" p r o p e r t y of points d i s p l a y e d i n the plane i s a
"local" definition of t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e (and a f o u r t h one a f t e r the "bumping p r o c e s s " of
4
3 , the shadows and s k e l e t o n s of
l'jeu de taquin" and p l a c t i c monoid of
5
5 and t h e
5 6).
P r o p o s i t i o n 11 i s t r i v i a l i n t h e c a s e of row (or c o l u m n ) c a n o n i c a l p e r m u t a tions ( s e e
9 4). T h u s
p r o p o s i t i o n 11 i s p r o v e d by showing the i n v a r i a n c e of t h e
m i n - m a x quantity u n d e r Knuth's t r a n s p o s i t i o n s . In f a c t , m o r e c a n be s a i d . One c a n i n t r o d u c e a n a l o g o u s l y "extendable" s u b g r i d s of skew Young t a b l e a u x (with " h o l e s " ) . T h e r e i s n o m o r e " i n t e r i o r " of a s u b g r i d but a n a n a l a g o u s definition i s p o s s i b l e f r o m the n e x t s e c t i o n . T h e c o r r e s p o n d i n g m i n - m a x f u n c t i o n p l a y s the r o l e of t h e f u n c t i o n in
4
introduce d k, 1 6 . Again one c a n deduce m o s t of t h e p r o p e r t i e s of t h e R o b i n s o n - S c h e n s t e d G
c o r r e s p o n d e n c e f r o m the i n v a r i a n c e of t h i s m i n - m a x function u n d e r the e l e m e n t a r y s l i d i n g s of the 'ljeu de taquin". T h e p r o o f s a r e m o r e c o m p l i c a t e d , but one h a s the s a t i s f a c t i o n of keeping the s y m m e t r y between r o w s a n d c o l u m n s , t h a t i s between c h a i n s and a n t i c h a i n s .
8
8
-
Extensions t o a r b i t r a r y p o s et s . In conclusion, we d i s c u s s ( v e r y b r i e f l y ) p o s s i b l e e x t e n s i o n of the Robinson-
S c h e n s t e d c o r r e s p o n d e n c e a n d of t h e g r i d c o n c e p t t o a r b i t r a r y p o s e t s . T h i s s e c tion i s m a i n l y b a s e d o n w o r k of G r e e n e [ 3 5 ] , F o m i n [lb], G a n s n e r [ 2 7 ] , F r a n k [ 2 3 ] a n d a f o r t h c o m i n g p a p e r of t h e a u t h o r [128].
Let [n] = (1,
P
be a p o s e t of c a r d i n a l i t y
. . . ,n ) . T h e
p e r m u t a t i o n of
8
n
.
cp
Let
be a l i n e a r e x t e n s i o n i n
p r o b l e m is t o a s s o c i a t e a s t a n d a r d Young t a b l e a u
s u c h t h a t t h i s t a b l e a u is
values
n
P ( u ) ( r e s p . Q(0))
, and cp
a(i) ( r e s p . indices
i n the c a s e
Y(P,cp)
P = P o s ( u ) with
is t h e l a b e l i n g of t h e v e r t i c e s of
u
POS(O)by the
i ).
A s shown by F o m i n [16], a solution c o m e s f r o m G r e e n e ' s t h e o r y ( t h e o r e m 1 and p r o p o s i t i o n 6 ) a n d the following l e m m a : if
P'
i s t h e p o s e t obtained by
45 1
Chain and antichain families deleting f r o m
P
G(P') a n d
Here
6
fined in
G(P)
n
a r e t h e G r e e n e ' s d i a g r a m s of
P'
1,
.
,
iG ;n],
let
P.
.,i . The F e r r e r s diagrams
i n t h e u n i q u e c e l l of
G(P ) =
6
but w i t h
P
and
G(P).
as d e -
b e t h e s u b p o s e t o b t a i n e d by t a k i n g t h e v e r t i c e s
( G(Pi) , 1 d i
u n d e r i n c l u s i o n . We c a n d e f i n e a s t a n d a r d Young t a b l e a u i
G(P')
( m a x i m u m value), then
2.
F o r any labeled
the vertex labeled
G(Pi)\ G(Pi-l)
(for any
cp
Y(P,cp)
i E [n],
). F i g u r e 24 gives a n example with the poset
form a chain
with t he convention P =Pos(a)
0
= 6 1 7 2 5 8 3 9 4
V e r y l i t t l e i s known a b o u t t h i s c o r r e s p o n d e n c e
of f i g u r e 1,
and Q-
P
different f r o m the two n a t u r a l l a l e l i n g s giving t h e
s y m b o l of t h e g e n e r i c p e r m u t a t i o n
by p l a c i n g
. .
(P,cp) + Y
Greene's pro-
p e r t i e s a b o u t k - m a t c h i n g of p e r m u t a t i o n s ( s e e § 4 ) h a v e b e e n e x t e n d e d by G a n s n e r [27], w h e r e h e g i v e s a l s o a d u al analogue with k - s c a t t e r s . A l t o u g h w e a r e g o i n g t o g i v e a n e x t e n s i o n of p r o p o s i t i o n 11 t o a r b i t r a r y posets, no simple algorithm, analogous to the Robinson-Schensted algorithm, h a s b e e n f o u n d f o r t h e c o n s t r u c t i o n of
Y(P,cp).
& 1
Figure 24 - The correspondence
(P,cp)-
2
Y
4
7
9
G. Viennot
45 2
P
Let
y
be a p o s e t . A s u b s e t p
c a n b e p a r t i t i o n e d into j , l
f o r any
~
j
of
'f
c
i s s a i d t o be a
( p , q ) - g r i d iff
{ c ~ , ~ ~ . . . < 1c, . 1 9 i 4 p , s u c h t h a t 1,q ) i s a n a n t i c h a i n of P
chains
~ [ cq. , 1.j " " '
P
.
prj
With t h i s e x t e n s i o n of t h e g r i d definition, i t i s e a s y t o s e e that l e m m a 10
P
i s s t i l l valid f o r any p o s e t
,
But f i g u r e 24 g i v e s a n e x a m p l e of a ( 2 , 4 ) - g r i d
in a p o s e t whose G r e e n e ' s d i a g r a m d o e s not c o n t a i n a 2 x 4 r e c t a n g l e . A s in $ 7 , we give a c h a r a c t e r i z a t i o n of g r i d s with s h a p e c o n t a i n e d i n t h e G r e e n e d i a g r a m . F o r t h a t , we u s e t h e following notion i n t r o d u c e d by F r a n k [Z3] Let
(a,,
. . . ,% P ] P ,
f a m i l y of the poset
be a c h a i n f a m i l y a n d
.
[B1, . . . , B )
be a n a n t i c h a i n 9 with t h e c h a i n s ( r e s p . a n t i c h a i n s ) s u p p o s e d p a i r w i s e
d i s j o i n t . T h e s e f a m i l i e s a r e s a i d t o be orthogonal iff they s a t i s f y t h e two conditions :
( U a,)U ( U ")
(i)
P =
(ii)
i f o r any
j i , 1 4i 4 p
, and
j
It i s e a s y t o s e e that the c h a i n f a m i l y
[ p , , . . . ,!3 ) ) 9 , 1d j & q )
mily
1
(a,,
. . . , aP 3
# fi .
( r e s p . antichain fa-
h a s m a x i m u m c a r d i n a l i t y . A l s o , the s u b s e t i s obviously a ( p , q ) - g r i d of
y
We s h a l l s a y t h a t a g r i d
e x i s t orthogonal f a m i l i e s of c h a i n s and s u b s e t s
, 1 6 j G q , ain B j
I s [p] , J 2 [q]
P
of
P
y =
.
[ai n B j '
i s c o m p l e t e l y extendable iff t h e r e
{ a l , . . . ,a ) P
and antichains
[ Pl, . . . , B q
3
such that Y =
{ai n B
j '
~ E I j, C
JI .
T h e concept of orthogonal f a m i l i e s i s e s s e n t i a l i n F r a n k ' s proof of G r e e n e ' s T h e o r e m . R e c a l l t h a t t h i s proof r e l i e s on l i n e a r p r o g r a m m i n g t e c h n i q u e s . F r o m F r a n k ' s proof of G r e e n e ' s T h e o r e m ( s e e t h e o r e m 3 of [23]), we c a n deduce
PROPOSITION 12 -
T h e G r e e n e d i a g r a m of t h e p o s e t
r e c t a n g l e iff t h e r e e x i s t s a
P
contains a
(p, q ) - c o m p l e t e l y extendable g r i d i n
P
px q
.
A l s o , using G r e e n e ' s T h e o r e m a n d r e s u l t s of F o m i n [16], we c a n d e d u c e f r o m p r o p o s i t i o n 12.
Chain and antichain families PROPOSITION 13 -
P
Y
and
T h e value
Let
P
45 3
be a p o s e t , cp : P + [n]
be a n a t u r a l l a b e l i n g of
be t h e s t a n d a r d Young t a b l e a u defined by t h e a b o v e c o r r e s p o n d e n c e
Y..(P) 'J
l o c a t e d in t h e
(i,j)
c e l l of
Y..(P) = min(max 'J Y
Y
i s equal to
V)
w h e r e the m i n i m u m i s t a k e n o v e r a l l c o m p l e t e l y e x t e n d a b l e max(Y)
d e n o t e s t h e g r e a t e s t l a b e l of the v e r t i c e s of
y
(p, q ) - g r i d s , a n d
.
I have not b e e n a b l e t o f i n d a d i r e c t c o m b i n a t o r i a l proof of P r o p o s i t i o n 12 and 13, not using l i n e a r p r o g r a m m i n g t e c h n i q u e s . T h i s p r o p o s i t i o n c a n be a p p l i e d i n t h e c a s e
P =Pos(0)
where
uEBn
.
It i s p o s s i b l e t o define the c o n c e p t of c o m p l e t e l y e x t e n d a b l e g r i d e x t r a c t e d f r o m a s k e w Young t a b l e a u , show i t s i n v a r i a n c e u n d e r t h e " j e u de taquin" a n d after
6 m a k e a n u l t i m a t e s y n t h e s i s of t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e
and t h e p l a c t i c monoid [127]. I n s t e a d of extending the R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e t o a r b i t r a r y p o s e t s , one c a n a l s o look f o r g e n e r a l i z a t i o n s involving o t h e r kinds of t a b l e a u x t h a n t h e Young t a b l e a u x u s e d a b o v e . In
6
3 we have m e n t i o n e d the s h i f t e d Young
t a b l e a u x . An a n a l o g of the "bumping p r o c e s s " h a s been defined by S a g a n [83], w h e r e he d e s c r i b e s a b i j e c t i o n between p e r m u t a t i o n s a n d p a i r s of " c o l o r e d s h i f t e d Young t a b l e a u x " having t h e s a m e s h a p e . T h i s b i j e c t i o n p r o v e s t h e a n a l o g of r e l a t i o n (1) f o r p r o j e c t i v e r e p r e s e n t a t i o n s of the s y m m e t r i c g r o u p . O t h e r g e n e r a l i z a t i o n s h a v e been m a d e by Z e l e v i n s k y [121] t o c e r t a i n kind of d i a g r a m s c a l l e d p i c t u r e s , i n connection with t h e i n t e r t w i n i n g n u m b e r of r e p r e s e n t a t i o n s of t h e s y m m e t r i c g r o u p , by Stanton a n d White
[loll, [I151
t o r i m hook t a b l e a u x giving a
c o m b i n a t o r i a l proof of t h e o r t h o g o n a l i t y of t h e c h a r a c t e r s of
Gn
a n d " r i m hook"
v e r s i o n of t h e "bumping p r o c e s s " a n d of t h e l'jeu de taquin", a n d by B e r e l e a n d R e g e v [4] t o " s e m i - s t a n d a r d tableaux" i n t w o s e t s of v a r i a b l e s i n c o n n e c t i o n with the r e p r e s e n t a t i o n t h e o r y of t h e g e n e r a l l i n e a r g r o u p
G L ( k , F ) , W e y l ' s "Strip"
t h e o r e m [114] and t h e t h e o r y of P. I . a l g e b r a s . A l s o we m e n t i o n e d a b o v e i n
4 4 the
Grassl-Hillman correspondence, rela-
t e d to t h e t h e o r y of p l a n e p a r t i t i o n s , a s u b j e c t t h a t we do not touch h e r e but c l o s e l y r e l a t e d t o the R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e ( s e e f o r e x a m p l e A n d r e w s [l] c h a p t e r 11, Mac Mahon [73], [74] c o l l e c t e d p a p e r s , c h a p t e r s 11 a n d 12,
G. Viennot
45 4
Stanley [ 9 8 ] ) . G a n s n e r [ 2 6 ] showed t h a t t h i s c o r r e s p o n d e n c e , when viewed through t h e F r o b e n i u s c o r r e s p o n d e n c e (between r e v e r s e plane p a r t i t i o n s and c e r t a i n p a i r s of t a b l e a u x ) , i s a n e x t e n s i o n of c o l u m n i n s e r t i o n , a n d c l o s e l y r e l a t e d t o t h e " B u r g e c o r r e s p o n d e n c e " [ 6 ] . In
[ill]
V o a n d White showed that
(a s l i g h t v a r i a t i o n , equivalent t o the o r i g i n a l , of) the G r a s s l - H i l l m a n c o r r e s p o n -
d e n c e i s j u s t row i n s e r t i o n in d i s g u i s e . T h i s c o r r e s p o n d e n c e h a s been e x t e n d e d to t h e t w o o t h e r f a m i l i e s of p o s e t s with hooklengths quoted i n
8
4 by S a g a n [ 8 5 ] .
We conclude t h i s p a p e r by mentioning b r i e f l y s o m e v e r y r e c e n t a n d i n t e r e s t i n g w o r k of E d e l m a n and G r e e n e [123], [124], In [125], S t a n l e y c o n j e c t u r e d P i s equal n of s t a n d a r d Young t a b l e a u x having t h e " s t a i r c a s e " s h a p e
that t h e n u m b e r of m a x i m a l c h a i n s i n the ( w e a k ) B r u h a t o r d e r of
f
t o the n u m b e r
W
IJ = (n-1, n - 2 ,
...,l).
the r e l a t i o n s
ud
(The weak Bruhat o r d e r on
Q
8
if
i s obtained f r o m
6
i s the o r d e r g e n e r a t e d by n u by a t r a n s p o s i t i o n of a d j a -
cent i n c r e a s i n g e l e m e n t s ) . An a l g e b r a i c proof i s given by Stanley [126]. A b i j e c t i v e proof i s g i v e n by E d e l m a n a n d G r e e n e . In f a c t , t h r e e b i j e c t i o n s a r e n e e d e d , using S c h i i t z e n b e r g e r ' s o p e r a t o r s [ 9 4 ] , [ 9 7 ]
Y + YJ
a p a r t i t i o n of
1
4
described in n
and
i s a labeling
dinct e n t r i e s m e n t of i t s
t.. 'J 1, 2, .
"hook"
( a n a l o g o u s to the dual
4), a n d t h e new c o n c e p t bf b a l a n c e d t a b l e a u x . L e t
X*
be
t h e conjugate p a r t i t i o n . , A b a l a n c e d t a b l e a u of s h a p e
of the c e l l s
. . ,n
A
( i ,j )
of the F e r r e r s d i a g r a m
, such that the value
H . . , where 'J
*
r . = 1. - i t 1 1j J
i s the
r,t.h
and where
t.
H. 1j
Fx w i t h d i s largest ele-
1J
1j
i s the m u l t i s e t
i s a balanced
t a b l e a u of s h a p e
1 = [ 5 ,2 , 2 , 1 ) .
One of t h e m o s t s u r p r i s i n g f a c t is t h a t t h e n u m b e r of b a l a n c e d t a b l e a u x of shape
A
is
f X , t h e n u m b e r of s t a n d a r d Young t a b l e a u x of s h a p e
1
(see
f o r m u l a e ( 3 a ) , (3b) a n d ( 3 c ) ) . Again t h e r e is d e e p c o m b i n a t o r i c s behind t h i s r e l a t i o n between t a b l e a u x a n d c h a i n s i n the B r u h a t o r d e r , r e l a t e d t o s o m e r e c e n t w o r k of L a s c o u x and S c h u t z e n b e r g e r [65], [ 6 6 ] a b o u t what they c a l l e d S c h u b e r t polynomials a n d
Chain and antichain families
45 5
the nilplactic monoid. T h i s m o n o i d i s d e f i n e d by a v a r i a t i o n of t h e r e l a t i o n ( 6 ) , ( m i x i n g p l a c t i c and C o x e t e r c o n g r u e n c e s ) , a n d c o r r e s p o n d s t o a modified Robinson-Schensted i n s e r t i o n a l g o r i t h m . S t a n l e y ' s c o n j e c t u r e c a n a l s o be d e d u c e d u s i n g t h e n i l p l a c t i c monoid. Acknowledgements ,
I t h a n k o n e of t h e r e f e r e e s f o r n u m e r o u s d e t a i l c o r r e c t i o n s of t h e o r i g i n a l manuscript.
-
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45 6
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45 7
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&
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G. Viennot
458
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*.
A . J . H o f f m a n a n d D . E . S c h w a r t z - O n p a r t i t i o n s of a p a r t i a l l y o r d e r e d B 2 3 (1977), 3-13. set, J . Combinatorial Theory,
m.
C . G . Jacobi - De functionibus al tern an ti b us 360-371 ( = W e r k e 3, 4 3 9 - 4 5 2 ) .
. ..,
J . C r e l l e , 22 (1841),
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PARTVIII ORDER AND EFFECTIVENESS
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- PARTIEVIII ORDRE ET EFFECTIVITE
Joseph G. ROSENSTEIN Recursive linear orderings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p . 465
Irhe GUESSARIAN - Maurice NIVAT About ordered sets in algebraic semantics, . . . . . . . . . . . . . . . . . . .p . 477 J.K. LENSTRA - A.H.G. RINNOOY KAN Two open problems in precedence constrained scheduling. . . . . . . p . 509
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Annals of Discrete Mathematics 23 (1984) 465-476 0 Elsevier Science Publishers B.V. (North-Holland)
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RECURSIVE LINEAR ORDERINGS Joseph G. ROSENSTEIN Department of Mathematics Rutgers- The State Universityof NewJersey New Brunswick, New Jersey 08903 U.S.A.
RESLJm.Dans ce bref discours sur les chaines rscursives, je voudrais vous pr6senter certaines questions qu'on sc pose et vous donner un 6chantillon des mgthodes et techniques qu'on emploie pour r6pondre h ces questions. Le contexte de ce discours est fourni par la question g6ni.rale suivante: C t a i t tlontzL' 2.m th'ort'rnc
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c m b i > i a t ~ i r ce J n c t m i u n t I e s LThainc.s, sst-ii Si n o i , cmrn-nt peut-on mesui=i;r 1 ' e f - f e c t i o i t P
Un exemple de th6orsme combinatoire vrai effectivement est le r6sultat de Cantor selon lequel toute chaine dsnombrable est isomorphe 2 un sous-ensemble des nombres rationnels. On peut modifier la d6monstration standard de ce r6sultat pour dBcrire un algorithme qui produit le th6orCme suivant:
Touts chainc rdcursivc est d c u r s i v e m e n t isornorplie a' un sousenscmbie rz'zursif des nnmbres rationneis. Remarquez que tout emploi du mot "recursif" doit Ctre prScis6ment dgfini; ainsi par exemple, la phrase "chaine recursive" entraine l'existence d'un algorithme au moyen duquel on peut determiner si a < b ou non, et la phrase "r6cursivement isomorphe" entraine l'existence d'un algorithme au moyen duquel on peut calculer les valeurs de la fonction. Dans l'exemple que j'ai cit6 ci-dessus, la d6monstration traditionnelle peut Ctre effectivisge; il y a des exemples oij cela n'est pas possible. Dansces cas-li on doit ou fabriquer un nouvel algorithme ou demontrer qu'il n'existe pas d'algorithme convenable. Par exemple, la dgmonstration standard quetoutensemble bien ordonn6 a une extension lin6aire bien ordonn6e ne peut pas s'effectiviser. Cependant, un autre algorithme, que j'ai dgcrit avec H. Kierstead, demontre que:
Tout ensemb7,e bier1 o r r h n 8 qui est re'cursif i unp extension line'aire qui est bien ordonne'e ct re'cursive. La question de savoir si tout ensemble ordonne qui est dispers6 et rdcursif a une extension lingaire qui est dispers6e et rgcursive reste ouverte. 11 n'y a pas toujours une manisre unique d'effectiviser une formulation combinatoire. Par exemple, une autre manisre d'approcher l'exemple mentionnd cidessus est de demander si tout ensemble ordonne qui est r6cursif et recursivement bien ordonn6 a une extension lin6aire qui est rgcursive et recursivement bien ordonnke. ("Recursivementbien ordonn6" signifie que l'ensemble ordonn6 n'a aucune suite infinie r6cursive qui soit dgcroissante). Cette approche mPne plutst 1 une solution nggative, due h moi-m&ne et h R. Statman: I 1 existe un ensemble ordonni q u i est re'rursif e t re'cursivement bien ordonne', mais qui n'a pas d'extension 7in6aire qui soit re'cursivc et re'cursivement bien ordonnie. AprSs avoir introduit la higrarchie arithm6tique de Kleene, nous rgpondons B l'autre moitig de la question g6nCrale: Etant donn6 un ensemble ordonn6 qui est recursif et r6cursivement bien ordonn6, quelle complexit6 l'extension lin6aire d6sir6e doit-elle poss6der ?
466
J.G. Rosenstein Tout ensemble ordonnd q u i e s t r e c u r s i f e s t re'cursivement ordonnd a m e extension Lindaire qui e s t rdcursivement bien ordonneeet e s t s i t u d e au niveau h 2 dans La hidrarchie arithmdtique de KZeene.
La dgmonstration de ce thSorPme e s t un exemple de l ' u t i l i s a t i o n des "argumentat i o n s d i a g o n a l e s " dans l a t h g o r i e de l a r g c u r s i v i t g .
Les v e r s i o n s e f f e c t i v e s de quelques a u t r e s r g s u l t a t s combinatoires s o n t d i s c u t g e s dans l ' a r t i c l e y compris l e thLorOme de Dushnik e t M i l l e r s e l o n l e q u e l t o u t e c h a i n e dgnombrable p e u t s ' a b r i t e r dans un sous-ensemble p r o p r e d ' e l l e - m i k e . Ces s u j e t s s o n t d i s c u t g s 3 fond dans mon l i v r e L i n e a r Orderings (Academic P r e s s , 1982) dans l e c h a p i t r e i n t i t u l 6 , comme on p e u t s ' y a t t e n d r e , L i n e a r Orderings and Recursion Theory.
*** The framework f o r t h i s survey a r t i c l e about r e c u r s i v e l i n e a r order i n g s i s provided by t h e f o l l o w i n g Reneral q u e s t i o n s : Given a c o m b i n a t o r i a l theorem about l i n e a r o r d e r i n g s , i s i t t r u e e f f e c t i v e l y ? I f n o t , how can t h e e f f e c t i v e n e s s of t h e r e s u l t be measured ? Among t h e s p e c i f i c c o m b i n a t o r i a l theorems considered i n d e t a i l are ( a ) Every c o u n t a b l e l i n e a r o r d e r i n g i s order-isomorphic t o a s u b s e t of t h e r a t i o n a l s ( C a n t o r ) , ( b ) Every well-founded ( r e s p . , s c a t t e r e d ) p a r t i a l o r d e r i n g h a s a well-founded ( r e s p ; , s c a t t e r e d ) l i n e a r e x t e n s i o n , and ( c ) Every c o u n t a b l e l i n e a r o r d e r i n g can be embedded i n t o a p r o p e r s u b s e t of i t s e l f (Dushnik and M i l l e r ) . I n t h i s b r i e f t a l k about r e c u r s i v e l i n e a r o r d e r i n g s , I would l i k e t o convey t o you some s e n s e of t h e q u e s t i o n s that a r e r a i s e d and some of t h e f l a v o r of t h e I w i l l not methods and t e c h n i q u e s t h a t a r e used i n answering t h e s e q u e s t i o n s . t r y t o be comprehensive. The framework f o r t h i s t a l k i s provided by t h e f o l l o w i n g g e n e r a l q u e s t i o n :
Given a combinatorial theorem about linear ordsrings, i s it true e f f e c t i v e l y ? If not, how can the effectiveness of the result be measured? I u s e t h e phrase " l i n e a r o r d e r i n g s " t h e way o t h e r s u s e t h e p h r a s e s " t o t a l o r d e r i n g s " o r "chains." Information about l i n e a r o r d e r i n g s can be found i n my r e c e n t book [ S ] on t h e s u b j e c t ; t h e c h a p t e r on r e c u r s i v e l i n e a r o r d e r i n g s also c o n t a i n s t h e a p p r o p r i a t e material on r e c u r s i o n t h e o r y . A l l l i n e a r o r d e r i n g s t h a t I d i s c u s s w i l l be c o u n t a b l e , and t h e r e f o r e w e need o n l y look at s u b s e t s of Q ( t h e r a t i o n a l numbers.) This i s c o r r e c t by t h e f o l l o w i n g theorem, due t o Cantor:
E v e v countable linear ordering i s (order) isomorphic t o a subset of Q . Is t h i s c o m b i n a t o r i a l theorem t r u e e f f e c t i v e l y ? Before answering t h i s q u e s t i o n . we must f i r s t f o r m u l a t e and e x p l a i n t h e e f f e c t i v e v e r s i o n of C a n t o r ' s theorem. Loosely speaking, t h e way t o a r r i v e a t a n e f f e c t i v e v e r s i o n of a c o m b i n a t o r i a l r e s u l t is t o add t h e word " r e c u r s i v e " a t a l l a p p r o p r i a t e l o c a t i o n s . Using t h i s h e u r i s t i c , we a r r i v e at t h e f o l l o w i n g statement:
E v e y recursive linear ordering is recureiveZy isomorphic t o a recursive subset of Q.
46 7
Recursive linear orderings
Each of the three usages of the term "recursive" requires an explanation. Let us think of a countable linear ordering as a structure
3 R 2 R 5 R l R 4
then If now 3 R 6 R 2 . the diagram below:
f(3)-
f(3)
<
f(2)
<
f(5)
<
f(1)
then we would choose f(6)
f(4).
between f(3)
f(5)-
f(2)-
<
-
f(1)
and
f(2).
f
as in
C
In order for f to be defined recursively, we must specify f(6) in a way which can be systematically applied to define the other values of f . The natural solution to this problem seems to be to define f(6) to be the midpoint of the interval, but this leads to some difficulty for the following reason. We have specified that the image of f be a recursive subset of Q so that we must be able to provide an answer to questions like "Is (f(5) + f(1))/2 in f [N]?" Now if for some a E N we have 5 R a R 1, then indeed the midpoint of the interval [f(5), f(l)] will be in Q, but otherwise it will not be. But the question of whether there is such an element a E N cannot be answered recursively, for the recursiveness of R only enables us to decide simple issues like the relative position of two elements, but not more complicated questions like the non-emptiness of an interval. Thus another definition of f(6) must be produced. Let us specify in advance a fixed canonical enumeration Iqnln E N) of 0 (which will be referred to elsewhere in this paper) and arrange the construction so that if qn is none of f(l), f(2), f(n), then qn is not in f [N] at all; then to determine whether or not b E f[N] we need only determine b's place in the enumeration and, assuming that b qm, carry out the first m applications of the algorithm, and at that time determine whether or not b E f[N]. To achieve this effect, we define f(6) to be the first rational which is between f(3) and f(2). and number on the list 96, 97, 98, similarly for each other value of n. This algorithm then has all the required properties.
...,
-
...
In the example above the traditional proof can be effectivized; there are examples where it cannot. In such cases, one must either produce a new algorithm or prove that no suitable algorithm exists.
J.G. Rosenstein
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A s a second example of a c o m b i n a t o r i a l r e s u l t , I w i l l , i n d e f e r e n c e t o t h e o r g a n i z e r s of t h i s c o n f e r e n c e , d i s c u s s e x t e n d a b i l i t y of p a r t i a l o r d e r i n g s . ( I am r e p o r t i n g h e r e on d i s c u s s i o n s w i t h H. K i e r s t e a d and R. Statman.) Consider t h e following statement:
If A
is a well-founded partial ordering, then A a well-founded linear extension.
Is t h i s t r u e r e c u r s i v e l y ?
If A
has
That i s , i s t h e f o l l o w i n g s t a t e m e n t c o r r e c t ?
A is a well-founded recursive partial ordering, then has a well-founded recursive linear extension.
Here of c o u r s e we a r e t h i n k i n g of a r e c u r s i v e p a r t i a l o r d e r i n g a s a r e l a t i o n on N f o r which t h e r e i s a s u i t a b l e a l g o r i t h m .
R
The s t a n d a r d proof of t h i s r e s u l t ( s e e f o r example Bonnet and Pouzet [ l ] ) proceeds by " l e v e l s . " That i s , we d e f i n e A0 to c o n s i s t of a l l minimal e l e m e n t s of A, and proceed i n d u c t i v e l y t o d e f i n e A, t o c o n s i s t of a l l minimal e l e m e n t s of A {Aglg
u
T h i s proof c l e a r l y cannot be e f f e c t i v i z e d . I n f a c t , we cannot even b e g i n t h i s way s i n c e t h e r e i s no way of t e l l i n g r e c u r s i v e l y which elements of A a r e minimal. There i s however a n o t h e r a l g o r i t h m which works. This a l g o r i t h m and t h e proof t h a t i t works a r e due t o myself and H. K i e r s t e a d . A t stage n, f o r a l l a , b < n, which e x t e n d s
..., ...,
we assume t h a t w e have a l r e a d y d e f i n e d whether a
...,
.1 [email protected][email protected]., 11, 2 , ..., n - l l g /
W e could p l a c e n anywhere between t h e l a r g e s t squared element and t h e s m a l l e s t c i r c l e d element. W e choose t o p l a c e n a t t h e l o c a t i o n i n d i c a t e d by t h e arrow; t h a t i s , n i s p l a c e d j u s t below t h e B-smallest m < n f o r which n < ~ m . Thus n is p l a c e d as high as p o s s i b l e among 11, 2 , n-llg.
...,
I t i s c l e a r t h a t w i t h t h i s a l g o r i t h m w e do end up w i t h a r e c u r s i v e l i n e a r e x t e n s i o n of A -- f o r t o determine whether a b, and a
Recursive linear orderings
469
c f o r which c < a and c
a.
Now suppose t h a t B i s not well-founded, s o t h a t t h e r e i s a s u b s e t of B S O t h a t b l > B b2 > B b3 > B We may assume w i t h o u t { b l , b2, b3, . . . I l o s s of g e n e r a l i t y t h a t b l < b2 < b3 < For e a c h i < j , e i t h e r b i >A b j b j , s o by Ramsey's theorem and t h e assumption t h a t A is w e l l or bi founded, we may a l s o assume t h a t bi ( A b j f o r e v e r y i and j . For each i > 1 we t h u s have b i ( A b l , b i > b l , and b i 1 t h e r e i s some c i such t h a t c i < b l and b i 1.
... .
IA
... .
I f we c o n t i n u e t h i s procedure i n d u c t i v e l y , we conclude t h a t f o r t h e o r i g i n a l t h e r e i s a sequence c1, c 2 , c 3 , such t h a t sequence b l > B b2 >B b3 >B f o r a l l i > j we have b i j i > j). i m p l i e s t h a t c i 3 c j , we know t h a t c 1 < c2 < c 3 <
...
...
... .
...,
We now c l a i m t h a t c 1 >A c2 >A c 3 >A c o n t r a d i c t i n g t h e assumption Indeed, s i n c e c j + l c j , c +I j + l . T h i s c o n t r a d i c t s t h e c h o i c e of c j since < c j . Hence c j + l
ii
d
As noted by s e v e r a l p e o p l e , t h i s a l g o r i t h m p r o v i d e s a new proof t h a t i f K i s an i n f i n i t e c a r d i n a l number and K A than there is a l i n e a r extension B of A f o r which K B.
4
It is i n t e r e s t i n g t o see what happens i f t h i s p a r t i c u l a r a l g o r i t h m i s a p p l i e d t o a p a r t i a l Ordering which i s n o t well-founded. L e t A be t h e f u l l b i n a r y t r e e , l a b e l l e d as i n t h e diagram:
J.G. Rosenstein
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It i s e a s y t o s e e t h a t i f t h i s tree were extended t o a l i n e a r o r d e r i n g u s i n g t h e a l g o r i t h m p r e s e n t e d above, then t h e even numbered p o i n t s would form a set of Indeed f o r each n , t h e Z n - l even numbers o r d e r t y p e rl (of t h e r a t i o n a l s ) . a t l e v e l n of t h e t r e e would, i n t h i s l i n e a r e x t e n s i o n , be l o c a t e d i n t h e 2"-1 i n t e r v a l s formed by t h e Z n - l - l even numbers a t h i g h e r l e v e l s of t h e tree. Of c o u r s e , one can extend t h i s p a r t i a l o r d e r i n g t o a s c a t t e r e d l i n e a r o r d e r i n g by u s i n g t h e o p p o s i t e a l g o r i t h m , namely, by l o c a t i n g each p o i n t a t t h e lowest p o s i t i o n p o s s i b l e . The example above a r i s e s when one a s k s whether every s c a t t e r e d r e c u r s i v e p a r t i a l o r d e r i n g has a s c a t t e r e d r e c u r s i v e l i n e a r e x t e n s i o n . The c o m b i n a t o r i a l v e r s i o n of t h i s q u e s t i o n has a p o s i t i v e answer, due t o Bonnet and Pouzet ( s e e [ l ] ) . and a l s o t o Galvin and MacKenzie. Whether t h e e f f e c t i v e v e r s i o n is t r u e i s an open q u e s t i o n .
