This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
, and J1 ). x . . z2 . .
V'
= 0).
l a y e r of
kth
+
F a c t 3 (Main Theorem, [ S l ) . Then
Let
J.
I.
J1
=
V
=
11.
~1
=
v
8 , v
111.
J1
=
V 8 B < b > ,
and
J1
a p p e a r s a g a i n and
V'
~1
=
-
be a f i n i t e
Fp
.
QE(pn-l) r i n g o f charac(Here
a nd d e t e r m i n e t h e m u l t i p l i c a t i o n ;
B <ar> w i t h t r i v i a l m u l t i p l i c a t i o n .
of t y p e
V
-ba ,
0
I , v-a
of type
V'.
J1 :
II
E
Fp, a 2
a-v
=
=
I,
V-a
=
pn-l,
0, a2
=
pn-1
V-b
=
a.V
=
a nd b2
=
i
=
or
tpn-1.
b-V
=
0,
p
E
tpn-l, pn-l,
IV.
are
Jk
F p. )
=
The s e t
J1
is i s o m o r p h i c t o o n e of t h e f o l l o w i n g :
J1
is a f i x e d n o n s q u a r e i n
Lpn-l
J k - < J k - 1 , pn-k>.
t h e r e is o n l y one t y p e r e a l i z e d i n
R
J1 as a vector space over
=
=
is t h e l a s t p l a c e where a n y t h i n g
J
We f o c u s f i r s t on t h e b o t t o m l a y e r
t e r i s t i c p , p odd.
Our a n a l y s i s s u c c e e d s by showing t h a t
a g a i n , s i n c e f o r homogeneous
ab
pi.)
Jk.
v -
=
t h e only r e l e v a n t l a y e r s f o r determining
t
pi
be t h e least i n t e g e r such t h a t
Intuitively, the
we write
b e t h e a n n i h i l a t o r of
Ji
of c h a r a c t e r i s t i c
J ( i i i ) Let V
let
1 5 i 6 n-1,
B <ar>, a i a j
=
ajai
=
0, i
*
j,
p
E
1 mod 4
3 mod 4 .
D. SARACINO and C. WOOD
210
V.
VI. b2
=
J1
-V
=
-
aibi
=
(p
3 only) J1
+3.
=
pn-l
If k > 1
-
1
aibj
-biai, 1 5 i 5 n, and =
Moveover, any
Thus if k
B <ar> B
<3"-1>
ai
B B where a2
as in I - VI
J1
=
we are done:
J
=
J1
+
=
bi
=
where =
bjai
0, ab = 3
0, i
= =
or trivial).
If k
>
j.
-ba,
is QE.
is known by Fact 3.
we can get additional information from [l]:
Fact 4 ([l]
i
1 then pk-l Jk
=
V and so VJ1
=
J1V
-
0.
Moreover V 2
If
k
=
J1V
=
0, so we have
> 1, then J1 is of type I, 11,
or 111.
(We shall see that all three possibilities can occur.)
is
Additional technical information about multiplication on Jk summarized in: Fact 6.
(113 or trivial).
If k
>
2 1 then Jk
We remark that more is claimed in [l] 2
that Jk
5 Jk-1
whenever
v
t
5 (Jk-1, pn-k>. If in addition
than what we state above, namely
p. 150 of 111, the additional hypothesis that p > 3
is required when V
is two- dimensional over Pp. As
2
Jk
we see in the theorem below, it can in fact happen for p
$ Jk-l * The goal of this paper, then, is the following:
=
3 that
Finite Homogeneous Rings of Odd Characteristic
Classification Theorem.
Let
J
be the Jacobson radical of a finite QE ring
of characteristic pn. n > 1 , p odd. Then J
21 1
Suppose k > 1
(i.e., J
i
J1 +
is isomorphic to one of the following (all of which are QE):
>
(Fp-dim V
A.
XiJk
v
=
JkXi
=
> 3)
2 or p =
J
<J1, X I ,
=
0, J1 Of type
..., xs,p>, where
I, 11, or 111, k S n/2,
e
pk-1 xi, i
=
...,
I,
=
S.
or
B.
(Fp-dim V (i)
=
2, p
( J Z
3)
=
J
0,
< J ~ ,x, 3>, x2
=
j3n-1, j
J1 type I, 11, or 111, k S n/2, V
=
<3k-1x> 8 <3"-l>.
xJ1
=
=
1, or - 1 ,
J ~ =x 0,
=
or (ii) (5:
xJ1
$
<3"-l>)
=
J1x
=
x2
=
<J1, x, 3> where
0, k
=
2, J1 of type I, 11, or of type I11 with the
additional restriction that ab
ba
=
=
E
<3n-2>
-
J
0, V
=
<3"-1>,
<3x> f3 <3"-1>.
This, then, together with Facts 1, 2, 3 provides the classification of finite QE rings of characteristic pn, hence of all odd characteristics. The proof breaks into cases in two different ways: size of V
and according to whether or not
p
=
according to the
3. There is considerable
overlap, but at little or no cost, in some of the cases--in particular, for small V
and
p > 3.
We choose to include arguments in some generality,
with an eye to possible future analysis. 2. Case A (F,-dimension
Let u
>
v
2 or
over
V
>
2
be the dimension of
p > 3).
or
V
p > 3, and so by Fact 4,
over v
Fp; by Fact 4, v 2 2.
We assume
is also the dimension of Jk/Jk-l
Fp.
We assume k > 1, J
finite as before.
show that in Case A , multiplication on Jk trivial.
Our goal in this section is to goes into
We work our way through the layers of
J
via a series of lemmas.
D.SARACINO and C. WOOD
212 Definitions.
( i ) Let
are i n d e p e n d e n t over
x, y
pk-lx,
if
("real
Jk'
E
pk-ly,
a nd
x and y
We s a y
Jk-elements").
are l i n e a r l y i n d e p e n d e n t
pn-l
Pp. ( i i ) Given x , y ( i i i ) G i ven
J,
E
-
x y = xy
x
XI,...,^,., y 1 ,
...,yr
yx.
J , we s a y ( X I ,
E
" h a s t h e same t y p e a s " ) p r o v i d e d t h e map x i
(read
+
-
...,x r )
yi, i
1,.
=
(y1,
...,y r )
...,r ,
p
+
p
d e t e r m i n e s a n i s om or p his m of t h e c o r r e s p o n d i n g s u b r i n g s o f J ( g e n e r a t e d by
IP, x i ,
..., X r ) ,
I f t h e r e is
well-defined. E
>
(v
Proof:
or
2
> 3).
p
Suppose n o t .
But now f o r any Z
Jk*
Z
= y + C, C
V'
Suppose
Let v
pk-lx E
x2
6
p
=
pk-2z2
3 we use
pk-2(x + y ) 2 and
pk-2
=
Jk' w e have y
E E
Jk-1,
=
and
E
k 2 3.
v
>
2
then
$.
Th u s
=
-9.pn-1,
E
Jkl
then
x2
E
V', p k - 2 ~ 2 = Qpn-1, Q k 0 mod p . y
E
J ' , pk-ly
J k ' w i t h pk-l
=
p k - 2 ~ 2 + pk-z (y c
=
pk-2y2
(y-2)
=
v , pk-2y2
= 0 , pk-2y2
~k-2. Then Qpn-l.
=
Lpn-'.
=
SO
t o choose =
x
If
E
for all
Lpn-l
pk- 2( x -y)2
(-x I y )
both cases.
~ k - 2+ < p n - ( k - l ) >
0
z
E
Lpn-1
=
x
If
J;.
and
pk-2x2
contradicting
+ cy + c 2 )
by F a c t 6 .
c o n t r a d i c t i o n by compa ring z a nd 22 : p k - 2 ( 2 2 ) 2 If
is
Jk-1/Jk-2
w e g e t t h a t e v e r y coset o f J ~ - I / J ~ - Z
V' t h e r e is
p k - 2 ~ 2 = pk-2(y + c ) 2
This gives
with
to
Jk/Jk-l
< ~ " - ( ~ - l ) > is i n t h e image o f
by h o m o g e n e i t y , f o r e v e r y
Thus
E
from
r$~
I 2p, a c o n t r a d i c t i o n .
S pv/2. p v
Lemma 2 .
x
V' a n d by h om oge ne ity o f
e x c e p t p o s s i b l y ones o f pv-p
resp. ).
By F a c t 6 , t h e S q u a r i n g map
Proof:
pk-2 x2
..., y r ) ,
Ip. Y ,
y =
> 3 we g e t a n i m m e d i a t e
p =
l l p k - 2 ~ 2 = 4P.p"-1
independent.
pk-2y2.
P. &
o
Lpn-1.
#
Now
*
T h u s pk-2(,
mod p . T h u s
x2
y) E
=
-1pn-l
~ k - 2 in 0
Finite Homogeneous Rings of Odd Characteristic Lemma 3 .
>
(V
Pro0
2
v = 2
If
S u p p o s e now
>
v
t h e n
> 3).
p
Or
and
If
> 3,
p
n
pk- l x>
n
p k - 2 ~ ~
pk-ly>
pk-lXr
pk-2 xy'
=
elements the
pv
pk-lx,
=
pk- 2x1y'
=
p2
v.
=
a contradiction.
v.
xy
independent, (pk-lx, (pk-lx,
pk-ly,
-
y)
pk-2Xy'
=
f o r independent
so are
x, y
x . pn-k
E
Jk-2.
and
+
pn-k
y
+
E
E
y
x
V
pk-'y) (pk-lx,
p k - 2 ~ ~ cpn-1,
v
B u t now
pk-2xc
Let
=
pk-2xy
(pk-lx, 2pk-ly,
x pn-k,
a nd
y.
X2
E
y'
Since
Jk-2.
x E
E
If
~ ~ - 2x ( ,y
Thus
X'Jk
+
5
y.
x
and
y
y'
pk-ly'
x, y pn-k) Jk-2.
mod J k - 1 ,
and
are with
2pk-ly
=
2y mod J k - 1 ,
Jk'.
Jk-1
x
9, c 0 mod p , a n d
Thus
xy
5
p".
6 y'
x + y'
T h u s t h e r e is
y'
>
p)
e x i s t s mod
for i n d e p e n d e n t
Now f i x a ny
-
2(pv
while
~pn-1.
SO
T h e s e elements must c o v e r
0.
Since
2pk-1y).
a nd we g e t t h a t
By Lemma 2 ,
=
pk-2xy'
=
BY
t h e r e are a t most p v
y'); in particular
p k - 2 ~ * 2 y= 2f,pn-l.
=
x ' mod J k - 1 ,
5
x , e x a c t l y one
E
x
a nd c o u n t y" s :
and
pk-lx>.
pk-lx> t h e r e e x i s t x' a n d y '
-
pk-2xy
J k-2.
say
J k-1,
E
=
We f i x
c
independent.
The f i r s t e q u a t i o n i m p l i e s
p k - 2 ~ ~ =1 ~ p n - 1 . !Lpn-l
pk-2 xy
B u t P ~ - ~ x ( +x y ' ) Thus
W e now show
If
v.
e l e m e n t s of
pk-2xy'
a nd
=
i t must b e t h a t f o r a g i v e n with
eas .y.
pk-2xty'
p k - 2 ~ ~ 1 since , for
-
5 Jk-2
-
E
5 Jk-2. 2 Jk
V
v
Jk
t h e n Lemma 2 i m p l i e s
homogeneity, t h e n , for e v e r y so t h a t
x
2 , and c h o o s e
2
then
k 2 3
213
and
so Xy
SO
E
Jk-2
are i n d e p e n d e n t E
~ k - 2 implies
A150
X +
pn-k
are i n d e p e n d e n t , s o
(X + p n - k ) ( y + pn-k)
E
T h i s p r o v e s t h a t a l l p r o d u c t s from
J k-2,
giving
( p n - k ) 2 E Jk-2.
Jk
are i n
J k- 2 ,
as desired.
0
D. SARACINO and C. WOOD
214 Lemma 4.
Jk-1
Proof: For
i
=
J1
We proceed i n d u c t i v e l y t o show t h a t t h i s is t r i v i a l :
= 2
M u l t i p l i c a t i o n by hence
Ji
>
>
(v
Proof: k
p
PJi+l
=
Lemma 5.
For
PJk.
+
k
=
JiIJi-1
Ji
PJi+i.
When
=
+
J C ~~1
Lemma 6 .
xy
J1
E
>
(v
2
or
p
By F a c t 6 ,
Proof: V
for a l l
x, y
we g e t
JEcV.
JlJk
V,
Jk.
E
u
E
J~
ux $ < p n - l , pk-lx>.
t h e r e is y from
Ji-1
=
Ji,
o
. For
k = 3 , i t is Lemma 3.
Since
Jk-1 v
with = =
Uy =
J1 + PJk
V,
-
3
Jk
SO
5 JlJk 5 V
VJk
xy
E
V, x
E
-
0
=
-v, a c o n t r a d i c t i o n .
xyxy
=
Thus 0.
Since
V.
Jk'
pk-ly.
t h i s implies
by Lemma 5.
by F a c t 6 ,
with
ux
In t h i s case for a l l
pk-lX
e t c . , u n t i l we g e t t o
We omit t h e g o r y d e t a i l s .
J1.
> 3).
by Lemma 5 , t h i s shows
Suppose t h e r e is Case 1 :
+
we a r e f i n i s h e d .
i = k
< ~ k - 3 , pn-(k-2)>,
t o squares i n
J: 5 Jk-2
JZ 5 ~ k - 3 and t h e n r e p e a t down t o
E
PJi+1
and so
pJi.
+
3 , we r e p e a t t h e p r o o f s of Lemmas 1-3, u s i n g Lemma 4 f o r t h e i n d u c t i o n ,
t o go from
xyx
Ji-1 = J1
t h i s is c o n t a i n e d i n F a c t 6.
2
J1 + p J i . 2 5 i S k.
=
Suppose
onto
PJi
+
J1 + pJ2.
=
Ji+l/Ji
P > 3).
or
2
For
maps Ji
+
J1
Ji-1
UX z
But t h e n
d
E
X E
V
-
< p n - l , pk-lx>
y mod J k - 1
Uy mod < p n - l > .
Thus Case 1 c a n n o t o c c u r .
Choosing
and V * -UX
Finite Homogeneous Rings of Odd Characteristic F o r a l l u a n d x , ux E < p n - l , p k - l x > .
Case 2:
mod < pn- l > .
Then f o r a l l
mod < pn- l > .
Since
-!Lpk-lx
I
(-u)x
Let
with
pk-ly
y
ux
u
by
uz
(since u
-u
!Lpk-lz mod
5
0 mod p , a nd so
5
ux
!2pk-lx
=
v , uy
=
t h i s s a y s t h a t whenever
i m p l i e s (-u)z T h u s 9.
Lpk-lx.
5
t h e r e is
V'
B u t now r e p l a c e
(-u)z d
again that
E
5
uJk-1
Ilpk-lz mod < pn- l >.
uz
v
F i x u.
215
-
L Pk-l Y
-u)
This says
< p n - l > , showing
E
5
JlJk
Similarly,
JkJl
5
hence
5
JkJk-1
0
Now we are r e a d y t o p r o v e Proposition A.
Proof:
>
(v
Case 1 :
pk-lx,
pk- l y>
xf,yl
with
2.
t h e n for a l l (pk-lx,
pk-ly,
=
v',
X'
x'y'
I
xy
mod < pn- l >.
xy
>
v
x'y'
v
or
2
E X
mod J k-1,
=
x (-y )
5
0 mod p.
a l l independent
x
y'
I
Thus
y
mod J k-1.
v'
=
v
xy
E
< p n - l , pk-lx,
B
y.
Now
I
=
~ ( p " - +~ y ) similarly T h i s s how s
E
< pn- l >
implies
xpn-k
E
s i n c e a l l m u l t i p l e s of
From x
pn-kx
lie outside
a n d so k 5 n / 2 , f i n i s h i n g Case 1 .
-apk-lx
+ E
xy
E
E
E
for
and
x + y
Likewise
This gives
hence
This
6pk-ly.
x)
there exist
t h i s implies
E
5i.x c < p n-l>. T h u s pn-k(pn-k
J i 5 < pn-l>.
-xy
x2
xy .k < p n - l ,
v', x', y'),
(x,-y),
0 mod p , a nd so
are a l s o i n d e p e n d e n t , a n d t h i s i m p l i e s
=
pk-ly>, say
-
x(x + y)
v
by LenUIIa 8 we h a v e
(x,y) =
If
n/2.
we see t h i s is i m p o s s i b l e :
pk-lx
+
Since
NOW
s
k
pk-ly>
(pk-lx, pk-ly,
But x(-y)
Similarly and
-
v, x, y)
a p k - l x + Bpk-l(-y).
implies a
and
v' E V -
apk- l x + Bpk-ly mod
5
T a ke x a n d y i n d e p e n d e n t .
But f o r
v + pk- l x mod < p n-l>.
!j
2 Jk
> 3).
p
X-Jk
5
E
we d e d u c e t h a t pn-kx = 0 , except
0.
Thus
n
-
k 2 k,
D.SARACINO and C. WOOD
216 Case 2: v
p
If
v = 2.
t h e r e is
V'
E
> 3,
y
with
pk-ly
we c o n c l u d e t h a t f o r a l l Y
z
E
z2
=
Y2apk-1,
w = 2 , J k = < x , pn-k,
Since
=
Jk',
E
Fp*, t h e n f o r a l l
v , y 2 z a pk-ly mod
=
apk-1,
By Lemma 8
But f o r z
mod
=
Y(crpk-lz) p a p k - l z mod
f o r a ny
Jk-1>
J i 2
follows readily t h a t
x
a n d u s i n g Lemma 8 i t
Jkl
E
k 2 n / 2 is a s i n
T he a r g u m e n t t h a t
Case 1 .
3.
YX,
=
we g e t a c o n t r a d i c t i o n :
Fp - ( 0 . 1 1
E
x2 s a p k - l x mod < p n - l > , a
0
Small V. I n what f o l l o w s we c o n s i d e r J k s u c h t h a t
v
t h e d i m e n s i o n of V
2, v
=
Pp, and show t h a t a l l p r o d u c t s from J k l i e i n t h e p r i m e s u b r i n g , h e n c e
over
< pnW k> . In p a r t i c u l a r , we p r o v e t h i s i n c a s e v = 2 , p
in
we r e a l l y a d d , i n l i g h t o f P r o p o s i t i o n A . a p p a r e n t l y no h a r d e r t h a n for sh o wi ng t h a t f o r
p
> 3
and
p
3, w h i c h is a l l
=
The p r o o f f o r a r b i t r a r y
p
is
3, a n d d o e s g i v e a n a l t e r n a t e r o u t e t o
=
2 Jk
v = 2,
5 < p n - l > , as a n e a s y c o r o l l a r y t o
J i c
N-o t a t i o n . V
We f i x some n o t a t i o n f o r t h i s s e c t i o n .
< pn- l >
=
@I
x t Jk'
with
pk-lx
J 1 is o f t y p e I , 11, o r 111, a nd we write
as i n F a c t 3 , h e n c e
Jk
=
or
J1
=
Since
or
V
=
2 , we write
U s i n g F a c t 5 we h a v e
v.
=
w
or
Th ro u ghout t h e p r o o f s i n t h i s s e c t i o n we a r g u e f o r t h e t y p e I11 case o n l y , a n d n o t e t h a t t h e o b v i o u s m o d i f i c a t i o n s (whe re
-ba
=
Lemma 1
=
(v
Proof:
axa
E
Lpn-l, =
2).
Since
ax, e t c .
w he re b2
Thus
=
x a , a x , x b , bx pa
=
we know
0
xaxa
=
o r pn-l
tpn-l E
and
b
are
W e s e t a2
=
pn-l,
according to p
E
1 o r 3 mod 4 .
a b s e n t ) g i v e t h e p r o o f f o r t h e o t h e r two t y p e s . ab
a
or both
b
V.
pxa
=
0, giving t h a t
0, hence xa
E
V.
xa
E
J1
a n d so
Similarly for 0
Finite Homogeneous Rings of Odd Characteristic Lemma 2
(v
=
Proof: exists pk-ly
Suppose
y E Jk =
with
xa
Thus
V'.
xa
6ba
+
(Y
=
( Y +6!L)pn-1 = 0.
P r o p o s i t i o n B.
Now v
-
=
=
Jk-lJk
contradicting
2pn-kx
E
+
a
d
k 5 n/2
x2b
p bx2
2
5
Jk
y
x2
and
=
x2a
-
(v,x) a(-v)
+
6
=
Then
we l o o k a t
(-v,y),
(x
+
E
0
0. so
=
E
By a
x2 = a p r x + ~ ? p " - ~ ,where
pk-r-1x2
From a(-v)
y2 y
=
pk-r
+
getting
gpn-r-l.
x2
4
pn-k>.
0 , a n d so x2
Suppose
This implies
Thus
2, a
E
Y t 0 mod p.
+ p J k ) J k = pk-r-lJIJk
mod < P " - ~ > . 0 mod p.
5 <Jk-1,
k I n/2.
1 S r 5 k - 1.
with
E
a , 5, Y , 6
2 Jk
2Y, hence
E
BY Lemma 2.4,
a x , x b , bx
where
we h a v e
0
=
and
by Lemma 2 , and so
0
=
divides
=
pk-r-l(J1
=
ax2
z
=
=
av
2 Jk
+
BP"-~-~.
-
apry
+
We
BP"-~,
-x mod Jk-1
we have
a v mod < P " - ~ > ,
pn-k)2
x2
E
and c o n c l u d e
n - k h k , so
But t h i s c a n o n l y happen i f
By Lemma 2 . t h e n , i t follows from
J i 5
Ya + 6 b
6!Z)pn-l.