Open question: Does every scattered recursive partial ordering have a scattered recursive linear extension? The a s t u t e r e a d e r w i l l perhaps have n o t i c e d t h a t i n g i v i n g t h e e f f e c t i v e v e r s i o n of our second example we d i d not q u i t e f o l l o w t h e p r e s c r i p t i o n l a i d down in our f i r s t example, namely, t o add t h e word " r e c u r s i v e " a t a l l a p p r o p r i a t e l o c a t i o n s . I n t h e theorem w e assumed t h a t A had "0 s u b s e t of o r d e r t y p e w* and showed t h a t then B had "0 s u b s e t of o r d e r t y p e w*. What i f we assume can we t h e n extend A o n l y t h a t A has no r e c u r s i v e s u b s e t of o r d e r t y p e w*; t o a r e c u r s i v e l i n e a r o r d e r i n g which has no r e c u r s i v e s u b s e t of o r d e r type w*?
L e t u s t h e n s a y t h a t A is r e c u r s i v e l y well-founded i f A has no r e c u r s i v e s u b s e t of o r d e r t y p e wR. The c o n c l u s i o n below, due t o myself and R. Statman, answers the q u e s t i o n above:
There i s a recursive partial ordering which i s recursively well-founded but has no recursively well-founded recursive linear extension. The proof of t h i s result--by c o n s t r u c t i n g a s u i t a b l e counterexample--has two s t e p s . The f i r s t s t e p i n v o l v e s c o n s t r u c t i n g a r e c u r s i v e b i n a r y tree--a subo r d e r i n g of t h e t r e e above--which though i n f i n i t e (and t h e r e f o r e c o n t a i n s a n i n f i n i t e p a t h by Konig's Theorem) c o n t a i n s no i n f i n i t e r e c u r s i v e p a t h . T h i s k i n d of c o n s t r u c t i o n is f a m i l i a r t o r e c u r s i o n t h e o r i s t s ; I w i l l s a y more a b o u t how one c o n s t r u c t s such a t r e e l a t e r in t h i s paper. The second p a r t i n v o l v e s showing t h a t such a p a r t i a l o r d e r i n g cannot be extended t o a r e c u r s i v e l y well-founded r e c u r s i v e l i n e a r o r d e r i n g . Assume t h e n t h a t A i s a r e c u r s i v e s u b t r e e of t h e f u l l b i n a r y t r e e which is r e c u r s i v e l y W e will well-founded and assume t h a t B i s a r e c u r s i v e l i n e a r e x t e n s i o n of A. show t h a t B has a r e c u r s i v e s u b s e t of o r d e r t y p e w* by c o n s t r u c t i n g a r e c u r s i v e w*-sequence of elements of B. This w i l l of c o u r s e prove t h e r e s u l t . The method of c o n s t r u c t i n g such a sequence might be c a l l e d a B - f i r s t s e a r c h through A, g e n e r a l i z i n g t h e n o t i o n s of b r e a d t h - f i r s t s e a r c h ( i . e . l e v e l s e a r c h ) and d e p t h - f i r s t s e a r c h (i.e. greedy s e a r c h ) through A which c o r r e s p o n d t o p a r t i c u l a r l i n e a r e x t e n s i o n s B of A. A t t h e end of s t a g e n w e w i l l have d e f i n e d a descending B-sequence { a l , a2, an} and a pool Pn of elements of A from which an+l w i l l be s e l e c t e d . We assume a s part of t h e i n d u c t i o n a,,) forms a t e r m i n a l p a r t of t h e tree A, t h a t hypothesis t h a t ( a l , a2, is, f o r each i, i f b is above a 1 on t h e t r e e , t h e n b = a j f o r some j < i, and t h a t t h e p o o l Pn c o n s i s t s of a l l immediate A-predecessors of an} e x c e p t t h o s e which are a l r e a d y i n e l e m e n t s of { a l , a 2 , {al, a2, an]. Thus a t s t a g e 1 we l e t a1 b e t h e t o p element of t h e tree
...,
..., ...,
...,
Recursive linear orderings
47 1
and l e t P1 be t h e A-predecessor of a l . Proceeding i n d u c t i v e l y we l e t an+l b e t h e B - l a r g e s t element of Pn and we o b t a i n Pn+l from Pn by d e l e t i n g a n + l and adding i t s A-predecessors. T h i s completes t h e c o n s t r u c t i o n of what i s c l e a r l y a r e c u r s i v e s u b s e t of A. To show t h a t t h i s s e t has o r d e r t y p e w* i n B , i t s u f f i c e s t o show t h a t a t + l < B a t f o r each t . But o t h e r w i s e a t + l > B a t , f o r s i n c e a t 4 Pt w e cannot have a t + l = a t . Also a t t l f P t - l f o r o t h e r w i s e we would have chosen i t t o be a t . But a t + l f Pt s o i t must be a n A-predecessor of a t , so t h a t a t + l
Open question: Does every recursively scattered recursive partial ordering have a recursively scattered recursive linear extension? Now t h a t w e have a c o m b i n a t o r i a l s t a t e m e n t about l i n e a r o r d e r i n g s t h a t i s not t r u e e f f e c t i v e l y , we go back and a s k how t h e e f f e c t i v e n e s s of t h e o r i g i n a l r e s u l t can be measured.
To d i s c u s s t h i s q u e s t i o n , we have t o i n t r o d u c e t h e c l a s s i f i c a t i o n of t h e a r i t h m e t i c a l sets. We s a y t h a t a s e t K of numbers i s a En-set i f t h e r e i s a r e c u r s i v e p r e d i c a t e R(x1, x2, x,, y ) s u c h t h a t
...,
where t h e n q u a n t i f i e r s a r e a l t e r n a t e l y 3 and b' , s t a r t i n g w i t h 3. S i m i l a r l y , w e s a y t h a t K i s a IIn-set i f t h e r e i s a r e c u r s i v e p r e d i c a t e R(x1, x2 xn, m) such t h a t
...,
m
E
K
f--+
--.,%,
( V x l ) ( 3 x 2 ) ( v ~ 3 )* * * (OnXn) R(x1, ~ 2 .
m)
d.
where a g a i n t h e q u a n t i f i e r s a l t e r n a t e , t h i s t i m e s t a r t i n g w i t h A set K i s a An-set i f it is both a En-set and a IIn-set. A set i s c a l l e d an a r i t h m e t i c a l set i f i t i s a En-set o r a IIn-set f o r some I . . That t h e s e s e t s form a h i e r a r c h y was shown by S.C. Kleene ( f o r r e f e r e n c e s e e [ 5 ] ) ; t h a t i s , i n t h e diagram below a l l i n c l u s i o n s ( i n d i c a t e d by a r r o w s ) a r e p r o p e r i n c l u s i o n s .
A1
...
For example, a set E is r e c u r s i v e l y enumerable i f t h e r e i s a Turing machine which w i l l produce a l l of i t s members; t h a t i s , i f f o r some t ,
47 2
J.G. Rosenstein m
E
E
+--+
(3s)
T ( t , s , m)
where T ( t , 6 , m) i s t h e r e c u r s i v e p r e d i c a t e which s a y s t h a t "Turing machine number t w i l l produce m w i t h i n s s t e p s . " Now f o r each f i x e d t , t h e p r e d i c a t e T t ( s , m) :T ( t , s , m) i s a r e c u r s i v e p r e d i c a t e R ( s , m), so t h a t m E E +--+ ( 3 s ) R ( s , m). Thus every r e c u r s i v e l y eIIUmeKable s e t i s a E l - s e t ; t h e converse i s a l s o c o r r e c t . A s e t is r e c u r s i v e i f and only i f both i t and i t s complement a r e r e c u r s i v e l y enumerable, or El-sets. Note t h a t a set is a En-set i f and o n l y i f i t s complement is a Hn-set, so t h a t t h e Al-sets a r e p r e c i s e l y t h e r e c u r s i v e sets. We l e t Wt {mi ( 2 s ) T ( t , says t h a t the enumerated,)
d e n o t e t h e t ' t h r e c u r s i v e l y enumerable s e t : Wt = (Hidden in t h e n o t a t i o n i s t h e Enumeration Theorem which r e c u r s i v e l y enumerable s e t s can themselves be r e c u r s i v e l y For which t i s Wt non-empty? T h i s s e t i s C 1 s i n c e s , m)}.
non-empty
Wt
+---+
( 3 m ) ( g s ) T ( t , s , m).
(In t h i s p r e d i c a t e , t h e two e x i s t e n t i a l q u a n t i f i e r s can be combined i n t o one by u s i n g t h e r e c u r s i v e f u n c t i o n f ( n , x ) = (x), = t h e power of t h e n ' t h prime i n X. Thus t h e above i s e q u i v a l e n t t o ( 3 x ) T ( t , ( x ) l , ( x ) ~ ) which can be s i m p l i f i e d i n f i n i t e ? This s e t i s H2 s i n c e t o ( 3 x ) R(t, x).) For which t i s Wt
wt
infinite
+--+
( V n ) ( 3 m) (m
which can be r e w r i t t e n i n t h e a p p r o p r i a t e
II2
>
n
m
E wt)
form.
One a d d i t i o n a l t o p i c t h a t we need t o d i s c u s s b r i e f l y i s t h e use of an o r a c l e . During a c o n s t r u c t i o n we may f i n d i t n e c e s s a r y t o r e f e r t o a set A which i s e x t e r n a l t o t h e c o n s t r u c t i o n , and i n q u i r e whether given numbers a r e i n A. In t h a t c a s e t h e set or f u n c t i o n c o n s t r u c t e d i s s a i d t o be r e c u r s i v e in A. I t t u r n s o u t t h a t i f A i s a '&-set o r a lIn-set, then sets r e c u r s i v e in A a r e a l l An+l-sets. Thus, f o r example, i f d u r i n g a c o n s t r u c t i o n we need t o know f o r i s i n f i n i t e , then whatever we c o n s t r u c t w i l l be a A3-set; each t whether Wt in t h i s case, t h e o r a c l e being c o n s u l t e d i s I t l W , i s i n f i n i t e } . The a r i t h m e t i c a l h i e r a r c h y p r o v i d e s a u s e f u l measure of t h e complexity of The v a l u e judgment i t propounds i s t h a t t h e lower g i v e n sets o r c o n s t r u c t i o n s . i n t h e h i e r a r c h y t h e outcome of a p a r t i c u l a r c o n s t r u c t i o n , t h e b e t t e r t h e construction. We now r e t u r n t o OUK p r e v i o u s q u e s t i o n . Given a r e c u r s i v e l y well-founded r e c u r s i v e p a r t i a l o r d e r i n g A , how complicated need a d e s i r a b l e l i n e a r e x t e n s i o n B of A be? I n Order f o r B t o be r e c u r s i v e l y well-founded, it is s u f f i c i e n t t o g u a r a n t e e t h a t no Wt p r o v i d e s a B-decreasing sequence of elements of A. How do we do t h i s ? Suppose t h a t a f t e r s s t e p s we have enumerated ( i n o r d e r ) t h e ak of Wt. I f a j B a2 >B >B ak and t h a t , even worse, t h i s p a t t e r n w i l l c o n t i n u e u n t i l t h e very end. So we must f i n d some a i and a j w i t h i < j ( e i t h e r from t h o s e elements s o f a r g e n e r a t e d o r from t h o s e n o t y e t g e n e r a t e d ) f o r which a t i s not y e t B-larger t h a n a j , Since a t each s t e p of t h e c o n s t r u c t i o n of B we make and d e f i n e a i
...,
...
.
Recursive linear orderings
473
The t r o u b l e is t h a t w e have t o do t h e same t h i n g f o r each W t and s o w e cannot wait around t o f i n d o u t i f Wt is infinite. I f i t i s , we w i l l e v e n t u a l l y f i n d a s u i t a b l e i and j , but i f i t is n o t , then o u r s e a r c h w i l l go o n f o r e v e r and no I3 w i l l be c o n s t r u c t e d a t a l l . I f a t a given s t a g e of t h e t h e n w e w i l l need t o know whether o r n o t c o n s t r u c t i o n we want t o d i s p o s e of Wt Wt i s i n f i n i t e ; i f i t i s , w e w i l l s e a r c h f o r t h e r i g h t a i and a . , and i f i t is n o t , we w i l l go on t o W t + l . Thus t h e c o n s t r u c t i o n of B he r e c u r s i v e i n a Il2 s e t . The r e a d e r may have observed t h a t we can u s e a s i m p l e r o r a c l e , namely, one t h a t answers whether W t h a s a n o t h e r element. S i n c e t h i s q u e s t i o n i s 11, t h e c o n s t r u c t i o n of B w i l l be r e c u r s i v e i n an 11-set. W e t h u s come t o t h e f o l l o w i n g c o n c l u s i o n :
will
i s a recursively well-founded recursive p a r t i a l ordering, then A has a r e c u r s i v e l y well-founded A2 l i n e a r extension.
If* A
By a s i m i l a r argument i t is p o s s i b l e t o show t h a t
i s a r e c u r s i v e l y scattered recursive p a r t i a l ordering, then A has a r e c u r s i v e l y scattered A2 l i n e a r extension.
If A
The proof w e p r e s e n t e d above is an example of t h e u s e of " d i a g o n a l arguments" i n recursion theory. Before going on t o t h e next example, l e t me pause t o p o i n t o u t how t o c o n s t r u c t an i n f i n i t e r e c u r s i v e b i n a r y t r e e which i s r e c u r s i v e l y well-founded; S t a r t generating the f u l l binary s u c h a tree w a s used i n an e a r l i e r argument. t r e e . A t a c e r t a i n t i m e , we may d e c i d e t o c l o s e o f f a c e r t a i n node; by t h i s we mean t h a t n o t h i n g new w i l l go below t h a t node, so t h a t below t h a t node t h e t r e e w i l l remain f i n i t e T W h e n do we make such a d e c i s i o n ? W e s i m u l t a n e o u s l y begin and i f i t happens a t some p o i n t t h a t we have enumerated t enumerating a l l W t , elements of a p a r t i c u l a r Wt and t h o s e elements form a descending c h a i n s o f a r on t h e t r e e , t h e n we c l o s e o f f t h e lowest of t h e t nodes. The r e a d e r should v e r i f y t h a t t h e t r e e t h a t r e s u l t s has t h e d e s i r e d p r o p e r t i e s . We now c o n s i d e r a t h i r d example of a c o m b i n a t o r i a l theorem about l i n e a r o r d e r i n g , t h i s one due t o Dushnik and Miller [ 2 ] .
Any countable l i n e a r ordering can be embedded i n t o a proper subset of i t s e l f . Can any r e c u r s i v e l i n e a r o r d e r i n g be r e c u r s i v e l y embedded i n t o a p r o p e r r e c u r s i v e s u b s e t of i t s e l f ? Those of you who have s e e n t h e proof of Dushnik and Miller's theorem w i l l r e c a l l t h a t i n t h e s c a t t e r e d c a s e t h e d e s i r e d embedding w a s t h e i d e n t i t y o u t s i d e of a s u b s e t of o r d e r type o ( o r a*) and was t h e s u c c e s s o r ( o r p r e d e c e s s o r ) map on t h a t s u b s e t . The t w i s t came i n t h e n o n - s c a t t e r e d case where a s u b s e t of order type n was s e l e c t e d and A was embedded i n t o t h a t s u b s e t u s i n g C a n t o r ' s theorem. More p r e c i s e l y , t h i s procedure was needed o n l y when A i s a dense sum of f i n i t e l i n e a r o r d e r i n g s . Thus we need t o f i n d a dense s u b s e t of a dense sum of f i n i t e l i n e a r o r d e r i n g s . Can w e make such a s e l e c t i o n r e c u r s i v e l y ? For example, suppose t h a t A i s a r e c u r s i v e l i n e a r o r d e r i n g of o r d e r type 2-11. The right-hand e n d p o i n t s form a s u b s e t of o r d e r type n b u t t h e r e i s no way t o t e l l r e c u r s i v e l y which p o i n t s a r e right-hand e n d p o i n t s . (Show t h a t t h i s i s a A2-set.) Nevertheless, t h e e f f e c t i v e v e r s i o n of Dushnik and Miller's theorem i s c o r r e c t i n t h i s situation.
414
J. G.Rosenstein Any recursive subset A of Q of order type Z'n can be recursively embedded i n t o a proper recursive subset of itself.
...
} in o r d e r of d i s c o v e r y and To prove t h i s we enumerate A = { a l , a 2 , a 3 , begin t h e d e f i n i t i o n of a f u n c t i o n P: A A f o r which a 1 4 P [ A ] by s e t t i n g P(a1) = a2. How do we d e f i n e P ( a 2 ) ? I f f o r example a2 > a1 we can w a i t f o r some am > a2 and then d e f i n e P(a2) = am. But t h a t w i l l g i v e u s problems l a t e r i f a, is t h e s u c c e s s o r of a2 b u t a2 i s not t h e s u c c e s s o r of a l . I n s t e a d w e wait f o r an a, which i s not t h e s u c c e s s o r of a2. HOW t o do t h i s s y s t e m a t i c a l l y we l e a v e f o r t h e r e a d e r . ( A more complete d i s c u s s i o n of t h e p r o o f s of t h i s and subsequent r e s u l t s i n t h i s paper can be found i n my book I S ] . ) ---+
The same argument works whenever A is a dense sum of bounded f i n i t e l i n e a r o r d e r i n g s . The r e a d e r might t r y h i s hand a t t h e a p p a r e n t l y open q u e s t i o n of what happens when A i s an a r b i t r a r y r e c u r s i v e dense sum of f i n i t e l i n e a r orderings. We s t a r t e d t h i s d i s c u s s i o n by mentioning how e a s i l y Dushnik and M i l l e r ' s theorem works in t h e s c a t t e r e d c a s e . This i s f a r from t r u e r e c u r s i v e l y , s i n c e one cannot i d e n t i f y s u c c e s s o r s o r determine t h e s i z e of i n t e r v a l s r e c u r s i v e l y The f o l l o w i n g r e s u l t , due t o L. Hay and even i f A has o r d e r type w. R o s e n s t e i n , is proved i n (51.
There i s a recursive subset A of Q of order type w f o r which there is rw recursive mzp from A t o A other than the identity. There a r e s e v e r a l i n t e r e s t i n g q u e s t i o n s which were r a i s e d in t h e above d i s c u s s i o n which merit f u r t h e r a t t e n t i o n . Does e v e r y non-scattered r e c u r s i v e s u b s e t of Q have a r e c u r s i v e s u b s e t of o r d e r t y p e n? The embedding of 2-II i n t o i t s e l f does y i e l d a r e c u r s i v e s u b s e t of o r d e r type n but t h a t i s t h e b e s t t h a t can be done. That i s , i t i s p o s s i b l e t o c o n s t r u c t a r e c u r s i v e (5 i s t h e o r d e r type of t h e i n t e g e r s ) which s u b s e t of Q of o r d e r type 5.17 h a s no A 2 dense s u b s e t (Lerman and R o s e n s t e i n [ 4 ] ) ; t h e b e s t one can t h u s s a y i s t h a t such a s e t has a II2 dense s u b s e t ( t r y t o prove t h i s ) . I s u s p e c t , b u t I cannot prove, t h a t t h e r e is a r e c u r s i v e n o n - s c a t t e r e d l i n e a r o r d e r i n g which has no a r i t h m e t i c a l dense s u b s e t .
Open question: 16 there a recursive non-scattered l i n e a r ordering which has rw arithmetical dense subset? n a r e c u r s i v e nons c a t t e r e d l i n e a r o r d e r i n g of o r d e r type E n * n which has no A2n r e c u r s i v e dense s u b s e t , and t h e n perhaps one of o r d e r type 5 which has no a r i t h m e t i c a l dense s u b s e t . A p o s s i b l e l i n e of a t t a c k would be t o c o n s t r u c t f o r each
S i n c e t h e r e i s no end t o t h e c o m b i n a t o r i a l r e s u l t s about l i n e a r o r d e r i n g s (and o t h e r s t r u c t u r e s ) , t h e r e i s a l s o no end t o t h e q u e s t i o n s of e f f e c t i v e n e s s of c o m b i n a t o r i a l r e s u l t s t h a t can arise. I w i l l d i s c u s s b r i e f l y two more t y p e s of q u e s t i o n s . The f i r s t i s t h e f o l l o w i n g : Given a r e c u r s i v e l i n e a r o r d e r i n g which has an automorphism, must i t have a r e c u r s i v e automorphism? In my book I c o n s t r u c t examples of r e c u r s i v e l i n e a r o r d e r i n g s of o r d e r type 5 and 2*r, which have no r e c u r s i v e automorphisms. (The l a t t e r c o n s t r u c t i o n u s e s a p r i o r i t y argument, which i s a more s o p h i s t i c a t e d t y p e of d i a g o n a l argument.) A r e c e n t t h e s i s by Steven Schwarz [ 6 ] c o n t a i n s t h e f o l l o w i n g d e f i n i t i v e r e s u l t : L e t a be a r e c u r s i v e o r d e r type ( i . e . t h e o r d e r type of a r e c u r s i v e linear o r d e r i n g ) . Then t h e r e i s a r e c u r s i v e s u b s e t A of Q of o r d e r type a which has no r e c u r s i v e automorphism, u n l e s s a has an i n t e r v a l of o r d e r t y p e n.
Recicrsiue linear orderings
41 5
F i n a l l y , I should d i s c u s s t h e o l d e s t r e s u l t concerning e f f e c t i v e v e r s i o n s of theorems about l i n e a r o r d e r i n g s . Consider t h e c l a s s i c a l f a c t t h a t every i n f i n i t e l i n e a r o r d e r i n g h a s e i t h e r a s u b s e t of o r d e r t y p e w o r a s u b s e t of o r d e r t y p e w*. What i s t h e c o m b i n a t o r i a l v e r s i o n of t h i s f a c t ? Tennenbaum showed a number of y e a r s a g o t h a t t h e r e is a r e c u r s i v e s u b s e t of 0 o f o r d e r t y p e w + a* which h a s no r e c u r s i v e s u b s e t of o r d e r t y p e w o r w*; s u c h a s e t c a n be t h o u g h t of a s e f € e c t i v e l y f i n i t e . Watnick [ 7 ] e x t e n d e d Tennenbaum's c o n s t r u c t i o n t o show t h a t f o r e v e r y r e c u r s i v e o r d e r t y p e a t h e r e i s a r e c u r s i v e s u b s e t of 0 of o r d e r t y p e w + 6.a + w* w h i c h h a s no r e c u r s i v e s u b s e t of o r d e r t y p e a o r w*. Looking i n a d i f f e r e n t d i r e c t i o n , i t is p o s s i b l e t o show t h a t e v e r y r e c u r s i v e l i n e a r o r d e r i n g h a s a r e c u r s i v e s u b s e t of o r d e r t y p e w, w*, w + w*, o r w 5-11 + w*; Lerman [ 3 ] showed t h a t t h i s was b e s t p o s s i b l e by c o n s t r u c t i n g a r e c u r s i v e l i n e a r o r d e r i n g of t h e l a s t o r d e r t y p e which h a s no r e c u r s i v e s u b s e t of any of t h e o t h e r t h r e e o r d e r t y p e s .
+
A s I h a v e m e n t i o n e d s e v e r a l t i m e s a l r e a d y , t h e s e r e s u l t s and c o n s t r u c t i o n s a r e d i s c u s s e d a t some l e n g t h i n my book [ 5 ] , i n t h e c h a p t e r t i t l e d , n o t s u r p r i s i n g l y , L i n e a r O r d e r i n g s and R e c u r s i o n T h e o r y .
I hope t h a t i n t h i s t a l k I have s u c c e e d e d i n c o n v e y i n g some of t h e f l a v o r of t h e s u b j e c t of r e c u r s i v e l i n e a r o r d e r i n g s .
REFERENCES B o n n e t , R. and P o u z e t , M., L i n e a r e x t e n s i o n s of o r d e r e d s e t s , i n : R i v a l , I. ( e d . ) , O r d e r e d S e t s , P r o c . of t h e NATO Advanced S t u d y I n s t i t u t e , B a n f f , Canada (1981). D u s h n i k , B. and M i l l e r , E.W., C o n c e r n i n g s i m i l a r i t y t r a n s f o r m a t i o n s of l i n e a r l y o r d e r e d s e t s , B u l l . h e r . Math.. SOC. 3 (1940). 322-326. [MR 1, p. 3181. Lerman, M., On r e c u r s i v e l i n e a r o r d e r i n g s , i n : Lerman, M., S c h m e r l , J. and S o a r e , R. ( e d s . ) . P r o c e e d i n g s , L o g i c Year 1979-80: The U n i v e r s i t y of o f C o n n e c t i c u t , L e c t u r e Notes i n M a t h e m a t i c s , Vol. 859, pp.132-142. ( S p r i n g e r - V e r l a g , B e r l i n and New York, 1981). Lerman M. and R o s e n s t e i n , J . G . , R e c u r s i v e L i n e a r O r d e r i n g s 11, i n : P r o c e e d i n g s , P a t r a s C o n f e r e n c e 1980, Amsterdam ( N o r t h - H o l l a n d P u b l . ,
1981). R o s e n s t e i n , J.G.,
L i n e a r O r d e r i n g s (Academic P r e s s , New Y o r k , 1982).
S c h w a r z , S . , Q u o t i e n t L a t t i c e s , I n d e x S e t s , and R e c u r s i v e L i n e a r O r d e r i n g s , Ph.D. T h e s i s , U n i v e r s i t y of C h i c a g o (1982). W a t n i c k , R., A g e n e r a l i z a t i o n of Tennenbaum's t h e o r e m o n e f f e c t i v e l y f i n i t e r e c u r s i v e o r d e r i n g s , t o a p p e a r i n J. Symbolic Logic.
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Annals of Discrete Mathematics 23 (1984)477-508 0 Elsevier Science PublishersB.V. (North-Holland)
411
ABOUT ORDERED SETS IN ALGEBRAIC SEMANTICS Irene GUESSARIAN et C.N.R.S.-L.I.T.P.
Maurice NIVAT UnivrrsitQ Paris 7
-
L.I.T.P.
U.E.R. dr MATHEMATIQUES Universitd Paris 7 2 , Place Jussieu 75251 PARIS CEDEX 05
Au Professeur E. COROMINAS, 5 l'occasion de s o n Emdritat.
RQsumd
La sgmantique algdbrique ddcrit les schemas de programmes par des systemes S
d'gquations rdcursives dont, intuitivement, les membres gauches repr.5-
sentent les "noms d e p r o c e d u r e " et les membres droits les " c o r p s d e p r o c e d u r e ' ' les d6finissant ; les dquations d'un tel systeme sont des dgalitds entre termes bien formds d'une me
S
Z-algsbre libre ordonnde. La fonction calculde par un program-
peut alors stre caractdrisde par la solution du systeme d'dquations
S
dans une Z-algebre ordonnde compliite libre : la compldtude de l'algsbre oii l'on rejoint
S
unique
TS
permet d'assurer d'une part que tout eel systiime S
,
admet une solution
dans l'alglbre libre, d'autre part que cette solution est calculable
par approximations successives. La compldtion la plus utilisde est une compldtion par id6aux de la Z-algiibre libre. La fonction calculde par le programme une algebre complete quelconque est alors l'image de
TS
S
dans
par un morphisme cano-
nique. On montre ensuite que la complgtion par iddaux s'avsre insuffisante
pour prendre en compte des varibtds, dventuellement non dquationnelles, d'algsbres et qu'elle contient "trop de bornes supErieures", dont certaines ne sont pas ddfinissables par systemes d'gquations. Ceci nous amiine 1 :
I. Guessarian, M. Nivat
418 (I).
D6finir des completions continues (intermgdiaires entre compl6tions par cou-
pures et par idEaux). (2).
D6finir des algPbres libres ordonn6es recursivement complPtes (consistant
exclusivement de solutions de schdmas de programmes). ( 3 ) . Etudier diverses vari6t6s d'algsbres ordonnges utiles du point de vue de l a
programmation et les alglbres libres complstes correspondantes (ce qui conduit 2 une Etude de cornpl6tions de quotients et P des combinaisons des points ( I ) et ( 2 ) ci-dessus).
Abstract : We show how complete ordered algebras are used in algebraic semantics. We introduce the notions of interpretation (or model, or algebraic structure), and of free and Herbrand interpretations. We define classes of interpretations. We
characterize and construct the free and Herbrand interpretations for a class. We define algebraic, equational and relational classes of interpretations and study the construction of the Herbrand interpretation in each case. We also give some hints on how this study can be applied to derive proof systems for program equivalences. I.
INTRODUCTION We here show how Universal Algebra methods, and more specially the
theory of ordered sets, can be applied to programming language semantics thus yielding an algebraic semantics. Problems in semantics can be expressed in terms of ordered algebras ; the theory of ordered algebras, within the theory of ordered
sets then helps i n providing partial answers for them. This in turn usually leads to new questions or different formulations of problems, hence new problems in the theory of ordered algebras thus resulting in a two way communication channel between the two disciplines. Algebraic semantics main goal has been to provide a clean and sound semantics of programming languages by splitting as much as possible the syntactic and semantic parts of a program. Using tools from Universal Algebra, one can then describe abstractly, i.e. independent of any interpretation, the syntactic properties of a program ; after what, one is well equipped for, given any concrete interpretation, translating the abstract or syntactic properties, via that interpretation, into concrete, or semantic properties, of the real program. Moreover, this can be done stepwise, introducing at each step the exactly needed amount of semantic knowledge. Algebraic semantics thus makes easier and more natural the concepts of modularization and abstraction, essential in software development. I n algebraic semantics, interpretations are nothing but certain ordered
479
Ordered sets in algebraic semantics algebras, or equivalently X-structures or models in the sense of
/GR/.
The main
tool in characterizing the syntax of a program is to have i t compute symbolically in a free interpretation, i.e. i n an absolutely free algebra of terms. The set of all possible computations then is naturally endowed with an ordering corresponding to the notion of successive approximations of a computation, or, equivalently, the relation "to be less defined than" for partial functions. For this ordering, the set of all finite symbolic computations has a lattice structure, hence can be represented by an infinite tree (or term) ; this tree belongs to an ordered algebra, complete with respect to the ordering "to be less defined than" ; moreover, this infinite tree (or term) characterizes the behavior of the program with respect to all possible interpretations : properties of programs are thus described by properties of the associated infinite tree (or term). One of the main problems one has to deal with in proving any program property, is to prove equivalences of programs : this approach shows that two programs are equivalent w.r.t. all possible interpretations iff their associated infinite trees are equal. However, equivalence w.r.t. all possible interpretations is by far too exacting to be of any real use. In practice one has to take into account some of the constraints or properties verified by the interpretations one is interested in ; this extra information should even be modularized according to needs. We thus have to look for alternate syntactic objects and structures for finding and describing the set of all possible symbolic computations under some given constraints. The constraints will be described by a class of interpretations C, and we will look for a generic any given class I)
-
or free, or Herbrand - interpretation H
for
C, together with effective ways of describing :
the free interpretation H
2) The equivalence w.r.t.
H ; this equivalence will in turn characterize
the equivalence w.r.t. the class C. This H
corresponds roughly to what is usual-
ly called a free complete C-algebra, or an algebra free relative to the class of complete ordered algebras. The description of this free interpretation H, its elements and its equivalence can then be fruitfully applied to prove various kinds of program properties, transformations, simplifications, etc... The main concepts of universal algebra used in our approach are :
-
complete ordered F-algebras
- free (and related) algebraic systems in various classes
-
classes of algebraic systems equational, quasi-equational (or inequational, or relational) and other
kinds of classes of algebraic systems. This paper is primarily a survey written with an intuition minded bias : we give numerous examples and informal explanations, but refer the reader to the literature and mainly /GU/ where most of the results here given are proved. It is organized as follows : section 2 introduces the preliminaries on algebraic semantics,
I. Guessarian, M. Nivat
48 0
in relation with ordered set theory ; ordered algebras. complete algebras, program schemes and their interpretations, specially the free and Herbrand ones are there defined. Section 2 introduces the "class of intcrpretations" approach, illustrates problems and questions inherent to i t and relates i t to the literature ; in sections 4 and 5 we study in a more detailed way the equational and first order definable
classes, the most interesting from a practical standpoint. 2.
PROGRAM SCHEMES
2.1.
-
SEMANTICS - FREE COMPLETE ALGEBRAS
Programs and program schemes We will briefly outline in this section the basic results of algebraic
semantics. The fundamental ideas of algebraic semantics are :
1) using universal algebra tools, characterize a program by a mathematical object which is an element of a free complete algebra /GU,GUI, N/. 2) Prove the existence of the free complete algebras needed and construct
them. 3) Use the mathematical representation of the program to obtain sound
proofs of its properties.
To these ends, programs are represented by program schemes, i.e. systems of mutually recursive equations whose right hand sides represent procedure names and left hand sides represent procedure bodies ; the equations of such a system are equalities between well formed terms of a free ordered algebra. Let us illustrate this by an example. Example I : Suppose we wish to compute nk using only addition and multiplication. The simplest idea, in view o f the binary representation, is to write down the definition of nk with 2 as explicit factors in the exponent, i.e. : 2 x [k/2]
nk
=
+ r (k)
n
, where [k/2] and r (k) denote the quotient and
the remainder of the integer division of k by 2 . Whence the recursive procedure : Recursive procedure G(n,k) G(n,k)
=
;f k=O
where p(n,b)
then 1 else
: =
; integer
n,lc ;
G(nx n , [k/2]) x p(n,r(k))
b=O then 1 else n .
(1)
}
p
Notice that the first line of this definition consists of procedure declarations inessential from the computational standpoint. We will forget that first line and concentrate on the last two lines which contain the procedure body. If we now abstract the previous program into a "program scheme", obtained by "forgetting" the nature and meaning of the base functions and data, we we obtain the scheme :
48 1
Ordered sets in algebraic semantics
S : G(n,k)
Scheme
g(a,k,b, m(G(m(n,n), d(k)), p(n,r(k)))) (2) gives program P back if we consider the interpretation Z having domain
=
S
N (i.e. the data belong to N), and s u c h that = 0, bI= A , mI= x , dI (k) = [k/2], gI(x,y,z,t) = if x = y __ then z I else t, p (n,b) = if b = 0 then 1 else n. Now, we consider the interpretation 3 , __ I having the same domain as I and giving the same meaning to the base function symbols
a
except for b and m which are respectively interpreted as bJ this case, GJ(n,k)
nxk.
=
=
0 and m
J
=
+. In
T h u s abstracting a program into a program scheme leads
to a greater generality. We now come to the problem of semantics
-
namely, givenaprogram scheme
S and an interpretation I , how to define the function T (S,G) computed by the scheme I is the least fixpoint of equation TI(S,G)
with respect to the interpretation I ?
(I)(in the algebra of partial functions N 2
defined than
T (S,G)
g).
I
-f
N ordered by: f
iff
f
is less
can be computed by successive approximations as follows : 0
is everywhere undefined which is abbreviated by :
the first approximation tI
to(n,k) = 1. Then, for q in N , the q+l-th approximation is defined by substiI tuting t; for G in the right hand side of equation ( I ) , whence :
and more generally :t
The sequence It:, q
=
!
nk
if
0< k
< Zq-'-l
1 otherwise
q € I N ] is increasing with respect to the ordering "to be less
defined than" ; the function {t:,
(n,k)
(n,k)
I-+
nk
is the least upper bound of the sequence
€IN}. This example suggests that interpretations be ordered algebras where
increasing sequences have least upper bounds, i.e. algebras which are complete in 0
some sense.
2.2 Complete F-algebras
Let us sketch more formally the algebraic semantics approach here. For more details see /GU/. Let
F
=
{f,g,h, . . . I
(resp.
= {G,H,K,
... I )
be a ranked alphabet of
base function symbols (resp. of function variables) ; the rank of a symbol denoted by p ( s )
and
F (resp. 0 )
(resp. function variables) of rank presentingthe"undefined";
algebra. Let
V
=
0
is
n. Let R be a special rank 0 symbol in F, re-
F i s asignature, or similarity type in terms of universal
{u,v,w,x,y,z, ... ) be a set of variable symbols (intended t o re-
present parameters or positions) of rank in
s
denotes the set of base function symbols
which may have arbitrary ranks.
0
, as opposed t o the function variables
48 2
I. Guessarian, M. Niuat An ordered F-algebra /ADJ/, or F-magma /N/, or algebraic structure
/AN, C , G R / , M
is an ordered set
a total and monotone mapping element of
DM. Ordered
(DM,
fM : D i
+
5)
together with for each
DM, and such that
RM
f
in
F
n’
is the least
F-algebras are actually a special kind o f algebraic
systems o f similarity type, or signature, F U
{
The class of a l l ordered F-algebras forms a quasi-variety (i.e. a class axiomatisable by quasi-atomic formulae) in the sense of /MA/. See also /GR/ p. 339, and /AN/(section 4 ) , /GU/, for quasi equational logic. Since quasi-varieties are epireflective in their similarity class, all the nice properties of similarity classes of algebraic systems are inherited by ordered F-algebras too. Recall now that a subset ted iff any pair
E
d,d’ of elements of
An F-algebra
M
of an ordered algebra is said to be direcE
is said to be
has an upper bound in
directed subsets (resp. countable chains) of f’s
are continuous, i.e. preserve l.u.b!s
M chains).
DM
have a 1u.b. in DM
and the
of directed sets (resp. countable
I n the sequel we will consider mainly
exactly similar for
E.
A-complete (resp. w-complete) iff all
A-complete magmas (the theory is
w-complete ones) and we will call them complete to shorten
notations. Whenever we consider a different notion of completeness, this will be mentioned explicitly. Notice that the notions of completeness here considered are slightly weaker than the usual ones in lattice theory : 1 ) n o t all subsets of
M
are required to have l.u.b!s.
2) no greatest lower bounds are required, hence the F-algebras we consider are not even lattices. Example 1 (continued) : here
F = In, a, b, d, r, p, m, g], where
p(Q) = p(a) = p(b) = 0 , p(d) = o(r) = 1 , p(p) = p(m) = 2 , p ( g ) = 4 . The F-algebra I has domain DI =IN u {l],ordered by the discrete (or flat) ordering which can b e represented by : 0
1
2
48 3
Ordered sets in algebraic semantics and the algebra operations o n
I
RI = I ,
aI
I
m (n,n') = I
=
1
if
b
and
n
if
b GIN
and
n 21
1
if
b = l
or
n=l
n
x
=
0
= I,
n,n' E M n = l or n ' = l
if
n'
I
bI
0,
n
if
z
if l # x = y # l
t
if 1 f x # y # I if
DI
are defined by :
x=lor
y=l
together with these operations is obviously a continuous F-algebra. Define the category of
0
(A-)complete F-algebras, or F-magmas by :
objects are (A-)complete F-algebras ; morphisms are the continuous homomorphisms, that is, in addition to being an F U { S } homomorphism they have to preserve lubs of directed sets too. We shall define Mm(F,V) freely generated by
V
in the category of
definition comes later. The category of
as being the algebraic system
A-complete F-algebras. A more precise
w-complete F-algebras can be defined simi-
larly.
It might be of interest to note that i t was proved in category of
/LP, P/ that the
w-complete F-algebras is reducible to a variety of partial algebras
in the sense of
/AN/.
This shows the strong relationship of algebraic semantics
with well investigated concepts of universal algebra. A
complete)
FUV-algebra
I is said to be free over generators V iff
for any (complete) FUV-algebra M, every set map from V extension into a morphism
v
into M
has a unique
: I +M.
I is also called the free (resp. free comp1ete)F-algebra generated by
V.
I. Guessarian, M. Nivol
48 4
The free and free complete F-algebras exist and can be constructed as follows : the free F-algebra
H generated by V
has as domain the set of finite,
well-formed (with respect to ranks) trees (i.e. terms) on the alphabet V +H
the set map that : (a) s2 generated by
V
An ideal of
and
<
is the identity. H is ordered by the least ordering
such
is the least element, (b) the algebra operations are monotone. The
free F-algebra generated by
E
F UV ;
V
is denoted by
M(F,V).
The free complete
F-algebra
is the ideal completion of M(F,V) cf. /B/ and is denoted by Mw(F,V).
M(F,V)
E
is a directed subset
of
M(F,V)
such that for any
d
in
is in E. Note that again the notion of ideal here i s slightly
d' < d , d'
different from the usual lattice theory notion. Recall tion of an algebra
M
/GU/
that the ideal comple-
is obtained by endowing the set of ideals of
by set-theoretical inclusion, with the obvious complete intuitively, Mm(F,V)
M , ordered
F-algebra structure. More
is described as the set of all finite and infinite trees
(terms) on F U V ; the ordering
<
extends the ordering on
M(F,V)
and can be
T <TI for any trees (terms) T,T' iff T' can be deduced from T by replacing s2b by arbitrary trees. This ordering corresponds to inclusion, namely T
ideal associated with
T'
.
2.3. Recursive program schemes A recursive program scheme, in short RPS, on
S
S : Gi(v
],...,
i
vn ) = ti
= I,
i
, G i E 4n ,
= ],...,XI
u
(2)
...,vn. 1 )
U 4 , {vl,
t i E M(F
i tree in M(F
(S,t), where
...,n
i where for
i s a pair
F
n equations :
is a system of
and
is a
t
4 , V).
It is associated with a schematic tree rewriting system (or context free
/BO, EN, GU, R / ) , defined by
tree grammar
i
= I , . ..,n
:
Gi(v,
,..., vn. )
, and which is also denoted by S. Let L(S,t)
=
+
ti + s2
, for
{t'/t'EM(F,V),
t
5
t')
S
be the tree language generated by It can be shown 1)
L(S,t)
2)
lub(L(S,t))
/GU/
S
with axiom
that
is a directed subset of
<)
,...,vP) = t E M(F, Gi(v,, ...,vn ) = t. + s2 Go(vI
T(S,t)
=
M(F,V)
is the least solution (with respect to inclusion, o r
equivalently the ordering
Let
t.
i lub(L(S,t)).
on
F(F,V)
,...,vP 1 ) for i = I , ...,n Iv,
of the system :
Ordered sets in algebraic semantics
485
Example 1 (end) : T(S,G(n,K))
here is the infinite tree defined as the 1.u.b.
of the chain of finite trees :
to
< t l < .. .< t < t < ... 9 q+l
= R
t
t1 =
a
Ag\ /”\ k
b
R
n/p\r
I
k For any
each
t
4
q
in N t h e tree t
q
is defined inductively by :
represents the formal computation of S
times the recursive definition of
tp
=
obtained by “unwinding”
S. For instance :
q
I. Guessarian, M. Nivat
48 6
The least upper bound
T ( S , G(n,k))
of the
t's can be pictured as the infinite q
tree
where
. ..
represents the
infinite branch of the tree. ?
m. I
,
When interpreted by I, t
k
,
I
q
naturally yields the function
.
t; previously defined; the
1.u.b. T(S,G(n,k))
of the
t's is then interpreted as 4
the 1.u.b. TI(S,G(n,k))
-
2 . 4 Interpretations
pretation is an interpretation of
; equivalently it is a pair
in P(F,V)
together with a valuation
induces, on any complete F-algebra
v m : Mm(F,V) + I
under the unique morphism T
of the tq' I S .
(1,~)
v : V +DI.
Mm(F,V), we can deduce the following :
TI : Iv +. I , such that for each valuation v : V Also, since each
nk
is a complete F-algebra ; a valuated inter-
F
F U V
consisting of an interpretation I From the freeness of
T
I+
Herbrand interpretations
An interpretation I of
any tree
: (n,k)
in P(F,V)
+
I , TI(\)) v
extending
I
,a
function
is the image of
is the least upper bound of the
0
in M(F,V!
which are below it, and these form a directed set, then it is clear that least upper bound of the
BI's.
We can thus infer : Theorem 1 :
(i) (ii)
for any
TI
T, T'
in Mm(F,V)
for all I I , TI
=
lub
iff
{e, I
T 0
< T'
< T 1
;
Let now (S,t) and (S',t') be program schemes, applying theorem 1 with T
=
T(S,t)
and
TI
=
T(S',t')
leads to:
T
. TI
is the
187
Ordered sets in algebraic semantics Corollary : Let
(i)
<.
for all
be two RPS6 :
(Sit')
T(S,t)I G I T(S',t')I T(S,t)
(ii)
(S,t) and
for all
I
iff
T(S',t')
I, T(S,t)I
lub
=
{@,/El
(ii) expresses the fact that the function computed by
(S,t)
w.r.t. I
i s defined as a lub of finite computations, by successive approximations ; and (i)
says that the infinite tree
T(S,t)
characterizes the behaviour of
all interpretations, which was goal # 1 Since the free complete F-algebra #'2
(S,t) w.r.t.
stated at the beginning of this section.
Mm(F,V)
has been explicitly constructed, goal
also has been achieved. As for goal #3,
let u s lay down the following refi-
nement of the notion of a free algebra :
Definition I : An interpretation H iff : for any
T,T'
TI Q I Ti
is said to be a Herbrand interpretation
in M~(F,v) : for all
I iff TH G ~ T ~ .
Then Theorem 1 expresses the fact that M"(F,V), identity valuation u(v)
=
v,
for all v
in V
,
together with the
is a Herbrand interpretation.
This fact can be applied to obtain proofs of programs properties. The Herbrand interpretation gives necessary and sufficient conditions for program equivalences to hold, thus leading to valid and complete proof systems (see a 2 . 5 ) . On the other hand, the free interpretation leads only to sufficient conditions for
such equivalences to hold. Here, Herbrand and free interpretations coincide, but this is not always true. I n section 4 . 2 , we shall see an example of a Herbrand interpretation H
which leads to a complete equational proof system for the corresponding
class of interpretations, even though that class is not a variety. Of course H does not belong to the class i t characterizes. From these remarks the reader might infer that Herbrand interpretations are the only ones of interest and might wonder why at all introduce free interpretations ; the reason is that free interpretations are usually much easier to construct, and might still suffice for many purposes. Note finally that Herbrand interpretations, like free interpretations, are not necessarily unique (cf. Example 3 below).
Notations due to typographic restrictions, the sign F the sign t-
represents semantical validity represents logical validity.
I, Guessarian, M . Nivat
488
2.5. Application to proofs of program equivalences
By characterizing the semantic behaviour of a program scheme by a syntactic object, its associated infinite tree, Theorem 1 provides us with the corner-stone of algebraic semantics. Let us illustrate this by two very simple, though interesting, applications.
7
Let
1
denote the class of all interpretations ; then, even though
is clearly not first-order, theorem 1 can be viewed as a Rirkhoff like completeness formalism of logics, TI 5 T i
I
in all models Ax
I , i.e.
for all
1F
is denoted by
Note first that, in the
T
T S T '
is semantically valid
. We
then can introduce a set T
of axioms and deduction rules for inequational logic such that
Ax b- T
< T' .
I F T
Hence theorem I can be restated as
Q
iff
.
iff Ax k T Q T'
T'
of axioms is defined by the axiom scheme :
The set Ax (3)
,
1
in
.
I
theorem for deriving all valid (in)quations in
t-nst
t
for any
in M"(F,v)
Consider then the following set of deduction rules : (4)
t
(5)
< t' ti < t;
(6)
for any
t < t
and
t
i
for
for any (7)
t'
Vi E N
< t" t = I
in M(F,V)
t
t S t"
,...,r
ti, ti
in M(F,V),
3j E W
/ t i S t' j
(reflexivity)
for any
t,t',t"
in M(F,V)
i =I
1-
,..., r,
lub(ti)
and
in F (monotonicity)
f
< lub(t!),J
(transitivity)
,..., ti)
f(tl ,..., tr)
/-
for any
ti,t;
in M(F,V)
(algebraicity and continuity). For
t
and
deducible from axiom iff
I-
t
t'
finite trees in M(F,V),
!-
let
using deduction rules ( 4 ) - ( 6 )
Ax
t
< t'
iff
; then clearly :
t Q t'
is
t
< t'. For T
and
T'
(possibly infinite) trees in Mm(F,V)
, it can be shown
that it is necessary and sufficient to add the induction rule (7) ; let 1 " T S T ' . Theorem 1 can thus be translated into two completeness theorems for finite and infinite trees :
, I F t < t' for T,T' in M~(F,v), T F T S T ' for
t,t'
in M(F,V)
< t'
iff
AX
C
t
iff
AX
I-"
T <TI
.
This makes clear how algebraic semantics can be applied to yield results in more model theoretic or logic minded approaches as in
/BL ,
BT, G /
.
This con-
nection between algebraic semantics and logics will be further investigated in the subsequent sections. Remark finally that, since
1F T
=
T'
iff
1 F T G T'
and
1 F T'
the above also provides us with a complete proof system for equational logic ; however this proof system does not immediately translate into a proof system using the deduction rules of equational logic involved in
/GU2/
.
/B,BT/
. See a
sample of the difficulties
48 9
Ordered sets in algebraic semantics
2.6 Some different standpoints Notice that, since a program scheme (S,t) is characterized by the associated infinite tree
T(S,t),
we may w.1.g. study infinite trees instead of
program schemes, although not every infinite tree is associated with a program scheme. However, the facts that : 1)
there are much more infinite trees than trees associated with program schemes
2 ) i t might then be tuo exacting t o require that all directed sets,
even those which are not associated with any program scheme, have lubs (in order to ensure completeness) led some authors t o introduce weaker notions o f interpretations. To this end, they define algebras which, though they are incomplete in the above sense, contain enough lubs to express the functions computed by the programs one is interested in. Let us briefly outline some of these approaches.
We need first recall some terminology from denotational semantics. Note first that if A
is any
F-algebra (not necessarily ordered or complete), we can
define as above a derived operation
tA(xl, ...,xn) = Wx
(t) for any t in ~ . * * ~ n Now, let A be an F-algebra
in M"(F,V)).
M(F,V)
(but not necessarily any
and
be a recursive program scheme defined by a system of equations ( 2 ) , let n i' -+ D ~ ) X... X(D~' -+D*) be the set 0 : n-tuples of mappings D~ - + D ~, for
u
=
S
nl (D*
t
-+ g
. .,n . Any
in D defines an F U @-algebra A(:) by : = (gl,.. .,gn) in F and G =g.forG. in 0 Hence, we can associate to 1 1 iA(3 each recursive program scheme S a mapping SA : D -+D defined by i = I,.
fA);(
=
fA
for
Now, an proper) iteration
.
f
F-algebra S,
An ordered
SA
A
is said to be iterative iff, for any ideal (or
has a unique fixpoint in
F-algebra
A
-
lub IS:
(aA,.. .,QA) /
GI, NE, T/.
is said to be
- regular iff, for any iteration is defined by
D /E,
S,
SA
has a least fixpoint which
n E N } / G , GPP, PP, T / .
]-rational iff, for any recursive program scheme
fixpoint which is defined by
lub { S i ( Q A , .
. . ,QA)
/ n €IN) /G/
S,
SA
has a least
.
By allowing for higher type schemes, Gallier also defines n-rational algebras which we won't consider here in order t o keep the notations simple. A subset of
DA
of the form
{Si ( G A ,
...,fiA)
/ n €IN}
for some itera-
tion (resp. RPS) S , is called a regular (resp. an algebraic) subset of
A
.
Regular
subsets are also called iterations. The following then holds
:/BG2, G, NE/
Theorem 2 : Any complete F-algebra (and any interpretation) is regular and rational. M"(F,V)
is iterative.
I. Guessarian, M. Nivat
490
Still different approaches consist in embedding metric spaces into cpo's/WE/, or endowing the set of infinite trees with a metric /ARNI, COM/ : one then studies computations of program schemes through their approximations by Cauchy sequences in metric spaces. 3 . CLASSES OF INTERPRETATIONS
We showed in the previous section that the characterization of a program in a
scheme (S,t) by an infinite tree, which is the function computed by (S,t)
Herbrand interpretation, can be rewarding. However, this approach is usually far too general : in practice, one never considers all interpretations I , but rather
I , where, e.g. some operation is associative and commutative, or
subclasses of
"if... then . . .*.
some base function symbol is always interpreted as an
.
.'I
Hence, if one wants to come any closer to real programs, one has to take into account some constraints on interpretations. For instance, equivalence with respect to all interpretations is far too exacting as is illustrated by the following : Example 2 :
Let
S : G(v)
(S,G(v))
be program schemes defined by
S'
f(v, f(a,v))
=
: G'(v)
S'
and
S
= f(f(v,a),v)
(S' , G'(v))
and
are not equivalent with respect to all interpretations
since T(S,G(V))
=
f (v, f (a,v))
T(S' ,C' (v))
=
f (f (v,a) ,v)
are incomparable
However, let us restrict our attention to interpretations where
in M"(F,V).
is associative, and let then
c
and
= {I/ fI(x,
fI(y,z))
For any
I
c ,
(S,
denoted by
(S',
G(v))
Kc
G(v))
and
, or
G'(v))
=
in DI}
TI(S',
G'(v))
(S',
T(S,G(v))
fI
I defined by
for any x.y,z
then : TI(S, G(v))
expressed by saying that (S,
be the subclass of
fI(fI(x,y),z)
=
in
c
G'(v)) are
c'
. , which
will be
C-equivalent, and
T'(S',
G'(v))
.
Formally, let us define : Definition 2 : A
class c
of interpretations is a subclass of
class of interpretations and
T,T'
in Mm(F,V)
iff for any
I
C
TI
%
T GCT'
iff for any
I in C
TI
=
C C < C' C << C ' C C'
and
C
T s+ T' For
M
C'
C'
in
, define
I
:
Ti
Ti.
two classes of interpretations, define :
zc
iff
'c,
5
iff
:ct
_C
-c
iff
'c,
=
iff
E~~
=
=
-C
. For c
a
Ordered sets in algebraic semantics Note that, clearly,
c < c' * c << c'
and
49 1
c c' * c 2 c' , counter %
examples for reverse inclusions will be shown later (see theorem 8 below and / G M / ) . Note also that
c
C C'*
C
< c' .
We will now try to generalize the approach of section 2 to classes of
C of interpretations, we first have to
interpretations : namely, given a class
characterize the behaviour o f a program scheme w.r.t. all interpretations in C
by
the function i t computes in Some Herbrand interpretation. We will show now that this is much less straightforward than in the previous completely free case : we have to accept a trade-off between a better modelling o f reality versus an increased complexity of proofs and results. We need one more definition :
C be a class of interpretations and 1 an interpretation ; I is said to be C-Herbrand iff C 2r {I} . A valuated interpretation (1,~) is Definition 3 : Let said to be
v' : V
+
v(v(v)) with
C-free (over generators V)
iff for any
DI', there exists a unique morphism 'P : I
=
w')
Remark 1
v'(v)
for any
--*
I' in C and valuation I' such that
in V , (i.e. the restriction of
v
~p
to
w(V)
coincides
. a) We are implicity considering classes of non-valuated interpretations (or functional interpretations). Valuated interpretations can be treated similarly /GU/. b) Note that our definitions of
C-free and
C-Herbrand are slightly
different from the classical notions of universal or free object in universal algebra or category theory /AN, C, GR, MA/ : we do not require that the class
C
. As
belongs to
C-free or
C-Herbrand interpretation belong to the
a result, it may indeed be the case that none of them
C
; for instance, Mm(F,V)
C(§
3 . 2 and 4 . 2 ) .
C-free interpretation belongs to C
C-free for any
C
.
We
C-Herbrand interpretations I
will see later various examples of which are not in
is
Note however that, whenever a
,
then (i) it is also
C-Herbrand,
(ii) it is the unique C-free interpretation belonging to C (up to isomorphism). c) We can also ensure uniqueness of the free interpretation by postulating the following stronger definition /AN/
:
Definition 3.1 Let c be the preordering defined on valuated interpretations by : ( 1 , ~ )_C (I',v') iff there exists a unique morphism ~p : DI +DI, such that : for any v
in V
iff (i) it is i.e. (Io,wo)
q(w(v))
=
w'(v)
C-free, (ii) for any is the largest
. Then
(Io,wo)
is said to be strongly
C-free
C-free interpretation (1,~) , ( 1 , ~ )_C (Io,wo)
C-free interpretation w.r.t. the preordering
5 .
d) Note finally that the notion of Herbrand interpretation can be extended for arbitrary atomic formulae of the form P(t arbitrary finite trees and
P
] , . . . , tn)
is a predicate symbol :
, where the t i ' s are
I. Guessarian, M. Nivat
492
Definition 3.2 : An interpretation I is said to be strongly C-Herbrand iff for c F@ 0 any atomic formulae 0 : I F 0 +f
This concept, even though it was never given a name, has been investigated in universal algebra (see e.g. / A N / ) .
3.1 Constructing C-free and
C-Herbrand interpretations.
For completeness sake, let u s state the following /GU page 7 9 / : Proposition I : For any class C a
of interpretations, there exists a
C-free and
C-Herbrand interpretation. and
C
The five element set
F
Example : Let F a=b or
c=d
.
=
{fi,a,b,c,d}
least element n , is both
be the class of algebras such that either
,
together with the discrete ordering with
C-Herbrand ; in general though, Herbrand
C-free and
and free interpretations need not coincide. See below. However, the proof of proposition 1 is highly non-constructive : it consists in taking some suitable subclass of the infinite product of all interpre-
. Hence
tations in C
this result is of no help and we have to find alternate characC-Herbrand interpretation. Let u s give first some
terizations of the C-free and terminology about preorderings.
Definition 4 : A preordering on an ordered F-algebra M sitive relation
T
containing
structure of M , namely : for any d.
TI
1
d!
1
...,dr)
imply
fM(dl,
imply
lub{e/e in E }
TI
f
d'
T
. When
in DM, for
in Fr, di, di
,...,di) .
f (d! M i in Dw, and any
for any directed set E
is a reflexive and tran-
s and which is compatible with the M = V(F,V)
i = I,
...,r
,
is said to be continuous iff
T
in DM , e
d'
F-algebra
T
d'
for all
e
in E ,
with the syntactic ordering
is said to be
-
...,T
substitution-closed if for any
t,t'
in M(F,{vI,
,...,Tn/vn)
...,vn})
and any
t'(T1/vl,...,T /v ) n n Equivalently, one may require that for any endomorphism h : Mm(F,V), t T t'
TI,
implies h(t)
t
< T,
t
,
in V(F,V)
T
T
t'
implies
t(Tl/vl
T
.
.
h(t')
algebraic
in M(F,V),
t
iff for any
T,T'
there exists some
in M"(F,V), t' < T I ,
T n T'
implies : for any
in M(F,V),
t'
such that
tnt'. For a preordering
T
on
Mm(F,V)
let
CT
=
11
/
T
C<{Il}
.
We can
now state the : Proposition 2 : /GU/ : C
-+%
and
'II
+Cs
define a Galois correspondence between
classes of interpretations and continuous, substitution-closed preorderings on p(F,V). One might then expect to use the above Galois correspondence in order to get somehow more tractable characterizations of the
c
-free and Herbrand interpre-
tations. This goal can be only partially fulfilled, as will be shown by the sequel.
49 3
Ordered sets in algebraic semantics For any ordered
M , and preordering
F-algebra
- Mm denote the ideal completion of
n
M , let :
on
M /B,CR,GU page 3 3 /
- M/n
denote the F-algebra obtained by factoring M through the -I equivalence IT = IT fl IT associated with n , and ordering the factor algebra by a / n . The equivalence class of an element t of M modulo
n
will be denoted by
Let
'II
be a continuous and substitution closed preordering on Mm(F,V).
c
We now try to find choice would be
.
[ tIn
I
I
c
-Herbrand interpretations. The most natural
.
fl(F,V)/n
=
However,
-free and/or
c
is usually neither
mediate reason for that i s that
C -Herbrand. The most im-
-free nor
I , being usually not complete, is not large
enough and does not contain enough lub;. However, there are more subtle causes for which
I
c
cannot be
I
-free or Herbrand : even when
is complete, lub; might
be there just by chance and might not be the right lub;. Hence : the operations on
I)
I may be non continuous
I are continuous, I may n o t be
2) even when the operations on
c
p :
-free because the unique mapping
may be not continuous, hence p
I +I'
into an
I'
cn
in
will not be a morphism.
This will be made clearer by the following example : Example 3 : We will show here continuous, substitution-closed and algebraic preorderings
such that
IT
Let Define b n h(b),
n
F =
Mm(F,V)/n
is neither
{n, a, b, c, h} r(n)
bn
c,
c IT h(c),
/ n E N}
f o r any n in N
h(c) =
hI
TI
c. Then
and
I
=
=
r(b) :
r(c)
=
k'n
Mm(F)/n
lub {hn(b)I
c
-Herbrand nor
as being the preordering generated by
lub {hn(a)I
-free.
= 0
r(h)
= 1
.
N , hn(a) nb, a n h(a),
€
is complete:
/ n E N}
=
cI , and
hr(c),c
.
However hI is not continuous since
> l~b{h~+'(a)~] f
c
= r(a)
=
[b],
=
bI
.
hI(bI)
=
[h(b)],
hI(lub{hn(a) n
=
Hence I is not even an interpretation.
n C
Figure 1 might help the reader's intuition. h2(d)
/ h(b)
/
I1 )
494
I. Guessarian, M. Niuat c
/
m m
1
Figurc 1 Now, even when the operations are continuous, I may still be not
c
-free. Let
TI' be the (continuous, algebraic, substitution-closed) preordering generated by II
.
together with the relation h(b) rr'b
I'
Then
=
the hn(a)I,'s
by mere chance, and this results in
v
instance, the unique
c
I' is
,-Herbrand and
:
I'
+
I"
I"
=
(M(F)/n)=
is both
Mm(F)/n'
is clearly a
bI, happens to be the lub of
complete F-magma, with continuous operations. But
I'
being not
CTI,-free.For
is clearly not continuous. However,
C,,-Herbrand and
c
,-free : the freeness
results from proposition 3 below, the rest is left to the reader. This example shows that neither M"(F,V)/n
c -Herbrand
C -free or
nor
(Mm(F,V)/~)m
in general, even for a very smooth choice of
can be
. We
IT
ne-
vertheless can state the : Proposition 3 : Let
IT
be a preordering on
Mm(F,V)
; then
(M(F,V)/IT)~ i s
C -free. 0 Example 4 : Note that, except when is usually not Let
CTI-Herbrand.Let
T . = lub {h:
clearly M(F) / TI
(il)), = M(F)
i =
for and
TI
is algebraic (cf. Theorem 3) (M(F,V)/TI)~
I n , h 1 ' h 21 with r(h,) = r(h2) = 1 . 1,2 , and IT be defined by : T I TI T2 . Then, F
=
is n o t
(M(F)/T)~ = F(F)
c
0
-Herbrand.
The previous example amply illustrate that the standard completion by ideals method cannot give us a
C-Herbrand interpretation. However, for any
continuous and substitution-closed preorder and
c
IT
, one can construct a c -free
-Herbrand interpretation by a more refined completion method : one has to
perform a "continuous" completion preserving lub; of algebraic subsets o f M"(F,V)/r
; by transfinite induction, one than gets the required
interpretation /CR,
C -Herbrand
G U 3 , MI/ thus yielding the :
Proposition 4 : For any continuous and substitution-closed preordering Mm(F,V)
we can construct a
c
-free and
c
IT
on
-Herbrand interpretation.
C -free and CT-Herbrand interpretation still exists by proposition 1, but cannotbe constructed explicitly.
i I n case TI does not satisfy these hypotheses, the
495
Ordered sets in algebraic semantics __ Proof : Let
Let
A
be a continuous and substitution-closed preordering on
II
, ordered by
denote the algebra Mm(F,V)/n
D
it exists, belongs to
.
For any subset D
namely the least closed ideal containing D
of
.
.
T/*
of
to be closed iff for any directed subsets D '
D
An ideal of
the 1.u.b. of
-
M
let
c(E)
The set A
.
Mm(F,V)
is said
A
D', whenever
denote its closure,
of closed ideals of
A
,
ordered by set inclusion, is a complete lattice. It has a complete F-algebra structure defined by :
f(D
,,...,Dn)
=
c({f(t,
,...,tn)
1
=
c({t})
=
{t'/t'
and
t' E A }
=
/ t E
{i(t)
i
=
I ,..., nj)
:
is a continuous and injective morphism.
-
Let u s define the continuous completion Ac
.A
ti E
is a complete F-algebra. Now, i : A + ' A
Henceforth, A i(t)
- / definedDi byfor
of
inductively by :
A
A) C A
a :
and for any ordinal
"
A a + I = Aa
{lubi E / E
is an ideal of
Aa} C A
I
is a limit ordinal. Since all A s ned in A , there exists an ordinal y
such that Ay
are contai-
= :Aa
Av ordered by inclusion and equipped with the restrictions of the - 1
f , for
operations
checks that A
c
-Herbrand. A
Mm(F,V).
Y Y
f
in F , is a complete subalgebra of
A
.
By induction, one
has the required universal property in order to be is thus the continuous completion of
c
-free and
and is also denoted by
A
For more details see / C R , GU3, MI/. This construction by transfinite induction is however only very slightly
more effective than the bare existential result of proposition 1 . I n particular, this construction can lead neither to a nice characterization of the function computed by an RPS w.r.t. a class C , nor to the faintest hope of getting a complete proof system for deducing valid inequations zation of
%.
t
5
t' , nor even to some characteri-
Hence, in order to get more manageable results, we will have to
consider somehow more specific classes of interpretations. Which, as we will see, can nevertheless cover most situations of interest in computer science. This is what we shall do extensively in the next sections. However, in order to get the flavour of oncoming results, let u s define here some classes of interpretations and state some of their properties. 3 . 2 . Algebraic and topological classes of interpretations A
class
C
is said to be algebraic iff
5
-
Applications
is an algebraic preordering.
The intuition behind the notion of algebraic class is the following : the behaviour of any infinite tree with respect to an algebraic class
c
can be characterized by
I. Guessarian, M . Nivat
49 6
c
the behaviour of its finite subtrees w.r.t. ; equivalently, all the computations of a program scheme are completely described by its finite computations : this makes sense since finite computations are the only ones which can be at hand. For any
Cn
=
F
dp
5
...
of elements of
...
of elements of
uI
c, s
0, the class of discrete interpretations. Clearly :
=
s
c2
Moreover, one has
c3
5
...
$ C m $ C w $ '
/CG,GU/
Proposition 5 : (i) (ii)
F
Cm
Q
Cw
%
I.