Also
xa
=
mod < p n - l > , c o n t r a d i c t i n g
-
pkwr-ly2
P k-r-1x2
To see
s x(-a)
y(-a)
x mod Jk-1.
(v = 2 ) .
-v, a n d
pk-r-1y2
=
y
E
This gives
to produce
-v
y(-a)
=
2, a f 0 mod p. and
pk-r-l
pk-ly
xa
F i r s t we show t h a t
Proof: E
In particular,
< p n - l > ; we a r g u e s i m i l a r l y f o r
similar a r g u m e n t w i t h
a , 13
,., ( - a , v . y ) .
we c o n c l u d e t h a t t h e r e
(-a,v),
r 2 1 , s i n c e by F a c t 6 we know t h a t
Here
Ya2
-
(a,v)
x2 = a p r x + f3pn-k
Now by Lemma 2 , x a =
Since
V'.
(a,v,x)
E
Write
Proof:
0
E
p J k , a n d so
+
mod p.
xa
E
v ; from t h e l a t t e r we c o n c l u d e t h a t
Jk-1 = J 1 E
x a , a x , x b , bx
2).
217
E
and
k I n/2. k 5 n/2
that U
Use
D. SARACINO and C. WOOD
218
4. Large V Classified.
>
w
We assume in this section that
2, and we show the Classification
Again we use the notation of Fact 3 f o r
Theorem's claims for this case. possible J1 ' s . Lemma 1
VJk
(v
>
=
0
Jkv
=
Choose x
-
x
But
y
x'y'
=
xy, so =
=
xy
=
0, giving
=
0.
JkJk-1
=
L
PJk
=
0 , by
-pk-ly, spn-l), there exist
pk-lx, pk-1~' = -pk-ly, xy
=
=
independent, and let xy = spn-l.
,.- (pk-lx,
Jk-1, hence x(y' + y)
E
x(x + y)
so
and
(pk-lx, pk-ly, spn-1)
XI, y' with pk-lxl x'
2
is of type I, then Jk
J1
by Fact 6. Therefore Jk-lJk
Proposition A . Since
If
2).
(x' - x)y
0. Now X2 = 0.
x
and
+
2
Thus Jk
This settles the picture for J1
0, giving
=
x
x'y'.
=
y
Now x'y'
y' + y =
xy'
=
and x(-y).
are also independent,
= 0.
0
of type I completely. We turn to types
I1 and 111: Lemma 2. x2
and
>
(u
2).
If J1 is of type 11, then there is x
E
Jk' with x x J1
z x J1
y x z any
=
J1
t
=
0. By adding a suitable multiple of y to z we can assume that
=
If either y2 or z2 is 0. fine.
0 also.
spn-1
for
s
+ o mod
p as w2 where w
=
ay
Otherwise we can represent +
E
Jkf for some a, 6.
BY
definability of J1, this says that all elements w of Jk' with nonzero square satisfy w (w
+
i
J1
=
0. Taking w with w2
a) x a = 0. But (w
Lemma 3 .
0
0.
=
Proof: Since w > 2 we can find y. z independent so that y =
=
(w
> 2).
if and only if
x2
+
a) x a
=
a
=
a2 we get (w
i
a
=
2a2
f
If J1 is type I1 or I11 and x =
0.
+
a)2
=
2a2
f
0 and so
0, a contradiction. E
Jk' then x x J1
=
0
0
Finite Homogeneous Rings of Odd Characteristic Case I :
Proof:
of type II. Here we know
~1
x
x
J1
y
I
J1 = 0 with y2
0 by Lemma 2 and homogeneity.
=
0. Since
f
(again by homogeneity) that y2 If
(y
Then
> 3
p
If
p
(t
=
l)a2
+
3 then y2
=
=
=
a2
and
says that +
(x
t
and
2a2
=
(y
x2
spn-1 * 0.
=
a)2
=
0, giving
Suppose there is x
(x')~ = 62spn-1
Since Jkv a2a2
+
=
VJk
fi2b2.
+
=
If
o
( x ' ) ~= x2 x2
o
=
z
+ i
p
=
3 we use
(x
-
o
have
0
imply
=
(x
+
t + 1 =
0, a contradiction.
2a2 0
Jk', x x a
are non-squares.
a x (y
=
6
x x b
=
0,
+
fib that
=
aa
+
0.
62 @ 0, 1 mod p
with
Necessarily a and 6 are not both 0. But
x , so
v
> 2 to produce x, y independent so that
=
0
=
=
and x' x b are not both
x' x a
a)
+
implies y2
-
0. Recall that here
=
Z x J1
But
-
a)2
~1
E
=
p > 3 we can find
x' x a
0 we know
f
x' x b
0, a contradiction.
=
Thus
p > 3.
x x J1 = y x J1 (x
0, 1, and so
implies x'
if
If
i
Jk* implies
E
we know for any x' = 6x
and solve for a t 6 so that (x')~ = spn-l. zero, since
x
a) x J1
+
a) x a
+
by homogeneity, again a contradiction. Thus y x J1 Case 2: J1 of type 111.
and
a non-square mod p.
ta2, t
-a2, so (y
=
aI2
+
o
=
To prove the other direction, suppose
so that both
t
we can choose
a)2
+
(x
x2
219
0,
yI2
=
+
a
+
bl2
z2
=
-3"-1.
Z2
x2
=
=
3"-l.
a2
b2
=
=
3"-'.
If x2
-3"-'
=
3"-l, so the Only elements of Jk' Similarly, if Notice that
says x x y = -x2
and
x2 = 3"-1
X f y
(x
-
E
Jk',
then z (X f
we get
which satisfy E
Jk* and
y) x J1
=
0.
y)2 = x2 says x x y = x2.
0 is the only possibility when x x J1
=
0.
This will complete our argument for Case 2 provided we can find some x
E
Jk'
This contradicts x2
with
x x J1
Jk/pJk:
0, and so x2
+
1)-dimensional subspace, and so the *-annihilator of J1
has dimension at least X
E
=
0. To do this we intersect the *-annihilators of a and b in
each is a ( v
in Jk/pJk suitable
=
i
Jk'.
v.
Since
v
>
2 , this subspace gives us 0
220
D. SARACINO and C. WOOD
then xJ1
>
(v
Lemma 4.
J1x
=
Proof:
Then f o r any v'
-
(or (v, a, pn-1) xa, (x'b
=
(v, a, pn-')).
Thus there is
xb), ( x ' ) ~= 0.
=
and so (x - x1I2
By choosing
xJ1
x2
y2
=
Proof: hence (x
>
(v
Lemma 5. with
0. Similarly
=
0, then
Take y)'
+
=
J1x
=
Notice x
-
v' e!
Jk'. =
pk-lxl
But (x -
0
=
v',
it
XI)
J1
0,
=
-
0.
=
x 0
x 0
and
y
*
y.
x
=
xy
yx
=
=
0.
a s above and notice that Let pk-lx
=
v, pk-ly
=
(x
v'.
(v',v,x',y').
xy, ( x ' ) ~= (y'I2
+
y) x J1
Then (v,v')
5
x mod Jk-1. and so by Lemma 4 and Lemma 2.4,
x'
=
y mod Jk-1, y'
xy
=
x'y'
yx
=
0 also.
-xy.
=
Thus xy
0, and from
=
x
i
=
y
=
0,
-
(v',v),
In particular,
v, x'y'
=
=
0. But now x - x'
=
yx
J1
we can be
v', pk-ly'
=
*
(v', a, b, pn-l)
so that
x'
Thus x - x'
and so there are x', y' with (v,v',x,y) pk-'xr
= 0,
If J1 is of type I 1 or 111, and x and y are independent
2). =
pk-lx.
Moreover (x - x')J1
0 by Lemma 3 .
=
=
V' we have (v, a, b, pk-')
E
sure that x and x' are independent.
gives
Jkl such that x2
E
0.
=
Take x as in the hypothesis, and let v
by Lemma 3 .
x'a
is of type 11 or 111 and x
If ~1
2).
0. But then
=
0
it follows that 0
We now prove the Classification Theorem, Part A , for the Case I I =
Fp-dim V > 2.
For
J1
=
V , Lemma 1 is all we need.
111, we see that Jk is generated by modifying any element and
b)
to get (y
+
y aa
+
For J1
~ 1 pn-k, , and elements with square 0, by
of
Jk'
by a suitable multiple of
6b)
*
=
J1
of type I1 or
a
(or of
a
0, then applying the previous lemmas.
The verification that the resulting J's
are homogeneous is routine.
Finite Homogeneous Rings of Odd Characteristic
5. Small V
221
Classified.
In this final section we consider V
of dimension
over
2
Fp, using
results in Sections 2 and 3 to obtain the rest of our Classification Theorem. 2
Recall that from Proposition B we know that Jk f
If p > 3
0.
=
Jk
(v
Lemma 1
Jkv
=
0.
2
J1 = V (type I) then Jk =
V
=
=
2x
mod Jk-1
+
From
Jk' such that (pk-lx, x2, x)
E
implies y x2
If
Notice that Jk-1
suffices to show x2 y
0.
=
> 3).
2, p
=
Proof.
=
we known more by Proposition A , namely
pJk
0.
=
annihilates Jk.
Let
x
Jk'.
E
It
(x2,pk-lx) * (x2,2pk-lx) we know there is ,- (2pk-lx,
and so
x2
=
y2
x2, y). 4x2.
=
But then pk-'y p > 3
For
=
2pk-lx
this implies
0.
0
Thus we have again that for type I J1's. multiplication is trivial, and
so we turn to types I1 and 111. (v
Lemma 2 such that
x x J1
Proof. x + a If
-
a(x + a)
(ii)
exists c
=
0.
of type 11: Let J1
=
=
I
x
=
-
-
a), then a2
=
Jk'
E
0. Since +
a)
=
+a(x - a).
-a2, which is impossible. From
a) we conclude that ax
-ax, hence ax = 0. Now a x x
=
=
0
0 as well.
=
J1 E
Then J1x
(a,-a) is definable modulo V, we get a(x
a(x
=
-a(x
=
gives xa
=
J1
(i)
is type I1 or 111, and suppose x
Suppose J1 xJ1
= 0.
x - a and
a(x + a)
J1
2, p > 3 ) .
=
J1
of type 111: Suppose J1
-
V
such that
c2
or
=
tc2
0.
=
tc2.
and so if d2
mod
=
cx
*
x
c2
(and so p
E
0. Since J1x
Choose d
Since =
=
v =
then
1 mod 4).
E
J1
2, fx dx
=
5
with
c x d
there =
0,
is definable
0 also, giving
J1x
=
Now every element of Fp*
0.
D. SARACINO and C. WOOD
222
u2
c a n be w r i t t e n as u,v
i
uc
with
0
giving
J1x
vd
+
v2t
-
pv
with
But t h e n
C.
0 , and i n p a r t i c u l a r t h e r e e x i s t
f
(uc
We are now r e a d y t o p r o v e t h e small
(v
>
2, p
=
From Lemma 2 we know
*
1, -I,
0,
-
(pk-lx,x2,x)
Write y
it
y
a
=
Ya
8
xa
=
dx
SO
xJ1 = 0
0,
=
also.
0
p a r t of Case A o f our Theorem
V
+
x2
so
0,
=
x2
Thus
for
Y, 6
y = Bx, x2 J
=
E
y2
=
< a , x , p>
y
Jkl,
E
8pk-lx, y2
=
Fp.
Then 6
82x2.
=
Fp, and c h o o s e
E
t h e r e is
8pk-lx,
is o f t y p e 111, t h e n c h o o s e
J1
a
=
T h i s g i v e s pk-ly pJk>
0.
=
-
pk-lx
6x mod
Y
J k ' s o t h a t a x x = 0 , J1 =
E
Let
0.
=
(gpk-lx,x2,y).
conclude t h a t If
ax
F ~ .Since
E
we g e t
0
=
and
0,
=
3):
I f J1 is o f t y p e 11, t h e n c h o o s e x
8
vd)x
+
I t f o l l o w s from J1 x x = 0 t h a t
0.
=
+
x2, y x a
=
=
0.
B and from
=
b2 * 1 , w e
Since
as i n t h e theorem.
*
x E Jk', x
J1
and a r g u e as
0
=
above, a g a i n u s i n g Lemma 2, t o g e t t h a t J
=
J k X = 0, k 5 1112.
=
Again, t h e c h e c k s t h a t t h e s e are homogeneous are r o u t i n e and we o m i t them. Notice t h a t a complication arises
We now t u r n t o P a r t B o f o u r theorem. h e r e s i n c e we do n o t know t h a t
VJk
=
Lemma 3.
x
Jk'.
ax
(v
=
(11) I f
J1
=
xa
=
bx
xb
=
(i) If
Vx
=
=
xv
=
(v, a, x)
a2.
-
-
(v, v
I
we have m
kv
=
+
vx
!La
1 mod
+
Vx
*
a
then
ax
x x b
=
0
=
x x a
0, then
=
ab
ba
=
xv
-
for all (v,v-a),
v
E
V.
y
xa
=
0.
0, then 0.
I f not, then
Choose
there exists
=
=
v E
E
V'
with
J k with
a, y). hence
mx
3.
=
(v.a)
vy y
it
we a r g u e e x a c t l y a s i n Lemma 2.
0
Since
x
is o f t y p e 111, and Moreover, i f
0.
=
<3n-1, 3k-1x>, so
=
Now
E
J1
=
vx
Let
(i) I f
Proof: V
2, p = 3 ) .
=
0.
=
vx, ( v
-
a)y = ax, (v
mod <3n-k> f o r some Thus from ( v
-
a)y
=
-
a) x y
k , II,
m
= E
0.
Z.
S i n c e vy
ax it follows t h a t
=
vx
Finite Homogeneous Rings of Odd Characteristic
vx
-
-
!La2
we g e t
*
v
-
x
!La x a
-
a2
a x = a x , hence
!La2 = -ax.
0 , a 2 -!La2
=
Thus
ax
( i i ) As i n ( i ) we need only c o n s i d e r when
vx
xv
=
Then
a2.
=
( v , v - a , -b, (v - a ) x y
y)
-
-
a , -b)
(v, a, b, x). y
0, -b
=
-
(v, v
0 , -by
=
bx.
Writing y
=
1 mod 3 we g e t ( v - a ) y
while (v
=
0 y i e l d s a2
so bx
s
0 mod 3.
xb
=
y
x
.a2
sb
and so a2
-
.a2
Now b x y
0.
=
Combining a l l t h i s g i v e s
- by
bx
=
ax
+
a ) y = ax, +
mx
- sab
xa
0.
=
-ax,
=
0 implies sb x b
=
0 , hence
=
with
0,
=
Similarly
we get
0, r
=
it follows t h a t
ab
-ba
a l s o , when
0
=
Vx
s b 2 = -bx.
-bx, hence
= #
We now p r o v e Case B of o u r theorem.
a s i n Case
-
bx, -bra
=
t h i s becomes
1
=
- bx
- sb2
-bra
s
+
kv
V'
with
ra
=
0.
= E
0
0.
=
Now from
Since
-
y
-
=
ax
=
v
vx, ( v
I n p a r t i c u l a r , vy =
xa
Choose
0.
#
( v , a , b ) , so there is
mod <3n-k> w i t h m
- a)
Vx
0 , and so
=
-
(v - a) x y
F i n a l l y , from
0.
=
223
ba
a x b
-
0
0.
If
J;
5 <3"-l>
then t h e d e s c r i p t i o n x2
( i ) is e x a c t l y a s f o r Case A , e x c e p t t h a t now
€3
From
0.
=
*3"-'
=
is
a l s o possible.
J ~ <3"-1> C
If xa
=
If r
xb
=
>
ax
=
bx
=
consider
2
Observe t h a t
so y
=
mx
and so r
(1
2 , x2
=
by Lemma 3 .
Let
0
m
-
-
3r-2
=
-
( 3 m , a, b )
=
-
v.
-
Thus x2
=
y2
E
=
y2
=
=
m2x2
+
m2) = 0 mod <3n-1-(n-r)>,
~ 3 " - ~a,
4x2
x x a
with
-
<
~ 3 " - ~ 1,
1 mod 3r-2
but
=
x x b = 0, hence
r S k , a $ 0 mod 3 . m2 f 1 mod 3 r - l .
and so there is y w i t h
z
ay
=
J1, and from ya
2mxv.
=
by = 0.
=
yb
=
0, e t c . we g e t y
But now ( 1
contradicting
-
m2)x2
&
m2
=
-
mx
E
V,
2mxv E <3"-'>
1 mod 3r-1.
Thus
0 mod 3 .
Now a p p l y t h e above argument w i t h x2
B
x2
x 2 , 3y = 3mx, ya = yb
3mx we g e t y - mx +
1 : m2
Jk*
E
(3mx, a. b , y ) , hence y2
=
x
(3x, a, b )
(3x, a , b, x )
From 3y
t h e n we choose
+ ~ X V , -3x2
=
4xv.
If
t o get
m = 2
4xv
-
0
then
y
=
2x
-3.3"-2
+
-
v
such t h a t
0, contradicting
D.SARACINO and C.WOOD
224 characteristic 3".
If 4xv
hence n - 2
=
+
k
- 1
#
n - 1.
0 then so k =
=
?3"-l, and so 3k-1*a3n-2 = t3"-l,
2. Thus we have k
=
2, x2
E
<3"-*> - <3"-l>
The rest of Case B (ii) follows from Lemma 3. Again we omit verification of homogeneity, as tedious and routine. This proves the Classification Theorem, and thus completes o u r description of finite homogeneous rings of odd characteristic.
References:
C. Berline and G. Cherlin, QE rings in characteristic pn, J . Symbolic Logic 48 (1983), 140-162. C. Berline and C. Cherlin, QE rings in characteristic p, in Logic Year 1979-80 (Storrs), Lecture Notes in Math. 859 (Springer, Berlin, 1981). G.
Cherlin and A. Lachlan, Stable finite homogeneous structures, preprint.
Lachlan, On countable stable structures which are homogeneous for a finite relational language, preprint.
A.
D. Saracino and C. Wood, Finite QE rings in characteristic p2, to appear in Annals of Pure and Applied Logic. D. Saracino and C. Wood, QE commutative nilrings, J. Symbolic Logic 49 (1984), 644-651.
LOGIC COLLOQUIUM '84
225
J.E.Paris, A.J. Wilkie.and G.M. Wilrners (Editors)
0 Elsevier Science Publishers B. V. (North-Holland), I986
SUBSTRUCTURE LATTICES OF MODELS OF PEANO ARITHMETIC
*
James H. Schmerl
Department of Mathematics University of Connecticut Storrs, Connecticut 06268 U.S.A.
1. INTRODUCTION The intersection of an arbitrary collection of elementary substructures of a model
N
N . N forms a lattice, denoted and referred to as the substructure lattice of N . The basic question
of Peano arithmetic
(PA)
is again an elementary substructure of
Hence, the collection of elementary substructures of by
Lt(N)
concerning substructure lattices is:
Which lattices can be realized as substruc-
ture lattices? That is, for which lattices L that L = Lt(N)? tions T
of
models of
hl of PA such
The question can also be asked for models of specific comple-
PA : Which lattices can be realized as substructure lattices of At present, there is no known way that this question distinguishes
T ?
between different completions T arithmetic).
is there a model
of
This peculiarity of
, provided
PA
TA
is an end extension of the minimal model of all other completions of
T # TA
(where TA
is true
is due to the fact that each model of TA
, whereas
TA
the minimal models of
PA have a rich collection of cofinal extensions. Fur-
thermore, we know of certain finite lattices which cannot occur as
Lt(!d)
when
N is an end extension of its minimal elementary substructure, but no such restriction is known for cofinal extensions. tice, also known as M5 : proved here) if
T # TA
if
, then
N b
TA
,
A specific example is the
1-3-1 lat-
then Lt(N) f M5 ; however (as will be
there is some
N
C T
such that
Lt(N) = M5
.
Therefore, it seems appropriate to study the more general intermediate G -
-~ ture lattice Lt(N/M) generated by
M
.
, where M 4 N , which
That is, Lt(N/M) = {M,
E
is the principal filter of Lt(N)
:
M
< Mo<
NI
.
text the basic question has the following reformulation: For each model PA
, which
Lt(N)
In this con-
M
of
lattices can be realized as intermediate structure lattices Lt(N/M)
Refinements to this question can now be made by restricting
N
tension, a cofinal extension, or some other type of extension of The main question to be considered in this paper is: *Research was supported in part by NSF Grant No. 8301603.
?
to be an end ex-
M
.
For a given countable,
J.H. SCHMERL
226
M
nonstandard model where
N
of
, which
PA
finite lattices can be realized as
M ?
is a cofinal extension of
Lt(N/M)
It is still not known that for any such
M
there are any finite lattices which cannot be realized in this manner.
PA
,
It will be proved, however, that for any countable, nonstandard model those finite lattices which can be realized as
M ,
final extension of
for any lattice
L
N
such that embeds
.
L
is the 1-3-1
M , there
From now on, all structures are models of will be working with a formalization of functions.