%
D < C3 < C4 < #
#
... < Cn < #
#
Cn+I <
... <
+
be defined inductively by
. Then
tn+l
%n
for all
and not
t
Corollary I :
, where n €IN
tn+, = f(v,tn)
< t,+,,
n tn
(tn+l
The classes
F
#
the proof of (ii) stems easily from the following : let
in F
is finite)
C, has a single element, the trivial interpretation reduced
c,
QE, and that
Proof :
and
f
to = 0 and
Corollary 2 :
F, Cm
C
Cw
F, Cm
or
and
.
is a rank 2 symbol
but a l s o :
.
Gcn+Itn)
Cn, for
For the proof of (i) see /GU page
in E4 , form a strict hierarchy for the
n
Col are algebraic and M"(F,V)
F
is
C-Herbrand for
for instance is a class of interpretations such that
F , as soon as F contains at least
no F-Herbrand interpretation can belong to
one base function symbol of rank greater than 1 . For, M (F,V) being the chain
c : Q
$
h(Q)
$
h2(Q)
...
$
F-Herbrand interpretation can be finite.
I is said to be algebraic iff its domain
Recall that an interpretation
is an algebraic complete partial order B
=
. Clearly, if
itI / t E M(F)}
F-Herbrand,
is infinite. c must thus be infinite in
F-Herbrand interpretation ; hence no
set
loo/.
0
This shows that
DI
tn
<.
ordering
any
has at
DI
i s finitel
Note that
=
5 .. . 5
d2
distinct elements} {I / any chain d l S 1 ... < d G I I P
= {I / D I
to
5
dl
n
most
cw =
in Dl, define :
n
11 / any chain
/B,GU,S/
with compact elements the
I is algebraic, the class
is alge-
111
braic, but the converse is false. For any preordering
IT''
t
IT
; and for any preordering
M(F,V) on
< T,
Mm(F,V)
by : for T,T'
there exists a
t'
on P'
Mm(F,V), on
in Mm(F,V),
in M(F,V),
t'
IT^
define its restriction
M(F,V)
T
< T' ,
to
define its algebraic closure
IT'^
T'
iff for any
such that
t IT' t'
t
.
in M(F,V),
0
49 I
Ordered sets in algebraic semantics
We now can state Theorem 3 : A class
c
/GU/ :
is algebraic iff i t satisfies the following equivalent
conditions :
(i)
(<;)a
GC
=
(ii) (iii)
<:)-
/
(M(F,V)
C-Herbrand
is
There exists an algebraic
0
C-Herbrand interpretation.
The importance of algebraic classes stems from the fact that the behaviour of infinite trees w.r.t. finite subtrees. Hence, suppose as in section 2.2, tions
t
5
a
i s characterized by the behaviour of their
GC
c
is an algebraic class and we have obtained,
complete deduction system C
holding between finite trees
t'
t,t'
for proving all valid inequain M(F,V) ; then comes
"for free" a complete deduction system for proving all valid inequations T GC T' holding between infinite trees T,T'
too : i t suffices to add to
an induction rule (7) : / G /
the rules of V
in Mm(F,V)
3 j E N
i E N for
ti,t! J
ti
< t' k j
in M(F,V)
< lub(t!)
lub(ti)
J
.
This is exactly what we did in section 2 . 2 . We shall see later on further instances of a similar situation (Proposition 7 , Theorem 6 ) . Note finally that T
sc T'
T,T'
is generally undecidable for infinite, or even only algebraic, trees
in Mm(F,V)
; hence usual deduction systems, such as equational logic
/C0/
/B,BT/, first-order logic, etc ..., will be incomplete in general and one will have to allow for some infinitary inference rule such as (7)
/G,GM/, the other alter-
natives being to consider still more restricted classes of interpretations (eg. first order classes of interpretations, see section 5 ) , or a smaller set of inequations, or both. Algebraic classes of interpretations present the advantages of being :
- general enough to include all usual
classes of interpretations
- well-behaved enough so that any inequation between infinite trees will be provable by computation induction : this is the reason why, by adding the single infinitary induction rule ( 7 ) one gets a complete proof system for deducing all valid inequations. Example : consider the schemes defines as follows /AK/ : S,
s;
:
G(n)
:I
H(n,r)
=
G'(n)
g(c(n),a,m(G(p(n)), =
d(n)))
H(n,a) =
g (c(n) ,r,H(p(n) ,m(r,d(n)))).
These schemes, when suitably interpreted, take
DI =IN
u
{o}
, with the
discrete ordering, aI = I , pI(x) = x - 1 , nI = x, gI(x,y,z) = if x = 0 then y else z , and cI equal to the identity function, give a recursive and iterative
I. Guessarian, M. Nivat
498
version of the factorial : TI(SI,G(n)) TI(S;, G'(n))
computes
C
.
(...(((I.n).(n-l)).(n-2))...1)
can prove that T(SI ,G(n)) conditions :
Let
computes (...(((l.l).2).3)...n)
:c
T ( S ; , G'(n)),
Then using our method, we
assuming only the very weak following
(i)
m(f2,r)
(ii)
m(m(x,y) ,z)
(iii)
m(g(n,x,y) ,z) = g(n,m(x,z) ,m(y,z))
=
and
f2 =
m(m(x,z) .y)
be the class of interpretations defined by the conditions (i), (ii) and
(iii). The proof uses the following lemma : L
m : let
Hq denote the q-th approximation of
T(S;, H(n,r)),
times the recursive definition of
S; , then for any
unwinding q
obtained by
Ec m(Hq(n,x),y). q in IN ; Hq(n, m(x,y)) Proof : for q = 0 the equivalence reduces to
f2
=
which holds by (i).
m(f2,y)
Suppose the equivalence holds for q = 0 , . ..,k-I, then, by definition : k H (n,m(x,y)) = g(c(n), m(x,y), Hk-l (p(n), m(m(x,y), d(n)))) = g(c(n),
m(x,y),
= g(c(n),
m(x,y),
Y)))
Hk-'(p(n), m(m(x,d(n)), k- 1 m(H (p(n), m(x,d(n))),
by (ii)
Y)) by the induc-
tive hypothesis m(g(c(n), x, Hk-'(p(n), m(x,d(n)))), k by definition. = m(H (n,x), Y) An immediate induction on q then shows that : =
G"(n)
Zc
Gq(n) (where Gq(resp. GIq)
(resp. T(S;, G'(n)))
. This is clear
integers, then by the definition af G"(n)
Y)
by (iii)
denote the q-th approximation of
0
T(SI, G(n))
for q = 0 ; suppose it holds for the q-I first GIq ,
=
Hq(n,a) = g(c(n) ,a,Hq-](p(n) ,m(a,d(n))))
=
g(c(n) ,a,m(Hq-' (p(n) ,a) ,d(n)))
=
g(c(n) ,a,m(Gq-' (p(n)) ,d(n)))
=
Gq(n)
by the above lemma by the inductive hypothesis
by the definition of
Gq
Hence, by the induction rule (7) : T(Sl, G(n))
Ec
0
T(S;. G'(n)).
3 . 3 Related approaches
Note that most o f the problems in constructing a C-Herbrand and/or
c-
free interpretation stemmed from completeness and continuity. Hence considering classes of
X-interpretations, where
X
is intended to be replaced by iterative,
regular or I-rational, one would expect to cancel some of these problems, since completeness and continuity are replaced by weaker conditions that only those lubs
499
Ordered sets in algebraic semantics
one effectively wants to compute should exist and be preserved by operations. This i s indeed partly the case.
An
X-morphism of
ving lubs of of
X-algebras is obviously defined as a mapping preser-
X-sgbsets ( o r unique fixpoints of ideal iterations). For a class
X-algebras, one can define as previously
X-C-free and
X-C-Herbrand interpre-
tations or algebras i n the present formalism (in the case when
5
simply has to replace free case, when Theorem 4 :
C
=
I
by
Zc)
.
C
X
=
iterative, one
First note the following for the completely
:
/BG2, G , GPP, GU, N I /
Let Alg(F,V)
(resp. Reg(F,V))
be the subset
m
of algebraic (resp. regular) infinite trees in M (F,V), namely trees of the form T(S,t)
for some RPS(resp. iterative scheme) (S,t)
and Herbrand I-rational algebra and
Then Alg(F,V)
is the free
is the free and Herbrand iterative
0
(and regular) algebra. Reg(F,V)
Reg(F,V)
.
is also denoted by
RTF(V)
in the A D J
terminology.
Notice that we still are going to obtain a Galois connection between classes of
X =
X-algebras and suitably defined preorderings (resp. congruences if
iterative)
on the free
X-algebra. Note now that, in the case where
X
iterative,
=
; these
we are going to run into troubles similar to the ones we had in section 3.1
troubles will no more be due to completeness but to unicity of solutions, which is at least as bad as completeness /CO,GPP/. This is illustrated by the following : Example 4 : we show here that factoring an iterative algebra, even by a very smooth congruence, one usually does not get an iterative algebra. Let with f(a,a)
r(a) 1~
= 2.
r(f)
= 0
f(f(a,a),a)
associated with
T
Then
S : z = f(z,a) t , = lub { S ;
n tion, and t2
and
T
f(f(a,a),a)
m
F
=
In, a,
be the preordering generated on Mm(F) 'TI
is generated by
(5
I = M (F)/:
f(a,a) f(a,a)
and let 5
f]
,
by
: denote the congruence
f(f(a,a),a)).
is not iterative since the ideal iteration
has the two distinct solutions
(a,)] =
Let
= lub In,,
f(a,a),
; and
f(n,a),, tl
2
f(f(C2,a),a)I,...) tp
, which is the least soluB = { w , a l , with
since the F-algebra
f (w,w) = f ( w , a ) = f ( a , w ) = w and f ( a , a ) = a i s a regular (and B B B B =o+t = a . complete) F-algebra, which is compatible with : and such that t
w Q a , aB = a ,
IB
See /GPP/ for more details. I' = Reg(F)
*B
/ : is similarly regular but not iterative. 0
In order to avoid such troubles we shall in the sequel limit X meanings "regular" o r "I-rational". The definition of an
to the
X-continuous preordering
will then clearly paraphrase definition 4 and we shall omit it. Now there clearly is Galois connection between classes C
of
X-algebras and
X-C-free interpretation by factoring the free
X-continuous, substi-
. We moreover can obtain the X-algebra A by % , namely :
tution-closed preorderings on the free X-algebra
A
5 00
I. Guessarian, M . Nivat
Proposition 6 : The X-C-free algebra H
is defined by
H
= A/GC ;
H is an
X-algebra belonging to c , hence is also X-C-Herbrand for any class
c
of
X-algebras. The same result can be achieved by categorical methods /BGI/. Note also that this result also holds for a higher-type notion of that
n-rationality
/G/, and
0-rationality corncides with regularity. Note finally that there is a trade-
off between the simplified construction of the X-C-Herbrand algebra H fact that H
may contain some meaningless lubs, e.g. bI
normal forms are also troublesome in H (cf.
and the
in example 2 where
H
=
I ;
/CO,G/).
4 . EQUATIONAL AND RELATIONAL CLASSES OF INTERPRETATIONS 4 . 1 Relational classes
Definition 5 : A class of interpretations is said to be relational iff it is of the form CR = {I/R
E
for some binary relation R
5 M(F,V)
x
M(F,V).
R
If
is
an equivalence relation, CR. is said to be equational. Relational classes are called algebraic varietal in
/M2/, varieties in
/BL/, and semi-varieties in /G/, who considers classes of rational algebras, defined by some relation R
possibly involving infinite trees. Relational classes are
those classes definable by a set of inequations between finite terms ; they are the most tractable classes of interpretations : for a relational class
%
an easy characterization of
c
restricted to finite trees, show that
, we will
c
get
is alge-
braic, hence obtain a complete deduction system within inequational logic to prove
c
all valid inequations in
(cf. sections 2.2 and 3.2).
Let u s state first a Birkhoff theorem characterizing relational classes in terms of closure operations : / W / . Theorem 5 :
c
is relational iff it is closed under products, continuous subalge-
bras and factor algebras, and ideal completions. A slightly simpler version of this theorem is proved in /G/ for semi-varieties of rational algebras and in /BL/ for varieties of ordered algebras. For a relational class preordering containing R abbreviate
<
by
-5
zR
C,
on M(F,V). and
z
by
define
as the least substitution-closed
Then, clearly, C, =R
=
c< . Finally, let us R
.
CR We can now state the main theorem of this section /GU/ : Theorem 6 : Let
R be a binary relation on M(F,V)
then :
(i)
5
is
(ii)
ZRf
= cR
(iii)
cR
CR-free and and
-R
=
CR -Herbrand
R
is an algebraic class.
0
and
HR = (M(F.V)/
,
Ordered sets in algebraic semantics
501
The importance of this theorem can be illustrated by its consequences. We simply state two of them.
c
Corollary I : A class class
cR
, i.e.
is algebraic iff i t is equivalent to some relational
C
iff
%
cR
(cf. definition 2 ) . 0
Recall that, as noted in section 2 . 2 , - t <-R
t'
is equivalent t o
cR F
- t <
t'
amounts t o s;.ying
that
5
t
t
t'
5 t'
can be deduced from
(R U <)
using the deduction rules ( 4 ) - ( 6 ) of inequational logic, completed by induction rule (7) and rule (8) expressing substitution closure : (8)
I-
t
5
v
: M ~ ( F , v )- + M ~ ( F , v ) .
t'
v(t)
5 v(t')
Hence the statement
zR =
for any
in M(F,V)
and endomorphism
(ii) of the above theorem can be res-
in
<:
t,t'
tated as the following completeness theorem : Corollary 2 (completeness theorem) : Let
1-
t 2 t'
for any
for any
T
t,t'
in M ~ ( F , v ) Then, for any
in M ( F , V )
be the axiom system defined by : AxR such that (t,t') is in R , and I - R 5 T
. t,t'
in Mm(F,V)
,
cR F
t
2 t'
iff AxR
1:
t
2 t'
using the deduction rules ( 4 ) - ( 8 ) . 0 One can obtain similar results when considering varieties (or relational classes) of regular /T/, rational /G/ or recursive /BGI/ algebras. Note however that i n the above cases, even though proposition 6 leads t o a similar complete deduction system, the inference rule for infinite trees i s obviously more complicated than a mere computation induction rule and must explicitly involve infinite lubs
/G/.
In the case of varieties of iterative algebras though, the existence and construction of a free or Herbrand interpretation for a variety of iterative algebras is more problematic /CO,GPP,PP/, because unicity of solutions is not preserved through factoring. Theorem 6 has many other applications which we merely list here for lack of space :
-
proofs of program properties (with or without induction) and program equivalences /G,GU,BK/.
-
program simplifications and transformations w.r.t. classes of interpretations and correctness proofs of such transformations /CO,EG,GU,K/
4 . 2 . Application to "if
...then...else
...I'
We will however detail some more the study of classes of interpretations
where a given base function symbol is interpreted as a test i.e. an "if...then... else..." . This will give some clues about how relational and equational classes can help in stydying non relational ones. Moreover, the tests (or conditionals,
I. Guessarian, M . Nivat
502
case or select functions,...) no matter what name they are given, arewellknownto be fundamental in computer science. Our approach replies on fundaments laid down
/MAC/, and pursued, among others, by
by
/BT,SE/.
B
We will start by recalling a result from /BT/. Let where
r(tt) = r(ff)
and
= 0
r(g)
=
3
.
and
tt
= {Q,
tt, ff, g }
ff are intended to be interpre-
ted as the constants "true" and "false" in any (not necessarily ordered) B-algebra I ; we will thus say that
(9)
for any
x,y.z
gI
is a
in DI
test
iff i t satisfies :
gI(x,y,z)
i
=
y
if
x = tt
z
if
x
nI
K
Let
K ,
=
=
I
ff
I otherwise
is a B-algebra satisfying (9)). It is well-known that
{I/I
being not closed under products, is neither equational nor relational. However,
the following completeness theorem has been shown in /BT/ : Theorem 7 : For any
-
A
t,t'
,K F
in M ( F , V )
t
5
t'
iff A 1 E- t
f
t'
, where
is the following set of equational axioms, in which g(x,y,z)
been abbreviated into
has
[x,y,z] :
A
-
A
I E-
means : t
t E t'
5
t'
is deducible from A
rules of equational logic /B,BT/, or, equivalently, t
5
tible and substitution closed congruence containing A
, which
= -A
*
using the deduction
is in the least compawe also denote by
o Intuitevely, this means that H
where A
K
t'
=
{I/I
= M(B.V)/EA
B-algebra satisfying A )
is a
K
5A
K K-Herbrand algebra which does not belong to is not equational ; hence
and
e=
is both
. Note that
H
"K
and
A-Herbrand",
is not in K
since
A , and this is an example of a
K.
Note that theorem 7 was obtained assuming two restrictive hypotheses, namely that one deals with unordered algebras, and that the unique operation allowed in the signature is the test F-algebras whose signature F
g
. We
can now extend this result to complete
contains base function symbols F'
provided those new symbols in F'
other than g
represent strict operations : this is a not too
stringent restriction since, in practice (see for instance abstract data types), the only base function symbol which is assumed to be non strict, and needs to be such, is the test
g
Let now F
. =
B
,
U F', for some set
F'
of base function symbols, and
Ordered sets in algebraic semantics
let
C
503
be the following set of axioms : for any
I
C
f
in
F t n ,n
,..., x i-1' n.
>
, and
1
i
1
,...,n
)
~[U,f(Xl,...,xi_l,v,xi+l,..., Xn) , f (XI,.* . f(xl
Let
A'
= A U
A' =
{I/I
C
:
xi+ ,,...,x ) E n
,,...,Xi-], [U,V,W1,Xi+] , . . . . x
f(x
=
'Xi-]'w'xi+l I *
and define :
is an
F-algebra satisfying
{I/I F A ' }
A') =
A:
=
A' f l 7
=
{I/I
is a complete F-algebra satisfying
A
=
A' n D
=
{I/I
is a discrete interpretation of
d
o
*.
A'}
F
satisfying
A']
.
And define similarly KS = K
n
{I/I
Ks = K
n
A'
is an F-algebra where, for any
f
FA , n
in
>
1
' fI
is strict) (recall an operation
fI
K:
DI whenever one of its arguments is QI)
=
is strict iff i t yields the result KS n 1
Ki
and
Then, extending in an obvious way the relations
ZC
Kz nD
=
and
RS
.
of definition
2 to classes of algebras which are not necessarily complete, one obtains the following
/GM/ :
A'
= A,'
(ii)
K:
Q
(iii)
K: Z A : ,
Theorem 8 : (i)
%
A:
KS% K:
, A;
K:
; i
K:
A:,
Q
, K' d #< A 'd '
0
This theorem has several consequences. It first yields a nice characterization of the equivalences valences coincide with generated by neral,
E
,
EC
for C
{Ks,
in
Ki, Ki,
A;
A;]
: by (i) those equi-
, which is by Birkoff's theorem, the congruence
ZA
.
E
A '
Note that, here, = ,, also coincides with E A l , whereas in ge-I -A0 -
CR with finite trees only).
3 ER
Hence we can state : s s A ' , Ks , KO , Kd}, zC
C in {A,',
Corollary I : For
closed congruence generated by Corollary 2 : K:
is not
A'
.
D-equational
This stems from :
is the compatible substitution-
d
K:
the only p o s s i b l e congruence is
C
A:
EA,
/GU/.
but
l(K:
, which
A:)
: by the first equivalence,
is excluded by the second inequiva-
lence. U Similar results hold slightly modifying lity test ; let
B'
=
{Q, 9')
satisfying (10)) , where :
, r(g')
=
4 , and let
B
in order to obtain an equa-
K'
=
{I/I is a
B'-algebra
I. Guessarian, M . Nivat
504
(10)
for any
x,y,u,v
in DI
nI PphY,U,V)
=
v
RI
x
if
x = y t n I
if
~ ~ + x + y + a ~
Then there exists an axiom system
=
or
y
=
/BT/ such that for any
.
M(B~,v) : K ' F t : t' iff 1 -E t = t' Letting ; i l = u C , where C = C U {(a),(b)l, with, for f in FA , nzl , and i=l,
A
RI
if
g(u,v,f(xl ,...,xi-l'u'xi+l,"' .Xn),Y)
(b)
g(u,v,g(f(xl
in
A
...,n
(a)
t,t'
:
:
g(u,v,f(x I , . . . ,x i ~ l , ~ , x i + l , . . . , x n ~ , Y ~
,. * .
'xi-I'u'xi+l,
-
f
,Xn),Y,Z,ZI),W)
g(u,v,g(f (x,, . * * ,x,i-l 'v'xi+l,*
* *
f
,xn) ,y,z,z').w)
The proofs and results of the previous case easily go through (see /GM/ for more details). 4 . 3 . Related approaches
As in section 3.3, we can define X-relational classes. I n the case when
X
= regular or I-rational, we will similarly obtain X-C -free and Herbrand interR pretations by factoring the corresponding completely free X-algebra by <;? ; when
X
=
iterative though, example 4 shows that no good solution can be expected. An alternate approach is to use category theory in order to build, without
any reference to either an ordering or a metric, free algebras in which all recursive program schemes have solutions, and which consist exactly of sets of solutions to recursive program schemes /BGI/
: this can be done via pushouts and coequali-
zers. Then, the factoring goes through very smoothly and guarantees the existence of a
C-free and
C-Herbrand algebra, where all RPSs have solutions and consisting
exclusively of solutions to RPSs, for any class
c
of algebras presented by gene-
rators and relations. This approach can also lead to an algebraic semantics of non determinism BO, MI/
/BG2/ and hopefully concurrency different from the ones in
/ARN,
and somehow closer to the one in /W/.
5 . FIRST-ORDER CLASSES OF INTERPRETATIONS
Because of the conditions that any algebraic subset (or iteration) have a lub (or a unique fixpoint), none of the classes of interpretations (or X-algebras) introduced
so
far are first-order definable ; hence, first-order logic will
be incomplete w.r.t. those classes. We introduce in the present section the last well-behaved classes of interpretations, nemely first-order definable classes. Definition 6 : Let A
be a set of closed first-order formulae
/BS/ and
C a
5 05
Ordered sets in algebraic semantics class of interpretations ; definr
. Mod ( A ) the set of . C(A) = c fMod l (A) and belong to
A
A class
A
some set
C'
models of
A
thr class of interpretations which are models of
c .
is said to be a first-order class iff
C'
=
Mod (A)
for
0
of closed first-order formulae.
The main result of this section is the following characterization theo-
rem
/GU/
Theorem 9 : in N
c
a class
and some set
A
is first-order iff i t is of the form
First-order classes are algebraic.
Corollary 2 :
For any
Cn
and
:
in N
n
Cn(A)
and set
A
Cn(A)
for some
n
0
of closed first-order formulae.
Corollary I :
classes
Cn)
(cf. section 3.2 for the definition of
o f closed first-order formulae, the
are algebraic. 2
this result was obtained in /GUI, CG/ together f = D) on with a decision procedure for deciding zu (recall that for n = 2 Note that, for
n
=
c
finite trees. Corollary 3 :
Any finite intersection of first-order and relational classes is
first-order and algebraic. Corollary 4 :
The classes
ucond(of discrete interpretations where a given base
function symbol is interpreted as a test, see
/GU/), K:
, KAs
(of section 4 . 2 )
are first-order and algebraic. Together with the fact that algebraic classes are closed under finite unions and products, Corollary 3 enables one to show that the classes one is interested in are algebraic. Note that this corollary is hard to improve since we can construct an algebraic class of discrete interpretations such that
c
cR
fl
is not algebraic
c
and a relation R
/GU/.
Note finally, that, first-order classes being algebraic, the following completeness result is clear : Proposition 7 : Let
c
=
Mod (A)
be a first-order class of interpretations, (i)
first-order logic is a complete system for deducing all valid inequations with
t,t'
in M(F,V),
from the set of axioms A
.
t
zC t',
(ii) first-order logic together with the induction rule (6) is a complete system for deducing all valid inequations
t <
-C
t'
from the axioms A
.
0
Acknowledgments : It is a pleasure to thank H. Andreka, I. Nemeti and I. Sain for helpful comments on a first draft of this paper. Thanks a l s o to S. Bloom, E. Nelson and A. Na'it Abdallah for valuable comments. Last but not least, thanks to the anonymous referee whose insightful comnents greatly helped in making this paper more readable.
I. Guessarian, M . Nivat
506
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509
TWO OPEN PROBLEMS IN PRECEDENCE CONSTRAINED SCHEDULING J.K. Lenstra Centre for Mathematics and Computer Science P. 0. Box 4079, 1009 AB Amsterdam
A.H.G. R~MOOY Kan Erasmus University
P.O.Box 1738, 3000 DR Rotterdam
The computational complexity of a machine scheduling problem can be affected in various ways if a partial order is imposed on the set of jobs that has to be executed. Some typical complexity results for such problems are discussed in the light of two prominent open problems in this area: the minimization of total tardiness for unit-time jobs on a single machine subject to chainlike precedence constraints, and the minimization of maximum completion time for unit-time jobs on three identical parallel machines subject to arbitrary precedence constraints.
RESUME La thbrie de l'ordonnancement traite de l'affectation temporelle de ressources limitks h des activitds. Dans ce contexte, des ordres partiels apparaissent de fawn naturelle: un ordre partiel sur l'ensemble des activites impose des contraintes sur l'ordre dans lequel les activitks doivent Ctre exkutkes et il rkduit ainsi l'ensemble des affectations admissibles. L'interCt du problkme consiste h incorporer ces contraintes de prdcddence de manibre aussi efficace que possible dans des algorithmes destinks a dkterminer une affectation admissible qui soit optimale par rapport a un certain critere. L'effet des contraintes de prWence peut Ctre double. Si le problkme sans contraintes peut Ctre rksolu par une methode efficace, alors leur adjonction conduira h modifier l'algorithme. Dans certains cas, cette adaptation n'affecte pas l'efficacitk de l'algorithme. Dans d'autres, l'efficacitk est diminuk au point
5 10
J.K. Lenstra, A.H.G. Rinnooy Kan
que parfois la nouvelle methode de resolution revient B humerer entikrement les affectations admissibles. Si le probleme sans contraintes de precedence est dkjh si complique qu’une approche par enumkration semble inkvitable, on peut tabler sur le fait que l’addition de contraintes de prkkdence peut rkduire le nombre d’affectations admissibles. Dans le premier cas, les contraintes rendent le probleme plus difficile; dans le second, le probleme devient un peu plus simple. En termes de complexiti de calcu(, l’addition de contraintes de prkkdence peut transformer un probleme bien soluble en un probleme NP-dur, ou encore rendre un probleme d‘ordonnancement NP-dur plus facile B resoudre en pratique. Nous nous concentrerons sur le premier phknomene et nous l’illustrerons dans le cas ou les ressources limitks correspondent a des machines MI,. . . ,M, dont chacune peut triter au plus l’une des activitks ou tdches J l , . . . ,J , h la fois. De nombreux problemes spkdiques ont pu Ctre formules et ktudies dans ce cadre gknkral. Le crit2re d’optimalitk qui doit Ctre minimis& joue un r61e proeminent dans ce contexte. Pour chaque ordonnancement admissible conduisant a un temps de fin d’exkution Cj pour J, (j = I,...,n), on fait l’hypothkse fondamentale que le critkre est une fonction non dkroissante de chacune des variables C1, . . . ,C,. Nous en donnerons des exemples plus loin. Panni les caractkristiques relatives aux tiches qui definissent des types particuliers de probkmes, on peut avoir des contraintes depricidence de la forme J j + Jk c’est-bdire que J j doit Ctre termink avant que Jk puisse commencer. De telles contraintes ont Ctk abondamment etudiks et certains types de contraintes de prkMence ont ktC mis en kvidence. En termes du graphe de p r W e n c e G dont les sommets sont I, ...,n et dont les arcs sont les paires ( j , k ) telles que Jj Jk, on s’est surtout intkressk au cas ou G est une collection de chaines, m e forit ou un graphe sirie-parallde. Beaucoup d’autres cas particuliers entre un graphe sans arcs et un graphe arbitruire ont aussi Cte examinks. En general, l’effort principal a ktC mis sur la dktermination d’une frontikre aussi bien dklimitee que possible entre les problkmes bien solubles et les problkmes NP-durs. Ceci a etk fait en determinant le cas le plus gkneral de contraintes de prkkdence qui peuvent Ctre traitks en temps polynomial et le cas le plus simple qui conduit h un probleme NP-dur. Dans cette note, nous nous concentrerons sur d ew problkmes ouverts connus: d’une part la minimisation du retard total pour des tiiches de durke unite h exkuter sur une machine avec des contraintes de prkckdence dkfinies par des chaines; d’autre part, la minimisation du maximum des temps d‘exkution sur trois machines paralleles identiques avec des contraintes de p r W e n c e quelconques. Les resultats connus pour des problkmes voisins seront passes en revue et gknkralisks ti l’aide de deux nouvelles preuves de NP -difficulte.
7i.00open problenis in precedence constrained scheduling
51 1
1. INTRODUCTION The theory of scheduling is concerned with the allocation over time of scarce resources to activities. In this context, partial orders arise in a natural fashion: a partial order on the activity set imposes constraints on the order in which the activities can be executed and as such delimits the set of feasible allocations. The challenge is to incorporate these precedence constraints as efficiently as possible in algorithms designed to determine a feasible allocation that is optimal with respect to some criterion. The effect of precedence constraints can be twofold. If the problem without precedence constraints can be solved efficiently, their addition will generally require the algorithm to be adapted. In some cases, this adaptation does not affect the efficiency of the algorithm; in other cases, it does, possibly to the point that the new solution method amounts to complete enumeration of all feasible allocations. If the unconstrained problem is already so difficult in itself that an enumerative approach seems unavoidable, one may capitalize on the addition of precedence constraints by exploiting the fact that they reduce the number of feasible allocations. In the former case, precedence constraints make the problem harder to solve; in the latter case, it becomes a little easier. The theory of computational complexity of combinatorial problems has served to formalize the preceding informal discussion. We will settle here for a very brief review of the main concepts of this theory and refer the reader for more details to [Cook 1971; Karp 19721 (the first two papers on the subject), [Garey & Johnson 19791 (a comprehensive textbook) and [Lawler & Lenstra 19821 (a survey likely to be readily available to the current readership). The size of a combinatorial problem is defined as the number of bits needed to encode its data, and the running time of an algorithm as the number of elementary operations (such as additions and comparisons) required for its solution. If a problem of size s can be solved by an algorithm with running time 0 (p(3)) where p is a polynomial function, then the problem is said to be well solvable; there are good theoretical and practical justifications for this notion. Many problems have been shown to be well solvable, simply by the construction of a polynomial-time algorithm. Only few problems have been proved to be not well solvable, but there is a large class of problems for which it is strongly suspected that this is indeed the case. These are the NP-hard problems, which share a notorious reputation for computational intractability as well as the property that a polynomial-time algorithm for any one of them would yield polynomial-time algorithms for all problems in an important subclass, the NP-complete problems - a very unlikely event.
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One establishes NP-hardness of a problem P by taking another NP-hard problem Q and showing that Q is reducible to P (Q e P ) , i.e., that for any instance of Q a corresponding instance of P can be constructed in polynomial time such that solving the latter will solve the former as well. This implies that Q is a special case of P , and since Q is NP-hard, P is NP-hard too. (This recipe obviously does not apply to theflrst NP-hardness proof - for this, see [Cook 19711.) Rephrased more formally, then, the addition of precedence constraints may turn a well-solvable problem into an NP-hard one, or may make an NP-hard scheduling problem easier to solve in practice. We shall focus on the former phenomenon, and illustrate it for the case , each of which that the scarce resources correspond to machines M . . . ,M,,, can handle at most one of the activities or jobs J 1, . . . ,J,, at a time. Within this general setting, many specific problem types have been formulated and studied. For a detailed problem classification and a survey of the complexity results in this area, we refer to [Graham et al. 1979; Lawler et al. 19821. A prominent role in this classification is played by the optimality criterion to be minimized. With every feasible schedule leading to a completion time CJ for J j 0 = l , ...,n), the basic assumption is that the criterion is a function of CI, . . . ,C,,, nondecreasing in every variable. We shall encounter various examples below. Among the various job characteristics that further specify a problem type, there may be precedence constraints of the form J j +Jk, sijpfying that JJ has to be completed before Jk can start. Such constraints have long formed a research subject in the area, whereby several types of precedence constraints have been distinpshed. In terms of the precedence graph G with vertex set { 1,...,n } and arc set {(j,k):J j + J k } , separate attention has been paid to the case that G is a collection of chains, aforest, or series-parallel. Many other special cases inbetween an empy and an arbitrary arc set have been investigated as well. In general, the effort has been to draw as sharp a borderline as possible between well-solvable and NP-hard problems, by identification of the most general type of precedence constraints that can be coped with in polynomial time versus the simplest type that leads to NP-hardness. For a review of the results obtained so far, we refer to [Lawler & Lenstra 19821. In this note, we concentrate on two prominent open problems in this area, while surveying known related results. 2. A SINGLE MACHINE PROBLEM
Let us assume that there is a single machine (m = 1) and that each of the n
Tcuo open problenzs in precedence constrained scheduling
513
jobs JJ (j = 1,...,n ) has to spend an uninterrupted processing time of pJ time units on the machine. Each JJ becomes available for processing at time 0 and incurs, upon its completion at time CJ,a tardiness cost T, = max{O,CJ - d J } , where dJ is a given due date. The criterion to be minimized is the total tardiness E;= I T / .
Thls is perhaps the most notorious open problem in single machine scheduling theory. It can be solved by dynamic programming techniques in O(n4EpJ)time [Lawler 19771; although the running time is obviously exponential in the problem size (which is O(Z(logpJ +log dJ))),the algorithm in question is called pseudopolynomial since the running time is polynomial in the problem data themselves. We will concentrate on the special case of unit-time jobs, i.e., pJ = 1 (j=1, ...,n ) . The cost cIJ of scheduling JJ in the i-th position is now given by qJ.= max(0,i - d J } , and the problem is to find a permutation u of { 1,...,n } rmnimizing X/”= caOh. If there are no precedence constraints, this is an ordinary linear assignment problem, which can be solved in O(n3) time (see, e.g., [Lawler 19761). If arbitrary precedence constraints between the jobs are allowed, the problem becomes NP-hard [Lenstra & Rinnooy Kan 19781. It is not known, however, what the effect of chain-like precedence constraints is, and this is our first open problem:
Given a directed graph G with vertex set { 1,...,n } in which each vertex j has an associated integer dj, indegree at most one and outdegree at most one, find a permutation u of { 1,...,n } satisfiing U(J) < u ( k ) whenever ( j , k ) is an arc of G, such that E,?= 1 max{O,u(j ) -a’, } is minimized. An optimality criterion related to the total tardiness I: T, is the number of late jobs XUJ, where UJ = 0 if C, < dJ, UJ = 1 if CJ > dJ. Since we know of no problem type for which minimizing XUJ is harder than minimizing XT, and since the problem of minimizing XUJ for unit-time jobs on a single machine subject to chain-like precedence constraints is NP-hard [Lenstra & Rinnooy Kan 19801, the most plausible conjecture is that the above problem will eventually turn out to be NP-hard. Three immediate generalizations of our open problem are worth considering: (1) The processing timesp, (j = 1,...,n ) are arbitrary nonnegative integers. The resulting problem is NP-hard (Theorem 1). (2) Each JJ (j = 1,...,n ) has to be completed no later than a given deadline eJ (not to be confused with the due date dJ). This problem is NP-hard as well (Theorem 2). (3) Each JJ (j = 1, ...,n ) becomes available for processing at a given release
J.K. Lenstra, A.H.G. IZinnooy Karl
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date r,. This problem is still open and, of course, also suspected to be NP-hard. As a preparation for the proofs of Theorems 1 and 2, we recall an NPhardness result for the total weighted tardiness criterion Zy= w, 5 , where w, is a given weight of J, (j = 1,...,n ) . Lemma 1 [Lawler 1977; Lenstra et al. 19771. The problem of scheduling jobs with arbitrary processing times on a single machine in the absence of precedence constraints so as to minimize total weighted tardiness Z w, Tj is NP-hard.
Proof [Lenstra & Rinnooy Kan 19801. We have to show that a known N P hard problem is reducible to the Z wjT, problem. Our starting point will be the following NP-hard problem [Garey & Johnson 19791: 3-PARTITION: Given a set S = {1,...,3t} and positive integers a1, . . . , a 3 t , b with lhb < aj < ‘hb for a l l j E S and X j E s a, = t b , does S have a partition into t 3-element subsets Si such that ZjEs, aj = b ( i =0, ...,t - l)? Given any instance of 3-PARTITION, we construct an instance of the Z w, T/ problems as follows: there are 4t - 1 jobs; - for each j ES, there is a job J, with processing time pi = a,, due date d, = Oandweightw, = a,; - for each i E { 1,...,t - l}, there is a job J’i with processing time p’i = 1, due date di = i ( b 1) and weight w’; = 2.
+
We claim that 3-PARTITION has a solution if and only if there exists a schedule with value Z w, T, C y , where y = & c j c k G 3 t ajak 4- %(t - 1)tb. This would imply that a polynomial-time algorithm for the Z w, Ti problem could be used to solve 3-PARTITION in polynomial time as well and therefore prove the theorem. Let us first ignore the jobs J’i ( i = 1,...,t - 1). Since dj = 0 for all j E S , we have X,Es wj T/ = 2,ES w j C j ; moreover, since pi = w, for all j ES,the value of X j E S wjCj is not influenced by the ordering of S. That is, for any schedule of the jobs J j (jES) without machine idle time we have
2 j ES
wjTj =
2
ajak* 1 cj 9 k 9 3 t
Let us now calculate the effect of inserting a job J’i in such a schedule. Suppose that Yi is completed at time Ci and define L’i = Ci -di . Since all jobs Jj (/ E S ) that are processed after J’i are completed one time unit later, the value of Xj ES w, Ti is increased by the total weight of these jobs, and we have
Two open problems in precedence constrained scheduling
2
+
wjT, wli T i =
j ES
=
2 2
+ 1 -d’, -L’i +21nax{O,L’j } a,uk +tb +I-d’i + IL’j I. a,ak +tb
1 < j
I =Zj< k
5 15
=Z31
More generally, insertion of all jobs J l i , resulting in completion times = dfi+L’i ( i = 1,...,t - 1) such that C’
x wj Tj
1-1
=
2
19j 9 k 931
a,uk
+ x ( t b +i
-d’<,)+ IL’<j)I)
i=l
1-1
i=l
irrespective of the permutation T . It follows that a schedule has value Z w, Ti < y if and only if there is no idle time and moreover the jobs J f i are completed at times Ci = d’i = i ( b 1) ( i = 1, ...,t - 1). Such a schedule exists if and only if the jobs J j (jE S ) can be divided into t groups, each containing 3 jobs and requiring b units of processing time, i.e., if and only if 3-PARTITION has a solution. 0
+
The proof of Lemma 1 provides the basis for our proofs of Theorems 1 and 2. We will specify reductions from 3-PARTITION to both Z T, problems in which the number of jobs created is Q ( t b ) and O(tb2)respectively. This may raise some eyebrows, as the size of an instance of 3-PARTITION is only O(t log b ) . However, 3-PARTITION has been shown to be NP-hard even when problem size is measured in a pseudopolynomial fashion as O ( t b ) [Garey & Johnson 19791, and hence the reductions below suffice to establish N P hardness.
Theorem 1. The problem of scheduling jobs with arbitrary processing times on a single machine subject to chain-likeprecedence constraints so as to minimize total tardiness Z Ti is NP-hard. Proof. Given any instance of 3-PARTITION, we first construct an instance of the Z wJ T, problem as in the proof of Lemma 1 and then transform it into an instance of the ZT, problem with chain-like precedence constraints as follows. Each job JJ with processing timepJ, due date dJ and weight wJ (whether it is a ”partition” job JJ (j E S ) or a ”splitting” job J’, ( i = 1,...,t - 1)) is replaced by a chain of wJ unit-weight jobs. The first job in the chain has processing timepJ and due date d J , the next wJ - 1 ones have processing times 0 and due dates dJ
*
J.K. Lenstra, A .H. G. Rinnooy Kan
516
The resulting problem instance has tb +2(t - 1) jobs. Given any feasible schedule in which the jobs of some chain are not scheduled consecutively, one can obtain another schedule by processing all the zero-time jobs of that chain directly after its first job. This schedule is still feasible, and its I: Ti value has not increased. Hence, each chain of length wj can be considered as a single job with weight w,, and we are back at our original construction. The reader who dislikes zero-time jobs could quite easily replace them by unit-time jobs and multiply the lengths of the other jobs by a factor polynomial in t and b such that the equivalence argument still carries through. 0
Theorem 2. The problem of scheduling unit-timejobs on a single machine subject to arbitrary deadlines ej and chain-like precedence constraints so as to minimize total tardiness I: T/ is NP-hard. Proof. Our proof is again related to the proof of Lemma 1, although it is not such a straightforward extension as the proof of Theorem 1. Given any instance of 3-PARTITION, we construct an instance of the Z Ti problem with unit-time jobs, deadlines and chain-like precedence constraints as follows: - there are n = tb2 t - 1 jobs; - for each j E S , there is a chain of ba, unit-time jobs:
+
4
with due dates and deadlines defined by
d j k ) = n ( k = 1,...,(b - l)aj), d,W-‘) = -I
(I=aj
- 1,
...,o),
e r ) = n (k = 1,...,baj);
-
for each i E { 1, ...,z - l}, there is a unit-time job J’i with due date and deadline defined by di = e’i = i(b2+1). We claim that 3-PARTITION has a solution if and only if there exists a feasible schedule with value Z Ti < z , where z = bZlGjike3, ajuk +%(t - 1)tb. Before we prove this claim, we make some introductory remarks about the way in which the job weights occurring in the proof of Lemma 1 have been simulated in the present construction. For each chain 0’ ES), the due dates have been specified such that in any schedule without machine idle time only the last aj jobs in the chain contribute to the criterion; if all these jobs are completed one time unit later, this adds aj units to Z Ti, which corresponds to the original weight wj = a j . For each job J’, ( i = 1, ...,t - l), we previously used a weight w ’ ~= 2 in combination with an upper bound y on I: wj Ti to enforce an implicit deadline di ; we now simply have an explicit deadline e‘i = d i . Consider any feasible schedule with value Z Tj < z . Without loss of gen-
6
517
Two open problems in precedence constrained scheduling
<
erality, we assume that the schedule contains no machine idle time, that each (j E S ) job J’, ( i = 1 , ...,t - 1) is completed at time d,,and that the chains do not preempt each other; the latter two statements can be proved by means of simple interchange arguments. The jobs J’, (i = 1 , ...,t - 1) do not contribute to the Z T/ value of the schedule. The contribution of the chains J/ (j E S ) consists of two terms. First, there is the total tardiness of all jobs in the chains when the chains are processed from time 0 onwards without interruption. It is not hard to see that this term is given by b E l , G k <3t a, ak , irrespective of the ordering of S . Secondly, there is the increase in total tardiness due to the insertion of the - l , d , ]= [ d , - , + b 2 , d ’ , ] ( i = 1 , ...,t - l), where jobs J’, in the intervals [d, = 0. Let S, C S denote the index subset of chains that are completed in + b 2 ] ,and let A, = 2, €s, a, ( i = O ,...,t - 1 ) . Note that bA, the interval [d,,d, is equal to the total length of all chains completed in the final interval [ d - I , d ’ , - 1 + b 2 ] , SO that A t - 1 2 b . More generally, we have that 2; -, Ah 2 ib ( I = 1 , ...,t - 1). Since all chain lengths as well as the interval lengths are integer multiples of b , we know that, if j ES, , the last b jobs of J/ and in particular the last a, ones (the only ones that contribute to 2 T,) must be processed in [a,,d’, +b2],so that J/ contributes ia, additional units to Z T/ . Thus, the second term is given by r-1
t-1
t-1
t-1
It follows that 2 T/ < z if and only if Ai = b (i =O ,...,t only if 3-PARTITION has a solution . 0
-l),
i.e., if and
3. A PARALLEL MACHINE PROBLEM We now assume that there are m machines and n jobs J j ( j = l , ...,n ) . The machines are parallel in the sense that each job can be assigned to any one of them, and they are identical in the sense that, when J j is assigned to some machine, it requires an uninterrupted processing time p i , irrespective of the machine. The criterion to be minimized is the maximum completion time Cm, - mal<j
J.K. Lenstra, A.H.G. Rinnooy Kan
518
cal parallel machines subject to precedence constraints, specified in the form of a directed graph G : (1) If m is arbitrary (i.e., specified as part of the problem instance) and G is an inforest (each vertex has outdegree at most one) or an outforest (each vertex has indegree at most one), the problem is solvable in O ( n ) time [Hu 19611. (2) If m = 2 and G is arbitrary, the problem is also well solvable; algorithms that have subse uently been developed require O ( n 3 ) time [Fujii et al. 1969, 19711, O(n ) time [Coffman & Grahman 19721, "almost linear" time [Gabow 19821, and O ( n ) time [Gabow & Tarjan 19831. (3) If m and G are arbitrary, the problem is NP-hard [Ullman19751. These results do not resolve the complexity status of the problem if G is arbitrary and rn is fixed but greater than 2. In particular, the case that m = 3 has withstood all attacks, and this is our second open problem:
P
Given a directed graph G with vertex set { 1,...,n }, Jind the minimum value of C for which there exists a function a:{ 1,...,n } + { 1,...,C } satisaing a(j) < a ( k ) whenever u , k ) is an arc of G and 10' E {1,...,n } : a(j) = t } l < 3 for all t E { 1,...,C}. In the course of research on this problem, progress has been made for several special types of precedence constraints other than forests. We mention the following results. Let the height h of G be defined as the number of arcs in a longest path in G . If m is arbitrary and h = 2, the problem is still NP-hard, and there exists no polynomial-time (approximation) algorithm that guarantees a relative error less than one third of the optimal C, value unless all NPcomplete problems are well solvable [Lenstra & Rinnooy Kan 19781. If both m and h are fixed, the problem is well solvable in O ( n h ( m - l ) 1+ l t h e [Dolev& Warmuth 3982Bl. Suppose G is an interval order: each vertex j corresponds to an interval [aJ. ,b.] on the real line and u , k ) is an arc of G whenever b, < a k . In J t h s case, the problem is solvable in O(n2) time [Papadimitriou & Yannakakis 19791. Suppose G is a level order: any two incomparable vertices with a common predecessor or successor have identical sets of predecessors and successors. If m is fixed, this problem is well solvable in O(n"'-') time [Dolev & Warmuth 1982Cl. Suppose G is an opposing forest, consisting of the disjoint union of an inforest and an outforest. If rn is arbitrary, this problem is NP-hard [Garey et al. 1983, Mayr 19811. If m is fixed, it is well solvable in O(n2m-210gn ) time [Dolev & Warmuth 1982Cl. If m = 3, there is an
Two open problems in precedence constrained scheduling
519
O ( n ) algorithm [Garey et al. 1983; Dolev & Warmuth 1982Al. These results give little additional insight into the complexity status of the three-machine problem with arbitrary precedence constraints. Recent rumors on a proof of its well-solvability have not been substantiated so far, but such a proof should be extendable to the case that m is any fixed constant. We believe that the problem stands a good chance to be the seventh one to be removed from the list of twelve open problems in [Garey & Johnson 19791.
4. CONCLUDING REMARKS The above discussion has illustrated that very detailed insights exist on the way in which partial orders on the job set affect the computational complexity of machine scheduling problems. The two problems considered in the preceding two sections figure prominently on the list of open problems that is produced by the computer program MSPCLASS [Lageweg et d. 1981,19821. This program keeps track of the complexity status of 4,536 machine scheduling problems, 390 of which are currently still open. Resolution of many of these problems, in particular of the two above ones, would seem to require the development of new algorithmic approaches or transformation techniques. ACKNOWLEDGEMENTS We are grateful to D. de Werra who translated our extended abstract into a risumt ditailli. Our research was supported by NSF grant MCS78-20054. REFERENCES E.G. COFFMAN, JR. & R.L. GRAHAM (1972) Optimal scheduling for twoprocessor systems. Acta Informal. I, 200-2 13. S.A. COOK ( 1971) The complexity of theorem-proving procedures. Proc. 3rd Annual ACM Symp. Theory of Computing, 151-158. D. DOLEV & M.K. WARMUTH (1982A) Scheduling flat graphs. Research Report RJ3398, IBM, San Jose, CA. D. DOLEV & M.K. WARMUTH (1982B) Sceduling precedence graphs of bounded height. Research Report RJ3399, IBM, San Jose, CA; J. Algorithms, to appear. D. DOLEV & M.K. WARMUTH (1982C) Profile scheduling of opposing forests and level orders. Research Report RJ 3553, IBM, San Jose, CA.
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M. FUJII, T. KASAMI & K. NINOMIYA (1969,1971) Optimal sequencing of two equivalent processors. SIAM J. Appl. Math. 17, 784-789. Erratum. 20, 141. H.N. GABOW (1982) An almost-linear algorithm for two-processor scheduling. J. Assoc. Comput. Mach. 29, 766-780. H.N. GABOW & R.E. TARJAN (1983) A linear-time algorithm for a special case of disjoint set union. Proc. 15th Annual ACM Symp. Theory of Computing, 246-25 1. M.R. GAREY & D.S. JOHNSON (1979) Computers and Intractability: a Guide to the Theory of NP-Completeness, Freeman, San Francisco. M.R. GAREY, D.S. JOHNSON, R.E. TARJAN & M. YANNAKAKIS (1983) Scheduling opposing forests. SIAM J. Algebraic Discrete Methods 4, 72-93. R.L. GRAHAM, E.L. LAWLER, J.K. LENSTRA & A.H.G. RINNOOY KAN (1979) Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discrete Math. 5, 287-326. T.C. HU (1961) Parallel sequencing and assembly line problems. Oper. Res. 9, 841-848. R.M. KARP (1972) Reducibility among combinatorial problems. In: R.E. MILLER, J.W. THATCHER (eds.) (1972) Complexity of Computer Computations, Plenum, New York, 85- 103. B.J. LAGEWEG, E.L. LAWLER, J.K. LENSTRA & A.H.G. RINNOOY KAN (1981) Computer aided complexity classification of deterministic scheduling problems. Report BW 138, Mathematisch Centrum, Amsterdam. B.J. LAGEWEG, J.K. LENSTRA, E.L. LAWLER & A.H.G. RINNOOY KAN (1982) Computer aided complexity classification of combinatorial problems. Comm. ACM 25, 817-822. E.L. LAWLER (1976) Combinatorial Optimization: Networks and Matroids, Holt, Rinehart & Winston, New York. E.L. LAWLER (1977) A "pseudopolynomial" algorithm for sequencing jobs to minimize total tardiness. Ann. Discrete Math. I , 331-342. E.L. LAWLER & J.K. LENSTRA (1982) Machine scheduling with precedence constraints. In: I. RIVAL (ed.) (1982) Ordered Sets, Reidel, Dordrecht, 655-675. E.L. LAWLER, J.K. LENSTRA & A.H.G. RINNOOY KAN (1982) Recent developments in deterministic sequencing and scheduling: a survey. In: M.A.H. DEMPSTER, J.K. LENSTRA & A.H.G. RINNOOY KAN
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(eds.) ( 1982) Deterministic and Stochastic Scheduling, Reidel, Dordrecht, 35-73. J.K. LENSTRA & A.H.G. RINNOOY KAN (1978) Complexity of scheduling under precedence constraints. Oper. Res. 26, 22-35. J.K. LENSTRA & A.H.G. RINNOOY KAN (1980) Complexity results for scheduling chains on a single machine. European J. Oper. Rex 4, 270275. J.K. LENSTRA, A.H.G. RINNOOY KAN & P. BRUCKER (1977) Complexity of machine scheduling problems. Ann. Discrete Math. I , 343-362. E. MAYR (1981) Well structured parallel programs are not easier to schedule. Report STAN-CS-81-880, Department of Computer Science, Stanford University. C.H. PAPADIMITRIOU & M. YANNAKAKIS (1979) Scheduling intervalordered tasks. SIA M J. Cornput. 8, 405-409. J.D. ULLMAN (1975) NP-complete scheduling problems. J. Cornput. System Sci. 10, 384-393.
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- PART IX ORDER AND SOCIAL SCIENCES
- PARTIEIX ORDRE ET SCIENCES SOCIALES
Jean-Pierre BARTHELEMY - Bruno LECLERC - Bernard MONJARDET
Ensembles ordonnes et taxonomie mathematique.
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p . 523
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Annals of Discrete Mathematics 23 (1984) 523-548 0 Elsevier Science Publishers B.V. (North-Holland)
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ENST, 4 6 r u e B a r r a u l t , 75634 PARIS Cedcsx 13 CAMS, 54 Bd R a s p a i l , 7 5 2 7 0 PAKIS Cedex 06 o*% U n i v e r s i t e P a r i s - V e t ChMS. J
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En hommage a u p r o f e s s e u r C o r o m i n a s , b l ' o c c a s i o n d e s o n Pmtritat. Le t e r m e t a x o n o m i e m a t h 6 m a t i q u e r e c o u v r e un e n s e m b l e d e p r o c t d u r e s d e s t i n b e s i o b t e n i r des c l a s s i f i c a t i o n s hic!rarchiques ou ( e t ) d e s g r o u p e s homoggnes d a n s u n e p o p u l a t i o n d " ' i n d i v i d u s " d e n a t u r e v a r i t e . Ces t e c h n i q u e s f o n t p a r t i e d e s d b v e l o p p e m e n t s r b c e n t s d e 1 " ' a n a l y s e d e s donnbes" e t s o n t l a r g e m e n t u t i l i s t e s d a n s l e s s c i e n c e s s o c i a l e s , Gconomiques e t d e g e s t i o n . L ' i m p o r t a n c e d e c o n c e p t s e t dc t e c h n i q u e s o r d i n a l e s s ' e s t m a n i f e s t e r a u moins d e p u i s l e s t r a v a u x d e J a r d i n e e t S i b s o n (Mathematical Taxonomy, 1 9 7 3 ) e t a donnc! l i e u i p l u s i e u r s c o n t r i b u t i o n s , ti d e s n i v e a u x v a r i t s (nutamment p a r Boorman, O l i v i e r , J a n o w i t z , S c h a d e r , Day, Mc M o r r i s e t l e s a u t e u r s d e c e p a p i e r ) . Nous n o u s p r o p o s o n s i c i d e d o n n e r une s y n t h h s e d e c e s c o n t r i b u t i o n s , e n p r 6 s e n t a n t l a t a x o n o m i e m a t h t m a t i q u e sous l ' a n g l e ( p a r t i e l ) d e s s t r u c t u r e s o r d i n a l e s q u ' e l l e met e n j e u . ABSTRACT A f t e r t h e i n t r o d u c t i o n ( s e c t i o n 1 1 , we b e g i n w i t h a n o v e r v i e w on t h e a i m s a n d t h e m e t h o d s o f m a t h e m a t i c a l taxonomy ( s e c t i o n 2 ) . A t a x o n o m i s t c l a s s i f i e s o b j e c t s d e s c r i b e d by "raw" d a t a ( ~ e n e r a l l yv a l u e s o f d i f f e r e n t a t t r i b u t e s , v a r i a b l e s , . .) t h a t c a n be c o n s i d e r e d as " i n p u t s " . A c h o s e n method of c l a s s i f i c a t i o n t r a n s f o r m s these inputs i n t o "uutputs" t h a t art' c l a s s i f i c a t o r y " m o d e l s " , a n d w e e m p h a s i z e t h e f o u r main s u c h models : p a r t i t i o n ; n - t r e e , c a l l e d a l s o h i e r a r c h y ( ' ; i i t r a r c h i e " , i n french) ; ordered t r e e ( " h i t r a r c h i e s t r a t i f i P e " i n french) ; valued tree ( " h i G i - a r c h i e i n d i c b e " in f r e n c h ) , o r t h e e q u i v a l e n t n o t i o n of u l t r a m e t r i c d i s t a n c e . We r e c a l l t h a t most ( b u t n o t a l l ) m e t h o d s of c l a s s i f i c a t i o n b e g i n w i t h t h e c o m p u t a t i o n of a d i s s i m i l . a r i t y c o e f f i c i e n t (a1 so c a l l e d d i s s i m i l a r i t y i n d e x ) h e t ween t h e o b j e c t s t o be c l a s s i f i e d , f r o m t h e raw d a t a . I n s e c t i o n 3 , we s y s t e m a t i c a l l y s t u d y t h e o r d i n a l s t r u c t u r e s o f t h e d a t a ( 3 . 1 . and 3 . 2 ) a n d of t h e m o d e l s of s e c t i o n 2 . A d i s s i m i l a r i t y c o e f f i c i e n t i s d e f i n e d as a map w i t h domain t h e s e t B , ( X ) o f a l l t h e p a i r s o f e l e m e n t s o f a s e t X ( t h e o b j e c t s t o b e c l a s s i f i e d ) and w i t h codomain a l i n e a r l y o r d e r e d s e t L , w i t h a l e a s t e l e m e n t ( u s u a l l y , L i s a s u b s e t of c o n t a i n i n g 0 ) . The s e t o f a l l L - d i s s i m i l a r i t y i n d i c e s i s o n e t o o n e w i t h t h e s e t of L - c h a i n s o f t h e l a t t i c e 8 ( 8 , ( X ) ) of a l l s y m m e t r i c r e l a t i o n s on X , w h e r e a L - c h a i n i s an i s o t o n e a p p l i c a t i o n F : L -B t ? ( p n ( X ) ) s a t i s f y i n g two a d d i t i o n a l c o n d i t i o n s ( o r , e q u i v a l e n t l y , a s n o t i c e d by J a n o w i t z , a r e s i d u a l map F : L + f ( f ' ? ( X ) ) ) . A f t e r a r e c a l l on t h e l a t t i c e o f p a r t i t i o n s , w e s t u d y l a t t i c e s of u l t r a o i e t r i c s , e m p h a s i z i n g t h a t t h e i r p r o p e r t i e s a r e s i m i l a r t o t h e p r o p e r t i e s o f p a r t i t i o n l a t t i c e s ; f o r example, t h e y a r e semimodular. Then, we p o i n t o u t t h a t t h e s e t of a l l " d i s s i m i l a r i t y r e l a t i o n s " ( i . e . t h e t o t a l p r e o r d e r s on 6 ) 2 ( X ) , "pr6ordonnance"in f r e n c h ) and t h e s e t of a l l u l t r a d i s s i m i l a r i t y r e l a t i o n s a r e " f r a m e " s e m i l a t t i c e s , t h u s m e d i a n s e m i l a t t i c e s ( a n o t i o n d e f i n e d by S c h o l n n d e r , i n 1954).
.
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I n s e c t i o n 4 , we p r e s e n t , f i r s t o f a l l , t h e a b s t r a c t f r a m e w o r k o f o r d i n a l m e t h o d s o f c o m p a r i s o n , f i t t i n g o r a g g r e g a t i o n . S o , t h e c o m p a r i s o n of two o r
524
J.-P.Barthelemy et al.
s e v e r a l e l e m e n t s of a n o r d e r e d s e t i s b a s e d on t h e " n a t u r a l " m e t r i c s d e f i n e d on i t . F i t t i n g o r a g g r e g a t i o n p r o b l e m s le,id t o d e f i n e maps be tw e e n o r d e r e d s e t s s a t i s f y i n g a l g o r i t h m i c , a x i o n i n t i c o r o p t i m i s n t i o n c o n d i t i o n s . Th a r e a p p l i e d t o o r d e r e d s t r u c t u r e s o f s e c t i o n 3. T h u s , we ~ i l i t ~ i i-n and sometirnc.s c h a r a c t e r i z e - many o l d - o r new - d i s t a n c e s u s e f u l t o compare p a r t i t i o n s o r 11t r e e s , by means o f r e s u l t s on m e t r i c s a s s o c i a t e d w i t h l o w e r o r u p p e r va 1tl;ttions on o r d e r e d s e t s . I n s e c t i o n 4 . 3 , we g i v e t h r e e e x a m p l e s of o r d i n a l me thods in o r d e r t o f i t some u l t r a m e t r i c o r u l t r a d i s s i m i l a r i t y r e l a t i o n t o ;i g i v e n c l i s s i m i l a r i t y c o e f f i c i e n t o r r e l a t i o n . The f i r s t e x a m p l e , d e f i n e d h e r e a s t h e " u l t r a m e t r i c a n t i c l o s u r e " ( o n t h e s e t of a l l L - d i s s i m i l a r i t y i n d i c ~ e s ) i s e q u i v a l e n t t o t h e well-known " s i n g l e l i n k a g e " method ; t h e o t h e r e x a m p l e s a r e due t o Zalin and S c h a d e r . Then ( s e c t i o n 4 . 4 . ) , w e i n t r o d u c e t h e J a n o w i t z o r d e r t h e o r e t i c model f o r c l u s t e r a n a l y s i s " ( 1 9 7 8 ) , t h a t , f o r i n s t a n c e , a l l o w s t o c h a r a c t c r i z e " f l a t " me t h o d s ("mGthodes 2 s e u i l " i n f r e n c h ) of c l u s t e r a n n I y s i s . I n s e c t i o n 4 . 5 , we p o i n t o u t t h a t a " G a l o i s " a p p r o a c h t o c l a s s i f y o b j e c t s d e f i n e d by " p r e s e n c e - a b s e n c e " a t t r i h u t e s , l e a d s t o f i x e d p o i n t p r o b l e m s on o r d e r e d s e t s . F i n a l l y ( s e c t i o n 4 . b ) , we t a c k l e t h e s o - c a l l e d " c o n s e n s u s proble nI" i n taxonoiiiy, i . e . t h e problem t o "aggregate" o r "summarize" a t b e s t s e v e r a l c l a s s i f i c a t i o n s i n t o a c l a s s i f i c a t i o n ( t h e similar p r o b l e m f o r p r e f e r e n c e s l e a d s t o t h e w e l l known Arrow t h e o r e m ) . One a l s o o b t a i n s " d i c t a t o r i a l " o r " o l i g a r c h i c " r e s u l t s , l i k e t h e M i r k i n t h e o r e m f o r p a r t i t i o n s , g e n e r a l i z e d t o u l t r a m e t r i c s by L e c l e r c . B u t , i n t h e c a s e of me d i a n s e m i l a t t i c e s , o n e c a n o b t a i n a x i o m a t i c o r o p t i m i s a t i o n c h a r a c t e r i z a t i o n s of more " p o s i t i v e " c o n s t r u c t i v e p r o c e d u r e s , 1 i k e t h e "median" p r o c e d u r e ( a g e n e r a l i z a t i o n of t h e well-known m a j o r i t y r u l e ) .
1 . INTRODUCTION. Les S c i e n c e s Humaines o n t h e s o i n d ' a n a l y s e r d e s "donnbe s " r e c u e i l l i e s e t c o n s t r u i t c s p a r d e s moyens v a r i C s a l l a n t d e l ' o h s e r v a t i o n " p u r e " h l ' e x p 6 r i r n e n t a t i o n en l a b o r a t o i r e . Les t e c h n i q u e s s t a t i s t i q u e s u t i l i s b e s j u s q u ' i c i o n t b t b rtcemment e n r i c h i e s p a r l ' a p p o r t d ' a u t r e s mGthodes p e r m e t t a n t notamment d e t r a i t e r d e s d o n n b e s m u l t i d i m e n s i o n n e l l e s n o m b r e u s e s , d a n s une p e r s p e c t i v e p l u s d e s c r i p t i v e e t e x p l o r a t o i r e q u ' i n f t r e n t i e l l e e t c o n f i r m a t o i r e . L e u r e s s o r e s t dCi t a n t h d e s p r o g r k s t h C o r i q u e s qu'au dbvel oppement c o n s i d b r a b l e d e s nioycns i n f o r m a t i q u e s d e c a l c u l . Comme r e p r 6 s e n t a t i f s d e c e c o u r a n t , on p e u t c i t e r 1"'Rcole F r a n s a i s e d ' h a l y s e d e s Donn6es" ( c f . notamment B e n z b c r i , 1973) OLI l e s d b v e l o p p e m e n t s a n g l o Sax o n s a u t o u r du " M u l t i d i m e n s i o n a l S c a l i n g " ( c f . , p a r e xe mple , C a r o l 1 e t A r a b i e , 1 9 8 1 ) . La ta x o n o mi e mat hPmat i que ( c l u s t e r a n a l y s i s ) f a i t p a r t i e de c e s m6thode s "n o u v ell e s", so n h u t s p G c i f i q u e t t a n t l a r e c h e r c h e d e g r o u p e s homoghnes d a n s une population d"'individus" de nature varibe. Eventuellement, ces groupes pourront Btre o r g a n i s C s d a n s u n s s t r u c t u r e h i t r a r c h i q u e , r e f l t t a n t p a r f o i s un phtnomtne G v o l u t i f , comme d a n s l e s c l a s s i f i c a t i o n s b o t a n i q u e s o u zoologiques. C ' c s t d ' a i l l e u r s 3 p r o p o s d e t e l l e s c l a s s i f i c a t i o n s que d e s n o t u r a l i s t e s comme Adanson f o n t i n t e r v e n i r l e s p r e m i h r e s c o n s i d b r a t i o n s m a t h b m a t i c o - l o g i q u e s a u 186me s i e c l e . Ma is c e n ' e s t que c e s d e r n i t r e s a n n 6 e s q u e l a t a x o n o m i e r n a t h h a t i q u e c onna 'it un d t v e lo p p eme n t c o n s i d t r a b l e ( c f . , p a r exernpl e, S okal e t S n e a t h 1963, Lerman1970, 1981, J a r d i n e e t S i b s o n 1971, BenzC cri 1973, Bock 1974, H a r t i g a n 1975, Jambu 1978, Diday 1980, Gordon 1981, Chandon e t P i n s o n 1 9 8 1 ) . L ' i m p o r t a n c e d e c o n s i d t r a t i o n s o r d i n a l e s e n t a x o n o m i e rnat hbmat i que a p p a r a ' i t au m oins d h s l e s t r a v a u x d e J a r d i n e e t Sibson, p a r f o i s d e m a n i t r e voilcie ( l a f o r m a t i o n h a b i t u e l l e d e s "taxonomistes" n e les p r t d i s p o s a n t pas B r c c o n n a y t r e e t 2 u t i l i s e r d e s s t r u c t u r e s d ' o r d r e s ) . Mais c e s s t r u c t u r e s s o n t e x p l i c i t e e s d a n s d e s t r a v n u x comrne c e u x d e Boorman e t A r a b i e ( 1 9 7 2 ) , Boorman e t O l i v i e r ( 1 9 7 3 ) , Degenne , Fla me nt e t V e r g e s ( 1 9 7 8 ) , Barth b l e my ( 1 9 7 9 ) , B art l i el emy e t M o n j a r d e t ( 1 9 8 1 ) , BerthClerny, L e c l e r c e t M o n j a r d e t ( 1 9 8 3 ) , S c h a d e r ( 1 9 8 1 ) , Day ( 1 9 8 1 ) . Mc M o r r i s e t Neumann ( 1 9 8 3 ) , l e s i m p o r t a n t s d b v e l o p p e m e n t s du m o d t l e de J a r d i n e e t S i b s o n p a r J a n o w i t z ( 1 9 7 8 , 1979, 1981) e t l e s c o n t r i b u t i o n s r c c e n t e s de R . L e c l e r c ( 1 9 7 9 , 1981, 1982, 1 9 8 3 ) .