PA
PA
.
lattice. Moreover,
is a cofinal extension
Throughout this paper we
that includes terms for all definable
Thus, all extensions and substructures of models will be assumed to
M 4 M . If M < N N generated by M u {a) .
and
be elementary. We adhere to the convention that then M(a)
of
Many finite lattices will be
Lt(N/M)
and any nonstandard model
N of M such that Lt(N/M)
M
is a co-
For example, every countable, nonstandard M
shown to be realized in this way. has a cofinal extension
.
depend only on Th(M)
, where N
Lt(N/M)
is the substructure of
a
E
N
,
In this paragraph the previous developments on the problem of characterizing the possible substructure lattices will be discussed. a unique
Mo
E
Lt(N/M)
M,,
end extension of
.
such that
M
M
3
N , then there is M and N is an has a
The fundamental theorem of MacDowell and Specker [9]
has a proper end extension. Gaifman [5] extended the MacDowell-
Specker theorem by showing that every Lt(N/M)
If
is a cofinal extension of
Therefore, it follows that every nonstandard M
proper cofinal extension. is that every
M,,
M
has a minimal end extension
N
( s o that
is the 2-element lattice), and Blass [l] showed that every countable non-
standard model has a minimal cofinal extension.
Later in [7] Gaifman showed that
other distributive lattices, such as all finite Boolean lattices, some infinite Boolean lattices, and some chains, could also be realized. Paris [ll] characterized those countable distributive lattices which can occur as substructure lattices as those which are complete and compactly generated. that for any finite distributive lattice L
N such that Lt(N/M)
^.
L
.
every model ,!d
Schmerl [15] proved has an end extension
Then Mills [lo] generalized this, showing that for
any complete w -like-compactly generated distributive lattice L and any M 1 there is N k M such that Lt(N/M) 2 L Paris [12] showed that Lt(N) need not
.
be distributive by proving that every model Lt(N/M)
embeds the
1-3-1
M
has an extension
lattice (or even the 1-u-1 lattice).
with an earlier observation of Gaifman [7] and Paris [ll] that if extension of
M
,
then Lt(N/M)
is not the 1-n-1
Wilkie [17] showed that the lattice he proved that every model M
Lt(N)
N
such that
This contrasts
N
is an end
lattice whenever 3 5 n ’< w
.
need not even be modular; specifically,
has an end extension
N
such that
Lt(N/M)
is the
Substructure Lattices of Models of Peano Arithmetic
221
pentagon lattice. He also showed in [17] that the hexagon lattice (and other related lattices)
cannot occur as
Lt(N/M)
N
when
is an end extension of
In outline the contents of this paper are as follows.
M .
In 12 we discuss a
certain type of lattice representation. This section is lattice-theoretic and contains the crucial definitions which will be used later.
Some examples of
these types of representations of lattices are presented in 53.
All of these ex-
In 14 we prove the main result
amples come from canonical partition theorems.
Then
which relates substructure lattices and the lattice representations of 52.
applying the examples from 13 we obtain in 55 all of our results about specific lattices which can be realized as substructure lattices. We conclude with 16 with some conjectures. 2.
REPRESENTATIONS OF LATTICES Given a set A
and
n
let
II(A)
be the set of all partitions of
, then we write
E
a if
{a, b}
in
n(A)
C
C
for some C
E TI
.
-
b (mod
A
.
If a,b
TI)
There are two extreme partitions QA
and
LA
, where a- b (mod LA)
A
E
a = b
iff
,
and a- b (mod OA) We partially order refinement of
vl
.
R(A)
so
for every
that if
r1,n2
Then we have for each
TI^
T I ~ V
Let 9* : n(B)
=
f3 : A +
n(A)
B
.
E
Il , then n1 5 n2
TI
E
n(A)).
{AoC_ A : A = A n A # 0 1 2 +
A
n(A)
ff
n2
s a
that
becomes a complete lattice (which is the dual of
the lattice one usually considers on then
E
n (lA.
OA( With this partial order, n(A)
a,b
In particular, if r1,n2 E n1 and A 2
0
for some
4
E
E
n(A)
be an injection. Then there is an induced function
such that for any
a,b
E
A
and
TI E
n(B)
,
,
TI^} .
a - b (mod O*(n))
228
J.H. SCHMERL
iff
@ ( a ) = Q(b)
and
@*(n
A S B let
VIA
1 and
Let
2
e*,(n)
L
=
-+
.
and
of
a
n1,n2
b e any f i n i t e l a t t i c e .
OL
by
representation
whenever
v n2)
8*(QB) = QA,
i s an i n j e c t i o n
II(A)
n
E
, we
II(B)
We d e n o t e t h e minimum and maximum
0
lL r e s p e c t i v e l y ( o r simply by
L
1
8*(1 ) =
B A’ In particular, i f
.
n(B)
E
1 i s t h e i n c l u s i o n f u n c t i o n , then f o r each
B
(L,A,v)
L
e l e m e n t s of
One e a s i l y checks t h a t
) = 8*(n
: A
=
.
(mod n)
) v O*(n
a : L
-+
and
.
1)
,
n(A)
A
such t h a t
and a ( x v y)
=
a ( x ) v a(y)
f o r any
x,y
E
L
N o t i c e t h a t w e do n o t r e q u i r e t h a t a r e p r e s e n t a t i o n s a t i s f y Thus, i f we c o n s i d e r b o t h
L
and
. a(xhy) = a(x)~a(y).
a s bounded upper s e m i l a t t i c e s , t h e n a
II(A)
r e p r e s e n t a t i o n is merely a n embedding.
a : L
Let
-+
(alB)(x) = a(x)IB
be a representation.
n(A)
# 2 whenever
Ia(x)I
.
x
E
L
.
If
B
Notice t h a t
C
alB
A
,
Then
is nontrivial i f
a
t h e n we l e t
alB : L
+
n(B)
where
may f a i l t o be a r e p r e s e n t a t i o n s i n c e i t
may n o t be an i n j e c t i o n ; however, i f i t i s a n i n j e c t i o n t h e n i t is a r e p r e s e n t a t ion.
We now come t o t h e c r u c i a l new d e f i n i t i o n .
D e f i n i t i o n 2.1. W e say that
then
alX
that
a
a or
Let
a : L
-+
II(A)
be a r e p r e s e n t a t i o n of t h e f i n i t e l a t t i c e
h a s t h e 0 - c a n o n i c a l p a r t i t i o n p r o p e r t y i f whenever
alY
i s a r e p r e s e n t a t i o n of
L
.
n-canonical
X C_ A
such t h a t
alX
p a r t i t i o n p r o p e r t y and
(For b r e v i t y , we s a y t h a t
a
is a n
i s a r e p r e s e n t a t i o n of a(x)IX = (alX)(x) n-CPP
,
A = X U Y
Proceeding r e c u r s i v e l y , w e say
h a s t h e ( n + l ) - c a n o n i c a l p a r t i t i o n p r o p e r t y i f whenever
t h e r e is some
L
=
nlX
L
E
, then
II(A)
having t h e
f o r some
x
E
L
.
r e p r e s e n t a t i o n whenever i t is a re-
p r e s e n t a t i o n w i t h t h e n-canonical p a r t i t i o n p r o p e r t y.)
Lemma 2.2.
If
is nontrivial.
a : L
+ n(A)
i s an
n-CPP
r e p r e s e n t a t i o n f o r some
n
, then
a
.
Substructure Latticesof Models of Peano Arithmetic
Proof: x # 0
We proceed by induction on
,
0-CPP =
For a contradiction suppose
.
either
(alX)(x)
alX or
and
an
(n-l)-CPP
(alB)(x)
=
Lemma 2.3.
(alB)(O)
If
a : L
is also an m-CPP
L
.
(alY)(x)
=
BIB
=
either
so
and this contradicts alB
+
n(A)
is an
n-CPP
Let
B C_ A
, and
lrlX or
B C_ A
B C_ X
a(O)IX = a(x)IX
=
Now suppose n > 0 , such that
for some y or
E
.
B C_ Y
L
E
is an (n-1)-CPP L
.
=
0 and B =
n > 0
{X,Y}
BIB is By the
Hence
,
n
5
L
then
an
.
.
0
a
Suppose
be n partition of
representation of
By Lemma 2.2, either
BC_ X L
L
or
.
A
.
and
BC_ Y ,
Thus a
so
is a
0
+
R(A)
n-CPP
is an n-CPP
representation, then for some finite
representation.
Proof: Use the Compactness Theorem and induction on n
.
0
We will be interested in those finite lattices which, for each n n-CPP
.
being a representation of
T I ~ ,Y respectively, is a representation of
If a : L
, C X ~ Bis
BIB
for some x
0-CPP representation.
Lemma 2.4.
(alX)(O)
representation and m
without loss of generality, let
be such that
lrlB = (alB)(x) either
u(y)IB
that
By the definition of
representation.
Proof: Clearly it suffices to consider just m A = X U Y
.
, so
X # A # Y
.
n = 0
Then there is B C_ A
and
,
[a(y) IBI # 2
,
a(x)IY
=
{X,Y)
TI =
representation of
inductive hypothesis
=
Assume
.
is a representation, yet
alY
(a(Y)(O) = a(0)IY
and consider the partition
{X,Y) , where
a(x) n
229
representations.
<
iIi
, have
In practice, such representations exhibit more uniform-
ity than is required, suggesting the following definition.
Definition 2.5. Then a
Suppose
arrows a2
a. : Li
+
Il(Ai)
(in symbols: al
1is an injection 0 : A2
-f
A1
+
a2)
are representations for iff whenever
such that
and
e*(n)
=
a2(x)
for some x
E
L 2'
TI E
Il(A1)
i = 1,2
,
.
then there
J.H. SCHMERL
230 Notice that if
a
B
-+
B is nontrivial, then a is a 0-CPP represen-
and
tation. The procedure for showing that a finite lattice has, for each n n-CPP representation will typically be as follows. Suppose that presentation of
L
where
,
3.
-+
an
+
... + al + a,,
is nontrivial. Then, by induction each
a.
tation. L
an+l
-f
In particular, if
then a
is an n-CPP
a
-+
a
is a re-
and that
for each n < w
...
, an
< w
a
, where
,
an+l
is an n-CPP
represen-
is a nontrivial representation of
a
representation of
L
for each n < w
.
SOME EXAMPLES In this section we will give some examples of
n-CPP
representations of
lattices, and also of representations which arrow other ones.
Example 3.1.
For
1 5 j < w
..., j-1)
j = { O , 1,
XC
w
j < n 5 w
,
define
x
B. and
E
a,b
I
E
a w b (mod an(x)) It is easily checked that trivial provided
B
be the set of all subsets of j is the Boolean lattice with 2’ elements. For
so that B j denote the set of increasing j-tuples from
, and
that if
so
[XI’
we let
, let
n 2 3
.
,
[n]’ iff
ai = bi
for each
, then
.
TI^
Example 3.2. J
.
Let
E
x
.
.
‘TI E
II([w]’)
w
-+
a w
.
In particular, the
there is an infinite X C_ w [XI’ = a w ( x ) I [XI’ Thus B has a k-CPP represenj Indeed, for each n < w there is m < w such that
.am a Thus, if k < w then a n sufficiently large finite n -+
i
is a representation of B , and that it is nonj The theorem of ErdBs and Rado [ 4 ] , which is the canon-
a
.
Now let
a
ical version of Ramsey’s Theorem, implies that x E B such that j tation for each k < w
.
then
ErdiSs-Rado theorem asserts that if and
X
is a
.
k-CPP representation of
D be a finite distributive lattice with
D be the set of join-irreducible elements of D
.
For
B.
I
for all
ID1 2 2
, and
2 5 n
w , define
5
let
23 1
Substructure Lattices of Models of Peano Arithmetic : D
a so t h a t i f
x
D
E
a-b
and
a,b
(mod a n ( x ) )
a
on I J ( , t h a t
w
+
.
a
so
Thus,
D
.
has a
k-CPP
t h e r e is m r e p r e s e n t a t i o n of
n E n(k)
kn
a
j
is a r e p r e s e n t a t i o n of
and
a,b
a r b (mod a n ( n ) )
that
whenever
j
< w
D
and
J
E
D
, and
j 5 x
.
t h a t i t is non-
r e p r e s e n t a t i o n f o r each
.
n(k)
, where
2 5 k
< w
iff
a
j
iJ
b (mod n) j
denotes t h e s e t of f u n c t i o n s from
is a r e p r e s e n t a t i o n of
For
n(k)
, and
n
f o r each to
k .)
j < n
.
It is e a s i l y checked
t h a t i t is n o n t r i v i a l provided
A s p e c i a l c a s e of t h e theorem of Prtlmel and Voigt [131 which is t h e
canonical v e r s i o n of t h e Hales-Jewett Theorem ( s e e 181) implies t h a t f o r each
m < w
n < w
t h e r e is
m-CPP
r e p r e s e n t a t i o n of
Example 3.4.
.
a + a Thus, i f m < w , then m n for a l l sufficiently large n < w
such t h a t n(k)
By u s i n g t h e f u l l theorem of
Promel and Voigt r e f e r r e d t o i n Example 3.3. w e can f i n d more examples of f i n i t e l a t t i c e s which have
m < w
.
representations for a l l
m-CPP
These l a t t i c e s tend t o g e t r a t h e r
complicated.
The s i m p l e s t one, o t h e r than
the partition l a t t i c e s
II(k)
,
is t h e 10-
element l a t t i c e described a t t h e end of [13] and p i c t u r e d a t r i g h t . pentagon l a t t i c e
P
Notice t h a t t h e
is an i d e a l of t h i s
l a t t i c e , so i t a l s o has an t a t i o n for each
m < w
.
m-CPP
< w
.
, then
kn
E
.
.
k < w
such t h a t a + a Thus, i f k m n for a l l sufficientlylnrge n < w
W e consider r e p r e s e n t a t i o n s of
that if
n L 2
= b
j
, define
< w
(Here,
a
There is a r a t h e r s t r a i g h t f o r w a r d proof, by induction
w 2 5 n < w
Indeed, whenever then a is a k-CPP
Example 3.3.
a
.
n L 3
J
Il(n )
, then
nJ
iff
I t is e a s i l y checked t h a t
t r i v i a l provided
1 sn
E
+
represen-
.
a
is an
J.H. SCHMERL
232 Example 3 . 5 .
W i l k i e [ 1 7 ] shows, e s s e n t i a l l y , t h a t t h e r e i s a n o n t r i v i a l r e p r e s e n -
tation
+
P
a : P
has an
Example 3 . 6 . each
of t h e pentagon l a t t i c e
n(w)
m-CPP
For e a c h
,
n
as a s u b s e t of t h e p l a n e A
kxk
,
E
a l i n e through
(il,jl)
kxk
morphism.
Clearly,
a
-t
1-n-1
(i2,j2)
lattice.
For
Okxk
and
R2
Think o f
kxk
such t h a t
t o be s u c h t h a t i f
9.
(il,jl)
and
and
(i2,j2)
which i s p a r a l l e l ( o r e q u a l ) t o Let
are e q u i v a l e n t i f f t h e r e i s ak : Mnk
i s a n o n t r i v i a l r e p r e s e n t a t i o n of
ak
Therefore,
which b e s i d e s
in
L
.
then
i s isomorphic t o t h i s s u b l a t t i c e .
Mnk
a
of t h e f o l l o w i n g s o r t .
, and f o r any l i n e
R2
be t h e
n(kxk)
, define the partition associated with
kxkl z 2
(il,jl),(i2,j2) some
such t h a t
.
Mn+2
let
we o b t a i n a c e r t a i n s u b l a t t i c e of
k < w
lkxk , c o n s i s t s o f a l l p a r t i t i o n s of IP.
,
3 5 n < w
P
m < w
r e p r e s e n t a t i o n f o r each
-*
n(kxk)
Mnk
L
.
Then
be an iso-
provided
k 2 3
.
The theorem o f Deuber, Graham, Promel and V o i g t [ Z ] , which i s t h e c a n o n i c a l v e r s i o n of t h e 2-dimensional
v e r s i o n o f v a n d e r Waerden's t h e o r e m , i m p l i e s t h a t f o r
r < w
each
k < w
is a
k-CPP r e p r e s e n t a t i o n o f
there exists
such t h a t
M,
a
+
ak
.
Thus, i f
k < w
for all sufficiently large
r < w
.
then
a
The
theorem o f [ 2 ] h a s h i g h e r - d i m e n s i o n a l v e r s i o n s , which y i e l d a d d i t i o n a l examples.
Example 3.7.
F i n a l l y w e m e n t i o n t h a t t h e theorem o f Deuber, Promel and V o i g t [ 3 ]
y i e l d s s t i l l more examples. L 0C L 1 C L 2 C
...
m < w
t i o n f o r each i n some
4.
Lk
From t h i s theorem t h e r e i s a n i n c r e a s i n g s e q u e n c e
of f i n i t e l a t t i c e s such t h a t each
.
Lk
has an
m-CPP
representa-
I t i s n o t e w o r t h y t h a t e v e r y f i n i t e l a t t i c e i s embeddable
'
THE M A I N THEOREM
The p r i n c i p a l r e s u l t of t h i s s e c t i o n is Theorem 4 . 1 which g i v e s a f i r s t - o r d e r c h a r a c t e r i z a t i o n o f t h o s e c o u n t a b l e n o n s t a n d a r d models tensions
N
such t h a t
Theorem 4.1.
lattice.
M k= PA
Let
hi
t h a t h a v e c o f i n a l ex-
i s some g i v e n f i n i t e l a t t i c e .
b e c o u n t a b l e and n o n s t a n d a r d , and l e t
Then t h e f o l l o w i n g are e q u i v a l e n t . has a cofinal extension
(1)
M
(2)
For each
Proof:
Lt(N/M)
n < w
,M
+ "L
N
such t h a t
Lt(N/M) = L
L
be a f i n i t e
.
h a s a n n-CPP r e p r e s e n t a t i o n " .
Throughout t h i s p r o o f any term which w e r e f e r t o is p e r m i t t e d t o h a v e
rameters from
M
.
pa-
Substructure Lattices of Models of Peano Arithmetic
=+
(1)
.
(2)
countable.)
Lt(N/M)
+
b e an isomorphism.
For e a c h
i
.
M
E
term
tij(x,y)
N
finable i n
N
such t h a t
such t h a t
X C_ M
b
t
ij
.
(b , b . ) = bivj i
(where by
XN
E
XN
J
let
L
E
b e s u c h t h a t h ( i ) = h4(b ) , and t h e n l e t b = bl For each i i s a term t i ( x ) such t h a t N + t i ( b ) = bi ; and f o r e a c h i,j E L
bi
M be
(The proof of t h i s d i r e c t i o n d o e s n o t r e q u i r e t h a t
h : L
Let
233
i
there
There is a bounded d e f i n a b l e
i s meant t h a t unique s u b s e t o f
by t h e same formula d e f i n i n g
L
E
t h e r e is a de-
N
M each o f t h e f o l l o w i n g
X ) and i n
sentences holds:
vx For e a c h x-y
i
(mod
a(i) =
TI^
E
, let
L
.
I- t i ( x )
M , that a
F i r s t w e show t h a t a(1) =
.
= lX
TI
x m y (mod n.)
, or,
x
J
r~
= t
, then
y (mod ni)
(ti(x),t.(x)) J
ij
To p r o v e
ij
TI ivj)
II
j
be d e f i n e d by
.
M
We now want t o show,
Clearly,
)
.
n
a(0) =
< w
.
= Ox
TI
and
i s a n i n j e c t i o n and t h a t
a
) and i To p r o v e t h e l a t t e r s t a t e m e n t , suppose
.
Then
x=y
,
(mod
, so
i,j
But
y (mod
and
L
E
.
x-
TI
.
t . (x) = t i (y)
( t ( y ) , t . ( y ) ) = tiVj(y) i 3
i s an i n j e c t i o n , suppose
a
ti ; t h a t is,
n(X)
+
n-CPP r e p r e s e n t a t i o n f o r e a c h
equivalently, t h a t i f
x z y (mod = t
a : L
is a representation.
x = y (mod
and
induced by
X
Let
So we need o n l y show t h a t
a(ivj) = a(i)va(j)
.
XI
c a n be made i n s i d e
a
is an
a
.
= ti(y)
This d e f i n i t i o n of
working i n s i d e
X[t (x) = 1
be t h e p a r t i t i o n of
ni
M
iff
TIi)
E
=
TI
tivj(x) TI
TIj
ivj)
.
*
Thus, f o r
ti(x) = t (y) i f f t . ( x ) = t . ( y ) D e f i n e a term t ( w ) as f o l l o w s : i 3 3 t ( w ) = z i f f t h e r e i s x E X such t h a t t i ( x ) = w and t . ( x ) = z Clearly, 3 S i m i l a r l y , bi E M(b 1 , so t h a t M(bi) = M(bj), N t= t ( b i ) = b j , so b3. E M(bi) j implying i = j This proves t h a t a is a r e p r e s e n t a t i o n of L
x,y
X
E
.
.
This representation i s nontrivial. yet
b.
E
M
.
a
To show t h a t
is a n
For, i f
5X
such t h a t
b
term which i n d u c e s t h e r e are terms
E TI
t'(z)
YN
.
and
a(i)lY = nlY
There i s some and
Ini[ = 2
, then
clearly
n-CPP r e p r e s e n t a t i o n f o r each s t a n d a r d
f i c e s t o show t h a t f o r e a c h d e f i n a b l e p a r t i t i o n Y
.
.
t"(z)
i
E
TI
of
f o r some L
such t h a t
i
,
# 0
it suf-
there is a definable
X
such t h a t
n
i
E
L
.
Let
t(x)
M(bi) = M ( t ( b ) )
t'(t(b)) = bi
and
be a
.
Thus,
t"(bi) = t(b)
.
J.H. SCHMERL
234 there is a definable Y C_ X
So
such that
b
M p vx
E
Y[t'(t(x))
M
E
Y[t"(ti(x))
YN
E
and
ti(x)]
=
and Vx
=
.
t(x)]
Therefore,
(2)
.