Nous t e n t o n s d a n s c e t e x t e une p r 6 s e n t a t i o n s y n t h G t i q u e d e s r e l a t i o n s a c t u e l l e s e n t r e ensembles ordonnbs e t taxonomic mathematique e t de l e u r s n p p o r t s m u t u e l s .
Ensembles ordonnes et taxonomie mathdmatique
525
I1 n ' e s t t!vidcmment p a s q u e s t i o n d e c r o i r e q u e l e s s t r u c t u r e s o r d o n n b e s s o i e n t l e s e u l i n s t r u m e n t - ou mGme l ' i n s t r u m e n t p r i v i l t g i b - p o u r r g s o u d r e l e s p r o b l k m e s d e s t a x o n o m i s t e s . On v e r r a d ' a i l l c u r s a p p a r a f t r e d a n s c e t e x t e l ' i n t t r g t d ' a u t r e s d i ~ m a i n e s m a t h & m , i t i q u e s comme l a t h g o r i e d e s g r n p h e s ou p l u s g t n b r a l e m e n t l a combin a t o i r c . , y c o m p r i s l e s p r o h l 6 m c s d ' o p t i m i s a t i o n ( c f . , s u r c e s p o i n t s , H u b e r t 1974, Hubert e t S c l i u l t z 1976, Hubert 1977, Matula 1977, Monjardet 1 9 8 0 ) . Mais l ' u n d e s p r o b l P m e s m a j e i i r s d u t a x o n o m i s t e c o n c e r n e l e c h o i x d e l a m6thode q u ' i l v a u t i l i s e r . A c e t t ! g a r d , l a p r o l i j - a r , i t i o n d ' a l g o r i t h m e s d e c l a s s i f i c a t i o n a u q u e l on a a s s i s t 6 ( e t q u i c o n t i n u e ) n ' e s t pas s a t i s f a i s a n t e . E l l e d o i t s'acconlpagner d ' u n e f f o r t d ' e x p l i c i t a t i o n d e s m s t h o d e s S O L I S - j a c e n t e sp o u r l e s c o m p a r e r e t l e s v a l i d e r . Une a p p r o c l i e " e x p b r i m e n t a l e " ( s t y l e procedures d e Monte C a r l o ) p e u t S t r e un e l b r n e n t d e r t p o n s e . Mais i l f a u t a u s s i r e c h e r c h e r l e s b a s e s t h 6 o r i q u e s d e s methodes u t i l i s G e s . Or l e s o h j c t s (clonn6es ou m o d $ l e s ) d e l a t a x o n o m i e m a t h b m a t i q u e o n t t r k s ( 7 . s o u v e n t u n e n a t u r e o r d i n a l e t a n t nu n i v e a u i n d i v i d u e l " ( u n e " h i b r a r c h i e s t r a t i f i b e l ' e s t un s u p - d e m i - t r e i l l i s arborescent) qu'au niveau " c o l l e c t i f " (l'ensemble des h i 6 r a r c h i e s s t r a t i f i t e s e s t un d e m i - t r e i l l i s h m g d i a n e s ) . D h s l o r s u n e m b t h o d e d e c l a s s i f i c a t i o n p e u t a p p a r a P t r e comme u n e a p p l i c a t i o n e n t r e e n s e m b l e s o r d o n n e s , g t r e GtudiGe d e c e p o i n t de vue e t , d a n s l e m e i l l e u r d e s c a s , S t r e c a r a c t t r i s b e a x i o m a t i q u e m e n t . On c o n s o i t q u e d a n s c e c o n t e x t e o n p u i s s e u t i l i s e r p r o f i t a b l e m e n t d e s r e s u l t a t s gbnbraux - o u s p e c i f i q u e s - s u r d e s a p p l i c a t i o n s e n t r e ensembles o r d o n n 6 s . De m&ne 1 3 t h t i o r i e d e s m k t r i q u e s sur l e s e n s e m b l e s o r d o n n t s v a Ctre u t i l e pour d b f i n i r e t p r G c i s e r l e s p r o p r i t i t t s d e d i s t a n c e s ( c o n d u i s a n t 2 d e s c o e f f i c i c n t s d ' n c c o r d s ) e n t r e c l a s s i f i c a t i o n s . On c o n s t a t e i c i l ' a p p o r t d e s s t r u c t u r e s o r d i n a l e s pour l a v a l i d a t i o n e t l a comparaison "axiomatique" d e mhthodes c l a s s i f i c a t o i r e s . Inversement, l e s a s p e c t s o r d i n a u x de l a taxonomie mathtmatique peuvent c o n d u i r e 3 d ' i n t b r e s s a n t s problgmes pour l e s s p g c i a l i s t e s d e s ensembles o r d o n n t s c o n t r i b u a n t a i n s i Zi l e u r d b v e l o p p e m e n t ( c f . , p a r e x e m p l e , l e s " F l a t problems" d e J a n o w i t z 1 9 7 8 a ) . Donnons m a i n t e n a n t l e p l a n d e ce t e x t e . Le p a r a g r a p h e 2 e s t u n e i n t r o d u c t i o n A p r t s a v o i r 6voque l e s sommaire 5 la p r o b l g m a t i q u e de l a t a x o n o m i e m a t h 6 m a t i q u e . d o n n b e s " b r u t e s " s u r l e s q u e l l e s l e " t a x o n o m i s t e " t r a v a i l l e , on m e t e n h v i d e n c e l e s q u a t r e p r i n c i p a u x modtles c l a s s i f i c a t o i r e s q u ' i l u t i l i s e : p a r t i t i o n , h i h r a r c h i e , h i t r a r c h i e s t r a t i f i6e (chaine etendue de p a r t i t i o n s , ultra-prbordonnance), h i h r a r c h i e i n d i c e e ( k + - c h a P n e d e p a r t i t i o n s , u l t r a m b t r i q u e ) . On r a p p e l l e l a mbthod o l o g i c l a p l u s c o u r a n t e q u i v i s e 2 o b t e n i r de t e l s m o d t l e s 5 p a r t i r d ' i n d i c e s de d i s s i m i l s r i t b s . Le p a r a g r a p h e 3 G t u d i e s y s t b m a t i q u e m e n t l e s s t r u c t u r e s o r d i n a l e s d e s d o n n P e s e t d e s rnodeles d u p a r a g r a p h e 2 . L e s i n d i c e s d e d i s s i m i l a r i t 6 h v a l e u r s d a n s un e n s e m b l e t o t a l e m e n t o r d o n n b ( a v e c p l u s p e t i t b l t m e n t ) L s o n t m i s e n c o r r e s p o n d a n c e b i - j e c t i v e n v e c l e s L-chaPnes d u t r e i l l i s d e s r e l a t i o n s s y m b t r i q u e s q u i s o n t a u s s i l e s a p p l i c a t i o n s r t s i d u e l l e s d e L d a n s c e t r e i l l i s . A p r h s un r a p p e l s u r l e t r e i l l i s d e s p a r t i t i o n s , on b t u d i e e n s u i t e d e s t r e i l l i s d ' u l t r a m b t r i q u e s duaux d e s t r e i l l i s d e s L-cha?nes de p a r t i t i o n s , e n m o n t r a n t q u e l e u r s p r o p r i b t 6 s s o n t v o i s i n e s d e c e l l e s du t r e i l l i s d e s p a r t i t i o n s (en p a r t i c u l i e r , c e s t r e i l l i s s o n t s e m i - m o d u l a i r e s s u p s r i e u r e m e n t ) . On c o n s i d k r e e n s u i t e l e s s t r u c t u r e s o r d i n a l e s d e l'ensemble des prkordonnances, d e s ultra-prtordonnanres et d e s h i e r a r c h i e s q u i s o n t t o u t e s d e s d e m i - t r e i l l i s d e " c h a r p e n t e s " e t done d e s demil a problbmatique genbrale t r e i l l i s 2 m b d i a n e s . 1.e p a r a g r a p h e 4 p r h s e n t e d ' a h o r d d e s m t t h o d e s o r d i n a l e s d e c o m p a r a i s o n , a j u s t e m e n t ou a g r e g a t i o n : a i n s i l a compar a i s o n de d e u x o u p l u s i e u r s t l t m e n t s d ' u n e n s e m b l e o r d o n n 6 v a r e p o s e r s u r l e s r n P t r i q i r e s n a t u r e l l e m c n t a s s o c i 6 e s 5 sa s t r u c t u r e d ' o r d r e ; l e s p r o b l 5 m e s d ' a j u s t e ment ou d ' a g r t g a t i o n c o n s i s t e n t h d 6 f i n i r d e s a p p l i c a t i o n s e n t r e d e u x ensembles ordonneis E e t F B p a r t i r d e c o n t r a i n t e s a l g o r i t h m i q u e s , a x i o m a t i q u e s o u / e t d ' o p t i m i s a t i o n . On p a r t i c u l a r i s e e n s u i t e c e t t e p r o b l k m a t i q u e a u c a s d e s s t r u c t u r e s o r d i n a l e s du p a r a g r a p h e 3 . En 4 . 2 . on r e t r o u v e p l u s i e u r s d i s t a n c e s e n t r e c l a s s i f i c a t i o n s et hventuellement leurs c a r a c t t r i s a t i o n s axiomatiques B p a r t i r de rbsult a t s s u r l e s v a l u a t i o n s - i n f b r i e u r e s oil s u p b r i e u r e s - d ' u n e n s e m b l e o r d o n n C . A p r h s a v o i r d b c r i t t r o i s m 6 t h o d e s o r d i n a l c s , on p r h s e n t e un " m o d t l e o r d i n a l " d e l a c l a s s i f i c n t i o n dG 2 - J a n o w i t z . Ce m o d t l e , a b s t r a c t i o n de c e l u i d e J a r d i n e e t S i b s o n , p e r m e t notamment d e d t f i n i r e t c a r a c t & r i s e r l a c l a s s e d e s p r o c P d u r e s
J.-P. Barthelemy et al.
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I!.
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p o i n t f i x e d a n s un e n s e m b l e o r d o n n 6 . On t e r m i n e p a r I e s probIi!mes d ' a g r G g a t i o n olii l ' a p p r c ~ c h e o r d i n a l e p e r m e t d ' o b t e n i r ou d e r e t r o u v e r d e s c n r a c t 6 r i s n t i o n s axionrat i q u e s d e p r o c e d u r e s r6sumant p l u s i e u r s c l a s s i f i c a t i o n s . S i g n a l o n s p o u r f i n i r q u e l a l i s t e d e s r c f c r m c c s , s a n s p r 4 t e n d r e :I l ' e x h n u s t i v i t P , c o n t i e n t p r e s q u e t o u t c e q u e n o u s c o n n c r i s s o n s s u r l e tli6me ensembles o r d o n t i c s e t taxonomie math6mntique.
N . B . Nous i n d i q u o n s c i - d e s s o u s q u e l q u e s n o t , i t i o n s o u definitions u t i l i s c e s t l a n s c k , texte. On n o t e X' l ' e n s c m b l e d e s c o u p l e s d ' c l h m e n t s d e l ' e n s e m b l e X , V(X) l ' e n s e m b l c d e s p a r t i e s d e X , 6',(X) l ' e n s e m h l e des p a r t i e s ?I d e u x 6 1 6 m e n t s d e X , nX l ' e n s e m b l e cles p a r t i t i o n s d e X. S o i t E un e n s e m b l e o r d o n n e a v e c p l u s p e t i t e t p l u s g r a n d ~ l 6 1 1 1 e n;t u n e C k ~ . i f n c c e s deux P l b m e n t s .
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Dans l ' e n s e m b l e o r d o n n k ( I ; , < ) line p a r t i e A d e E e s t : - une pcci.t%e finissantc: s s i ( s i e t s e u l t ~ n i e n t s i ) [ x E A A1 ;
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2 . DONNEES, MODELES ET METHODKS. 2.1
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Le p r o c e s s u s q u i c o n d u i t B d e t e r m i n e r d e s c l a s s i f i c a t i o n s d ' u n e n s e m b l e d ' o b j e t s c o m p o r t e p l u s i e u r s & t a p e s . Nous s u p p o s o n s i c i r c s o l u s l e s p r o b l t m e s - f o n d a m e n t a u x - d e c h o i x e t d e r e c u e i l d e s d o u n h e s e t q u e celles-ci s e p r 6 s e n t e n t s o u s l a forme d e tableaux du t y p e s u i v a n t :
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Dans l e t a b l e a u d e l a F i g u r e 1 , I e s t l ' e n s e m b l e d e s " i n d i v i d u s " ( d e n a t u r e q u e l r o n q u e ) , J e s t l ' e n s e m b l e des " v a r i a b l e s " ( c a r a c t k r e s , a t t r i b u t s ) I e s d C c r i v a n t , a . . e s t l a " v a l e u r " d e l a v a r i a b l e i p o u r l ' i n d i v i d u i . Ce t y p e d e donneies e s t 1J t r h s f r C q u e n t : t a b l e a u x d e m e s u r e s , d e n o t e s , d e r e p o n s e s c o d e e s B un q u e s t i o n n a i r e , d e p r b s e n c e ou a b s e n c e d e c a r a c t g r e s , d e r e l a t i o n e n t r e d e u x e n s e m b l e s . L a F i g u r e 2 e s t un e x e m p l z 6 1 C m e n t a i r e d e t e l t a b l e a u a v e c d i x i n d i v i d u s ( p l a r t s c c t t e f o i s en l i g n e ) e t d e u x v a r i a b l e s x e t y . On p e u t d o n c i c i r e p r b s e n t e r c e s i n d i v i d u s comme d i x p o i n t s d u p l a n , c e q u i e s t f a i t 2 l a F i g u r e 3 , c i - a p r h s . C e t exemple - a r t i f i c i e l - nous s e r v i r n d ' i l l u s t r a t i o n d a n s l a s u i t e .
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Figure 3
Un a u t r e t y p e d e t a b l e a u d e donnGes f r 6 q u e m m e n t r e n c o n t r 6 e s t c e l u i ob I = J . Un t e l t a b l e a u c o r r e s p o n d 1'' l a d o n n e e d ' u n e r e l a t i o n b i n a i r e v a l u t e , comme on en o b t i e n t p a r exemple d a n s l e s metbodes d e coinparaison p a r p a i r c s . 2 . 2 . MODELES. 1.e b u t d e l ' a n a l y s e d e s t a b l e a u x d e d o n n 6 e s c o n s i d 6 r 6 s c i - d e s s u s e s t i c i d ' o b t e n i r u n e t y p o l o g i e d e l ' e n s v m b l e d e s i n d i v i d u s , 011 d e l ' e n s e m b l e d e s v a r i a b l e s , o u d e s d e u x . Dans l a s u i t e , n o u s a p p e l l e r o n s X l ' e n s e m b l e d e s 6 1 6 m e n t s 5 c l a s s e r , q u ' i l s ' a g i s s e d e s i n d i v i d u s ou d e s v a r i a b l e s . Nous a l l o n s m a i n t e n a n t p r e c i s e r les m o d e l e s m a t h 6 m a t i q u e s d e t y p o l o g i e l e s p l u s couramment u t i l i s 6 s . A c e t e f f e t n o u s r e p r o d u i s o n s a u x F i g u r e s 4 e t 5 c i - d e s s o u s , des s o r t i e s d e programmes u s u e l s d e c l a s s i f i c a t i o n a u t o m a t i q u e , p o u r l e s d o n n 6 e s d e l ' e x e m p l e p r 6 c C d e n t ( F i g u r e s 2 e t 3).
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La sortie du programme de classification hierarchique CLIH, Figure 5 , est souvent appelee dendrograrrirne o u arbre de cZassi~ication.Nous lui avons donne une forme un peu differente B la Figure 6. L'information apportee par ces figures comporte differents niveaux qui examines successivement, vont nous fournir trois autres modgles typologiques. 1") I1 est clair que la Figure 6 est une representation du diagramme de Hasse d'un sup-demi treillis arborescent (sup-demi treillis oh tout element different de 1'ClCment universe1 est couvert par un Clement unique). Ce sup-demi treillis a pour elements minimaux l e s dix QlGments B classer ; B tout autre Clement, numerot6 de 1 1 B 19, correspond une partie de X, formee des elements minimaux inferieurs B cet element. Par exemple, B 17 correspond la partie { 1 , 2 , 3 , 4 , 5 1 . Aux 19 "noeuds" de l'arbre de classification correspond donc une famille de parties de X, contenant X, les parties B un element de X et verifiant la condition que deux parties incomparables ont une intersection vide (autre manigre d'ecrire que l'ensemble de ces parties, ordonne par inclusion, est arborescent). Une telle famille de parties sera appelee une h i i m r c h i e sur X et notee HX. Ses Plements (i.e. les parties de X appartenant B H ) seront appeles les classes de la hierarchie. Un X algorithme de classification hierarchique fournit toujours une telle hierarchic. Dans l'exemple traitb, on a H = ~1,2,3,4,5,7,8,9,10,12,78,910,125,678,1245, X 67C91O , X } . 2 " ) Les Figures 5 et 6 comportent une echelle numeriquc graduee de 0 B 100. Cette echelle permet d'attribuer B chaque noeud, (i.e. B chaque classe A de la hierarchic H ) un indice h(A). Par exemple, X(11) = h({1,2)) = 16 ; X X(18) = X({6,7,8,9,101) = 58. On en deduit un prPordre total s u r HX : A 5 B s s i
h(A) 5 A ( B ) . Ce preordre est compatible avec l'ordre d'inclusion sur P(X) : A c B implique A < B . De maniere generale, on appellehie'rarchie stratififie'e sur X, une hierarchie sur X munie d'un prbordre total sur s e s classes, compatible avec l'ordre d'inclusion.
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On a p p e l l e / ~ ~ ~ I Y ~ F C / L! ' ?n CL i f ( ~ t ; cs,u r X , une h i h r a r c h i e munie d ' u n e t e l l e a p p l i c a t i o n I 1 c s t c l a i r q u ' h line h i ~ i r a r c h i e i n d i e b e c o r r e s p o n d u n e h i P r a r i s o t o n e d a n s JR". c h i e s t r a t i f i P e unique e t qu'une h i g r a r c h i e s t r a t i f i e e e s t " r e p r 6 s e n t a b l e " p a r une i n f i n i t ; de h i b r a r e h i e s indic4t.s. Le den6ropramme de l a F i g u r e 5 e s t donc une r e p r b s e n t a t i o n d ' u n e h i 6 r a r c h i e i n d i c 6 e e t f o u r n i t a f o r t i o r i , une h i 6 r a r c h i e s t r a t i f i b e e t u n e h i 6 r a r c h i e s u r X . T o u t e f o i s , i l e x i s t e e n c o r e une a u t r e f a s o n , l a p l u s u s u e l l e , d ' i n t e r p r C t e r c e dendogramme e t n u u s l a d b c r i v o n s m a i n t e n a n t .
4 " ) Coupons le dendrogramme p a r une h o r i z o n t a l e d ' o r d o n n b e X . L e s c l a s s e s d e l a h i b r d r c h i e c o r r e s p o n d a n t a u x n o e u d s s i t u C s j u s t e a u - d e s s o u s ou s u r c e t t e h o r i z o n t a l e , forment une p a r t i t i o n 7x d e s 616ments de X . Par exemple, F i g u r e 6 , n50 = i 1 2 4 5 , 3 , 6 7 8 , 9 1 0 ) . TI\
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u n e c l a s s e d e IT*. P a r e x e m p l e , X = 38 p o u r 1 e t 4 . On e s t a i n s i c o n d u i t h l a not i o n d e R tc h a 9 n e d e p a r t i t i o n s d e X , c ' e s t - 2 - d i r e d ' u n e a p p l i c a t i o n i s o t o n e d e IR" d a n s l e t r e i l l i s TIX d e s p a r t i t i o n s d e X t e l l e que F-'(TI ) = b e t q u e p o u r t o u t x # y i l e x i s t e X minimum a v e c F(X) 3 { x , y } . L ' i mage d ' u n e t e l l e a p p l i c a t i o n F c o n s t i t u e u n e c h a i n e d e I I X , a l l a n t d e l a p a r t i t i o n l a p l u s f i n e n o , $ l a moins f i n e une chaEne Ctendue d e JIX).
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formbe d e s c l a s s e s maximales d ' i n d i c e i n f C r i e u r ou & g a l b A ; s i F e s t une I R k h a f n e , i l l u i c o r r e s p o n d l a hierarchic i n d i c C e f o r m b e d e s c l a s s e s d e t o u t e s l e s p a r t i t i o n s d e l a c h a i n e F(iR+) , l ' i n d i c e d ' u n e c l a s s e C t a n t c e l l e d e l a p l u s f i n e p a r t i t i o n l a c o n t e n a n t . D ' a u t r e p a r t , s o i t HX une h i 6 r a r c h i e i n d i c C e ; n o t o n s
V l ' o p b r a t i o n d e sup-demi t r e i l l i s d e H X , e t p o u r x , y C l b m e n t s d e X p o s o n s
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I n v e r s e m e n t h t o u t e ultra mC-
t r i q u e s u r X , on p e u t f a i r e c o r r e s p o n d r e u n e h i b r a r c h i e i n d i c b e ou u n e R 2 c h a i n e d e p a r t i t i o n s ( c f . 3 . 2 . 3 . ) . L e s t r o i s n o t i o n s d e h i b r a r c h i e s i n d i c b e s H X , RZchaEne de p a r t i t i o n s d e X e t d ' u l t r a m b t r i q u e s u r X , s o n t donc b q u i v a l e n t e s e t c o n s t i t u e n t l e
mzme modhle t a x o n o m i q u e . Dans l e c a s d ' u n e h i b r a r c h i e s t r a t i f i b e , on a b g a l e m e n t l ' h q u i v a l e n c e a v e c d eu x a u t r e s n o t i o n s ; c e l l e d e c h d n e e ' t e n d u e d e p a r t i t i o n s d e X , e t c e l l e d'uZtra-
J.-P. Barthelemy et al.
530
pr6ordonnunce de X, c'est-i-dire d'un prkordre total sur l'ensenible des paires d'kl6ments de X v6rifiant l'analogue de la condition ultrametrique (cf.3.3.2.). En particulier les ultra-preordonnances B deux classes correspondent aux partitions de X. En conclusion, nous considkrons que nous disposons de quatre modhles taxonomiques pour classer des objets : partition, hibrarchie, hihrarchie stratifibe ( o u chafne etendue de partitions ou ultraprbordonnance) , hierarchic indic6e ( o u k? cha'ine de partitions o u ultram6trique). 2.3. METHODES
.
Le prohleme est maintenant de passer des donn6es B l'un - ou plusieurs - des modeles taxonomiques recherch4s. Par exemple, comment est-on pass6 du tableau de la Figure 2 au dendrogramme de la Figure 5 ?
Certaines methodes dites parfois "directes", partent du tableau de donnees lui-mcme ou d'informations issues directement de ce tableau. Par exemple, dans la mbthode des partitions centrales (Regnier 1965), on associe B un tableau I x J de donnees les partitions de relkvement sur l'ensemble des ohjets B classer, puis on agrege ces partitions. Ainsi si l'on veut classer l e s elements de I, on considhre l e s IJI partitions T . de I d6finies par les variables de J : i et i' sont J
dans la mcme classe cle
71
= a i l j ; puis on agrgge ces IJI partitions ij (cf . 4 . 6 . ) .
ssi a
j'
en une partition "centrale"
Toutefois les methodes les plus usuelles passent par l'intermediaire d'une "proxirnite'" d6finie sur l'ensemble des objets b classer. L'id6e est que si l'on veut classer les elements de I b partir du tableau I x J, deux individus i et i ' proches - c'est-5-dire tels que les lignes i et i' le soient - doivent Etre classes ensemble. I1 faudra donc &valuer numbriquement la proximit6 de deux lignes du tableau, c'est-b-dire d6finir un indice de proximite ou ce qui est 6quivalent et plus courant un i n d i c e de dissimilurit6 d(i,i'), Le choix de l'indice utilis6 conditionne evidemment les resultats ohtenus et ce choix pose de nombreux probl5mes que nous n'ahorderons pas ici (cf. l e s ouvrages de taxonomie math6matique cites dans l'introduction). Pour l'exemple Figure 2, on a pris comme dissimilarit6 la distance euclidienne entre les points, ce qui conduit au tableau de la Figure 7 ci-dessous :
1
2
3
4
5
6
7
e
9
1
0.00
2 3
1.00 3.61
2.83
0.00
4
2.24
3-16
5.10
0.00
5
2.00
2.24
3.00
2.24
6
5.00
5.10
4.24
4.47
3.00
0.00
7
7.00
7.07
5.83
6. 32
5.00
2.00
0.00
e
8.06
8.25
7.21
7.07
6.08
3.16
1.41
0.00
9
10.20
111.05
8.06
9.85
8.25
5.39
3.61
3.61
0.00
10
8.54
8.25
6.00
8.60
6.71
4 - 2 4
3.16
4.00
2.24
10
0.00
0.00
0.00
Figure 7 Ggngralement, une dissimilarit.6 est une application de X2 dans IR' verifiant d(x,y) 2 0 , d(x,x) = 0 et d(x,y) = d(y,x). On peut aussi, plus Qconomiquement, l a
Ensembles ordonnes et taxonomie niathematique c c i n s i d c r e r comme m e a p p l i c a t i o n d e f',(X)
dnns
531
k?+.
S i on a c o n s t r u i t une d i s s i m i l a r i t 6 s u r l ' e n s e m b l e X d ' o b j c t s B c l a s s e r , i l e s t n a t u r e 1 d e r a p p r o c h e r cl ' abord l e s o b j e t s d e f a i b l e d i s s i m i l a r i t 6 , p u i s d ' a u g -
E R+,on pose :
mcnter p r o g r e s s i v e m e n t l e s e u i l . A i n s i , pour
R(h)
RO)
est
=
!(x,y)
I n r e l a t i o n au s e u i l
E X2
> ,p o u r
t e l quc
d(x,y)
5 !!
l a d i s s i m i l a r i t 6 d . C ' e s t u n e r e l a t i o n sym6-
t r i q u e e t on a : h
<
),'
implique
R(A)
5
R(A')
Une t e l l e a p p l i c a t i o n d e R+ d a n s l e t r e i l l i s bool bim d e s r e l a t i o n s b i n a i r e s s y m 6 t r i q u e s s u r x s ' a p p e l l e r a line R*c 2 t 2 1' 1 ; s o n image e s t u n e chaTne 6 t e n d u e d e r e l a t i o n s s y m t t r i q u e s . La b i j e c t i o n e n t r e d i s s i m i l a r i t 4 s e t IR*chaTnes s e r a p r k c i s 6 e a u p a r a g r a p h e 3 . 1 . 1 . La d o n n t c d ' u n e d i s s i m i l a r i t 4 s u r l ' e n s e m b l e X pe rme t d ' b v a l u e r s i d e u x 6 1 6 m en ts s o n t p l u s p r o c h e s q u e deux a u t r e s , ( i . e . s i d ( x , y ) 5 d ( z , t ) ) . Dans c e r t a i n s c a s , c ' e s t u n i q u e m e n t c e t o r d r e e n t r e l e s d i s s i m i l a r i t 6 s ( e t non l e u r s v a l e u r s a b s o l u e s ) q u i e s t s i g n i f i c a t i f e t q u i d o i t @ t r eu t i l i s 6 p o u r r e c h e r c h e r u n e c l a s s i f i c a t i o n . O n e s t a i n s i c o n d u i t 2 l a n o t i o n d e yr~S~i2.~ion,zLrnci, sur X, i.e. d ' u n p r 6 o r d r e t o t a l s u r X2 d o n t l a c l a s s e m i n i m a l e e s t form6e d e s c o u p l e s (x,x). On p e u t a u s s i d 6 f i n i r u n e p r 4 o r d n n n a n c e comme un p r 6 o r d r e t o t a l s u r 8 (X). Nous
2
notons P l a pr4ordonnance a s s o c i 6 e canoniquement 2 l a d i s s i m i l a r i t 6 d : ssi d(x,y) 5 d ( z , t ) . (x,y) Pdfz,t) On p e u t m a i n t e n o n t p r b c i s e r l a m 6 t h o d o l o g i e g 4 n 6 r a l e d e l a t a x o n o m i e mathbmat i q u e . A p a r t i r d e s donn4es " r e c u e i l l i e s " s u r l ' e n s e m b l e d e s Clbments B c l a s s e r , o n c o n s t r u i t d e s d o n n 6 e s d e r e s s e m b l a n c e e n t r e c e s 616me nts ( d i s s i m i l a r i t & , prbordonnance, p-uple de p a r t i t i o n s , ) p u i s on u t i l i s e une procbdure l e s t r a n s f o r mant en l ' u n d e s modilles t a x o n o m i q u e s d 6 f i n i s c i - d e s s u s . Ces p r o c 6 d u r e s s o n t e x t r z m e m e n t n o m b r e u s e s e t d e m a n d e r a i e n t elles-msmes h @ t r e c l a s s i f i 6 e s . Nous n o u s c o n t e n t e r o n s d e d o n n e r q u e l q u e s p r i n c i p e s d e c l a s s i f i c a t i o n , notamment c e u x m e t t a n t en l u m i k r e d e s a s p e c t s o r d i n a u x .
...
1 . On p e u t d ' a b o r d d i s t i n g u e r l e s p r o c 6 d u r e s s u i v a n t l a n a t u r e d e s d o n n 6 e s d e ress e m b l a n c e e t d e s mod6les t a x o n o m i q u e s q u ' e l l e s p r e n n e n t e n c o m p t e . En ne r c t e n a n t p o u r l e s d o n n b e s q u ' u n e d i s s i m i l a r i t 6 ou u n e p r b o r d o n n a n c e e t p o u r l c s m o d e l e s q u e les q u a t r e d 6 f i n i s p l u s h a u t ( p a r t i t i o n , h i 4 r a r c h i e , h i 4 r a r c h i e s t r a t i f i b e ou i n d i c 6 e ) on o b t i e n t d C j 2 h u i t t y p e s e t e n p a r t i c u l i e r l e s de ux l e s p l u s frkquemment r e n c o n t r k s : t r a n s f o r m a t i o n d ' u n e d i s s i m i l a r i t 6 en s o i t une p a r t i t i o n , s o i t une hi6rarchie indic6e. 2 . Les e n s e m b l e s d ' o b j e t s - d o n n b e s ou modhl es - d ' u n c e r t a i n t y p e r e n c o n t r b s e n taxonomie math6matique, o n t d e s s t r u c t u r e s o r d i n a l e s q u i s e r o n t dbveloppdes au p a r a g r a p h e s u i v a n t . On p e u t a l o r s d i s t i n g u e r l e s p r o c b d u r e s q u i u t i l i s e n t e x p l i c i t e m e n t ces s t r u c t u r e s ou l e u r s d 4 r i v b e s ( f e r m e t u r e s , m 6 t r i q u e s ... ) a u n i v e a u d e s r & g l e s d e c o n s t r u c t i o n ou d ' e x i g e n c e s a x i o m a t i q u e s ( p a r e x e m p l e , G t r e i s o t o n e ) . On en v e r r a d e s e x e m p l e s e n 4 . 3 . a v e c l ' o u v e r t u r e u l t r a m b t r i q u e ou l a m6thode d'ajustement de Schader. 3 . Une a u t r e d i s t i n c t i o n e s t f o n d b e s u r l e s p r o p r i 6 t i . s d ' " i n v a r i a n c e o r d i n a l e " d e l a p r o c 6 d u r e . Nous n e c o n s i d b r e r o n s i c i q u ' u n e d e c e s p r o p r i 6 t b s d ' i n v a r i a n c e b i e n q u ' o n p u i s s e en d 6 f i n i r p l u s i e u r s ( c f . 5 c e p r o p o s , B e n z 6 c r i 1 9 6 7 , J o h n s o n 1967, Lerman 1 9 7 0 , S i b s o n 1 9 7 0 , 1972, e t l e s t r a v a u x d e J a n o w i t z ) . D i s o n s q u e deux d i s s i m i l a r i t 6 s sont 6 q u i v a l e n t e s s i l e u r s pr6ordonnances associt5es P e t Pd, sont d 6gales : d(x,y) 5 d ( z , t ) s s i d ' ( x , y ) 5 d ' ( z , t ) , ( i l r e v i e n t a u &me d e d i r e q u ' i l e x i s t e u n e i n j e c t i o n i s o t o n e cp d e k+ d a n s k?+ t e l l e que d ' = c p d ) . S o i t
J . 2 Barthelerny et al.
532
une procedure transformant une dissimilarit6 en une autre dissimilaritl (par exemple, une ultrametrique). On dira que @ est globalerient i n v a r i a n t e si 1'4quivalence de d et d' implique celle de @(d) et O(d'). Cette approche ordinale permet de d6finir une classe de procedures comprenant 6videmment toutes celles dont l a donn6e de depart est une preordonnance, plus celles qui ne dependent en fait - d e facon 6vidente ou non - que de la pr6ordonnance associee i la dissimilarit6 utilisee. On en donnera trois exemples ci-dessous. 4 . Donnons enfin une autre distinction valide uniquement pour l e s procldures de classification hibrarchique, i.e. celles conduisant i une kkchafne de partitions. On peut, d'une part, utiliser un critere "global" pour calculer cette k k h a i n e , par exemple en cherchant l'ultram&~rique " B 6loignement minimum" de la dissimilarite initiale. On peut, d'autre part, obtenir cette IR?cha'ine, pas 2 pas, en calculant s e s partitions successives B partir soit de la partition triviale TI '(pro-
c&iure
agregative ou ascendante) soit de la partition grossiere
TI
(proc6dure
divisive ou descendante). Par exemple, i chaque &tape d'une proc6dure a?re.gative, on calcule la partition suivante en optimisant un critere local ou global parmi les partitions couvrant celle qu'on vient d'obtenir. Dans la procedure la plus rbpandue (algorithme agregatif de Johnson, 19671, on se donne un indice de dissimilarite entre parties de X, et la nouvelle partition est celle obtenue en fusionnant les deux classes de dissimilarit6 minimum de la partition precedente (critkre local). En choisissant comme dissimilarit6 entre deux parties respectivement le minimum, le maximum ou la mkdiane des dissimilarites entre leurs elbments, on obtient des procedures appel6es respectivement lien simple, lien complet et lien m6dian. I1 est clair que ces trois procedures sont globalement invariantes au sens defini ci-dessus. On verra d'autres d6finitions de celle du lien simple au paragraphe 4 . 3. LES STRUCTURES ORDINALES DES DONNEES ET DES MODELES. 3.1. INDICES DE DISSIMILARITE ET L-CHAINES DE RELATIONS. 3.1.1. Dissimilarites. La donnee de dCpart est le plus souvent (cf. 2.3.) un indice de dissimilarit6, c'est-i-dire une application d : $ , ( X I --t L, OL L est une partie de I R ' contenant le z6ro. Ici, nous retiendrons seulement que L e s t un ensemble totalement ordonne avec un p l u s petit element note 0 . due essentielleUne transformation des indices de dissimilarit6 ,)d'(E''L ment B Janowitz (1S78a) reprenant la problematique de Jardine et Sibson (1971) met l'accent sur l'aspect ordinal du problkme de la classification. Nous presentons d'abord cette idee, que l'on trouve Bgalement s o u s une forme moins 6laborCe et dans le cas particulier des ultram6triques (8.3.2.2.) dans l'article de Boorman et Olivier (1973) : A d E L8z(x) et 2 un seuil X E L, on associe la relation sym6trique R(d,A) = {e E p,(X)/d(e) 5 X I , avec les proprietes imediates :
Pour d fix6, on dBfinit une application f : L--t8(p2(X)) par fd(X) = R(d,X). f est une indiciation par L de la chafne Im f L ) ={ R(d,X)/A E L l du treillis d fd booleen 6(p,(X)). On dira que fd est une L-chafne de ce treillis. I1 est immediat que les L-chafnes f sont caract6risees parmi les applications isotones de L dans par les conditions (LCI) et (LC2) suivantes :
6'(6',(X))
Ensembles ordonnis et tawonomie mathematique (LC1)
€-l(
533
92(X)) # 0
(1.~2) ( V e E
P,(x))
I X E L/e E f(h)I
a un minimum.
quand L est fini. Des formes differentes de (LC2) ont CtC On a (LCl)*(LC2) donnGes par Boorman et Olivier et par Janowitz. Toutes ces fornies sont bquivalentes pour L = R+. Une application f : L + 6'(6',(X)) v6rifiant (LC1) et (LC2) dPfinit un indice de dissimilarit6 d . f ' df(e) = Min { A E L/e E f(X)l. (V e E f',(X)) Les applications d + f et f + df sont inverses et, d'aprhs la proposition 1, antitones. Elles Ctablissenf une correspondance bijective entre les indices de dissimilarit6 i v,ileurs dans L et les L-chafnes.
3.1.2. Rtsiduation. Soient E, F deux cnsembles ordonnks. On note l s o ( E , F ) l'ensemble des applications isotones de E dans F. On dit que f E Iso(E,F) est i-e'sidue'e -resp. re'sidueZZe- ssi, pour tout y,G F , f-l({y€F/y y o } ) est un ideal princiual de E (resp. E Y/y > yo}) cstun filtre principal de E. Si f est rbsidube-rCsiduelle-, f-'({y il existe une application residuelle - residube - unique g E I s o ( F , E ) telle que le couple (f,g) &tablit une correspondance de Galois entre E et l'ensemble ordonne F* dual de F - entre E" et F - (Blytli et Janowitz 1972). Les indices de dissimilarit6 se ramenent aux fonctions residuelles en prenant E = L, F = 8(82(X)). Proposition 3.2. est rhsiduelle.
Une application f : L + P ( I P ~ ( X ) ) verific (Lc1) et (Lc2) ssi elle
Cette Gquivalence, qui reformule celle Ctablie par Janowitz (1978a), reste vraie si L est un sup-demi-treillis avec un 6ltment nul. L'application rCsidu6e g correspondant 5 f est dkfinie par : (V
R C@?(X))
g(R)
=
max {df(e)/e c1 R ] .
Dans I'etude des structures des ensembles de classifications intervient la question suivante : F Ptant un demi-treillis, l'ensemble Res(E,F) des applications r6siduelles de E dans F est-il stable pour 1'opPration de demi-treillis induite dans FE ? @and E est u n ordre total et F un inf-demi-treillis, la stabilitt pour l'infimum de Kcs(E,F) est imm6diate. Rlyth et Janowitz proposent en exercice de montrer que la stabilitt pour l e supremum est egalement v6rifiPe lorsque F est un ordre total. Un autre cas de stabilitb pour le supremum est celui oh F est un sup-demi-treillis fini (mais elle disparaft si E n'est plus un ordre total). Ce dernier cas est celui des L-cliaTnes de p ( p , ( X ) ) , ce qui entrafne le fait que l a correspondance d + fd est un anti-isomorphishie de treillis :
Proposition 3.3.
f d v d , = f d h f d , , et fd A d l
= fdV
fd'
3.1.3. Applications 5 la taxonomie. Soit F 5 &'(4:2(::)) un ensemble de relations symbtriques sur X ayant de bonnes proprietes classificatoires. En posant Y = y:F y , on vbrifie que l'on a, pour f F ( (P(y))L telle que Imf(L) 5 F :
-
f F Res(L,F) f E Kes(L,p(Y)). Si Res(L,F) n'est pas vide, on a Y E F, donc Y = yu: maximum de F. En gbnbral, aucun Clement de f f ? ( X ) n'est exclu a priori des classifications consid6rhes et l'on a Y = P2(X) Alors aux L-chafnes de F sont associGes des indices
.
J.-P. Barthelemy et al.
534
d c clissiini1;irith
p a r t i c u l i c 3 r s ; on n o t c d F I 'ensemble dc' c e s i n d i c e s .
Deux u s a g e s d e 1 3 c o r r e s p o n d a n c e F
+
d
I?
se
:
prhsentcnt.
- L ' u n d e s b u t s d e l a t axonomi c mnt l i &mat i que e s t tlc c o n s t r u i r e une bunnt3 9,(XI
:ipplication
a
: L
+
d F , c e q u i p e u t G t r e ramen&
Ii
l a rc~ctii.rclii. d ' u n e a p p l i c ; i -
t i o n 0 ' : Res ( L, p ( B , ( X ) ) ) + Kes ( L , F ) . Janowitz ( 1 9 7 8 a ) p o s e a i n s i l e p r o b l t n i c d e l a r e c h e r c h e e t d e 1' ; t i i de d e s iiietliudes d e c l a s s i f i c a t i o n dans d e s t c r m e s t o u t - i - f a i t o r d i n a u x . C e t t e a p p r o c l i c C s t ;ihordCe a u p a r a g r a p t i e 4.4.
- P o u r G t i i d i r r l a s t r u c t u r e d ' o r d r e dt, (IF, u n u t i l i s e l e f . i i t q u ' c ~ ll e <'st d u n l e d e c e l l ? d e R e s ( L , F ) . P a r e x e m e l c , s i F e s t u n t r e i l l i s , R L - s ( I . , F ) e s t un s o u s - t r c i l l i s d e I s o ( L , f ? ) , donc d e F d F e s t d o n c u n t r e i l l i s , q u i p"ut c u n s e r v e r c e r t a i n e s p r o p r i t t g s d u t r e i l l i s F . Au p a r a g r a p h e 3 . 2 . 2 . , c c t t e d&rnarc~tie(1st appliqu&t, h 1 ' & t i l d e d e s t r e i l 1 i s d ' u l t r a n i 6 t r i q u e s .
.
3 . 2 . PARTITIONS e t ULTMMETR1C)UES : TKEIT.I,TS SEMI-MODULAIKES.
.
3.2.1
P a r t it ions
.
s L e s p a r t i e s c ~ e63, (x) I C S p1iis t y p i q u e m e n t c l n s s i f i c n t o i r e s s u n t ~ e Cquivalences stir X. On n o t e r a nX ( o u n) l e t r e i l l i s d e s C q u i v a l e n c e s s u r X ( i d i , n t i f i 6 e s aux p a r t i t i o n s d e X a v e c l ' < i r d r e de f i n c x s s e ) , s o n ciltmeiit n u l , q u i e s t l ' b q u i v a l e n c e
:I
1x1
=
i[
=ff2(X)
s o n niaximuni, T O
n classes.
Le t r e i 1 1 i s g t om; t r i q u e ( rl t omi q u e c' t s emi -niodu 1a i r e i n f C: r i e 11 renie n t ) d c s p a rt i t i o n s e s t b i e n connu ( c f . p a r c'xernple R i r k h o f f 1967, Bnrhut e t M o n j a r d e t 1 9 7 0 , Crawley e t D i l w o r t h 1973, G r z t z e r 1 9 7 8 ) . Sa c o m p l e x i t & a Ct6 h t a b l i e a v e c l a d t m o n s t r a t i o n d r l a c o n j e c t u r e d e Whitman ( 1 9 4 6 ) p a r Pudlrik c t Tcimii ( 1 9 8 0 ) .
3.2.2.
UltramGtriqucs.
I.es 111 t r m i t t r i q u e s o n t p r i s u n e g r a n d e i m p o r t a n c e en ta xonomic d e p u i s 1 ' o h s e r v a t i o n s i m u l t a n b e p a r .Julinson ( 1 9 6 7 ) e t B e n z e c r i ( 1 9 6 7 , r e p r i s dans Benzccri 1973) d e l ' t q u i v a l e n c e s u i v a n t e :
P r o p o s i t i o n 3.5. ( 1 ) ( V h C L) (2)
(V x,y,z
E X)
8, (XI
Les d e u x c o n d i t i o n s s u i v c i n t e s s o n t t q u i v a l e n t e s p o u r d E L R ( d , E ) C rlX
d(x,z)
5
max ( d ( x , y ) , d ( y , z ) )
I:n d ' a u t r e s t e r m c s l ' c n s e m b l e d a n s I. ( L . - u l t r a m t t r i q u e s
$,x
(ouu) d e s u l t r a m t t r i q u e s s u r X
s u r X) e s t i d e n t i q u r 3 dT,
u s u e l , e s t un t r e i l l i s a n t i - i s o m o r p h e d e s t a n t i - i s o m o r p h e :i n .
[ i n t j i a l i t t (iltram6triquel.
h Re
(par.3.1.4.),
(TA,n), f:n
I:
valeurs
e t , avec l ' o r d r e
p a r t i c u l i c r , p o u r I*
=
IO, 1 t ,
n
p o u r 1 ' C . t u c l e dtis p r o p r i c t c s d e s t r e i l i s u e t K c s ( L , n ) , on p e u t d ' a b o r d s u p p o s e r I, f i n i . Dans c e c a s , l e s t r e i l l i s f i n i s I s o ( L , n ) o n t p o u r r e l a t i o n d e r o i ~ v e r t u r el a r e s t r i c t i o n d e c e l l e d e : ( V f , f ' t n'a) f > f ' Q "3 h o E I.) f ( h ) > f ' ( h o ) l A [(V h E L - I h 1) €(A) = f'(h)l. Les t r o i s t r e i l l i s n l ' , I s o ( l , , n ) e t r c s ( l , , n ) o n t l a m e m e g r a d u a t i o n = g ( f ( h ) ) e t s o n t s e m i - m o d u l a i r e s s u p & r i e u r e m e n t (Moniez 1 9 8 2 ) , AEL [,cur lonpueur commune e s t I + ( I L I - 1 ) ( 1 x 1 - 1 ) . Le t r c i l l i s f i n i U Gt:int g(f)
sous-suo-denii-trei
I 1 i s d e L p 2 ( X ) c u n t e n a n t s o n 6lGment n u l . Y i n d u i t
un
L I ~ C ouvc'r-
Ensembles ordonnes et taxonomie mathematique
535
L'application lilt e s t idempotente, isotone, anti-extensive [ult(d) 5 d] image e s t u . L'ultrambtrique ult(d) -0cii.e d a de nombreuses appellations traditionnelles : ultramtitrique inftrieure maximale (ultimax), s o u s dominante, distance du moindrc saut, fernirture transitive floue de d (par dualiti.) ... Elle correspond h I'une des methodes de classification les p l u s connues dont on rappellc quelques propri6tii.sau paragraphe 4 . 3 . ) . e t son
L'ouverture ultram6trique esiste encore lorsque L n'est p a s fini. Pour l a construire, on utilise le fait que L
est un treillis d'applications dbfinies P2 (X) sur un ensemble fini, en posant, pour d t L , L~ = Imd(82(x)) et '2(x)
e, (x)
ult ( d ) , oil u l t d cst I'ouverture ultrambtrique dans I. . De facon d d analogue, on montre que le treillis U vhrifiela condition de semi-modularitb d e Dubreil-Jacotin, Lesieur et Croisot (1953) (Cf. Leclerc 1979). ult(d)
=
Une Ctude plus combinntoire des ultram6triques et de leur treillis a b t b faite par Leclerc (1979, 1981). Reprenant une idbe d e Hubert ( 1 9 7 7 1 , on y gbnbralise la coatomicitb de n par l'hcriture de t o u t e ultrambtrique comme supremum d'au plus n-1 ultrametriquessup-irrbductibles. Si L a un hlbment universel, il y a une ghn6ralisation analogue de 1'atomicitP de n . Finalement, les proprietbs ordinales des treillis d'ultrambtriques apparaissent trks proches de celles d e s treillis des partitions. Pour L fini, le dbnombrement d e s L- ultram6triques de msme graduation serait intbressant pour la statistique des classifications. Cn donne plus loin une facon de dhnombrer toutes les L-ultrambtriques. 3.3.ULTRAPREORDONNANCES ET HIERARCHIES : DEMI-TREILLIS A MEDIANES. 3 . 3 . 1 . Demi-treillis b mhdianes.
Un demi-treillis 5 mhdianes est un inf-demi-treillis vhrifiant les deux conditions suivantes (Sholander 1954 ; cf. Bandelt et Hedlikova 1983) : (Ml) Tout idbal principal de M est un treillis distributif. (M2) Pour tout triplet x,y,z de M tel q u e x v y , y v z , z v x existent, x v y v z existe. On dbfinit dualement un sup-demi-treillis 5 mbdianes. Sholander montre notamment qu'un demi-treillis B medianes M admet un plongement avec de bonnes propribths dans un treillis distributif M'. I1 suffit en fait de prendre pour M' le treillis d e s parties commencantes des irrbductibles de M : M est une partie commencante de M' et tout element de M' est supremum d'une partie finie d e M. I1 en resulte que le plongement est isom6trique. Le fait pour un ensemble d'avoir cette structure peut donc a priori faciliter l'obtention d e solutions aux problkmes d'ajustement et d'agrhgation. Des exemples en seront donnbs aux paragraphes 4 . 3 . et 4.6. 11s utilisent notamment le fait que tout triplet x,y,z E M posshde une mediane unique (x A y) v (y A z ) v ( z A x). Pour M fini et k impair, tout k-uple a egalement une mbdiane. De plus, celle-ci conserve les propribtbs mbtriques &tablies par Barbut (1961) pour l a mbdiane dans les treillis distributifs (Bandelt et Barthblemy 1983).
Les demi-treillis 5 mbdianes qui vont Stre considerhs ici sont des demitreillis de charpentes (Bandelt 1982). Soit E un ensemble ordonnh fini, Min E(Max E ) l'ensemble de s e s Clbments minimaux (maximaux). Une partie C de E une charpente ("frame") de E ssi les deux conditions suivantes sont verifihes
est
J,-P.Barthelemy e t al.
536 par C : (Cl) (Min E) U (Max E ) 5 C ( V x E Min E ) ( V y E Max E ) (C2)
(z
E C/x 5 z 5 y) est totnlement ordonnh.
L'ensemble e ( E ) des charpentes de E, ordonn6 par l'inclusion, est un demitreillis 5 m6dianes (avec pour Cl6ment nu1 Co = (Min E) U (Max E)). On dira de deux charpentes C et C ' qu'elles sont presque disjointes lorsque C n C ' = Co. L'exemple le plus imm6diat de demi-treillis de charpentes est don& par l'ensemble des chafnes Ctendues (c'est-$-dire contenant le maximum et le minimum) d'un treillis fini. Si E est un treillis distributif, et F l'ensemble ordonn6 des bl6ments sup-irrbductibles de E, l'inf-demi-treillis des chaines btendues (charpentes) de E est dualement isomorphe au sup-demi-treillis des pr6ordres totaux compatibles avec F. Celui-ciest donc h medianes. En particulier l e supdemi-treillis des preordres totaux sur un ensemble fini S est h mtdianes. 3.3.2. Ultrapreordonnances. Une prbordonnance (dissimilarity relation, PR) sur X est un pr4ordre total P sur P z ( X ) . Soient S 1 , ...,Sk 5 f'z (XI les classes de P indicCes dans 1 'ordre induit par P, C en posant Ro
=
P
= (Ro,
8, Ri
=
Rl,
..., %)
la chafne Ctendue de
S 1 , Rz = S i U S z , .
..R.1
= S.U 1
associbe
ff(82(X))
Ri-l,...,\
=
6)2(X).
i P
On
reprend la demarche du paragraphe 3.2.2. en s'intbressant aux prbordonnances P pour lesquelles les blbments de la chaine Cp ont de bonnes propribtfs, et plus prtcisbment au cas oij ce sont des equivalences sur X. Proposition 3.6. Les deux conditions suivantes sont equivalentes pour une priordonnance D sur X : (1)
La cha?ne CD est une chaine de
nX'
(2) Pour tous x,y,z E X, on a : ({x,y},{x,z})ED
ou ((x,y),(y,z1)€
D.
Une pr4ordonnance vbrifiant ( 1 ) et (2) est une ~rltruprdordonnance (ultradissimilarity relation, UPR) sur X. Les conditions de la proposition 3.6. correspondent respectivement h celles de la proposition 3.5. : le prbordre D(u) induit sur 6 ' 2 ( X ) par une ultrambtrique u est une ultrapreordonnance sur X tandis que l'image de L par laL-chaine fU est la chaine des partitions C (privbe de D(u) R O si fu(0) # no). L'ensemble
d3X
des UPR sur X est un sup-demi-treillis B mbdianes, dualement
isomorphe au demi-treillis des charpentes (chaines) de T l . De plus, d'aprgs l a condition (2) ci-dessus, c'est une partie finissante du demi-treillis px des prbordonnances sur X et, donc, un sous-sup-demi-treillis absorbant (Schader 1980). On dispose dc formules de recurrences pour le c a l c u l du nombre D des -n,k ultraprbordonnances B k classes (de mgme graduation) et du nombre total D des -n ultraprbordonnances sur X (avec n = 1x1). S dCsignant les nombres de Stirling de seconde espece -9P Proposition 3 . 7 .
Les nombres D -n
et
vCrifient :
n- 1
D -n
=
=
i=l
sn,; D i
(Lerman 1970)
n- 1
D -n,k
=
E S D i=1 -n,i -i,k-1
(Schader 1980)
Ensembles ordonnis et taxonomie mathematique
avcc
O n iin t l e d u i t , p o u r I. f i n i , I c nombre U -11,. " = IL :
531
des L - u l t r a m ~ t r i q u c s s u r
1x1
soit,
U
-n,V
i=l 154 u l t r a m k t r i q u e s sur 1 1 , 2 , 3 , 4 3 h v a l e u r s d a n s
I 1 y a , pCir exemple, {O, 1,2,3 t.
3.3.3.
lli6rnrcliies.
P o u r iine u l t r n p r G o r d o n n a n c e D s u r X e t p o u r l a cii;i?ne & t e n d u e CD = ( T ~ , T T ~ , . . . , ~ a~ s)s o c i g e , on p e u t c o n s i d b r e r I ' e n s e m b l e H = HD d e s c l a s s e s d e s C q u i v a l e n c c s d e CD : i l e s t n a t u r e 1 en c l a s s i f i c a t i o n d e s ' i n t h r e s s e r a u x c l a s s e s .
H vPrifie lcs propribtes : X E H b ?! f l (H3) ( V x E X ) [xi E H (H4) ( V h , h ' E H) h n (H1)
(H2)
Une note
11'
E {h,h',@!
hiirarchie s u r X e s t uii h l g m c n t d e P ( p (X)) v k r i f i a n t ( H l ) 5 (H4). On
VX l ' e n s e r n b l e
des h i e r a r c h i e s s u r X.
Pour H
EgXe t
Y
5 X,
l'ensemble
{ h F H/Y 5 h l est totalement ordonnt p a r l ' i n c l u s i o n , c e q u i a deux consequences : e s t I'ensemble dcs charpentes de
p(X) - i01
d e m i - t r e i l l i s arborescent, c'est-;-dire h l v 11: c 111 v h3 ou h l V ti2 5 hz minimale i n i q u e d e H c o n t e n a n t h n h ' . Soit alors D
H
V
; t o u t e h i t r a r c h i e H e s t un s u p -
v t r i f i a n t : (V h , , h , , h i € H) h ; . Pour h , h ' E H , h v h ' e s t l a c l a s s e
l e preordre p a r t i e l d e f i n i s u r ( { x , y j , { z , t } ) E DH
Q
x v y
5
P?(X) p a r
:
z v t.
I,e p r b o r d r e DH v e r i f i e l a c o n d i t i o n ( 2 ) d e l a p r o p o s i t i o n 3 . 7 . , mais n ' e s t p a s t o t a l . P a r c o n t r e , t o u t e p r k o r d o n n a n c e s u r X c o n t e n a n t DH e s t u n e u l t r a p r t o r d o n n a n c e s u r X . 011 a s s o c i e a i n s i 2 t o u t e hierarchic u n e c l a s s e d'UPR : S o i e n t kI u n e h i t r a r c h i e s u r X e t D un p r t o r d r e t o t a l s u r $?(X) Proposition 3.8. c o n t e n a n t DH e t m i n i m a l avec c e t t e p r o p r i g t i . . A l o r s D e s t u n e u 1 t r a p r ; o r d o n n a n c e s u r X e t HD = H .
Les h i g r a c h i e s s o n t d o n c e x a c t e m e n t l e s e n s e m b l e s d e c l a s s e s d e f i n i s p a r l e s u l t r a p r e o r d o n n a n c e s nu l e s u l t r a m e t r i q u e s s u r X ( c e s o n t les f a m i l l e s d e b o u l e s de c e s d e r n i k r e s ) ; d'oh l e s a p p e l l a t i o n s c l a s s i q u e s , B l a s u i t e de Benzecri, de h i G r a r c h i e s s t r a t i f i g e s pour les u l t r a p r 6 o r d o n n a n c e s , e t d e h i e r a r c h i e s i n d i c h e s p a r les u l t r a m e t r i q u e s ( r e s p e c t i v e m e n t " o r d e r e d t r e e s " e t "valued trees" chez Boorman e t O l i v i e r ) . Une e t u d e d e s d e m i - t r e i l l i s d e h i e r a r c h i e s , o r i e n t g e v e r s l a t y p o l o g i e e t l a c o m p a r a i s o n d e c e l l e s - c i , a e t b f a i t e p a r L e c l e r c ( 1 9 8 2 ) . On y met notamment l e demi-treillis e n r e l a t i o n avec l ' o r d r e rnodulaire d e s p a r t a g e s d e I ' e n t i e r n - 1 e n e x h i b a n t un m o r p h i s m e d ' o r d r e , p r e s e r v a n t l a c o u v e r t u r e , d u p r e m i e r s u r l e s e c o n d . Le p r o b l P m e d u dbnombrement d e s h i e r a r c h i e s r e m o n t e B S c h r i j d e r ( 1 8 7 0 ) . Comtet ( 1 9 7 0 ) a dCnombr6 l e s h i e r a r c h i e s d e m&ne g r a d u a t i o n p a r l e s n o m b r e s d e S t i r l i n g 2 - a s s o c i e s d e s e c o n d e e s p t c e , e t e n a d b d u i t un c a l c u l p a r r b c u r r e n c e e t un G q u i v a l e n t a s y m p t o t i q u e d e s n o m b r e s ( d u q u a t r i k m e p r o b l k m e ) d e S c h r o d e r .
xX
Les h i e r a r c h i e s maximales ( d i t e s gbneralement h i e r a r c h i e s b i n a i r e s ) o n t 2n-1 b l b m e n t s . P o u r l a s t a t i s t i q u e d e s h i e r a r c h i e s , l a r e p o n s e B c e r t a i n e s q u e s -
J.-P. Barthelerny et al.
5 38
tions surl'intersection des hiCrarchies serait intCressantc. Plus gbneralement, les problemes classiques d'intersertion de ch,iPnes peuvent stre posGs pour l e s intersections de chdrpentes. Par exemple, pour un ensemble ordonn6 F fini, on pc'iit chercher l e nombre maximum de ses charpentes maximales presque disjointes deux 1' deux. Si E a un plus petit et un plus grand flfment c'est le nombre maximum d c chaPnes maximales presque disjointes de E. Une rCponse est fournie p a r l e th6ot-Cme de Menger (1927, cf. Berge 1 9 7 0 ) . 4 - COMPARAISONS, AJUSTEMENTS, AGREGATION : METHODES ORDINALLES.
4.1. Situations. Nous nous intkressons, ici, d u s mCthodes de classification qui conduisent construire une fonction 9 : E + F, E et F Ctant deux ensembles ordonnes (dans l a pratique, il s'agira, bien sSr, des ensembles ordonnbs, treillis ou demi-treillis dCcrits au paragraphe 3 ) . Une telle fonction sera appelCe Line prucd&irt di,
classification. Deux cas se rencontrent essentiellement : 2 F. On dira que 9 est un ajustement. Un exemple classique est : E = L "(') (ensemble des indices de dissimilarit4 bur X h valeurs dans I.) et F = L,X (ensemble des ultramktriques sur X, 2 vnleurs dans L). Un autre serait E = 8 ( 8 , ( X ) ) (ensemble des relations reflexives et sym6triques sur X) et F = r'' X (ensemble des kquivalences, ou des partitions, de X). Dans le premier cas on "approche" une dissimilarit6 par une distance ultrametrique de manihre B construire une hi6rarchie indicke ; dans le second on "construit" une partition 5 partir d'une "relation de ressemblance" quelconque. (1) E
u
(2) E = U FV. On dira alors que 9 est une agre'gaiation. Ce c a s se rencontre, par exemple,"dans la situation suivante : l'ensemble X est dkcrit par v variables, chacune d e ces variables induit sur X une partition : deux blkments pour lesquels elle prend la mOme valeur sont dans l a m@me classe. I1 s'agira, des lors de rksumer, en une seule, ces v partitions (on a donc, ici, F = I$,,). Bien scr la fonction 9 ne sera pas n'importe quai. On exigera qu'elle vCrifie certaines "contraintes" (au sens le plus large d e c e terme). On peut regrouper ces contraintes en trois rubriques (d'ailleurs largement empibtantes). a) .Rhgles On indique explicitement l'art et la manihre de . . . . . . .de . . . construction. ........... construire la fonction 9. On ohtient ainsi les proce'dures a2gorithrnique.s (par exemple, la construction d'une hierarchic indicbe, ou stratifiee, en fusionnant, h chaque niveau, des classes, de maniere B optimiser un critgre). On se restreint aux fonctions 0 verifiant des b) ContrainteS-"axjomatiques". conditions jugees indispensables du point de vue oh l'on se place. On dCbouche alors sur des problemes d'existence et le cas &heant, d'unicite o u sur la caract6risation de toutes les "solutions". ___________S__-T_____-------
On s'intkresse aux fonctions 9 qui permettent c) Contraintes_dloetjmjsatjon_. --'-r-------"--T-r--n------Par exemple, lorsque est un ajustement, d 6tant d'optimiser un critere une distance sur E, on demande que 0(r) soit une solution de Min d(r,s).
.
sEF
Cette r6partition en (a), (b), (c) appelle un certain nombre de remarques : Remarque 1. Les ensembles ordonn6s interviennent fort diversement dans ces trois cas. Si les constructions algorithmiques (cas (a)) ne sont qu'exceptionnellement fond6es sur des crithres ordinaux (la methode dite du "lien simple" que nous rencontrerons un peu plus loin constitue une telle exception), les "axiomes" (cas (b)) le plus souvent rencontr6s sont fortement libs aux ordres de E et F. Enfin, pour le cas (c), les criteres qu'utilise le taxonomiste sont parfois lies B ces structures ordonn6es.
Ensembles ordonnes et taxonomie mathematigue
5 39
Remarque 2 . 1.a r u b r i q u e ( c ) ( e t d a n s une c e r t a i n e m e s u r e ( a ) ) i m p o s e l ' e x t e n s i o n de l a notion de procedure de c l a s s i f i c a t i o n . I 1 n ' y a , en e f f e t aucune r a i s o n de p r c s u p p o s e r 1 ' r i n i c i t c d ' u n h l c m e n t d e F q u i o p t i m i s e un c r i t t r e . On e s t a i n s i conclriit 311x < ,< l -L i icati:or;, q u i s o n t d e s a p p l i c a t i o n s d e E d a n s 1 ' e n s e m b l e cles p a r t i e s d c F . I'
Keni;irqur. 3 . L ' c x e n i p l e q u ' o n R d o n n e p o u r ( c ) m o n t r e q u ' i l p e u t 8 t r e n G c e s s a i r e d ' i i t i l i s e r une d i s t a n c e (ou un i n d i c e d e d i s t a n c e ) s u r E . Indgpendamment d c t o u t s o u c i d ' a j u s t e m e n t , d e t e l l e s m c t r i q u e s s o n t f o r t u t i l e s pour comparer d e s c l a s s i f i c a t i o n s . A c e p r n p o s , nous r e t r o u v o n s nos t r o i s r u b r i q u e s . Les r t g l e s d e c o n s t r u c t i o n p c rme t t e t i t d e c a 1c u 1 e r r f f e c t ivemen t un e d i s t a n c e Le s c o n t r a i n t e s a x i o n i a t i q u e s c u n d u i s e n t 5 ne c u n s i d e r e r q u e c e l l e s q u i v 6 r i f i e n t c e r t a i n e s c o n d i t i o n s , j u ~ : G e s " n a t u r e l l e s " clans l e c o n t e x t e oh l'on t r a v a i l l e . E n f i n , on p e u t t'xiger q u ' u n c r i t G r e s o i t o p t i m i s r ( e n t r a v a i l l a n t , p a r exemple, a v e c d e s " d i s t a n c e s d u p l u s c o u r t rhemin")
.
.