==+ (1)
N = M(b)
and
Lt(N/M) = L
.
Our object is to construct a type over M
realizes this type, then N
b
Let < n k : k < a)
M
b "L
+ "L
has an
is countable.
M
the proof where the countability of Suppose M has an a
E
r-CPP
M
M
n-CPP
is used.)
representation" for each standard n
representation".
M
be such that
Let
"a : L
X C M
r
E
.
such that
M
be a bounded definable sub-
0 -
-*
n(Xo)
is an
r-CPP representa-
...
We will define a decreasing sequence X 3 X 3 X 3 of subsets of 0 1- 2 such that in M , ak : L + n(%) is an (r-k)-CPP representation of L ,
tion". M
and
(This is the only place in
is nonstandard, by overspill there is a nonstandard
set and let
M
be an enumeration of all the definable partitions of
Such an enumeration exists because M
Since hl
such that if
is a cofinal extension of
is an
(r-k-1)-CPP
Let
Z(x)
5#
and let
nk
Since ak
.
0
representation and
< w
,
M k v x
Furthermore, Z(x)
be the partition of is, in particular, a
OX^+^ .
~ l ~ l X= ~ + ~
Thus, either
is the formula x # a
$(x) E
Z(x)
.
5 $(x) .
E
Therefore,
,
5%
Since ak
$(x)
i
"k I %+1 E
L
.
(allowing parameters from M) such Z(x)
is a type since
For, consider any formula $(x)
+
(mod TI ) iff M $(x) k representation, it follows that
M where x = y 0-CPP
is an
such that
for some
Certainly,
is complete.
or -$(x)
then since
Z(x)
.
Xk
a (i) = nkIXk+l k
be the set of formulas $(x)
that for some k
$(x)
Suppose we already have such an
representation, there is a definable %+1
(r-k)-CPP
each
.
ak = alXk
where
is in > 1
is nonprincipal.
,
Z(x)
.
* $(y).
Notice that if
it must be that
Substructure Lattices of Models of Peano Arithmetic Let Xo
b
realize
N
i s bounded,
Lt(N/M)
.
= L
For e a c h
.
bi = t i ( b )
Let
~ ( x ) i n a n e x t e n s i o n of i E L
ti(x)
let
M
M
is a c o f i n a l e x t e n s i o n o f
, and
.
If
M
Mo 4 N
(b)
If
i,j
E
.
,
L
f o r each t(x)
Let
i
,
M,,
then
then
= M(b.)
f o r some
M ( b i ) 4 M(b )
iff
j
terms
and
M so t h a t
t=
N
i=-
so
Since
i 5 j
v x E X,+,[t'(t(x))
) = bi
Conversely, some
k
E w
, M
r e f i n e m e n t of
.
i 5 j
E
E
L
. E
i E L
such t h a t
Lt(NIM)
Mo
of t h e form
.
There is a p a r t i t i o n
,
~ l ~ l X= ~cxk+l(i) + ~
ti(x) = ti(y)]
.
i
= c
Let
.
Hence,
.
.
rk
Therefore, there a r e
t(x)
J
that
,
, proving
(a)
.
z
a(j)
be a t e r m s u c h t h a t f o r any which c o n t a i n s
M(bi)
< M(bj) and l e t
= ti(x)]
i 5 j
= t(X)]
M(bi) = M(c)
a(i)
M(bi) 4 M(bj)
X,[t(t.(x))
a(i)IX, , so
proof of t h e theorem.
Mo
= t (x)A t"(ti(x))
Therefore,
suppose
+vx
c = t(b)
N
= t ( y ) ++
t ' ( c ) = b i A t"(bi)
1
Since
such t h a t
is t h e equiva1,ence c l a s s of
N t== t ( b
i
is f i n i t e , w e can f i n d
L
But t h e n f o r some
\+l[t(x)
E
t"(z)
For ( b ) , s u p p o s e
t(z)
.
t(x)
vx,y
M
t'(z)
.
L
E
be a term such t h a t
which is induced by Clearly,
.
= M(b)
be a t e r m s u c h t h a t
To p r o v e (a), l e t u s s u p p o s e t h a t t h e r e is some M(c)
N
let
W e w i l l now show t h a t
The f o l l o w i n g two t h i n g s need t o b e shown:
(a)
Mo # M(bi)
235
.
.
z
.
E
,
Then,
. N f= t ( b j )
=
Clearly, then,
bi
.
Then f o r
is a
a(j)l\
T h i s p r o v e s ( b ) and c o m p l e t e s t h e
0
The f o l l o w i n g v a r i a n t of Theorem 3 . 1 w i l l be u s e f u l i n t h e proof of C o r o l l a r y 5.8.
Theorem 4.2. lattice.
Let
M
PA
b e c o u n t a b l e and n o n s t a n d a r d , and l e t
Then t h e f o l l o w i n g are e q u i v a l e n t .
L
be a f i n i t e
J.H. SCHMERL
236
(1) For e a c h n o n s t a n d a r d a E M , M N /=b < a and Lt(N/M) = L
has a c o f i n a l extension
N
there is a cofinal extension
N
.
such t h a t
(2)
For e a c h n o n s t a n d a r d
and an isomorphism
t
N
where
an
a
<
and
For e a c h
(3)
Proof:
bi
h : L
M
E
such t h a t f o r each
,
L
has an
F i x some
representation".
i
.
h ( i ) = M(bi)
n < w
(1) 4 ( 3 ) .
n-CPP
a
Lt(N/M)
+
n-CPP
.
n < w
L
E
E
N
,
We know from Theorem 4 . 1 t h a t
"L h a s
b4
L
r e p r e s e n t a t i o n s of
n-CPP
n-CPP
.
bl
By
represen-
i n t h e real w o r l d .
L
4
bi
representation.
o v e r s p i l l , t h e r e is some s t a n d a r d o n e , and t h i s one must be a n
(3)
M
of
t h e r e is
I n f a c t , from c o n d i t i o n (1) i t f o l l o w s t h a t i n
t h e r e are a r b i t r a r i l y small n o n s t a n d a r d t a t i o n of
M(b)
=
(2).
J u s t m i m i c t h e proof of
.
xo
out with a s u f f i c i e n t l y s m a l l
=+ (1)
(2)
i n Theorem 4 . 1 , s t a r t i n g
0
There i s a n a n a l o g u e o f h a l f o f Theorem 4 . 1 f o r end e x t e n s i o n s .
Before
s t a t i n g t h i s a n a l o g u e we w i l l n e e d some d e f i n i t i o n s . For a l a t t i c e to
L
L
let
M
minimal e x t e n s i o n of M(a)
N
1
a new l e a s t e l e m e n t .
N
is cofinal i n
.
i s a n almost-minimal
Lt(N/M) = 1 @ L
if
be t h e l a t t i c e o b t a i n e d from
@ L
If
N
,
M 4 N
M
i s a n end e x t e n s i o n o f
f o r some l a t t i c e
L
M
.
and
by a d j o i n i n g
i s an almost-
and f o r any
Gaifman [ 6 ] o b s e r v e d ( s e e ( 6 . 1 . 1 )
e x t e n s i o n of
L
N
then (following [16])
a
E
N\M
,
o f Lemma 6 . 1 ) t h a t i f
is f i n i t e , t h e n
Lt(N/M)
The f o l l o w i n g d e f i n i t i o n s are due t o Gaifman 161, and are made w i t h r e s p e c t t o some f i x e d c o m p l e t i o n plete mula
1-type) a (u)
8
For any model
T
s u c h t h a t whenever
M
T
t
, where
LM(x) o v e r
M
M
A definable type
Z(x)
e a c h model of
i s almost-minimal.
T
i f f o r every formula
if
N
Z(x)
@ ( a , x ) t ZM(x) =
M(c)
$(u,x)
t h e r e is a f o r -
i s a c o n s t a n t term,
each d e f i n a b l e type
X(x)-extension of
type
A t y p e (by which w e mean a com-
of Peano a r i t h m e t i c .
is definable
Z(x)
, where
c
h a s a c a n o n i c a l e x t e n s i o n to a
iff
M
a8(a)
.
Then
is a n element r e a l i z i n g
i s s a i d to b e & - e x t e n s i o n a l
i f every
N
is a ZM(x)
C(x)-extension
. of
Substructure Lattices of Models of Peano Arithmetic
237
The following theorem is proved in much the same manner as was (2) 4 ( 1 ) in Theorem 4.1.
Theorem 4 . 3 .
2 PA
Suppose T
is a complete theory and
that
T ~ ~ x ( "has L an x-CPP representation").
type
Z(x)
such that whenever M
Lt(N/M) = 1 @ L
N
T and
.
L
a finite lattice such
Then there is an end-extensional is a
E(x)-extension
Proof: We will only construct the definable type Z(x)
, leaving
of M
, then
to the reader
the verification that it does what it is supposed to do. We work inside the minimal model
M
of
T
.
Since M k=
representation"), we are able to find a formula $(u,x)
t(u,v) which M :M $(u,a)} For each u and each i E L ,
have the following properties.
For each u
,
let Xu = (a
Then u
<
implies
<
b
t(u,i)
is a partition of
Then
a
U
w
A
a
X A
E
b
2
is a u -CPP
X
E
W
Xu
.
Let
a
: L
a
representation of
Our object is to define a sequence ($ let
Z(x)
+
.
II(XU)
(x) : n
, where
< u>
.
E
a (i) = t(u.i)
.
of formulas, and then
be the type these formulas generate.
Let ($n(x,y,z) are among
L
.
bx("L has an x-cpp
and a term
: n <
x, y, z ,
LO)
be a sequence of all formulas whose free variables
such that for each
an equivalence relation on M
.
z
the formula $n(*
,
*
,
z)
defines
, we are also going to define formulas Bn(u). will generate a minimal type, and the construction
Along with the formulas $,(x) (The sequence (en(u)
:
n
< W}
of this sequence will be exactly the same as the construction of a minimal type.) Let
e,(u)
be
u = u
.
Each of the following sentences should be in T :
J.H. SCHMERL
238 Let
$,(x)
, and
en(u)
In order to get
M
defined by
$n+l(x)
n,z
IYu = au(i)IYU
Then
5.
$n+l(x)
,
$n(*
YuC_ {x : bn(u,x)} TI
.
be the formula 3 u $(u,x)
Suppose we already have
$,(x)
and
that the following sentences are in T :
iff
, first obtain , z ) . For each
such that for each and x
E
aUIYU Y
is a
Bn+l(u)
.
Let
such that
u
TI
n,z 9n+l(u)
be the partition of
,
there is
z < u there is i E L such that 2 (u -(n+l)u)-CPP representation of L
for some u
.
.
0
CONSEQUENCES Using the theorems in 84 and the examples in 83 we can obtain interesting re-
sults about intermediate structure lattices.
Before stating these we make two
remarks about the examples from 8 3 , the first being relevant when applying Theorem 4.1 and the second when applying Theorem 4 . 3 .
Remark 5.1.
Suppose n
and
< w
L
is a finite lattice.
The statement "L has
an n-CPP
representation" can be formalized in the language of
sentence.
Thus, if in fact
L
an n-CPP
representation".
For example, we get from Example 3 . 3 that for each
n,k
< w
,
able if
PA b"II(k)
has an n-CPP
has an n-CPP
"L has an n-CPP
PA
representation, then
representation".
as a PA +"L
z1 has
It would be quite remark-
representation" were consistent with
PA without
actually being true.
Actually, more than what is indicated in Remark 5.1 is true, and this is the point of the second remark.
An inspection of the proof of the theorem of Prijmel and Voigt [ 1 3 ]
Remark 5.2.
which was used in Example 3 . 3 reveals that it can be carried out on the basis of PA
.
Thus, whenever
tion")
.
25 k
carried out on the basis of lattice
< w
, then
PA
t/x("ll(k)
has an x-CPP
representa-
Similarly, the proof of the theorem involved in Example 3 . 2 can be D
,
PA k'Jx("D
PA
.
Thus, we get that for any finite distributive
has an x-CPP
representation")
.
Substructure Lattices of Models of Peano Arithmetic
239
A s consequences of Example 3.2 and Theorems 4 . 1 and 4.3, w e o b t a i n t h e f o l -
The second one is e x p l i c i t l y s t a t e d i n [15]; t h e f i r s t i s not
lowing two r e s u l t s .
but could a l s o be derived by t h e method of [15].
Corollary 5.3.
M be countable and nonstandard, and l e t L be a f i n i t e d i s M has a c o f i n a l extension N such t h a t Lt(N/M) = L
Let
tributive lattice.
Corollary 5.4.
Let
a unique atom.
Then
Lt(N/M)
2
L
.
.
Then
M be any model and L a f i n i t e d i s t r i b u t i v e l a t t i c e having M has an almost-minimal end extension N such t h a t
From Example 3.3 and Theorem 4 . 1 we can d e r i v e t h e next c o r o l l a r y .
Corollary 5.5. L e t M be countable and nonstandard, and l e t M has a c o f i n a l extension N such t h a t Lt(N/M) II(k)
.
2
In p a r t i c u l a r , w e can improve upon P a r i s [12] by t a k i n g the
1-3-1
2 5 k
< w
.
Then
k = 3 to realize
lattice.
Using Theorem 4.3 with Example 3.3 y i e l d s t h e following.
Corollary 5.6.
Let
M be any model and 2 5 k < N such t h a t Lt(N/M) 2 1
minimal end extension
w
.
Then
@ n(k)
.
M has an almost-
The c e l e b r a t e d theorem of Pudllk and TSma [14] says t h a t every f i n i t e l a t t i c e i s embeddable i n some lowing c o r o l l a r y .
II(k)
.
From t h i s theorem and Corollary 5.6 we g e t t h e f o l -
(Example 3.7 could have been used i n s t e a d of Example 3.3, but
t h e PudlPk-TSma theorem would s t i l l be necessary.)
Corollary 5.7.
Let
M be any model and L any f i n i t e l a t t i c e . Then M has an N such t h a t Lt(N/M) is f i n i t e and embeds L
almost-minimal end extension
.
Using Theorem 4.2 and t h e Compactness Theorem, we can o b t a i n t h e following corollary.
240
J.H. SCHMERL
Corollary 5.8.
Let
N
and
x E L
Corollary 5.9.
Let
be any l a t t i c e .
embeds
A sublattice
Lo
of
a < x < b
then
be a c o u n t a b l e s t r u c t u r e ,
hl
.
L
If
bl
Let
Lo = { x
.
But s i n c e
a l s o has a c o f i n a l e x t e n s i o n
NO
x
L
E
Lo
.
N
such t h a t
such t h a t
and l e t
h : L
M :h ( a )
, we
such t h a t
has a co-
a f i n i t e l a t t i c e , and
has a c o f i n a l e x t e n s i o n
L : a 5 x 5 b}
E
L t ( h ( b ) / h ( a ) ) = Lo
M
i s convex i f whenever
L
,
Then
.
L
, then M has a c o f i n a l e x t e n s i o n No
Lt(N/M) = L
Proof:
L
Lt(N/hl)
a r e such t h a t
convex s u b l a t t i c e of
Then
be nonstandard and
such t h a t
be a l a t t i c e .
L
a , b E Lo
bl
Let
f i n a l extension
+
Lt(No/M) = Lo
Lt(N/M)
Lo
.
be an isomorphism.
g e t from Theorem 4 . 1 t h a t
Lt(No/Cl) = Lo
.
a
M
0
The l a t t i c e p i c t u r e d a t r i g h t i s a convex s u b l a t t i c e of t h e l a t t i c e pict u r e d i n Example 3 . 4 .
Therefore, every
countable nonstandard
M
extension
N
such t h a t
has a c o f i n a l Lt(N/M)
is
isomorphic t o t h i s l a t t i c e .
We f i n a l l y mention t h a t Example 3 . 6 t o g e t h e r w i t h t h e technique of t h e proof of Theorem 4 . 1 and w i t h an a d d i t i o n a l o v e r s p i l l argument y i e l d s t h e following.
Corollary 5.10. extension
6.
N
Let
M
be countable and nonstandard.
such t h a t
Lt(N/M)
is the
1-w-1
Then
M
has a c o f i n a l
lattice.
CONJECTURES
The problem of which l a t t i c e s , o r even which f i n i t e l a t t i c e s , can occur a s Lt(N)
remains u n s e t t l e d .
where
4 2 n < w
, can
t h e hexagon l a t t i c e . sible t o realize.
.
1-n-1
N k- TA
Lt(N)
, where N t=-
lattices,
be r e a l i z e d as s u b s t r u c t u r e l a t t i c e s , nor do w e know about
Y e t , t h e r e i s no f i n i t e l a t t i c e which i s known t o be impos-
There a r e f i n i t e l a t t i c e s t h a t cannot be r e a l i z e d a s
when
jecture."
A s an example, we do n o t know i f t h e
Lt(N)
,
I n regard t o t h e q u e s t i o n of which f i n i t e l a t t i c e s can occur a s TA
, Wilkie
[17] s a y s
"
... t h e r e
is n o t even an obvious con-
We propose h e r e t o go o u t on a limb and make such a c o n j e c t u r e .
Substructure Lattices of Models of Peano Arithmetic
Let
1
that
be a f i n i t e l a t t i c e and l e t
L
.
C
E
We s a y t h a t
t h e f o l l o w i n g manner.
, where
D = IMo
E
Lt(N/M) : N
N
E
D)
(so, i n p a r t i c u l a r ,
.
be a l i n e a r l y o r d e r e d s u b s e t s u c h
i s a ranked l a t t i c e .
(L,C)
I( -< N
Let
CL
C
Ranked l a t t i c e s a r i s e i n
is f i n i t e .
Lt(N/M)
i s a n end e x t e n s i o n o f
Then l e t
24 1
Let
MOl
Lt*(N/M) = (Lt(N/M),D)
,
which i s a
ranked l a t t i c e . Suppose
h : (L,C)
c o f i n a l e x t e n s i o n of
,
h(x) each of
is a p r o p e r t y of
z
, set -
L
E
h(x)
iff
Lemma 6.1.
x
, and
-
Suppose
and
E
and
E
.
L
Whether o r n o t
h(y)
h(y)
is a
i s a n end e x t e n s i o n of
To see t h i s make t h e f o l l o w i n g d e f i n i t i o n :
C : z 5 cl)
.
Then
h(y)
i s a n end e x t e n s i o n of
h(y)
(L,C)
x < y
whether o r n o t
.
(L,C) min([c
=
y ;
=
Lt*(N/M)
z
h(x)
i s a f i n i t e ranked l a t t i c e and
for
is a cofinal extension
h(x)
iff
x
=
y
x
A
(L,C) = Lt*(N/M)
.
.
Then t h e f o l l o w i n g two p r o p e r t i e s h o l d :
and
(6.1.1)
If
x,y
L
(6.1.2)
If
w,x,y,z
E
-
w = w A z , t h e n
x=7
and
L
E
,
-
then
x
A
are s u c h t h a t
x
V
-
y = x
.
w = y
V
w = z
,
x
A
w = y
A
w
x = y .
P r o o f : The p r o o f s are e a s y and c a n be found i n [ 1 5 ] . t i o n 1 . 5 o f [ 1 5 ] , and f o r
For
see Lemma 2 . 6 of [ 1 5 ] .
(6.1.2)
(6.1.1)
see P r o p o s i -
IJ
A s a n example of how Lemma 6 . 1 i s
implemented, w e w i l l show t h a t t h e diagrammed 6-element Lt(N/M)
M
.
lattice
For, suppose
.
Either
otherwise
(6.1.1)
0,l
E
C
Similarly,
y
C
E
c a n n o t be r e a l i z e d as
L
, where N
is a n end e x t e n s i o n o f (L,C) = Lt*(N/M) x
E
or
C
w
E
C
But t h e n ( 6 . 1 . 2 )
E
C
since
is contradicted.
or
w
6
l i n e a r l y o r d e r e d , so e i t h e r Therefore
w
, where
, and
so
C
.
x
But
4C
C or
C = {O,w,l>
is contradicted.
is
y
.
4C .
W
J.H. SCHMERL
242
The main r e s u l t of [ 1 5 ] i s t h a t i f l a t t i c e satisfying
(6.1.1)
N
mentary e x t e n s i o n
such t h a t
such t h a t
i s a f i n i t e ranked d i s t r i b u t i v e
(L,C) 0
C
E
Lt*(N/M)
, then
= (L,C)
e a c h model
.
M has an e l e -
Only minor changes i n t h a t
proof a r e needed t o prove t h e f o l l o w i n g theorem.
Theorem 6.2. fying
Suppose t h a t
(6.1.1).
i s a f i n i t e ranked d i s t r i b u t i v e l a t t i c e s a t i s -
(L.C)
M
and suppose t h a t
has an elementary e xte n sio n
N
is a c o u n t a b l e n o n s t a n d a r d model.
such t h a t
Lt*(N/M)
Consequently, f o r ranked d i s t r i b u t i v e l a t t i c e s
(6.1.2)
implies
M
Then
.
(L,C)
2
(6.1.1).
I t i s q u i t e e a s y t o g i v e a d i r e c t proof of t h i s . A l l known examples o f f i n i t e l a t t i c e s t h a t c a n n o t be r e a l i z e d by
N
where
,
TA
f a i l b e c a u s e of Lemma 6 . 1 .
,
Lt(N)
Encouraged by t h i s and by Theorem
6 . 2 , w e make t h e f o l l o w i n g b o l d c o n j e c t u r e s .
C o n j e c t u r e 6.3. (6.1.1) Then
and
Suppose t h a t (6.1.2),
(6.1.2)
lattice that
L
.
2
(6.1.1)
and
L
,
(L,{l))
M
i s a c o u n t a b l e n o n s t a n d a r d model.
such t h a t
Lt*(N/M) = (L,C)
.
is a ranked l a t t i c e s a t i s f y i n g
(6.1.1)
So, i n p a r t i c u l a r , C o n j e c t u r e 6 . 3 would imply t h a t f o r any f i n i t e
, every
Lt(N/M)
C o n j e c t u r e 6.4. Then
i s a f i n i t e ranked l a t t i c e s a t i s f y i n g
M h a s an e l e m e n t a r y e x t e n s i o n N
For any f i n i t e l a t t i c e and
(L,C)
and suppose t h a t
L
nonstandard countable
.
Suppose t h a t
(6.1.2)
where
(L,C) 0 E C
M has a c o f i n a l extension N
such
i s a f i n i t e ranked l a t t i c e s a t i s f y i n g
, and
suppose t h a t
M h a s a n e l e m e n t a r y end e x t e n s i o n N
such t h a t
is any model of Lt*(N/M)
^.
(L,C)
.
PA
.
REFERENCES
2
[l]
Blass, A . , On c e r t a i n t y p e s and models f o r a r i t h m e t i c , J. Symb. Logic (1974), 151-162.
[2]
Deuber, W . , Graham, R.L., PrLlmel, H.J. and V o i g t , B . , A c a n o n i c a l p a r t i t i o n theorem f o r e q u i v a l e n c e r e l a t i o n s on Z t , J. Comb. Th. (A) 24 (1983), 331-339.
[3]
Deuber, W . , Prtimel, H.J. and V o i g t , B . , A c a n o n i c a l p a r t i t i o n theorem f o r c h a i n s i n r e g u l a r trees, i n : C o m b i n a t o r i a l Theory, L e c t u r e Notes i n Mathem a t i c s 969, S p r i n g e r - V e r l a g , 1983, pp. 115-132.
Substructure Lattices of Models of Peano Arithmetic [41 [51
243
Erdos, P. and Rado, R., A combinatorial theorem, J. London Math. SOC, 5 (1950), 249-255. for conGaifman, . H... On local arithmetic functions and their applications .. structing types of Peano's arithmetic, in: Mathematical Logic and Foundations of Set Theory (North-Holland, Amsterdam, 1970), 105-121. Gaifman, H., A note on models and submodels of arithmetic, in: Conference in Mathematical Logic, London '70, Lecture Notes in Mathematics 255, Springer-Verlag, 1972, pp. 128-144. Gaifman, H., Models and types of Peano's arithmetic, Annals Math. Logic (1976), 223-306.
9
Graham, R.L., Rothschild, B.L., and Spencer, J.H., Ramsey Theory, Wiley, New York, 1980. MacDowell, R., and Specker, R., Modelle der Arithmetic, in: Methods (Pergamon Press and PWN, Warsaw, 1961), 257-263. Mills,
G.,
Infinitistic
Substructure lattices of models of arithmetic, Annals Math. Logic
16 (1979), 145-180.
Paris, J., On models of arithmetic, in: Conference in Mathematical Logic, London '70, Lecture Notes 255, Springer-Verlag, 1972, pp. 252-280. Paris, J., Models of arithmetic and the 1-3-1 lattice, Fund. Math. 195-199.
95
(1977),
Prtrmel, H.J., and Voigt, B., Canonical partition theorems for parameter sets, J. Comb. Th.(A) 35 (1983), 309-327. Pudldk, P., and Tams, J., Every finite lattice can be embedded in the lattice of all equivalences over a finite set, Algebra Universalis lo (1980), 74-95. Schmerl, J.H., Extending models of arithmetic, Annals Math. Logic 89-109.
(1978),
Shelah, S . , End extensions and number of countable models, J. Symb. Logic (1978), 550-562.
43
Wilkie, A . , On models of arithmetic having ncn-modular substructure lattices, Fund. Math. 95 (1977), 223-237.
LOGIC COLLOQUIUM '84 .I.B. Paris, A.J. Wilkie, and C.M. Wiliners (Editors) Elsevier Science Pirblishers B. V. (North-Holland), 1986
245
DECIDABLE THEORIES OF VALUATED ABELIAN GROUPS P. H. Schmitt Universitat Heidelberg Im Neuenheimer Feld 294 6 9 0 0 Heidelberg W. Germany
We introduce the class of tamely p-valuated abelian groups and prove its decidability by a relative quantifier elimination procedure. SECTION 0: INTRODUCTION Somewhere in the seventies the concept of a valuated Abelian group emerged and quickly established itself as a ubiquiteous and promising research topic in Abelian group theory. For a prime p a p-valuation v is a function from an Abelian group G into the ordinals plus the symbol m (this situation will be refined a little in Section 1 below) satisfying the axioms: (1)
v(pg)
> v(g)
if
v(g) <
(2)
v(g-h) L min{v(g) ,v(h)}
.
-
These are generalizations of the axioms characterizing the p-height function. A good idea of this concept is conveyed by the theorem proved in [ 6 1 that for every p-valuation v on G there is a supergroup H of G such that v coincides on G with the p-height function on H. For more information see 1 3 1 , [ 5 1 , t 6 1 . We want to start in this paper the model theoretic investigations of p-valuated Abelian groups, which we view as two-sorted structures (G,a U {m],v) with Q. an ordinal and v a function from G into Q. U I - } . In [ 7 1 we already obtained some undecidability results. Here we introduce the notion of a tamely p-valuated Abelian group (G,a,v) by the requirements: (1)
G is torsion free
(2)
v(pg)
(3)
for all primes q and all s 2 1 qSG is a nice subgroup of G i.e. for every g E G there is some h such that v(g+qsh) is maximal among v (q+qSh')
=
v(g) + 1
for
v(g) <
.
m
246
P.H. SCHMITT
These axioms were chosen in a minimal way to prevent an undecidability proof by the methods of [ 7 ] and we obtain indeed as Theorem 5.3 below: The elementary theory of tamely p-valuated Abelian groups is decidable. By adding countably many unary predicates to the value sort we can eliminate group quantifiers in favor of value quantifiers (relative quantifier elimination Theorem 3.1). Decidability is obtained from this by observing the decidability of the theory of well-orderings with countably many unary predicates (Theorem 5.2) and by determining which of these structures arise from tamely p-valuated Abelian groups (Lemma 5.1). The relative quantifier elimination result also leads to an axiomatization of the class of direct sums of p-valuated rank one groups satisfying v(pg) = v(g) + 1 for v(g) < m and of the class of free valuated Zp-modules (introduced in [ 6 ] ) . We also consider the class of valuated pS-groups which are defined as direct sums of copies of cyclic groups of order ps with a p-valuation w satisfying: If for 0 < m < s and w ( p g ) = y then there is some h satisfying pmh = pmg and y
=
w(h)
+ m.
Valuated pS-groups occur basically as quotients G/pSG of tamely p-valuated Abelian groups. We obtain in the course of the proof of Theorem 3.1 also a relative quantifier elimination Theorem for this class and subsequently decidability of its theory. SECTION 1: PRELIMINARIES We use standard notation concerning group theory as established e.g. in L. Fuchs' monograph on infinite Abelian groups. [ X I denotes the cardinality of the set X. proof.
signals the end of a
Let p be a prime. Definition: A p-valuated Abelian group is a two-sorted structure (G,a U {co,cl},v) where G is an Abelian group
a U {cO,cl} is a linearly ordered set with a a well-ordering and for all y E a : y < co < c1 v is a mapping from G into a U {co,cl~ satisfying the axioms:
Decidable Theories of Valuated Abelian Groups
241
(VI) v(pg) > v(g) for all g such that v(g) < co (V2) v(g-h)
>=
minIv(g) ,v(h)}
(V3) v(g) = c1 iff g = 0. In the following "group" will always mean Abelian group. We will write (G,u,v) for p-valuated groups the presence of the additional elements co,cl being implicitely understood; or even (G,v) when CY is clear from the context. two-sorted language Lv which we will be using contains group variable x,y,z,... group function symbols and constant +,-,O value variables S , U , ~ I binary relation symbol < , constant symbols co,cl (111) the function symbol v.
Terms built up from group function and group constant symbols are called group terms. The constants co,cl and strings v(t) for any group term t are called value terms. We denote by L, the language of linear orders containing the symbols listed under (111). As an immediate consequence of the axioms we obtain: Lemma 1.1. (a) if v(g) < v(h) then v(g+h) = v(g) (b) v(n-g) = v(g) for (n,p) = 1.
0
The most important notion in analysing the elementary properties of p-valuated groups G is the m-part v, of v for every natural number m > 1. Definition: vm(g) = min{y : there is no h E G such that v(g+mh)
2
yj.
In order that vm(g) be defined for all elements g we add a new value denoted by c2 such that c1 < c2. Lemma 1.2. (a) vm(g) > v(g) (b) vm(g) = c2 iff g is divisible by m (c) vm(g+mh) = vm(g) (dl
vm(g-h) 2 min vm(g) ,vm(h)
(e) If vm (g) < vm(h) then v,(g+h)
=
vm(g)
P.H. SCHMITT
248
If (n,p) = 1 then vmen(n-g) = vm(g)
(f)
(9) If (m,n) = 1 then vm(ng) = vm(g)
(h) If (m,n) = 1 then ~,.~(g)= min{vn(g) ,vm(g)} (i) vm(g) is a limit number or one of the elements co,c1,c2 if (m,p) = 1. Proofs : (a)
-
(e) are immediate from the definition.
(f) By Lemma 1.1 (b) we have vm(g) inequality is obvious.
2 vman(n-g) while the reverse
(9)We have in any case vm(n-g) 2 vm(g). If now v(ng+mh) y then choose k,r such that 1 = kn + rm and we obtain from v(nk-g+mk-h)Ly:
v(g+m(k-h-r.g)) 2 y. (h) Of course vnmm(g) 5 vn(g),vm(g). If on the other hand there are elements h, ,h2 such that v (g + nhl) 2 y and v (g + mh2) 2 y then we get for kn + rm = 1 v(g + nm(rhl + kh2) ) 2 y. (i) It suffices to show for each fi < c1 that v(g+mh) 2 fi implies the existence of some h' such that v(g+mh') 2 B+1. Let 1 = kp + rm then v(pkg+mpkh) > B and we may take h' = pkh - rg. Remark: The m-part v, of a p-valuation need not be again a p-valuation, axiom (Vl) may be violated, even for m = p. instead of vm for m = ps. P*S Definition: A subgroup H of a p-valuated group (G,v) is nice if for every g E G there is some h E H such that for all h' E H
Notation: We write v
v(g+h) 2 v(g+h'). Remark: Nice subgroups are exactly the kernels of homomorphisms in the category of p-valuated groups. Definition: For a p-valuated group (G,v), m > 1 and y a value, we set: GV(y)
=
{ g E G : v(g)
GV(m,y) = Ig E G
:
g y)
vm(g) 2 Y}
.
When there is no danger of confusion we will simply write G ( y ) and G(m,y).
Decidable Theories of Valuated Abelian Groups
249
Lemma 1 . 3 . (i)
pSG i s a n i c e subgroup of
(G,a,v)
iff Vp , s ( g )
i s never a l i m i t v a l u e .
( i i ) For m w i t h
(m,p) = 1
mG i s a n i c e subgroup of
(G,a,v)
iff v m ( g ) i s never a l i m i t of l i m i t s . Proofs. obvious.
0
Remark: The e l e m e n t co i s c o n s i d e r e d a l i m i t v a l u e , c 1 , c 2 a r e n o t .
co i s c o n s i d e r e d a l i m i t o f l i m i t s i f t h e r e i s no g r e a t e s t l i m i t number i n a. D e f i n i t i o n : A p-valuated group (G,a,v) i s c a l l e d tamely p-valuated if (Tl)
G is torsionfree
(T2)
v ( p g ) = v ( g ) + 1 f o r v ( g ) < co
(T3)
qSG i s a n i c e subgroup of
( G , a , v ) f o r a l l p r i m e s p and s 2 - 1.
H e r e y + l d e n o t e s t h e s u c c e s s o r of t h e o r d i n a l y , w i t h t h e conven-
t i o n s co+l = c , and c l + l = c 2 and y + l < c o f o r e v e r y y < co.
Lemma 1 . 4 .
L e t G b e a tamely p - v a l u a t e d g r o u p .
( P Y ) = v m ( g ) + 1 if ~ ~ . ~ (
v
Vp , s ( P 4 )
’ vp,,(g)
if v p , s ( g ) < co.
For g E n {pnG : n
1 ) w e have v ( g )
2 co.
For a l l p r i m e s q d i f f e r e n t from p k if v ( q g ) = y and k < s qls
t h e n t h e r e e x i s t s some h s a t i s f y i n g v
k
q (g-h) E qSG.
q.s
( h ) = y and
k ( p g ) = y and k < s PIS t h e n t h e r e e x i s t s some h s a t i s f y i n g v (h) P,S k p (g-h) E pSG. If v
+ k
=
y and
Proofs : ( i )v (pg) v m ( g ) + 1 i s o b v i o u s . NOW l e t v ( p g ) = y + 1 which Pam p-m i s by (T3) and Lemma 1.3 ( i ) a l w a y s t h e case f o r ~ ~ . ~ (
w e have f o r some h: v ( p g
+
phm) = y and t h e r e f o r e by ( T 2 )
250
P.H. SCHMITT
+ 1
v(g+mh)
=
y . Thus vm(g)
L y. If ~ ~ . ~ ( p>= gco) then we must al-
= gc2) by (T3) and (TI). ready have ~ ~ . ~ ( p
(ii) immediate consequence of (T3), Lemma 1.3 (ii) and (Vl). (iii) Assume v(g) < co, say v(g) = A+k for X a limit number and 0 5 k < w . By assumption there is some h satisfying g = pk+’h which yields the contradiction v(g) = v(h) + k + 1 . (iv) Let us first consider the case y = w (B+l) < co. There is some ho such that v(q k g + q s ho) 2 w(B+l). Thus vq,s(qkg) = vq,s(q kg+gsgo)>= >= ~ ~ , ~ (s-k g + ho) q > v(g+qS-kho) = v(qkg+qSho) 2 0 . 5 + 1 . Since (g+qS-kho) = y and (g+qS-kho) can only be a limit number v v qrs q,s we set h = g + qs-kho. The argument for y = co in case there is a kg) = c1 then largest limit number < co is analogous. If v (qkg + qSho) = v (g + qS-kho) = co and we must have (g+qS-kho) = c2 is impossible. Thus v (~~+q~-~h = ,c)1 since v qts
9 1
s
we use h = g + qs-kho. In case v h E qSG.
qIs
(qkq) = c2 we may take any
.
(v) analogously to (iv) We observe that vps(q) possible for g f pG.
=
c1 is only
If is of course possible to iterate the process that leads from the p-valuation v to its m-part vm and consider mappings vm;n . The next lemma records that nothing new is obtained in this way. Here we deal only with the qs-parts for primes q different from p. The corresponding results for ps-parts are contained in section 4, Lemma 4.2. Lemma 1.5. (i)
vk;m.n(g)
=
(ii)
vm;,(g)
c3 for (m,n)
=
minlvk;m(g)lvk;n(g)l for (m,n) = 1. =
1.
(iii) vq,s;q,t(9) = vq,s(g) + 1 if t
2
s.
In tamely p-valuated groups we have in addition
Proofs : (i) obvious. (ii) It suffices to notice that by definition ~,;~(g)= c3 if there exists an element h such that g + nh E mG. (iii) It suffices to observe that vq,s(g) > y implies
25 1
Decidable Theories of Valuated Abelian Groups > y+l which is a trivial consequence of the definition q,s;q,t (g (9) >= y which follows and second that vq,s;q,t(g) > y implies v qIs from t >= s
V
(qs-tg) = y . Then we get by (a) and (b) TO continue let v qrs Y & V q,s;q,t(9) y+l. Let us first assume y < co, thus y = w(D+l) for some $ . Let 6 be some ordinal satisfying w e @ < 6 < y . For some h (g+qth) > 6. Thus we must already have vq, (g + sth) 2 Y we have v qls and therefore v (9) = y+l. Similarly v (qs-tg) = co is q,s;q,t qrs (qS-tg) = c1 or c2 the result follows dihandled while in case v qIs rectly from (a) and (b). The dimensions r ( y ) = dim(G(q,y)/G(q,y+l)) will turn out to be imq portant elementary invariants for tamely p-valuated groups. We extend the basic language Lv to :L which will allow us to speak of these dimensions in quantifier-free formulas. :L contains in addition to the symbols of Lv: unary value predicates
r qrn unary function symbols vm a constant symbol
for all primes q, n for m
5
1
2 denoting the m-part of v
c2
-
We need only explain the interpretation of r q,n’ (G,v) I= rq,n(y) iff r (y) >= n in case y < c1 9 and r (c2) is definied to be false for all n. qrn and c2. Similarly LT is L< enriched by r qrn For a p-valuated group (G,v) we denote ist LG-expansions by (G,v)* and its value-part by Val*(G,v). Lemma 1.6. :L
is a definitional extension of
h.
Proof: clear. We let TV* denote the L:-theory
P
of tamely p-valuated groups.
Let { (Gi,ai vi): i E I} be a family of p-valuated groups and i E I}. The direct sum (G,a,v) = @ C {(Gi,ai,vi):i E I} c1 = sup{ai is obtained by taking G to be the direct sum of the family (Gi : i E I of groups and setting for g E G: v(g) = min{vi(gi) : i E I}
.
P.H. SCHMITT
252
It is easily verified that Lemma 1.7. (i)
The class of tamely p-valuated groups is closed under direct sums.
(ii) For (G,a,v) = @
1 { (Gi,ai,vi) :
i E 11,all primes q and
Lemma 1.8. For tamely p-valuated groups (G,v): (i)
For all primes q different from p and s 2 1, y 5 co: G(qS,y)nqS-lG/G(qS,y+l)nqs-lG N G(q s+ 1 ,y)nqSG/G(qS+l ,y+l )nqSG.
(ii) For all s 2 1 and y < co: G (pS,y)npS-lG/G(pS,y+l)npS-lG
Y
G(pS+l ,y+l)npSG/G(pS+l,y+2)npSG
Proof: (i) By Lemma 1.2 (f) & (g) the required isomorphism is induced by multiplication with q. (ii) The required isomorphism is induced by multiplication with p. - Use Lemma 1.4 (ii). 0 The following Lemma is an easy consequence of the definition of v
4,s’ Lemma 1.9. For tamely p-valuated groups (G,v), all primes q,s and y < co G (qs,y+l) ”qS-’G/G (qs,y+2)nqS-lG
N
2
1
G(y)nqS-lG/G(y+l ) + (qSGnG(y) .o
The dimension of G(y)/G(y+l) is certainly an elementary invariant. The next lemma shows how it can be expressed in terms of the invariants we chose. Lemma 1.10. For tamely p-valuated groups, y < co: (i)
G(y+l)/G(y+2) = G(y)/G(y+l)
(ii) If y
=
@
.
G(P,Y+~)/G(~,Y+~)
A + k with A a limit number, 0 5 k < w then
~ ( y ) / ~ ( y + l=) @
1
OZiLk
G(p,X+i+l)/G(p,h+i+2)
-
Proof: (i) G (y+l ) /G(y+2) is a group of exponent p, thus G(y+l) n pG/G(y+2) n pG is a direct summand. Using 1.9 its complementary summand is seen to be isomorphic to G(pry+2)/G(pry+3). Finally G(y+l) n pG/G(y+2) n pG is by axiom (T2) isomorphic to G(y)/G(y+l). (ii) Iterate (i) and notice that G(A)
n
pG/G(A+l) = 101 for A a
Decidable Theories of Valuated Abelian Groups
253
limit number.
0
The next lemma shows that also the Smielew-invariante G/qG are expressible in terms of the predicates r q,n' Lemma 1.11. For tamely p-valuated groups we have for all primes q
1 r,(y) (mod m ) YLCl (i.e. either both cardinals are finite and then equal or both are infinite). dim(G/qG)
=
Proof: For every y 5 c1 let (hl : i E IY 1 be representatives for a basis of G(q,y)/G(q,y+l) considered as a vector space over the field F with q elements. We may assume that v(hi) + 1 = y, if q = p and 9 v(hr) + o = y if q # p. We claim (1)
Ho = (hl : y
2 c l , i E IY 1 is q-independent
.
1 nl-hl E qG with not all i,y coefficients nx divisible by q. Let y be the smallest value such that nl*hr E G(y) + pG, for some i E Iy q does not divide nr. Then 1 iEI ny-h; E G(q,y+l) which is a contradicrion. This establishes i.e. 1 iEIy already (1) and the inequality 2 in the claim of the lemma. If Ho is infinite we are thus finished. If Ho is finite we claim
Assume this were not the case and we have
(2)
G = H
+
pG where H is the subgroup generated by Ho
.
For g E G we find by repeated application of the definition of Ho elements 0 = ho,hl,...,hi E H such that yi+l = v (g-hi) > yi. Since 9 we assumed Ho to be finite also the range of vq is finite and we will have for some i v (g-hi) = c2 i.e. g-hi E qG. q From (2) we get by modularity G/qG c* H/H ll pG. Since H is a q-pure subgroup of G we have in addition H/H ll pG c* H/qH. Since Ho is a basis for H we are finished. 0 Lemma 1.12. Let (G,v) be tamely p-valuated. For every g with v(g1 < co there is some n such that v (g) = v(g) + 1 for all m 2 n. PIm (ii) If for some limit number A < co, G I= r ( A + w ) for a prime qtl y < A+@ . q # p, then G satisfies rp,l(y) for some y , A
(i)
Proof: Because of vp,n+l (9) 5 vP,,(g) there must be some n such that for all m 2 n Vp,m(g) = vp,n(g) = 6. Assume vp,m(g) > v(g) + 1 for some m g n. For some m' 2 m there is thus an element h such that F' If v(g+pm'hm,) = 6 > v(g) + 1, in particular v(g) = v ( p h,,). v(g) = A + k with A a limit number, 0 5 k < w this is for m ' > k a contradiction to Lemma 1.4 (v).