Reniarque 4 . Cornme s o t i v e n t , p l u t 8 t q u e 1 ' C t u d c d e ('11, ( b ) , ( c ) p r i s i s u l e m e n t , un p o i n t t o u t 11 f a i t e s s e n t i e l s e r a l a d G t e r r n i n a t i o n d e r a p p o r t s e n t r e c e s t r o i s c a s . P a r e x e m p l e , Line p r o c e d u r e v e r i f i a n t u n e c o n t r a i n t e d ' o p t i m i s a t i o n s e p e u t c l l e c o n s t r u i r e I: l ' , l i d e d ' u n n l g o r i t h m e ? q u e l l e s s o n t s e s p r o p r i G t 6 s ? ( c o n f r o n t a t i o n aux a x i o m a t i q u c s d ' a u t r c s p r o c G d u r e s : q u e c o n s e r v e - t - o n , q u ' a b a n d o n n e t-on ? ) . P e u t - o n n l l e r j u s q u ' 2 lcas c a r a c t 6 r i s e r a x i o m a t i q u e m e n t ? D;ins c c t t e qlJatrii.nie p a r t i e , n o u s G t u d i e r o n s s u c c e s s i v e m e n t l e s c o m p a r a i s o n s , a j u s t e m e n t s e t a,qrGgntions d e c l ~ s s i f i c a t i o n s , : I l ' a i d e d ' e x e m p l e s , sous l ' 6 c l a i rage des t r o i s rubriques ( a ) , ( b ) , ( c ) . 4 . 2 . Compnraisuns de c l , i s s i f i c a t i o n s . 1.e s ~ 1 6 < ~ ~ & s iS~U~I !E< ce o: n s t i t u e n t d e s m G t r i q u e s f o r t l i 6 c . s h sa s t r u c t u r e ordonnee. L k t e l l e s d i s t a n c e s s ~ n td c s " d i s t a n c e s du p l u s c o u r t chemin" d n n s l e g r a p h e ( n o n o r i e n t h ) d e c u u v e r t u r e d e E , u n e a r S t e uv d e c e g r a p h e e t a n t 6valui:e p a r I . b ( u ) - $ ( v ) I , 9 G t a n t une f o n c t i o n , 5 v a l e u r s r e e l l e s , s t r i c t e m e n t c r o i s s i i n t c siir E . O n t r o u v e r a , clans l ' n r t i c l e d e s y n t h s s e d c M o n j a r d e t ( 1 9 8 1 ) , l e s d i v e r s e s p r o p r i 6 t i . s a c t u e l lenient c o n n u e s d e c e s d i s t a n c e s a v e c l e s i n d i c a t i o n s b i b l i o g r a p h i q u e s c o r r e s p o n d a n t e s . En p a r t i c u l i e r , u n e h y p o t h t s e s u r $ ( v a l u a t i o n i n f e r i e u r e ou s u p 6 r i e u r e ) f o u r n i r a une c o n d i t i o n n g c e s s a i r e e t s u f f i s a n t e p o u r q u e l a d i s t a n c e g G o d 6 s i q u e dd, ( d o n c d e f i n i e p a r une c o n t r a i n t e d ' o p t i m i s a t i o n ) p u i s s e S t r e c a l c r i l t e p a r l a "formule" :
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l e p r e m i e r c a s , on p e u t c i t e r : A r n h i e e t Doorman ( 1 9 7 8 ) , Boorman ( 1 9 7 0 ) , I3oorm;in e t A r a b i c ( 1 9 7 2 ) , BarthGlemy ( 1 9 7 9 ) , Day ( 1 9 8 1 ) . D a n s l e s e c o n d c u s : Robisucl e l Hobisud ( 1 9 7 2 ) , H u b e r t o t 13;iltcr ( 1 9 7 7 1 , Lec l e r c ( 1 9 8 P ) , Mnr}:ush ( 1 9 8 2 ) , S o h 1 e t RohlF ( 1 9 6 2 ) , Watermnn e t S m i t h ( 1 9 7 8 1 , Wil1i;irns e t C l i f f o r d ( 1 9 7 1 ) . N o u s v e n o n s d e v o i r i n t e r v e n i r l e s r u b r i q u e s ( 3 ) ('t ( c ) . Q u ' c n e s t - i l de ( h ) ? O n s e r a t t n c h e i i i U I I problGme dia Iri c o r a c t 6 r i s ; i t i o n n x i m a t i q u e de d i s t a n nnns l e r a s d e s r e l a t i o n s b i n r i i r e s c t ' theme reniontc Kcnicny ( 1 9 5 9 ) d o n t l a tiGniarc h t . a ;tt r e p r i s e p o u r l a c l a s s i f i c a t i o n p;ir M i r k i n e t Cliernyi ( 1 9 7 0 ) q u i rardL't 6 r i s ; S r e n t l a d i s t a n c e d e I n d i f f G r e n c e syni;triqrie e n t r e d e s r e l n t i o n s d ' 6 q u i v a l e n c c . N o t o n s q u ' i l s'agit, en f n i t , d e I n d i s t a n c e g & o d t s i q u e s u r !Ix p o u r P 2 TI = {Al, A ] ) . Le ccls d e s I i i G r a r c t i i e s ;i c t c G t u t l i G par , l ( n ) = 1 [ A i l (avoc i =1 P L e c l e r c ( 1 9 8 2 ) e t Plargush ( 1 9 8 2 ) . Remarquons e n f i n ( R a r t h 6 l e n i y 1979h) q u e C E S c n r a c t 6 r i s n t i o n s a x i o m a t i q u r s dependent 6 t r o i L e m e n t d e l a s t r u c t r i r e orc1imni.e de E e t sous d e s l i y p o t h t s e s c o n v e n a b l c s ( I n s e m i - m o d u l a r i t 6 , p a r r x c 9 n i p l e ) a p p a r a i s s e n t s o u v e n t comme des a p p l i c a t i o n s d e rssul t a t s b e a u t o u p p l u s gt!nGr;iux.
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4 . 3 . A j u s t e m e n t s I : t r o i s cxxemples. 1 , ' e x e m p l e l e p l u s c l n s s i q u e d ' a j u s t e m e n t e s t s a n s d o u t e l a tr.2 sivipTc ( " s i n g l e l i n k a g e " ) , C e l l e - c i a g t t ! abordcc. d e d i v e r s e s m a n i b r e s e t a r r s u d i v e r s noms ( c f . 3 . 2 . 2 . ) . h n s une p e r s p e c t i v e d c c l u s s i f i c a t i o n a s c e n d a n t e tiit!r a r c h i q u e , on f u s i o n n e deux c l a s s e s d o n t deux 6 l b m e n t s s o n t l e s p l u s p r o c h e s v o i s i n s ( d ' o h son n o m ) .
I-n d ' a u t r e s t e r m e s , i l s ' n g i r a d ' a p p r o c h e r u n e d i s s i m i l a r i t 6 p a r u n e u l trcim t t r i q u e , o u , c e q u i r e v i e n t a u m@rne, ( c f . 3.1.3.) d e t r a n s f o r m e r Line a p p l i c a t i o n r g s i d u e l l e f : T,+ ff(e,(X))e n uiie a p p l i c a t i o n g : L - , f ( I [ x ) . La m t t h o d e d u l i e n simple c o n s i s t e , p o u r uiie d i s s i m i l a r i t 6 d , <' c a l c u l e r I ' u l t r a m C t r i q u e supremum d e s u l t r a m 6 t r i q u e s i n f c r i e u r e s B d . I1 s ' n g i t 1,'i d e I ' o u v e r t u r e u l t r a m b t r i q u e de 3 . 2 . 2 . . C e t t e d e r n i h r e se c o n s t r u i t e f f i c a c e m e n t en e x t r a y n n t du g r a p h e c o m p l e t s u r X , v a l u 6 p a r d , un a r h r e m i n i m a l . C e t t e r e m a r q u e i l l u s t r e l ' a s p e c t ( a ) ( r 6 g l e s d e c o n s t r u c t i o n ) . L e mot " o u v e r t u r e " , quaiit 5 l u i , r e n d c o m p t e d e ( b ) . En e f f e t , c h e r c h o n s B d 6 t e r m i n e r l c s f o n c t i o n s 11,: E + F , o t ~E = F = I, B ' ( x ) d(x,y) < d'(x,y)),
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une d i s t a n c e u l t r a m 6 t r i q u e ( c o n d i t i o n c l n s s i f i c a t o i r e )
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O(d) a u g m e n t e o v e c d ) .
On r e m a r q u e q u e U *, U j , U s o n t d e s p a r a p h r a s e s d e , r e s p e c t i v e r n e n t : @ e s t 4 i d e m p o t e n t e , t e s t a n t i e x t e n s i v e , @ e s t i s o t o n e . C'est donc e n c o r e l ' o u v s r t u t - e u l t r a m G t r i q u c q u e l ' o n r e n c o n t r e . On t r o u v e r a d a n s J a r d i n e e t S i b s o n ( 1 9 7 1 ) unc c a r a c t 6 r i s a t i o n a x i o m a t i q u e d e c e t t e o u v e r t u r e oh i n t e r v i e n n e n t d e s considerations topologiques.
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N o t e r q u ' o n p e u t s ' i n t ~ ! r r s s c - r ail p r o b l s m e d u a l , b e a u c o u p p l u s d i f f i c i l e , d e m i n i m i s a t i o n de 6 ( d , u ) p o u r l e s u l t r a m k t r i q u e s s u p G r i e u r e s ou & g a l e s h d ( c f . D e f a y s 1 9 7 5 ) . Lc-s s o l u t i o n s , non n & c e s s a i r e r n e n t u n i q u e s , s e t r o u v e n t p a r m i 1 ' e n semble d e s u l t r n m P t r i q u e s s u p c r i c u r e s m i n i m a l e s d e d , d o n t l a c a r a c t G r i s a t i o n e t l ' o b t e n t i o n s o n t d e s p r o b l h m e s o u v e r t s ( l a m c t h o d e du l i e n c o m p l e t , k v o q u b e e n 2 . 3 . 4 . p r o d u i t line t e l l e u l t r a m G t r i q u e ) . Deux n u ~ r e st e c h n i q u e s d ' a j u s t e m e n t s o n t , :I d e s d e g r 6 s d i v e r s , l i k e s a u x s t r u c t u r e s c i ' o r d r e : 1 3 mettiode dc- Zalin ( 1 9 6 4 ) e t c e l l e d e S c h a d e r ( 1 9 8 0 ) .
La p r e m i t r e c o n s i s t e :I a j u s t e r u n e r e l a t i o n s y m g t r i q u e S p a r une r e l a t i o n d ' G q u i v a l e n c e IT. P o u r r e f a i r e , on c h o i s i t u n e G q u i v a l e n c e q u i m i n i m i s f 6 ( S , n ) , 6 P t a n t l a d i s t a n c e d e l a difference s y m g t r i q u e e n t r e d e u x r e l a t i o n s s y m b t r i q u e s . I1 s ' a g i t d o n c d ' u n e m u l t i p r o c G d u r e . C ' e s t p a r I ' i n t e r m G d i a i r e d e 6 , q u i e s t d C f i n i e p a r line v a l u a t i o n s u r f ' ( 6 ' ? ( X ) ) , q u e s e f a i t 1 e l i e n a v e c l e s ensembles o r d o n nGs. I'our le p o i n t ( a ) m e n t i o n n o t i s q u e 1 ' G q u i v a l e n c e d e Zahn e s t s o l u t i o n d ' u n programme l i n t a i r c en n o m b r e s e n t i e r s . Le p o i n t ( b ) e s t , q u a n t 5 l u i , un p r o b l t m e ouvert. C e t t e tprhnique f a i t p a r t i e de la v a s t e c l a s s e des "proctdures mtdianes" (Bartti6lemy-Monjardet 1981). 1.a mGthode d e S c h a d e r e s t t r t s u r d i n a l e d ' e s p r i t , e t d e m i s e e n o e u v r e . I1 s l a g i t d ' a j u s t e r une p r t o r d o n n a n c e p a r une u l t r n p r C o r d o n n a c e . Les p r 4 o r d o n n a n c e s s u r X f o r m e n t un s u p - d e m i - t r e i l l i s s e m i - m o d u l a i r e s u p t r i e u r e m e n t ( 3 . 3 . ) , d o n c g r a d u k p a r u n e v a l u a t i o n s u p C r i e u r e @ ( M o n j a r d e t 1 9 8 1 ) . La d i s t a n c e du p l u s c o u r t c h e m i n y e s t donc donnPe p a r l a t o r m u l e d ( u , v ) = 2 $ ( u v v ) - $ ( u ) - $ ( v ) . L ' e n semble d e s u l t r a l , r ; . o r d o n n a n c e s sur X f o r m e 1111 r G s e a u d a n s l ' e n s e m b l e o r d o n n k d e s prbordonnances. Schnder propose d ' a s s o c i e r 5 l a pr4ordonnance P 1' u l t r a p r t o r donnance (en f i i i t e l l e e s t u n i q u e ) s o l u t i o n d e :
DX
+
I1 s ' a g i t d o n r d'une p r o c P d u r e d P f i n i e p a r u n e c o n t r a i n t e d ' o p t i m i s a t i o n . Une c n r a c t b r i s a t i o n a x i o m a t i q u e d e c o u l e i m m k d i a t e m e n t d e l a r e m a r q u e g k r 4 r n l e c i - d e s s o u s , ob I ' o n c o n s i d t r e une d i s t a n c e " a s s o c i k e " 5 u n e f o n c t i o n $ ( : E + l R , s t r i c t e m e n t c r o i s s a n t e ) p a r l a f o r m u l e 1" du p a r a g r a p h e 4 . 2 . , ou sa g h b r a l i s a t i o n au cas n o n l a t t i c i e l ( c f . M o n j a r d e t 1 9 8 1 ) . Proposition 4.1. S o i t E un e n s e m b l e o r d o n n 4 a v e c un p l u s g r a n d 6 l C m e n t . S o i t A un r e s e a u f i n i s s a n t d e E e t s o i t d u n e d i s t a n c e " n s s o c i k e " h une f o n c t i o n (1 b v a l e u r s r e e l l e s e t s t r i c t e m e n t c r o i s s a n t e s u r E . L'blernent d e A q u i m i n i m i s f d ( a , r ) , p o u r r € E , e s t u n i q u e e t e s t l ' i m a g e d e r p a r une f e r m e t u r e $ s u r E . Le f a i t q u e l a p r o c 6 d u r e d e S c h a d e r e s t u n e f e r m e t u r e p e r m e t , o u t r e u n e c a r a c t 6 r i s a t i o n a x i o m a t i q u e , d ' o b t e n i r un a l g o r i t h m e f o r t s i m p l e . On r e n d compte a i n s i d e s p o i n t s ( a ) , ( b ) , ( c ) . I1 f a u t cepcrndant n o t e r q u e c e t t e rnbthode s e &v&l e peu d i s c r i m i n a n t e .
4 . 4 . A j u s t e m e n t 1 1 . Les p r o c k d u r e s h s e u i l e t l e m o d s l e o r d i n a l d e J a n o w i t z . La mCthode d u l i e n s i m p l e , d G c r i t e a u p a r a g r a p h e p r e c b d e n t a p p a r t i e n t 5 l a c l a s s e d e s p r o c k d u r e s i s e u i l ( " f l a t m e t h o d s " ) . Ces d e r n i g r e s , i n t r o d u i t e s p a r Jardine e t Sibson ( 1 9 7 1 ) o n t kt4 e t u d i k e s ( e t g6nGralisbes) dans l e cadre d'un modsle purement o r d i n a l p a r J a n o w i t z .
5 42
J . 2 Barthelerny et al.
I.eur p r i n r i p e p e u t s e resumer con~me s u i t : u n e m4thode d e c l a s s i f i c . a t i o n t r a n s f o r m e une d i s s i m i l a r i t 6 d en une d i s s i m i 1 ; i r i t b t t ' ( d ) ( o n o u b l i e i c i 1 'casi>:encc s u p p l 6 m e n t a i r e : & ( d ) e s t une u l t r a m G t r i q u e ) . A c h a q u e s r u i l X E L c o r r c s p o n d I n r e l a t i o n s y m C t r i q u e f d ( X ) = t I x , y l t e l q u e c l ( x , y ) 5 A , } . On demandc q u e f ~ ; , ( ~ i ) ( I ) ( i . c . " @ ( d ) a u s e u i l A " ) ne d6pcnde que de f d ( A ) ( i . e . tie " d a11 s e u i l A"). Te c h n iq u e me n t : i c h a q u e d i ssi ni i I n r i t e d c o r r c s p o n d I J I.-clinine f d E R e s( L, 8 ( p,(X)) ( 3 . 1 . 2 ) . On c i i t a l o r s clue I n p r o c 6 d u r e d e c l n s s i i c a t i o n '11 : I, i p 2 ( X ) + I . p 3 2 ( X ) e s t L: scL4i/ s ' i l e x i s t e t e l l e q u e , pour t o u t e L - d i s s i m i l n r i t & d : a l o r s npye1i.e u n e j b n i r t i o n
line
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a p 7 l i c a t i o n y : x'(P,(x))+ fd = 1,a [ u n c t i o n y
P( P , ( x ) ) rst
Dks l o r s s e p o s e n t deux q u e s t i o n s (les " f l a t proble ms " tle J n n o w i t z 1 9 7 8 ~ ) . 1) c a r a c t t r i s e r I e s p r o c P d u r e s h s e u i l , 2) c a r a c t b r i s e r l e s f o n c t i o n s s e u i l . J a n o w i t z l e s t r a n s f o r m e e n p r o h l 6 m e s plus g b n e r a u x e t pure me nt o r d i n n u x e n r e m p l a c a n t L p a r un s u p - d c m i - t r e i l l i s ;aver u n p l u s p e t i t 616nrent e t ff( B~(x)) p a r deiix e n s e m b l e s o r d o n n 6 s q u e l c o n q u e s M e t N , a v e c un p l u s g r a n d 6 l b m e n t . Dans c e c a d r e , u n e f o n c t i o n s ' 1 s e r a une a p p l i c a t i o telle q u c , pour t o u t g F Re s( L,M ) , y o g E R e s ( L , N ) . De rnSnie ;in<' [ I m c ' 7 s e r a line a p p l i c n t i o n @ : Res(L,M) + Res(T,,N) t e l l e qu'il existe y : M + N vi'rifiant ,?(f) = y o f . O n t r o u v e clans J a n o w i t z l a c a r a c t i ' r i s a t i o n c o m p l t t e d e c e s proc-6tiures :I s e u i l e t de c e s fonctions s e u i l (dans l e premier c s , selon l a n a t u r e de L : ch aPn e o u non c h a f n e o n o b t i e n t d e s r b s u l t a t s d i f f r e n t s ) . Une c ons i'que nc c rc ma rquable des r 6 s u l t a t s de Janowitz e s t l ' i s o t o n i e d e s proc6dures L s e u i l e t d e s f o n c t i o n s s e u i l ( p o u r c e s d e r n i i i r e s , l o r s q u e L ri'est p a s une c h a i n e :i d r u x Clements).
Les p r o c e d u r e s 2 s e u i l o n t donnb l i e u i d i v e r s a f f a i b l i s s e m e n t s t e l s clue l e s " s e m i - f l a t me t h o d s" ( J a n o w i t z 1978b) e t l e s "monotone e q u i v a r i a n t m e t h o d s " ( J a n o w i t z 1 9 7 9 a ) . C e s d e r n i k r e s a p p a r a i s s e n t comme une c l a s s e b i e n c a r a c t 6 r i s t ! e , d'un p o i n t d e v u e o r d i n a l , d e s mktlrodes h i 6 r a r c h i q u e s a s c e n d a n t e s .
E n f i n , i l c o n v i e n t d e m e n t i o n n e r i c i q u e c e t y p e d e proc G dure r e l t v e d e c o n s i d t r a t i o n s l i C e s i la t h b o r i e d e s g r a p h e s ( l e s f o n c t i o n s s e u i l 6 t a n t d e s t r a n s f o r m a t i o n s d e g r a p h e s ) . P a r exemple, la mCthode d u l i e n s i m p l e r e v i e n t :i a s s o c i e r , h chnque r e l a t i o n ( g r a p h e ) s e u i l , l ' k q u i v a l e n c e obtenue p a r f e r m e t u r e t r a n s i t i v e ( d o n t l e s c l a s s e s s o n t l e s c l a s s e s c o n n e x e s d e c e g r a p h e ) . L e thPme " g r a p h e s e t ta x o n o mi e " e s t d 6 v e l o p p 6 notamment d a n s H u b e r t (1974 a e t b), Ma tuln ( 1 9 7 7 ) , Day ( 1 9 7 7 , 1 9 7 8 ) , M o n j a r d e t ( 1 9 8 0 ) .
4.5. A j u s t e m e n t 111 - L ' a p p r o c h e g a l o i s i e n n e . Nous a v o n s e u l ' o c c a s i o n d e c o n s t a t e r q u e l e s f e r m e t u r e s ( e t l e u r s d u d e $ l e s o u v e r t u r e s ) i n t e r v i e n n e n t en t a x o n o m i e . L ' a p p r o c h e g a l o i s i e n n e 1 e s u t i l i s e d t : l i b c r g m e n t , a i n s i q u e l ' i m p o r t a n t e n o t i o n d e p o i n t f i x e d a n s un e n s e m b l e ordonnC. L ' i d b e d e s c l a s s i f i c a t i o n s g a l o i s i f n n e s e s t d ' o b t e n i r une p a r t i t i o n d o n t l e s c l a s s e s s o n t formCes d ' o b j e t s " p o s s P d a n t t o u s e t p o s s b d a n t seuls un ou p l u s i e u r s c a r a c t k r e s communs" ( L a l a n d e , V o c a b u l a i r e d e l a p h i l o s o p h i c , c f Degenne, Fla me nt e t V e r g t s 1978).
.
On p a r t d ' u n t a b l e a u en 0 , l c r o i s a n t un e n s e m b l e d ' o b j e t s X e t un e n s e m b l e d e c a r a c t 6 r e s Y . Un t e l t a b l e a u t r a d u i t une r e l a t i o n R e n t r e X e t Y e t on p e u t l u i a s s o c i e r une c o r r e s p o n d a n c e d e Galois ( B i r k h o f f 1 9 4 0 ) , ( n , B ) e n t r e P(X) e t I'(Y) : a ( A ) = {y/y F Y e t (x,y) B ( B ) = {x/x E X e t ( x , y )
t R p o u r t o u t x E A1 E R p o u r t o u t y F B}
Ensembles ordonnes e t taxonomie mathematique
t!~i
est
iin
dit
q u ' r ~ n c - p a r t i t i o n r F [Ix
:ilL)rs
5 43
c'st
f e r m b pour 1 3 f e r m e t ~ i r e f i a l o i s i e n n e
1.i~.
P l u s ::bnCrnlement, c o n s i d C r o i i s Line f e r m e t u r e ip s u r 1 ' e n s e m b l e p(X) e t c h e r clions I c s p a r t i t i o n s I: d o n t c h a q u c c l n s s e est u, - f e r m 6 e ( n 5 i p ( 8 ( X ) ) . C o n s i d & r o n s l a f o n c , t i o n f : ) I x + nx d c f i n i c p a r : f ( n ) = l e s " c c ~ m p o s n n t e s c o n n e x e s " d e s
ip(Ai) des
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s m t l c s cl.isscs d e
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TI
lion d i s j o i n t s .
L ' a p p l i c . i t i o n f esL i s u t o n r ' e t e x t e n s i v e . On s a i t a l o r s ( F l a m e n t 1978) q u ' i l e x i s t c sur tine ftmrnieturc ', t c - l l e qire : y(nX) = 1 ; 1 F 1 ! ~ / f ( ~ r )= II} ( p o i n t s f i x e s c1c7
f ) . c)r,
TI
c -
ip(
P(x))
s i e t seiilenient s i f ( n )
= 71.
I . ' u t i l i s , i t i o n d ' u n "tliGortmc. d e p o i n t f i x e " a donc p e r m i s d ' G L a b l i r l ' e x i s tc'nce d ' u n e c l a s s i t i c a t i o n g a l o i s i e n n e , e t s u n p r i n c i p e i t 6 r a t i f c o n d u i t t o u t n;i t l i r e 1 1 ement ;I un a I g o r i t lime
.
4 . 6 . Agrcgation (vu c o n s e n s u s ) , P o u r line a g r b g a t i o n , i l s ' a g i r n d e c o n s t r u i r e , p o u r chaque e u t i e r v , u n e a p p l i c a t i o n ~1~ = EV E . On r e t r o i i v e 111 un d o m n i n c c l a s s i q u e d e s m a t h 6 m a t i q u e s s o c i a l e s : l a tliCcirie d u c t r o i x r ~ i l l c c t i t . L ' n p p r o c h e a x i o m a t i q u e , i n a u g u r s e e n c e d o m a i n r p a r Arrow ( 1 9 6 2 ) a C t ; r c a p r i s e p a r M i r k i n ( 1 9 7 5 , c f . a u s s i 1 9 7 9 ) p o u r ' r des r e l a t i o n s d ' e ! q u i v n l e n c e . L e s c o n d i t i o n s a x i o m a t i q u e s q u ' o n i m p o s e R U X p r o c b d u r e s d ' a j i r 6 g a t i o n p e u v e n t , e n t a i t , s't3crii-e d a n s n ' i m p o r t e q u e l ensemb l e o r d o n n C . C o n s i d c r o n s d u n c , d ' u n e m a n i h r e j i b n 6 r a l e , un e n s e m h l e o r d o n n t E e t u n e n t i e r p o s i t i t v , unc a p p l i c a t i o n : EV + I: e t n o t o n s V I ' e n s e n i h l e [I, vl --f
...,
+
pour t o u t i , r n t r a i n e s lorsque : ( r l l ,..., r;),
5
'0
(I:).
LW
i I s
(NM) p o u r t o u t
1 ' i nc 1 u s i o n :
ri =
(r
,...,r v ) t i I s S
c t ri'
5 r.i 5 (rI)
=
irl,
5
,..., r '
ti I s'
implique
pour t u u t s E F, 1'bgalitG e n t r a T n e qiie :
r.']
5
rli1
s'
5
),pour tout s , s ' F F, entrai'ne que :
P (7').
T o u t e f o n c t i o n F - n e u t r e m o n o t o n e e s t F - d & c i s i v e . L o r s q u e F = E , on d i r a d t c i s i v e ( r e s p . n e u t r e monotone) a u l i e u d e F - d e c i s i v e ( F - n e u t r e monotone). C o n s i d G r o n s m a i n t e n a n t line f a n i i l l c f i n i s s n n t e F d e p a r t i e s d e V. On d i r a q u e 0 e s t F - r q j ( i l i t a i rc 1o r s q u e : p o u r t o u t r) = ( r ,,...,r v ) , l ' e c r i t u r e : V
IEF
A
r.
i 6 1
a un s e n s d a n s l ' e n s c m b l e o r d o n n e E ( c e s e r a l e c a s , e n p a r t i c i i l i e r , I o r s q u e E e s t 1111 t r e i l l i s ) e t c o i n c i d e RVL'C 0 (n). L o r s q u e 6r c o i ' n c i d e a v e c l e f i l t r c e n g e n d r k p a r H 5 V , on o b t i e n d r a line a p p l i c a t i o n B - m a j o r i t a i r e s i e t seulement s i E est un i n f - d e m i - t r e i l l i s . Cellec i s e r a a l o r s d e l a fornie :
5 44
J.-P.Barthelemy et al.
On dira alors que 0 est une fonction H-olic{urchique. Si P e s t un ultrafiltre 0 est une projection (alias une "procbdure dictatoriale")
.
Les caractkrisations des fonctions parbtiennes et monotones actuellement connues conduisent toutes a des fonctions F-majoritaires qui peuvent @tre de formes tr$s varibes suivant l'ensemble ordonnG considbrc!. Ainsi : Proprosition 4.2. (Monjardet 1982). Si E est un inf-demi-treillis r'i mtdianes ct si F est l'ensemble des sup-irreductibles de E, (D : Ev + E est F-neutre inondtone et parbtienne si et seulement si il existe une f a m i l ~ efinissante F telle que @ soit F-majoritaire. En taxonomie, ce rbsultat s'applique aux hibrarchies, aux hierarchies strntifibes et aux ultraprbordonnances. Dans le cas ob E = IIx, on obtiendra une version (aux hypothtses affaiblies) d'un rbsultat d e Mirkin. Proposition 4.3. (Mirkin 1975) 'D : II;
+
est decisive et par6tienne si et s e u -
IIx
lementsi il existe un sous-ensemble tl de V tel q u e
0
soit 13-oligarchique.
Cette derniPre proposition illustre le fait, que comme en agrbgation des pr6fCrences, des rhgles d'apparcnce "raisonnables" peuvent conduire 5 une procbdure "triviale" (Noter q u e le cas H = 0 , correspond h la fonction constante, quij tout 11 associe la partition grossitre I! 1. On peut maintenant considerer cette proposition comme un cas particulier d'un resultst de B. Leclerc (1983) portant sur l'agr4gation des relations transitivement valubes. On trouvera dans Barth&lemy, Leclerc, Monjardet (19831, l'bnoncb de ce resultat dans le cas particulier des ultramGtriques : sous des hypothtses paretiennes et d e decisivitt, l'agrbgation d'ultrametriques est une "oligarchie gh6ralis6e". Donnons maintenant un exemple de multiprocbdure 0 contrainte d'optimisation : les partitioris c e ntraze s
:
Ili
+
IIx
vbrifiant une
de Regnier (1965). Pour V
E 6(n',ni), i=l 6 btant la distance de la difference symCtrique sur RX. L'aspect ordinal d e cette multiproc4dure vient du fait que, ainsi q u e nous l'avons mentionn4 en 4.2., 6 est definie par unc valuation inferieure sur 0 =
( n 1 ,...,II) ,
O(q)
est l'ensemble des solutions de
Min
II'E
n
".
TWL
comme la methode de Zahn, cette multiprockdure appartientb la classe des
procedures medianes et lelecteur est renvoye B Barthelemy et Monjardet (1981) pour la description de son environnement et 1'4noncG de quelques-unes d e ses propriCtCs ( c f . notamment, Regnier 1965, Regnier et de la Vega 1976, Michnud et Marcotorchino 1979). Par ailleurs, si des propribtes du type "confrontation aux conditions d'Arrow" sont bien Gtablies, le probleme de l a caracterisation axiomatique d e s partitions centrales demeure ouvert. Notons enfin que, dans un inf-demi-treillis 5 medianes E, gradu4 par 9, la mkdiane d u v-uple (r ,,,..,r ) (v impair), v A r ( F d4signant les I C F i E I v+ 1 parties d e V a - 6lbments) est l'uciaue solution de : 2 Min d (s,ri) $ s F E i=l
Ensembles ordonnes et taxonomie mathematique
5 45
(tLirb u t( 1 9 3 0 ) p o u r l e s t r e i l l i s d i s t r i h u t i f s , B a n d e l t e t b a r t h 6 l e m y ( 1 9 8 3 ) p o u r l e s d e m i - t r e i l l i s ?i m t d i a n e s ) . D a n s l e c a s p a r t i c u l i e r d e s h i 4 r a r c h i e s , on r e t r o u v e a i n s i u n r P s u l t a t d e Margush e t Mac M o r r i s ( 1 9 8 1 ) . En o u t r e , e n p a r t i c u l a r i s a n t l e s c o n d i t i o n s d e 13 p r o p o s i t i o n 4 . 2 . on o b t i e n t d e s c a r a c t b r i s a t i o n s a x i o m a t i q u e s d e l a m6dinne ( M o n j a r d e t , 198'). On r e n d a i n s i c o m p t e , d a n s l e c a s d e l a mGdiane, d e s t r o i s p o i n t s a ) , b ) , c ) . C e p e n d m t , 1,) n k d i a n e d ' u n e famil l e d e % sera s o u v e n t l a % h i 6 r a r c h i e t r i v i a l e ( c e ph6noniene f i t a n t e n r o r e a c c e n t u G p o u r l e s h i t r n r c h i e s s t r a t i f i 6 e s ) . C e t t e rernarque t e n d r e s t r e i n d r e , d a n s c e c o n t e x t e , l'usage d e l a r g g l e m a j o r i t a i r e ,I l a r e c h e r c h e cte compromis e n t r e d i v e r s e s h i 6 r a r c h i e s s u f f i s a m m e n t " v o i s i n e s " .
ARABIE, P. e t BOORMAN, S . A . , b etween p a r t i t i o n s , i
M u l t i d i m e n s i o n n a l s c a l i n g of m e a s u r e s of d i s t a n c e s Z., 17, ( 1 9 7 8 ) , 31-63.
iilz'dzcaz valzies, 2nd e d . , J. Wile y & S o n s , NewYork,
1962, ( p r e m i s r e C d i t i o n , 1951).
BANDELT,
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H.J.,
BANDELT, M.J. e t BAKTHELEMY, J . P . , !'isci?. :"lcrtl!. 1 9 8 3 .
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Medi ans i n media n g r a p h s , t o a p p e a r i n A p p l .
e t HEDLIKOVA, J., Median a l g e b r a s , Discrete bkztlr. 4 5 , ( 1 9 8 3 )
1-30.
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BARTHELEMY, J . P , t'xprie't6s r?fQi,iz,k[ues des erisefiibZes ardonnks. L'orrpamison e t tr(irc5:rutiurz d r r e / a t i o ? z s b-inaires, T h e s e d e D o c t o r a t d ' E t a t d e M a t h b m a t i q u e s , U n i v e r s i t C d e Uesanc;on, 1979a. BARTHELEMY, J . P . , C a r a c t r r i s a t i o n s n x i o m a t i q u e s d e l a d i s t a n c e d e l a d i f f b r e n c e s y m r t r i q u e e n t r e l e s r e l a t i o n s b i n a i r e s , Matl!. S c i . hiin!. 67, ( 1 9 7 9 b ) 85-1 1 3 . BARTHELEMY, J . P . , FLAMENT, C . e t MONJARDET, B . , O r d e r e d s e t s a n d s o c i a l s c i e n c e s , i n I . R I V A L ( e d . ) Ur'lcred s e t s , L ondon, D. REIDEL, 1982, 721-758. BAP,TFEI,EEF!, J . P . , LECLERC, B . e t MONJARDET, B. Q u e l q u e s a s p e c t s du c o n s e n s u s e n c l a s s i f i c a t i o n , i n Dutu Analysis anti Informatics 111,N o r t h - H o l l a n d , 1984. BARTHELEMY, J . P . e t MONJARDET, B . , The medi an p r o c e d u r e i n c l u s t e r a n a l y s i s a nd s o c i a l c h o i c e t h e o r y , i % z t h . Soc. Sci. 1 , ( 1 9 8 1 ) 235-267. BENZECRI, J . P . , D e s c r i p t i o n math6matique d e s c l a s s i f i c a t i o n s , cles donnhes, Tome I , I,a Tazinomie, Dunod, P a r i s , 1973.
1967 i n L ' A n a l y s e
B E N Z E C K I , J . P . , e t C o l l a b o r a t e u r s , L ' a n a l y s e d e s donne'es, Tome I , Tome 11, L'Ancrlyse des correspandances, Dunod, P a r i s , 1973. BERGE, C .
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B : ) ~ L ( ? c : ~ oj"
e t ARABIE, 1'.
,
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~ ~ ~ I t i i j i r i i ~ t ~ s i oSr icnaaI /i n ~ ~Vol , . 1 , T h c o r y , S t m i n a r Press, New-York, BOORMAN, S.A. e t O L I V I E R , D . C . ,
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DAY, W . H . E . ,
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