254
P.H. SCHMITT
(ii) The assumption r ( X + w ) implies the existence of some element 4r1 h such that X <= v(h) < X + w . By (i) we have v (h) = v(h) + 1 for Prn some n. If n = 1 then we are done; otherwise we may assume V p,n-l(h) > v(h) + 1 which yields the existence of an element h' satisfying v(h+pn-'h') > v(h). In particular v(h) = v(pn-'ht) = = v(h') + n-I. If v (h') > v(h') + 1 then v(h' +ph") > v(h') for P,n some h" which implies v(pn-lh' + pnh") > v(h) and therefore the contradiction v(h+pnh") > v(h). Thus vp(h') = v(h') + 1 and G satis0 fies r (v(h') + 1). PI 1
SECTION 2 : GROUPS WITH MORE THAN ONE VALUATION The purpose of this section is to show that admitting more than one p-valuation, be it for different primes or the same, yields undecidability even for very restricted classes. We first present a hereditarily undecidable auxiliary class AUX which is designed so that it can easily be interpreted in theories of valuated groups. The language of AUX contains three unary relation symbols X1,X2,X3 and three binary relation symbols <,R1,R2. AUX will be the class of structures (A,X1,X2,X3,<,R1,R2) satisfying the following conditions: (1)
(A,<) is a well-ordering.
(2)
Xi is strictly less then Xi+l for i = 1,2 i.e. for all a E Xi, b E Xi+l we have a < b.
(3)
XI
U
X2 U X3 = A and for every i (Xi,<) is of order type
5 - w(4)
For all a,b in A R1 (a,b) implies a E X I , b E X2 and R2(a,b implies a E XI, b E X3.
(5)
Ri is minimurnclosed for i = 1,2, i.e. for all a,b,c,d R. (a,b) and Ri(c,d) implies Ri(min(a,c) ,min(b,d)).
Lemma 2.1. AUX is hereditarily undecidable. Proof: We aim to interpret the theory of two equivalence relations El,E2 in AUX using the formulas (~~(x,y) = 3z(Ri(x,~)&Ri(y,z)&Vu(Ri(x,u)&Ri(y,u)+ u<_z)vx=y)& L%
XI (X)&X1(y)
.
We notice that in very model A of the theory of AUX $i defines an equivalence relation on Xl(A). Let on the other hand two equivalence
Decidable Theories of Valuated Abelian Groups
255
relations E1,E2 on the set B be given. We may assume w.1.o.g. that B is countable and B = {n : n < k} for some k 5 w. Let { C . . : i < k . ) 3 ,= 3 for some j = 1,2 , k . < w be enumerations without repetitions for I = all E j equivalence classes. We construct a model A of AUX by taking (Xl,<)to be (k,<) and (X2,<) (resp. (X3,<)) to be ( k l , < ) (resp. (k2,<)). Thus the universe of A is the ordinal k + k., + k 2 and the order relation of A is the natural ordering. The relations R. for 3 j = 1,2 are defined on A by: R1 (n,m) iff n < k, m = k+i for some i < kl and i < 1 where C1,l is the El equivalence class of n. R2(n,m) iff n < k, m = k + k l + i for some i < k 2 and i < 1 where C2,1 is the E2 equivalence class of n. It is easily seen that the structure A defined in this way satisfies requirements ( 1 ) - ( 5 ) . Furthermore we have for j = 1 , 2 and n1,n2 < k A I= Jlj(n1-n2) iff
Ej(nlnn2)
.
Lemma 2 . 2 . Let p1,p2 be primes, different or not. Let K be the class of all structures (G,vl,v2) where for j = 1,2 (G,v.) is a tamely 3 p.-valuated group. Then K is hereditarily undecidable. 7
Proof: We will interpret AUX in the theory of K . We will use a1,a2,a3 to interpret X1,X2,X3. Since some of these sets may be finite we first fix formulas a4 which will later be narrowed down to a j . a; ( y ) = " 3 8 ( 0
5
B < w
a i ( y ) = "3B(w
5
B < w-2
a;(y)
= "3B(w-2 5 -
& y =
w*E)"
& y =
B < 0.3
&
w-B)"
y = w.B)"
The ordering relation of AUX will be interpreted by the restriction to a. = a 1 v a 2 v a 3 of the ordering on the ordinals. Finally the relations Rj will be interpreted by cp. as follows: 3
'pl
(a,E) =
~ 2 ( a r E )=
ai(a)
&
o i ( E ) & 3x(vl(x) = a
~ { ( a &) o;(E)
a1 (a)
= 3B(cp1 ( a , E ) v
02(~1)
= 3BQ1 ( E r a )
a3(a)
= 3Bv2(B,a)
&
~ x ( v ~ ( x =)
&
v2(x) = R )
&
v ~ ( x )= 13)
v2(a,13))
For a given countable structure A in AUX we may assume w.1.o.g. that
P.H. SCHMITT
256
for j = 1,2,3
x. c 7 -
The construction of a valuated
[w(j-l),w.j).
group G in K satisfying A = ( u o ( G ) , a l ( G I ,a2 (G) ,u3 (6),< ,cpl (GI ,cp2 ( G ) 1 is greatly facilitated by the fact that K is closed under direct sums and the following observation: Assume that for every i in some index set I there is a group Gi in K such that for all ordinals a,B Gi != c p . ( a , R ) implies 7 Rj ( a t 6 1 * If furthermore G is the direct sum of the family (Gi : i E I) then we still have G C c p . ( a , O ) implies R.(a,B), since the rela7 3 tions Rl,R2 are minimumclosed. It thus suffices to find for every pair a,B of ordinals satisfying R 1 ( a , 5 ) in A a valuated group G in
(i)
G t
cpl
K such that
(a,@)
(ii) for all y , 6 and j = 1 , 2
G
!= cp,(y,6)
implies R.(y,6) 3
and likewise for pairs satisfying R2(a,B). But this can easily be achieved by taking G to be the integers Z with vl(l) = a and v2(l) = 5.
0
SECTION 3 : RELATIVE QUANTIFIER ELIMINATION FOR TAMELY p-VALUATED GROUPS The main result of this section is the following relative quantifier elimination theorem. Theorem 3.1. Every LG-formula is in TV* equivalent to a formula P without group quantifiers. Corollary 3.2. For every L$-sentence cp there is a Lr-sentence J , such that for every tamely p-valuated group (G,a,v) (G,a,v) C
cp
iff
Val*(G,a,v) C
J,
.
Proof of Corollary 3.2. By the theorem there is a L:-sentence I)' without group quantifiers which is in TV* equivalent to cp. We only P have to get rid of atomic formulas in J,' involving the group constant 0, which can easily be done; we replace, to give just two examples V(0) = rl by rl = c1 and 0 + 0 = 0 by co = co and so forth. Corollary 3.3. For two tamely p-valuated groups (Gl,vl), (G2,v2): (G1,vl)
(G2,v2) iff
Val*(Gl,vl)
Proof: follows from 3 . 2 and Lemma 1 . 6 .
Val*(G2,v2)
. 0
Decidable Theories of Valuated Abelian Groups
257
Lemma 3.4. In order to prove theorem 3.1 it suffices to find for every L:-formula of the form (3.5) where
3 Xcp 'p
is a conjunction of formulas of the form v(t(x)) = rl
,
vm(t(x)) = rl
for t(x) group terms involving x and T- a value-variable or one of the constants co,c1,c2 an TV*-equivalent formula without group P quantifiers. Proof of Lemma 3.4. We prove theorem 3.1 by induction on the complexity of LG-formulas. The only difficult step arises when we are given a formula of the form 3x'po, where 'po does not contain group quantifiers and we are to eliminate 3x. We may certainly assume that all equations between group terms occuring in 'po are of the form t = 0. These we replace by v(t) = c l . Let S be the set of all group terms in 'po involving x and M the set of all natural numbers m > 1 such that vm occurs in 'po. Then 3xqo is equivalent to:
where q; arises from 'po by replacing each occurence of a value term v(t) by the variable and likewise vm(t) by .c: Notice that x does 0 no longer occur in cp;.
ct
Lemma 3.6. In order to prove theorem 3.1 it suffices to find an TV*-equivalent formula without group quantifiers for every L*-forP V mula of the form (3.7)
3x$
where $ is a conjunction of formulas of the form vm(t(x)) =
T-
with varying m 2 2, group terms t(x) and rl a value variable or one of the constants co,c1,c2. Proof: We will in a series of reductions replace a formula of type (3.5) by a Boolean combination of formulas without group quantifiers or of type (3.7). In order to keep notation down to managable size we will arque rather informally. In particular we will
-
frequently usethe trick to replace 3x'p by the conjunction of vQ, a consequence of the axioms 3x(v&Qi) for l z i l k with $, v
...
P.H. SCHMITT
258
-
not display assumptions which do not involve the quantified variable after they have been explained once
- switch to semantic verifications without extra warning. Let us first fix in greater detail our point of departure: A v(n.x+t.) = ni & A vmi(nix+ti) = 0 . ) iEI1 iEI2 where ti are terms not involving x and q i are value variables or one
(3.8)
3x(
of the constants co,c1,c2 and I1 f 0. Using torsionfreeness, Axiom (T2) and Lemmas 1 . 1 (b), 1 . 2 (b), 1.4 (i) we may assume that for all i E I1 U I2 : ni = n. Replacing nx by x and adding vn(x) = c2 we may even assume for all i E I1 U I2 ni=l (using Lemma 1 . 2 (f), (g)). At this point we see that we may assume ni < c 1 for all i E I ~ .~f n i = c 1 occurs, we eliminate x by replacing all occurences of x by the term -ti. If n i = c2 occurs the formula is contradictory and thus equivalent to, say co # co. By distinguishing cases we may assume that some order between the ni is fixed. Let 17 = max{qi : i E I,}. For i E I1 with n i < we have by Lemma 1 . 1 (a) v(x+ti)
=
ni
iff
v(x+t)
=
n
&
v(t-ti) =
ni
.
Since v(t-ti) = n . does no longer contain x we may assume w.1.o.g. that for all i E I1 : ni = n . Using Lemma 1 .2 (b) and 1 .4 (i) again we may assume furthermore that for all i E I2 mi = m. Now the arguments of vm are terms si(x) which will in general not be of the form x + t i , but this will no longer be of importance here. It seems appropriate to give an update of the formula (3.8) after all the above reductions have been performed:
By a variable transformation x
+
x+ti we may assume that for some
i E I 1 ti = 0. Thus (3.9) is equivalent to the disjunction of the following two formulas:
.
A V(X+ti)=q& A Vm(Si(X))=ni&Vm(X)="l) iEI1 iEI2 We will show that (3.10) is equivalent to the conjunction of the following two formulas:
(3.11)
3X(V(X)=n&
Decidable Theories of Valuated Abelian Groups
259
.
A V(Si(X)) = lli & Vm(X) > ll+l) iE12 One of the implications of this equivalence is trivial. For the reverse implication we assume that g,, resp. g1 is a witness for the existential quantifier in (3.12) resp. in (3.13).
(3.13)
3X(
Since vm (go ) > n+l is true we have v(go-nh) 2 - n+1 for some h. Thus A
iEIl
v(nh+ti) =
n
&
v(nh) = n
.
Since also vm(gl) > n+l we may by Lemma 1.2 (c) assume that v(gl)
2 n+l. Thus g1 + nh will be a witness for (3.10).
Let us next deal with (3.12). We may assume that n 5 co, since otherwise (3.12) is contradictory. Since for i # j v(x+ti) = q & v ( t i - t . ) > n implies v ( x + t . ) = rl 3 7 and v(x) = n & v(tj) > n implies v(x + t . ) = n we may assume 7 (3.14) for all i,j E I1 with i # j: v(ti-tj) = & v(t.) = 0. For i E I1
v(x) =
r)
and v(ti) =
q
already imply v(x
+ t1. ) -2 n.
Thus
vm(ti) = n+1 and vm(x) > r l + l would imply vm(x + ti) = n + 1 and therefore v(x+ti) = n . In this case we would drop i from 11. Thus we may assume (3.15)
for all i E 11: vm(ti) > n + l
.
This leads us to the conclusion that (3.12) is equivalent to (3.16)
11~1 .
I G ( ~ )n mG/G(n+l) n mGI >
To prove this let g be a witness for (3.12), then we have v(g-mgo) 2 - n+l for some go and by (3.15) also elements ty for i E I1 such that v(ti-mty) 2 q+l. Using the properties of g and (3.14) we see that mgo,mtp for i E I, represent different elements in G ( n ) n mG/G(n+l) n mG. If on the other hand (3.16) is satisfied we are guaranteed to find an element "go in G(Q) n mG, mgo
B
G(n+1) such that
mgo + G(n+1) # -ti + G(n+l) for all i E I1. Thus v(mgo) = Q & A v(mgo + ti) = n & vm(mgo) = c2 > n+l iEI1 By Lemma 1.10 (3.16) can be expressed by an L:-formula quantifiers.
. without group
Now we take up the further reduction of (3.11). We may again assume (3.14). But instead of (3.15) we have this time
P.H. SCHMITT
260
(3.17)
for all i E 11: vm(ti)
=
.
q+l
For assume to the contrary that vm(ti) > n+l for some i E 1 1 , then there would be go with v(ti +mgo) 2 n+1 and v(mgo) = v(ti) = 0 . Because of vm(x) = n+l we must have v(x+ngo) = n thus v(x+ ti) = n. We would consequently drop i from 11. We finally claim that (3.11) is equivalent to (3.18)
3x(
A
iEI1
vm(x+ti)
=
n
&
A
iEI2
vm(si(x)) = n
&
vm(x)
=
q+l)
.
One implication is easy using (3.17) and Lemma 1.2 (e). If on the other hand g is a witness for (3.18) then v(g+mgo) = for some go. Since by Lemma 1.2 (c) q + mgo is also a witness of (3.18) we may have assumed v(g) = I- right away. Since v(ti) = n , we have 2 r j for all i E I1. But since v(g+ ti) > would contradict v(g+ t.) 1 vm(g + ti) = n we must have v(g + ti) = n . 0 Lemma 3.19. To prove theorem 3.1 it suffices to eliminate the group quantifier in L:-formulas (3.20)
of the form
3Xbq
where Jiq is a conjunction of formulas of the form (ms+t) = n qrs for a fixed prime q, which may be equal to or different form p and varying m,s 2 1, group terms t without x and rl a value variable or one of the constants co,c1,c2.
v
Proof: We start with a formula of type (3.7). Using the same transformations as in the first step in the proof of Lemma 3.6 and Lemma 1.2 (h) we may restrict attention to formulas (3.21)
A Jigj 021$ already meet the requirements of (3.20). where the Ji qj It remains to be shown that the existential quantifier distributes over the conjunction. Let s be the highest exponent for which v qrs appears for some prime q in (3.21). Let n be the product of all q;, j 5 k and n . = n/qs. Then there are integers m . with 7 3 7 1 = mono + + mknk. If q . are witnesses for
3X
...
3
3xJiqj= let 9
=
monogo + V
qjns1
jX
~A E
...
Iqjrsi ~ (x + ti) =
rl.
+ mknkgk. We have by Lemma 1.2 (f), (g)
(m . n . g .+ m . n . t . )= 3 3 3
7 7
=
q.
26 1
Decidable Theories of Valuated Abelian Groups
for all j and all i E Ij. Since njo is divisible by qs for every 3 jo # j we obtain by Lemma 1.2 (c) v , (g+ti) = ni for all j and q~ rsi 0 i E I. 3'
In the remainder of this section we will eliminate the group quantifier in formulas of type (3.20) for a prime q different from p. The equal prime case will be dealt with in the next section in slightly greater generality. Starting from the formula ri (3.22) 3x A vq,si(miq x + ti) = 'li iEI with (mi,q) = 1 for all i E I we use Lemma 1 .2 (b), (f), (9) to obtain for all i E I : s . = s and mi = m. The x-free terms ti will of course change in this procedure. Now we may w.1.o.g. replace m by 1. Indeed if some element b satisr. fies for all i E I v (q b ' + ti) = ni then we have 1 = nlm + n2qs qts for appropriate n1,n2 and hence ~ ~ , ~ri (n l m bq+ t i ) = ni by Lemma 1.2 (c). We will use in the following for notational simplicity w to denote V
q,s' Thus it suffices to consider formulas of the form (3.23)
3x( A O$i<s
A
tEMi
W(qiX+t)
=
Qi,t)
.
By introducing possibly new existential value quantifiers as we did in the proof of Lemma 3.4 we may assume w.1.0.g. 0 E Mo
and
q M i s Mi+l
for all
.
i, 0 5 - i c s-1
By distinguishing cases we may assume that some linear order is fixed among the value-variables and constants ni,t. With respect to this . = 11i,to and rl i,tl < ni order we define ni = max{nilt: t E Mi}. If n 1 for to,tl E Mi then w(qix+tl) = ni,tl iff w(qix+to) = ni,to and w(to-tl) = ni,tl. Thus tl may be dropped from Mi. (3.24)
A W(qiX+t) = Qi) 3x( A O$i<s tENi where N. = {t E Mi : ni,t - ~ i } and
5
~1
5
- - *
5
ns-l
-
To see why the given chain of inequalities may be assumed, take some t E Ni for some i, 0 6 i < s-1. Because qt E Mi+l we get i+l xqt) = qi+l,qt 5 ni+l. If in the assumed ordering ni = Q1,t s w(q of the ni we have ni+l < q i then we are facing a
262
P.H. SCHMITT
contradictory formula and its group quantifier can trivially be eliminated. We also assume n i # co since otherwise ( 3 . 2 4 ) is again contradictory. To prepare the most important reduction step we first collect those i and j together for which n i - nj. Formally we define: ko
= O
{
the least element in { i
:
k ]= .< i 5 - s-1 and
k.
=
r
= the least j such that kr+l = s
K. 3
is not empty s otherwise
= [k.,k. )
5
3 I+? Furthermore we set
{O
,...,S-1)
W(qiX+t) = Il* & A A 7 i€Kj tENi for 0 <= j < r, 13. = k J+l . +1, for and T an arbitrary element from fixed for the remainder of this
(b) 'pr =
A
A
iEKr tENi
w(qix+t)
=
< qil
if this set
The formula ( 3 . 2 4 )
W(qlJX+T) >
.I;
all i E Kj, rli = q * j N1. which will be kept 3 section.
ns
where for all i E Kr : rli = Theorem 3 . 2 6 .
kj
.
(a) cpj =
(3.25)
n
n;.
is equivalent to
A
o<=j <=r
3xVj.
Proof: One implication is obvious. For the other assume that 3xcpj is satisfied for all j, 0 5 j 5 m r that this implies
r. We will prove by induction on m,
0
.
For m
=
3X( A A W(qiX+t) = ni) kr-mii<s tENi 0 this is 3x'pr while for m = r this is just ( 3 . 2 4 ) .
For the induction step from m to m+l we assume that there are elements h,g with (3.27)
A
A
kj+l_li<s tENi
w(pih+ t) = q i
with
j+l = r-m
and cp.(g) which we write in greater detail as: 3
(3.28)
A
k.
A
tENi
W(qig+t) = Qi
&
>
T l l
.
k. and T E N1+l 1+1 From w(ql+lg+r) > n 1 and w(q'+lh+.r) = =
W(q'+lg+T)
nl+,
>
n l we get
Decidable Theories of Valuated Abelian Groups
263
w(q1+l (g-h)) > q l . By Lemma 1.4 (iv) there is an element c such that (1)
w(g-h+c) > q 1
(ii) ql+lc = qsc' for some c'. Now (3.28) and (i) yields by Lemma 1.2 ( e )
.
A A w(qi(h-c) + t ) = q i k.
3X
h
A
< i<s kr-m-1 -
tENi
W(qiX+t) = q i
We may rephrase what we have achieved by saying that it suffices to eliminate the group quantifier in formulas of the form: (3.29)
A A W(qi)X+t) = T) & W(ql+'X+T) > rl ktill tENi where the last conjunct is missing exactly when 1 = s-1.
3x
We may assume that: (3.29) (a) for all i,j,k 5 i 5 j 5 1 and t E Ni, t' E N. I w (4 t') > q and w(q l-i+'t-T) rl since otherwise w(qx) L w(x) and Lemma 1.2 (d) would yield a contradiction. 0 The further treatment of (3.29) is split into two cases. s-i Case I: for all i,k 5 i 5 1 and t E Ni w(q t) 5 rl Case 11: complement of Case I. (t) 2 rl+l which by We may assume for every i and every t E Ni w q,i s-i Lemma 1 . 5 (iv) is equivalent to w(q t) 2 q. Thus the assumption of Case I actually gives us w(qs-lt) = 0 for all relevant i and t. Claim (3.30): Under the assumption of Case I formula (3.29) is equivalent to (a) 3x(w(q1+lx+-r) > (b)
1
0)
3x(w(q x + to) 2 q )
if
1+1 < s
if 1+1 = s , with to an arbitrary element from N1.
Proof of Claim 3.30. Only one implication of the claimed equivalence
P.H. SCHMITT
264
is non-trivial. Let g be an element satisfying w(q'+lg+T) > I-). Let tl be an arbitrary element of Nk. From 3.29 (a) we have > q which yields w(ql+'g + ql-k+ltl) 2 n Lemma 1.4 w (ql-k+lt 1 - T ) (iv) provides us with an element go such that w(g0) 2 rl and ql-k+l(9, - tl) E ql+l (mod qs). By this last congruence we find an 1+1 element u such that qku F (go- ti) (mod qs) and also ql+lu q g (mod qs). Therefore we have w(q'+lu+~) > n . But now we have in addition w(qku+ t1) = w(go) 2 n. By 3 . 2 9 (a) this gives w(qiu+ t) 2~ for all i,k 5 i 5 1 and all t E Ni. But w(qiu + t) > 0 would give
.
Wq,i(t) >
n+l which is by Lemma 1.5 equivalent to w(qit) > q , contradicting the assumptions of Case I. Thus u is a witness for ( 3 . 2 9 ) . 0
By definition 3x(w(q1+'x + T ) > n ) is equivalent to wq,l+l ( T ) > q+1 which is by Lemma 1.5 (iv) equivalent to w(qS-'+lr) > n+1, an LC-formulas without group quantifiers. Likewise 3x(w(q1x + to) 2 0 can be replaced by w(qs-'to) T-. We may thus assume from now on Case 11. Let ko be the least number i r k 5 i 5 1 such that for some t E Nko w (qs-kot) > n . We fix some to E Nko with this property. We claim that ( 3 . 2 9 ) is equivalent to (3.31)
3x(
A A W(qiX+t) = & w(q'+'X+T*) > l) k o ~ i ~ tEN; l where of course the last conjunct is omitted for 1+1 = s l-ko+l and N* = {t-qi-koto : t E Nil, T * = T - q to .
TO verify this claim we first note that by the choice of to and Lemma 1.5 (iv) there is some t; such that w(to - qkot&) > rl which i-kOt0-q1tA) > r- for all i,ko 5 i 5 1 and implies w(q w (ql-ko+'t 0 - ql+l t;) > q. From this we see that h = g + t& is a witness for ( 3 . 3 1 ) . If on the other hand we are given h satisfying ( 3 . 3 1 ) then g = h - t & certainly satisfies (3.32)
A
ko5izl
A
tENi
W(qig+t) = ll
&
W(q'+lg+T)
> 11
.
But what about the conjuncts for i < ko. Let tl be some arbitrary element in Nk. As in the verification of claim ( 3 . 3 0 ) we find using Lemma 1.4 (iv) some element u with qkou 3 gkog (mod q s ) and w(q ku + t,) 2 q. Thus u satisfies ( 3 . 3 2 ) in place of g and as before we find
265
Decidable Theories of Valuated Abelidn Groups
Considering the way we obtained Ni from Mi (see ( 3 . 2 4 ) ) we notice that for all i,k 5 i < 1 qNi 5 Ni+l. This yields (3.33)
(i)
qN:
(ii)
o
E
5 Nr+l for all i,ko 5 i < 1
~c~
(iii) for all i,j,ko 5 i 5 j 5 1, t E NT, t ’ E NT 3 w(qJ-it - t’) 2 q. This allows us to reduce ( 3 . 3 1 ) further to 1 (3.34) 3X( A W(q X + t ) = Q & W(ql+lX+T*) > 0 ) tEN where we have used N to abbreviate N* 1 ‘ Indeed assume that g is a witness for ( 3 . 3 4 ) . Since 0 E N we have in particular w(q 1g) = I-.Using Lemma 1 . 4 (iv) we may assume w.1.o.g. w(g) = I-,which gives by ( 3 . 3 3 ) (iii) at least w(qlg+t) 2 I- for all i,ko 5 i < 1 and all t E Ni. But w(qig + t) > I- would imply 1-i w(qlg+ql-it) > I- which contradicts q t E N.
0
We may further assume (3.33)
(iv) w(t)
= I-
for all t E N, t # 0
since we have in any case w(t) r- and those t E N, t # 1 w(t) > T- is true may be dropped since w(q x + t) = 0 iff w(qlx) = Tl & w(t) > n.
0
for which
If I- = c2 then we may assume that the last conjunct in ( 3 . 3 4 ) is missing and the remaining formula is equivalent to the purely group theoretic formula
By the results in [ I ] this formula free formula Qo in a language with dicates for the Szmielew invariant q‘. But Qo is in TV* equivalent to P Lemma 1.2 (b) and Lemma 1.11.
is equivalent to a quantifier divisibility predicates and preB(G) = dim G/q‘G for all primes a quantifier-free formula by
In case n 5 c 1 ( 3 . 3 4 ) is equivalent to the formula (3.35)
q 1x #
-t & q1+’X = 7 tEN in the group H ( Q ) = G(qs,q)/G(qs,r)+l) with the bar denoting the canonical homomorphismus from G(qS,n) in H ( Q ) . 3x(
A
Again using [ I ] we know that ( 3 . 3 5 ) is equivalent in the theory of groups which are direct sums of copies of E(qS) to a quantifier-free
P.H. SCHMITT
266
formula
JI, i n t h e l a n g u a g e c o n t a i n i n g p r e d i c a t e s f o r t h e r e l e v a n t
Szmielew i n v a r i a n t s , which i n t h i s c a s e i s j u s t t h e d i m e n s i o n o f H ( q ) [ q ] . But JI1 i s i n TV* e q u i v a l e n t t o a q u a n t i f i e r f r e e f o r m u l a P s i n c e r q ( q ) 2 n i f f dim H(q)[q] n by Lemma 1 . 8 ( i ) f o r q 5 co and s i m p l y checked f o r SECTION 4 : VALUATED
L e t p be a prime, s
q =
c1.
0
GROUPS 2
1 an i n t e g e r . A p - v a l u a t e d g r o u p ( G , w )
is
c a l l e d a v a l u a t e d pS-group i f G (as a g r o u p ) i s a d i r e c t sum of c o p i e s of
(Kl)
Z(ps
(K2) For e v e r y m , 0 < m < s and e v e r y v a l u e B 5 c o , g E G i f w(pmg) = 6 t h e n t h e r e i s an element h E G such t h a t B = w ( h + m and pmh = p g . One r e a s o n f o r s t u d y i n g t h i s c l a s s of p - v a l u a t e d Lemma 4.1.
g r o u p s i s g i v e n by:
( G , v ) i s a t a m e l y p - v a l u a t e d g r o u p t h e n (G/qSG,v
If
i s a v a l u a t e d pS-group.
PIS
)
-
= (G/pSG,v ) a renaming of P,S t h e c o n s t a n t s o f Lv i s needed: cz i s o m i t t e d , it n k v e r i s a v a l u e
Remark: I n d e f i n i n g t h e s t r u c t u r e ( H , w )
cy
for v
cz and cG2 = cHl .
=
Proof: S i n c e G i s t o r s i o n f r e e (Kl) i s t r u e . 1.4
( K 2 ) f o l l o w s from Lemma
( v ) . That v
i s a p - v a l u a t i o n f o l l o w s from Lemma 1.4 ( i i ) . [3 PtS S i n c e w e d o n o t r e q u i r e t h a t i n v a l u a t e d pS-groups w ( g ) i s n e v e r a
l i m i t t h e r e are v a l u a t e d pS-groups which d o n o t d e r i v e from t a m e l y p - v a l u a t e d g r o u p s i n t h e way g i v e n by lemma 4 . 1 . Examples. (El)
V a l u a t e d p-groups
are v a l u a t e d v e c t o r s p a c e s o v e r t h e f i e l d
w i t h p e l e m e n t s i n t h e s e n s e of
[2].
L.
F u c h s ' d e f i n i t i o n i s more
g e n e r a l by a l l o w i n g a r b i t r a r y c o m p l e t e l i n e a r o r d e r i n g s a s s e t of values. (E2)
L e t g b e a g e n e r a t o r f o r Z ( p s ) . W e d e f i n e w by w(pig) = a + i
for 0 < i < s and w ( 0 ) = c l . Then ( Z ( p s ) , w ) i s a v a l u a t e d pS-group. (E3)
Direct sums of v a l u a t e d pS-groups a r e a g a i n v a l u a t e d pS-groups.
I n s t u d y i n g v a l u a t e d pS-groups w e w i l l u s e t h e e x t e n d e d l a n g u a g e L** = Lv U { c 2 1 U I d n ( ) V
: n
2
1)
Decidable Theories of Valuated Abelian Groups
261
where t h e i n t e r p r e t a t i o n of d n i n t h e v a l u a t e d pS-group
(G,w)
is
g i v e n by: G k dn(n)
d i m ( G ( q ) n p S - l G / G ( n + l ) n pS-’G)
iff
where G ( n ) i s t h e subgroup { g E G : w ( g ) W e need n o t i n c l u d e i n L *:
n 5 co
L n}.
T V ( p , s ) d e n o t e s t h e Lv-theory of v a l u a t e d pS-groups, L$*-theory.
for
TV*(p,s) t h e
names f o r t h e mappings wm
s i n c e t h e y c a n b e e x p r e s s e d i n t e r m s of w w i t h o u t u s i n g g r o u p quant i f i e r s as t h e n e x t lemma shows.
Lemma 4 . 2 .
For v a l u a t e d pS-groups
(G,w)
t h e following are t r u e f o r
a l l g E G and m, 0 < m < s : (i)
~ ~ , ~ = (w g ( g )) + 1
(ii) wn(g) = c 2
for
for all
n
n
L
s
prime t o
p
( i i i ) i f w p , m ( g ) i s a l i m i t number, c o , c l o r c 2 t h e n Wp , m ( g )
= Wp,m+l ( P g )
(iv)
if w ( g ) i s a s u c c e s s o r number < co t h e n Plm W p,m+l ( P 9 ) = Wp,m(g) + 1
(v)
~ ~ , ~+ s( -gm -) 1 = ~ ( p ‘ - ~ g )
.
Proof: ( i ) t r i v i a l s i n c e pmh = 0 f o r a l l h E G . ( i i ) t r i v i a l s i n c e g i s d i v i s i b l e by a n y n prime t o p.
B e f o r e p r o v i n g ( i i i )w e n o t e : I f w p , m ( 4 ) < Wp,m+l (Pg) t h e n w(pg + p m + ’ h ) 2 ~ ~ , ~+ 1( fgo r) some h. By axiom ( K 2 ) w e have f o r some go E G
w(go)
g o - g + p m h E pS-lG.
2
w ( g ) + 1 and pgo = p g + p m + ’ h . By ( K l ) PIm Thus ( g o - g ) E P G and w e o b t a i n t h e c o n t r a -
d i c t i o n ~ ~ , ~ = ( wpg, m)( g o ) > w(go) Wp , m ( g )
5
2 ~ ~ , ~+ 1( . gS i )n c e
W p , m +(~p g ) i s t r u e f o r a n y p - v a l u a t i o n w e t h u s have Wp , m ( 4 )
I
Wp,m+l ( P 9 )
5 Wp,m(9) + 1
-
(iii)I f w
( 9 ) = f3 i s a l i m i t number o r co t h e n t h e r e c a n be by Plm ( K 2 ) no h s u c h t h a t w(pg + pm+’h ) = B . Thus w p,m+l ( P 4 ) = Wp,m(g)* If W p , m + l ( p g ) = c 2 t h e n pg E pm+’G which i m p l i e s by ( K l ) g E P G , i . e . Wp,m(g) = c 2 . (iv) Let w ( 9 ) be R+1. Then t h e r e i s s o m e h s u c h t h a t w(g+pmh) = R Prm and t h u s w ( p g + p m + l h ) 2 B+1, i . e . w (pg) 2 B+1. p,m+l 0 ( v ) F o l l o w s from ( i i i ) & ( i v ) .
268
P.H. SCHMITT
Lemma 4 . 3 . (i)
For all
n
< co:
G(n)/G(q+l) is a direct sum of cop es of Z(p)
(ii) G(n)/G(q+l) = G(n+l) (iii) If n
=
fl
plG/G(q+l+l
O'w+r, 0 < r < 1 then G ( n )
(iv) G ( C ~ )n P~G/G(C,) n p 1G
= 10)
n plG Il
for 1 >
plG/G(n+l) n plG
o
= {O}
.
Proof: (i) clear, since w(pg) > w(g) for w(g) < co, (ii) Since p 1G(n) c_ G(n+1) n p1G the mapping g * plg is a homomorphismus from plG(q) into G(n+l) fl plG/G(n+l+l) Il plG. By (K2) it is both injective and surjective. (iii) By (K2) w(p 19) n implies already w(p 1g) 2 r- + (1-r). (iv) By (K2) w(pg)
2 co implies already w(pg)
=
c l , i.e. pg = 0.
To apply the results of this section to complete the proof of Theorem 3 . 1 we need the following observations connecting the invariants r and dn.
Pt*
Lemma 4.4. Let (H,v) be a tamely p-valuated group, (G,w) = (H/pSH,vprs).Then (i)
in (G,w) dn(n) is false for all n 2 1 and q of the form = O . w + r , 0 5 r < s or = CI;
n
(G,w) C dn(n + 1) iff (H,v) C r ( n + 1-s+l); Prn (iii) in (G,w) dn(co) is false if s > 1 and (G,w) C dn(co) is equivalent to (H,v) C rp (cl) if s = 1. ,n Proof: (i) By (K2), (ii) by Lemma 1.8 (ii), (iii) by definition. (ii) for 1
2
s:
Theorem 4.5. Every L:*-formula is in TV*(p,s) equivalent to a formula without group quantifiers. Lemma 4.6. In order to prove theorem 4 . 5 every L:*-formula of the form (4.7)
it suffices to find for
3xcp
where cp is a conjunction of formulas of the form w(t) = for group terms t involving x and tl a value variable or one of the constants co,cl an TV*(p,s)-equivalent formula without group quantifiers. Proof: same as for lemma 3 . 4 .
0
Decidable Theories of Valuated Abelian Groups
269
The elimination of the group quantifier in formulas of the form (4.7) greatly parallels the reductions performed in section 3 from ( 3 . 2 3 ) onward where the role of the prime q is now taken over by p. The main difference lies in the fact that we have now w(pg) > w(g) while in section 3 w(qg) = w(g) was possible. For this reason we will give a very sketchy proof of theorem 4.5 indicating only the major steps in the reduction and refering to the corresponding parts of section 3 for the trick to be used. Starting from (4.7) we obtain as a first reduction (4.8)
3x( A A W(piX+t) = lli,t) Oci<s tEMi with t group terms not containing x and 0 for 0 5 i < s-I
.
Mo, pMi 5 Mi+l
E
This uses only the fact that w is a p-valuation and the possible introduction of new value quantifiers. Set r)i = max{qilt (4.9)
t
:
E
Mi}. Then ( 4 . 8 )
is equivalent to
3X( A A W(piX + t) = ni) OCiCs tENi with Ni = {t E Mi : TI. = nil and l,t (see ( 3 . 2 4 ) ) .
'lo < q l <
...
< T ? ~ - ~
Set ko
= 0
kj+l = r
=
K.
{
the least element in {i : k . < i 7 if this set is not empty s otherwise
&
qk
j
+ i-k' 3 < 11.3 1
the least j such that kr+l = s
.
[k.,k. ) 7 3+1 By this definitions we have for i 3
5 s-1
=
if also i+l E K . 3
if i+l 9 K . 7+1 Furthermore we set (a) cp. =
E
K. 7
ni + 1
then
qi+
=
then
ni+
> Q i + l .
A W(piX+t) = q j & W(plJ+T) > Q j 1 tENi + 1 and 'I is an arbitrary element where 0 5 j < r, 1, = k . 7+1 from N1 which will be kept fixed for the remainder of this j section.
A
iEKj
P.H. SCHMITT
270
(b) 'P,
=
A
iEKr
A
tENi
i W(p X + t ) = nj
.
As in (3.26) we show (4.9) equivalent to
A
ozjzr
3x'P. J '
This uses (K2). From now on it suffices to deal with formulas of the form
with ni+, = ni + 1 for all i,k 5 i < 1. The least conjunct is missing exactly for 1 = s-1 (compare (3.39)). At this point the further procedure is split into the cases Case I:
for all i,k 5 i 5 1 and all t E Ni
w(p s-it) 5 q i + s-i
Case 11: complement of case I. We remark that by lemma 4.2 the inequality specifying case I could equivalently be written as wp,i (t) 5 ni + 1. If case I applies then (4.10) is equivalent to (4.11) 3x(w(p1+1 X + T ) > n l + 1 if 1 < s-1 W(PtO)
2 n1 + 1
if
1 = s-1
where to is an arbitrary element from N1 (corresponds to claim 3.30). Lemma 4.2 tells us that also (4.11) is equivalent to a formula without group quantifiers. From now on we are working in Case 11. By the same type of argument as used in verifying (3.31) and (3.32) we equivalently replace (4.11) by a formula of the following form
with the last conjunct missing just for 1+1 = s and for all to,tl E N, to # tl (4.12) (a) w(t)
=
w(to- t,)
= rl
.
(4.12) (b) w (t) > n+l P,l For 1 = s-1 (4.12) is equivalent to dn(q) with n such that pn 2 IN1 +l.
he smallest number
Let us assume 1 < s-1. We may assume n < co since otherwise (4.12) would be contradictory. If W(T) > q + l then (4.12 is equivalent to (4.13)
3X(
A
tEN
W(plX+t)
= Tl
& W(P1+'X)
> Q+1)
.
Decidable Theories of Valuated Abelian Groups
27 1
This is certainly a consequence of (4.14) 3X(
A
tEN
W(pS-lX+t)
= rl)
.
For assume g is a witness for (4.14) then p s-1-1 g would be a witness for (4.13). If on the other hand g is a witness for (4.13) then 1
w(pl+’g) > T- + 1 implies by (4.2) w p,s-l (p 9) > T- + 1, thus > q + 1 for some h and h would be a witness for (4.14). w(p1g+pS-lh) Thus (4.13) and (4.14) are actually equivalent and we are reduced to the case of (4.12) with 1
=
s-1
.
We assume therefore from now on (4.12) (C) W(T) = II + 1
.
We also note that (4.12) (d) wp,l+l ( T ) > T - + 2 is implied by (4.12). We define No = {t E N : w(pt-r) > rl+1]. Claim 4.15.
(4.12) is equivalent to dn(q) with n the smallest number
such that pn 1. I NoI + 1.
Proof: Assume first that g i s a witness for (4.12) in some valuated pS-group (G,w). For all t E No we get w(p(plg + t)) > rl + 1 . Thus there are by axiom (K2) elements c; such that pc; = p(plg+ t) and w(c;) = w(p(plg+ t)) - 1 > 0. Setting ct = ctt -pig+ t we get pct = o and w(ct) = q . By axiom (Kl) we must have ct E pS-lG. For to,tl E No with to # tl we have w(cto - ctl) = q , since w(cto - ctl) > rl would imply w(cio - cil + to - t l ) 2 q + 1 and therefore w(to- t1) > q + 1 contradicting (4.12) (a). Thus I G ( q ) fl pS-lG/G(u+l) n pS-lGI ->INo[+l which implies dn(q).
...,
If on the other hand c l , cm are m = IN0 I + 1 representatives Of different cosets in G(q) I7 pS-lG modulo H = G(u+l) fl pS-lG. Let ci E G be such that p’c; = ci. By (4.12) ( d ) we find T~ such that W(P To + T ) 2 q + 2 which gives w ( p l + ’ ~ ~ =) q + l by (4.12) (c). 1 Axiom (K2) allows us to assume w(plTo) = q. Since p T~ + H, p1(-rO+ci) + H 1 5 i 5 m are (m+l) different cosets, we find some
c in {ci : 1 5 i 5 m} U { O } such that for all t E No W(P’(T~+C) + t) = q while pl+’ ( T ~ + c = ) pl+’~, still guarantees
W(P l+’T0+T)
> ll+1.
Finally consider t E “No
for which we certainly have
P.H. SCHMITT
212
w(p 1 ( T ~ + C+)t ) w(p'+l~~+pt) > n (4.12).
rl.
+
But strict inequality would imply
1 contrary to t
Thus T O + c is a witness for
No.
A s easy consequence of theorem 4.1 we obtain
Corollary 4.16. For every L**-sentence cp there is some Lr*-sentence V $ such that for all valuated pS-groups (G,w) (G,w) k cp
iff
Val**(G,w)
.
C $
Corollary 4.17. For any two valuated pS-groups (G1,w?), (G2,w2): (G,,wl)
(G2,w21
iff
Val**(Gl,wl)-Val**(G2,w2)
.
SECTION 5: DECIDABILITY RESULTS The major problem in proving decidability of the class of all tamely p-valuated groups, namely which Lf-structures do occur as the value part of a tamely p-valuated group, is solved in the following lemma. Lemma 5 . 1 . Let a be a well-ordered set, co,c1,c2 element such that subsets of CY U {co,c1,c2) then the folloa < c < c1 < c2 and r qIn wing conditions are equivalent: (I) there is a tamely p-valuated group (G,a,v) such that ) N Val*(G,v); q,n for all q and n 2 1 and all y:
(a U {co,cl ,c2),<,r
(11) (i)
rq,n+l( v ) implies r (Y) qln (ii) for all y: r ( y ) implies y # c2 and y is not a limit PI 1 number (iii) for all primes q # p and all y: r
q, 1
(y)
implies that y is
a limit value or c 1 but not a limit of limits (iv) if for q # p l(i<w.
rq,l(X+o) then r
PI 1
( X + i ) for some i,
Proof: Necessity of I1 (i) is obvious, follows for I1 (ii), (iii) from axiom (T3) and Lemma 1.3. Necessity of I1 (iv) derives from Lemma 1.12 (ii). To prove sufficiency we define for every prime q and value y: 0
n w
if if if
Y P rq,l Y E rq,n\rq,n+l y E rq,n for all n
Decidable Theories of Valuated Abelian Groups
273
By I1 (ii) it suffices to find a tamely p-valuated group (G,a,v) G such that for all values y and all primes q: rq(y) = fq(y). By Lemma 1 . 7 it suffices to find (Gl,a,vl) and (GX,a,vX) for and every limit number X < co such that range(vl) = (co,cll and range(vA)
=
(A
+i
:O<
i <w1 U
=
0
{ell
and
< ~ + . w rGX(y) = fq(y) for all y , A < y 9 If a largest limit number X exists in a we adopt the convention A + w = co. For G1 we have rG1 ( c , ) = dim Gl/qGl: so we pick a group q G1 with dim G1/qG1 = fq(cl) and valuate it trivially by v(g) = co if g # 0 and v(g) = c2 for g = 0. For G A and q # p we have fq(y) = 0 for all y # X + w by I1 (iii) and rGA(A+w) = dim GA/qGA by 1 . 1 1 . 9 Furthermore by I1 (ii) f (X+w) = 0. Let yo be A < yo < A + 0 such P that f (yo) > 0. If no such yo exists then by I1 (iv) f (A+o) = 0. P 9 Otherwise we define
(b)
= Z = (r/s : r,s E Z, (p,s) = 11 P Yfi for y , A < y < A+w, 0 5 i c f (y) for y # yo and 0 < i < fp(yo)
G
P
with the valuation v
given by vy , ( 1 ) = y
.
y,i Let (Go,a,vo) be the direct sum of all these groups. We have which we call dim Go/qGo = 0. It remains to choose (G Yo,g'a~vYoro)' (G2,a,v2) for short, to satisfy (b). Let ( Z a,w) be the additive P' group of the p-adic numbers valuated by w( 1 dipi) = yo + i O
. q # p, 0 5 i < fq(h+o) & (m,q)= 11 u {em : (m,p)= 1 1 q,i,m. where Z is the canonically embedded copy of the integers in ?fp with z
U {e
: q # p, o 5 i < f ( A + o ) I generating element say e and {el U Ie q,i,l q are linearly independent, m - e = e and m-e, = e. q,i,m q,i,l 0 Now (GA,a,vX) = (Go,a,vo) + (G2,a,v2) satisfies (b).
Theorem 5 . 2 . The theory W, of the class of all well-orderings with countably many additional unary predicates is decidable. Proof: By replacing the given predicates Pn by the collection of all finite intersections of Pn's and complements of Pn's it is easily
P.H. SCHMITT
214
seen that we may w.1.o.g. assume that the predicates Pn are mutually disjoint. Call the resulting theory wd. Let N, be the structure
b has at least length 4
(ii) the last three elements of b are 101 and there is no occurrence of a subsequence 101 in b except the one at the end (iii) all entries in b except the first one, b(O), are in { O , l l . For an element a of T, different from the empty sequence, we denote by e(a) the tail of a, obtained by omitting the first entry, a(0). We claim ( A ) For every countable well-ordering (X,C,Pn) with countably many
mutually disjoint predicates Pn there is a subset C 5 B such that (i) for a,b E C e(a) = e(b) implies a = b (ii) there is an isomorphism f : (e(C),c) + (X,c) (iii) for c E C c(0) = 0 iff f(e(c)) 9 Pm for all m c(0) = n+l iff f(e(c)) E Pn
.
That Co _c e(B) can be found to satisfy (ii) is already proved in [ 4 1 . From C, we easily find C 5 B such that (i)-(iii) hold true. From the decidability of SoS we get in particular that the firstorder theory of the countable structures (X,c,Pn) considered in ( A ) is decidable. But by the Lowenheim-Skolem Theorem for first-order logic this theory coincides with W,.d 0 Remark: The above proof shows that even the weak second order theory of the class of all well-orderings with countably many disjoint unary predicates is decidable. Theorem 5 . 3 . The theory of tamely p-valuated groups is decidable. Proof: We describe a recursive procedure that determines for every L,-sentence cp in finitely many steps whether cp is a theorem of TVP
Decidable Theories of Valuated Abelian Groups
215
or not. First we determine the sentence J, in Lt that is equivalent to cp in the sense of corollary 3 . 2 . Let Qo be the finite set of primes q and N the largest natural number such that some predicate rq,n, n 5 N occurs in $. Let $ * be the conjunction of the finitely many sentences in condition (11) of Lemma 5.1 where r for q E Qo qfn and n 5 N occur. We claim
.
9 TV I- cp iff W, t- $ * P We need only remark on the implication from right to left. Assume that (X,c,r ) is a model of Wu which satisfies $ * & -r$. We may qIn assume that (X,z) is a well-ordering, since changing the predicates r for q f Qo and q E Qo with n > N does not affect the truth of qfn $ * & T $ we may assume that the hypothesis of Lemma 5.1 are satisfied and we get a tamely p-valuated group (G,a,v) such that Val*(G,v) = (X,a,rq,n1 . By choice of J, we have G F i c p and thus not TVp I- Q . The procedure now continues by an apeal to Theorem 5.2. +
In a very parallel way we obtain: Theorem 5.4. The theory of valuated pS-groups is decidable. We conclude by Theorem 5.5. A p-valuated group (G,a,v) is elementary equivalent to a direct sum of torsion-free rank-one p-valuated groups satisfying v(pg) = v(g) + 1 iff (G,a,v) is tamely p-valuated and satisfies for every prime q # p and every limit number X < co (including X = 0 )
Proof: Necessity of the conditions is simply checked. For sufficiency we note that under the present hypothesis we may in the construction preformed in the proof of Lemma 5.1 dispence with the direct summand Gy0,o which was the only not rank one summand. 0 In [61 a free valuated Zp-module was defined to be a direct sum of cyclic Z -modules G with a p-valuation v satisfying v(pg) = v(g) + 1 P for every g # 0 . Theorem 5.6. A p-valuated group (G,a,v) is elementary equivalent to a free valuated Zp-module iff (G,a,v) is tamely p-valuated and qG = G for every prime q # p. Proof: Necessity is simply checked. For sufficiency we see that in Lemma 5.1 only Z -modules are used. 0 P
276
P.H. SCHMITT
We conclude with the following open problem: Give a criterion on the pair G 5 H such that the restriction of the p-height function of H turns G into a tamely p-valuated group.
REFERENCES [ I ] Eklof, P . and Fischer, E.: Elementary properties of Abelian groups. Annals of Math. Logic 4 ( 1 9 7 2 ) , 115-171. [ 2 1 Fuchs, L.: Vector Spaces with Valuations. Journal of Algebra 35 ( 1 9 7 5 ) , 23-28. [ 3 1 Hunter, R. and Walker, E.A.: Valuated p-groups; in: "Abelian Group Theory", Proc. Oberwolfach 1 9 8 1 , Springer Lecture Notes in Math. 8 7 4 , 350-373. [ 4 1 Rabin, M.O.:
Decidability of second order theories and automata on infinite trees. Transaction A M S 141 ( 1 9 6 9 ) , 1 - 3 5 . [ 5 1 Richman, F.: A guide to valuated groups; in: "Abelian Group Theory", 2nd New Mexico State University Conference 1 9 7 6 , Springer Lecture Notes in Math. 6 1 6 , 73-86. [ 6 1 Richman, F . and Walker, E.A.: Valuated abelian groups. Journal of Algebra 56 ( 1 9 7 9 ) , 145-167. [ 7 1 Schmitt, P.H.: Undecidable theories of valuated abelian groups. to appear in Memoirs de la societg mathematique de la France.
LOGIC COLLOQUIUM '84 J.B. Paris, A.J. Wilkie. and G.M. Wilmers (Editors) 0 Elsebmier Science Publishers B. V. (North-Holland). 1986
277
COMPLETE UNIVERSAL LOCALLY FINITE GROUPS OF LARGE CARDINALITY Simon Thomas Department of Mathematics Yale University New Haven, Connecticut U.S.A.
Mathematisches Institut Albert-Ludwigs-Universitat 7800 Freiburg i. Br. West Germany
1. INTRODUCTION A group
(i)
is in the class ULF
G
of universal locally finite groups if:
is locally finite.
G
(ii) Every finite group is isomorphic to a subgroup of (iii) Any two isomorphic finite subgroups of Hall introduced the concept of a
ULF
are conjugate in G .
group in [l] and showed that: which is unique up to isomorphism.
(iv)
There is a countable ULF
(v)
Every locally finite group of cardinality x x.
of cardinality embedded in
group
G
C
G.
is contained in a
ULF
group
In particular, every countable locally finite group is
C.
In [ 5 ] , answering questions of Kegel and Wehrfritz [ 3 ] , Macintyre and Shelah proved : (vi)
x >
For every
cardinality
w
,
there are
2x
nonisomorphic ULF
x.
(vii) There exists a locally finite group H every
x 2 w1
groups of
there exists a
ULF
of cardinality
group of cardinality x
w1
such that for
in which H
does
not embed. Macintyre and Shelah used Ehrenfeucht-Mostowski models to construct their nonisomorphic ULF
groups.
Consequently we do not have a very clear idea of the
structure of these groups, and it remains an interesting problem to construct 2' nonisomorphic
ULF
groups which are nonisomorphic for simple "group-theoretic"
reasons. Hickin solved this problem for x = w1 nonisomorphic complete ULF
in [ 2 ] , where he constructed
groups. He also showed that no locally finite
278
S. THOMAS
group of c a r d i n a l i t y
o1
is inevitable.
be i n e v i t a b l e i f i t embeds i n e v e r y equal t o
IHI
.
ULF
(A l o c a l l y f i n i t e g r o u p
H
is s a i d t o
g r o u p of c a r d i n a l i t y g r e a t e r t h a n o r
Macintyre introduced t h i s n o t i o n i n [ 4 ] where, assuming
h e showed t h a t t h e r e are no i n e v i t a b l e a b e l i a n g r o u p s o f c a r d i n a l i t y Similar r e s u l t s w e r e obtained f o r
x
=
2w
by S h e l a h
0
,
wl.)
[a].
I n t h i s paper, we s h a l l p a r t i a l l y extend H i c k i n ' s r e s u l t s t o a r b i t r a r y successor c a r d i n a l s .
Our main r e s u l t is:
THEOREM Let
h z w
.
ULF
groups of c a r d i n a l i t y
(a)
if
(b)
(G.C.H.)
S
5 G5
{GSl 5 <
Then t h e r e e x i s t s a f a m i l y
h+
2
k}
of nonisomorphic
satisfying:
is a s o l u b l e s u b g r o u p , t h e n
IS1 5 A.
i s a complete group.
GS
The methods o f H i c k i n , M a c i n t y r e a n d S h e l a h r e l y h e a v i l y on t h e f a c t t h a t t h e f r e e p r o d u c t w i t h amalgamation o f two f i n i t e g r o u p s i s r e s i d u a l l y f i n i t e . There i s no a n a l o g u e o f t h i s r e s u l t f o r i n f i n i t e l o c a l l y f i n i t e g r o u p s , as Neumann [ 6 ] h a s shown t h a t amalgamation f a i l s i n t h i s c a t e g o r y . t h e r o o t of t h e d i f f i c u l t y o f c o n s t r u c t i n g l a r g e
ULF
This f a i l u r e is
groups w i t h r e s t r i c t i o n s
on t h e i r s u b g r o u p s . The work i n t h i s p a p e r w a s c a r r i e d o u t w h i l e I w a s s t u d y i n g f o r a Ph.D. a t Bedford C o l l e g e , London, w i t h f i n a n c i a l a s s i s t a n c e from t h e S c i e n c e R e s e a r c h Council.
I would l i k e t o t h a n k W i l f r i d Hodges f o r some v e r y h e l p f u l d i s c u s s i o n s
and s u g g e s t i o n s .
Thanks are a l s o d u e t o t h e A l e x a n d e r von Humboldt S t i f t u n g f o r
s u p p o r t i n t h e form of a r e s e a r c h f e l l o w s h i p i n 1983/84.
2.
THE CONSTRICTED SYMMETRIC GROUP
I n t h i s s e c t i o n , we s h a l l p r o v e t h e t e c h n i c a l lemmas which form t h e h e a r t of t h e c o n s t r u c t i o n . DEFINITION
Let
G
b e a l o c a l l y f i n i t e g r o u p and l e t
group on t h e set
G.
Sym(G)
The c o n s t r i c t e d symmetric g r o u p ,
be t h e f u l l symmetric S(G),
i s t h e subgroup o f
Complete Universal Locally Finite Groups of Large Cardinality c o n s i s t i n g of those
Sym(G)
of
F,
such t h a t
G
u which s a t i s f y : U
I t is e a s i l y s e e n t h a t in
S(G)
paper, p
for all
= x F
(x
279
t h e r e e x i s t s a f i n i t e subgroup x € G.
i s a l o c a l l y f i n i t e g r o u p , and t h a t
S(G)
via t h e r i g h t r e g u l a r r e p r e s e n t a t i o n
G
embeds
(Throughout t h i s
p : G G S(G).
w i l l always d e n o t e t h e r i g h t r e g u l a r r e p r e s e n t a t i o n . )
A proof o f t h e
f o l l o w i n g lemma may b e found i n Kegel and W e h f r i t z [ 3 ] . LEMMA 1
Any two f i n i t e isomorphic subgroups o f
Gp
are c o n j u g a t e i n
S(G).
0
We b e g i n w i t h a s i m p l e o b s e r v a t i o n . LEMMA 2
Let
H
be l o c a l l y f i n i t e groups.
G
completed so t h a t f o r e a c h
h € H
and
T
Then t h e f o l l o w i n g diagram can b e € S(H),
h f ( T ) = hT
.
PROOF
Let
1 € I.
I
b e a set of l e f t c o s e t r e p r e s e n t a t i v e s of
For
€ S(H),
T
x € I
and
(x h)f(T) I t is e a s i l y checked t h a t
f
=
h € H,
s a t i s f i e s our requirements.
We s h a l l e x p l o i t t h i s i n t h e n e x t two lemmas.
1 € I.
LEMMA 3.
With t h e n o t a t i o n o f lemma 2 , suppose t h a t : (i)
a € G\H.
(ii)
7
c s(H)\H~.
G , chosen s o t h a t
x hT.
N o t i c e t h a t t h e r e are many c o m p l e t i o n maps
always c h o o s e
in
define
representation.)
I.
H
(It i s j u s t a r e p e a t e d 0
f,
depending on t h e c h o i c e of It i s u n d e r s t o o d t h a t w e
S. THOMAS
280 (iii)
Coset representatives have been chosen for H
(iv)
There exist b, c € G < K, a
>$
in K
C
G.
such that < K,a,b >
2
.
Then f can be chosen so that
PROOF Let g"
h € H
satisfy '1 = h.
Since
T
f Hp, there exists
g € H
such that
# gh. We still have freedom in the choice of coset representatives for bH,
baH, cH and
caH. We select
b, ba, c and
-1
cag
respectively. Then,
regardless of the choice of the other coset representatives, we have (bh)f(T)
-1
p(a)f(")
(,,)f(T)-'p(a)f(T)
=
bah
=
-1
7
.
0
This lemma will be used to restrict the size of soluble subgroups in the construction. The next lemma enables us to build many nonisomorphic ULF
groups.
LEMMA 4 . With the notation of lemma 2 , suppose that: is an involution.
(i)
a 6 G\H
(ii)
z c s(H)\H~.
(iii) there exists h € H Then
f
such that ah-lhT a ! , H.
can be chosen so that
PROOF.
-1 7 By hypothesis, haH, ha h h a H hah-l, hah-lhTa h-l
and H
are distinct cosets. We choose
as coset representatives. Thus
Complete Universal Locally Finite Groups of Large Cardinality
CASE 1.
hah-'hTa
h-'h"a
t?
h"a
28 1
H.
We a l r e a d y know t h a t (ha)~(a)f(")~(a)f(") =
(h" a)f(")
F h" a H.
This h o l d s r e g a r d l e s s of whether we s t i l l have a c h o i c e of c o s e t r e p r e s e n t a t i v e for
Thus w e have
h" a H.
(ha) d a ) f (7) d a ) f ( d
Case 2 .
a h - l h" a €
hah-'h"
Clearly
h" a H
h"aH = ha h-lh"
a H.
+
(ha) f ( 7 ) p ( a ) f ( T ) p(a)
h" a H.
# H and by h y p o t h e s i s Then t h e r e e x i s t s
h" a H
g F H
hah-l h" a h-l h"
# haH.
Suppose t h a t
such t h a t
= h" a g.
The assumption i n t h i s c a s e i m p l i e s t h a t h" aga € Thus
ga € H
and so
h" a H.
a € H, a contradiction.
h" a H
# hah-l h" a H.
w € H
such t h a t
A second a p p l i c a t i o n of t h e assumption y i e l d s an element
hah-l h" a h-l h" a Since
z 6 Hp,
there e x i s t s
as c o s e t r e p r e s e n t a t i v e f o r
z € H
h" a H .
=
Once a g a i n , w e have
We conclude t h a t
= h" a w .
such t h a t
z"
Thus
h" a z-lz"
.
# zw.
W e choose
h" a 2-l
282
S . THOMAS
We now u s e lemma 3 t o prove: LEMMA 5 Let
x
be a l o c a l l y f i n i t e group of c a r d i n a l i t y
G
f i n i t e isomorphic nonconjugate subgroups of f i n i t e group
3
< G,T >
(i)
F:
(ii)
I f u E < G , z >\G,
5
.
w
F1,
Let
F2
G.
Then t h e r e e x i s t s a l o c a l l y
GI1
<
be
such t h a t :
G
= F ~ .
then
I{g E G
1 ga
E
.
PROOF. may be expressed a s t h e union of a s t r i c t l y i n c r e a s i n g smooth c h a i n ,
G
G =
(a)
5u< x
G
< F1,
(b)
lG51
(c)
If
<,
satisfying:
F2
t
=
.
5
G
151
+w
= p. < A ,
IG51
5
for a l l
G5+1
then H',
i n c r e a s i n g smooth c h a i n w i t h
< A
= G
. H:
k Up .
=
i s t h e union of a
strictly
5'
( I n l a t e r s e c t i o n s , t h i s c h a i n w i l l be chosen t o s a t i s f y f u r t h e r c o n d i t i o n s . ) We s h a l l show t h a t t h e r e e x i s t s requirements of t h e lemma.
such t h a t
z C S(G)
The a c t i o n of
z on
< Gp, z >
satisfies the
w i l l be d e f i n e d i n d u c t i v e l y ,
GS
by completing diagrams a s i n lemma 3 .
By lemma 1, t h e r e e x i s t s
zo E S(GO)
such t h a t
'col Ff zo = F i .
< = q + l Let
=
I G q+l
= p..
We may assume t h a t t h e c h a i n ,
has been c o n s t r u c t e d so t h a t f o r each
Let
X = (G
\
Gq)
x
( < Gp,
+1+1 well-ordering, X = < xi
I
z >\Gp
q r l
q
0 < i < p.
i < p
Then
).
,
there exist
1x1
=
so t h a t i f
p
bi,
G
q+l ci
U H?1' i
=
and we may choose a
x . = < a,a >
then a C
"7.
Complete Universal Locally Finite Groups of Large Cardinality
283
We shall use lemma 3 to complete the diagram P
G
P1
:f
GT
P
> S(+
in :H
Assume inductively that coset representatives for G
T
so that, regardless of further choices, we will have
have been chosen
f(n)-'p(g)f(n)
f GG1
for each x. = < g, n > with j < i.
By lemma 3, we can choose coset
representatives for GT
that
J
xi
=
5
< a,a >
.
in l + : H
We define
so
T
y+1
!$ GW1,
where
f(Tn).
=
is a limit ordinal Assume inductively that we have defined elements
i< 5
so that if
i< j< 5
7.
then g
7 T~
P
f(U) -1p(a)f(U)
6
S ( G ) is defined by
5.
the image of
7.
g
7
= g
for a l l
zi € S(Gi) g 6 Gi.
for all Then
5
=
g
for
g 6
Gi. Notice that we have taken
under the direct limit mapping
and that every element of
< GE,
T~
>
has a preimage in some < G f , zi >
for
i < 5 . Thus
< G
T >
satisfies the requirements of the lemma, where
7 = 7
0
A '
3.
KILLING OUTER AUTOMORPHISMS Suppose that
n
is an outer automorphism of the uncountable ULF
group
In this section, we shall show that there exists a locally finite group H such that n
cannot be extended to an automorphism of H.
2
G
G.
284
S . THOMAS
LEMMA 6
Let
b e a s i m p l e g r o u p of c a r d i n a l i t y
G
b >o
.
Suppose t h a t
pi
€ Aut G
s a t i s f i e s t h e condition: t h e r e exists for a l l
H
5
and
G
g € G
such t h a t
IHI < b
xnH = g x H
and
x E G\H.
Then
x
= g x g
-1
for all
x
G.
PROOF
Choose any e l e m e n t hg # 1.
Suppose t h a t (hg)
class
Then
k
y
G
.
Since
= < H,g >.
yx
Then
x"
=
f o r some
g x h
h € H.
is s i m p l e , i t i s g e n e r a t e d by t h e c o n j u g a c y
G
Hence t h e r e e x i s t s and s o
G1
!! Go
x
y € G
such t h a t
y
-1 hg y
d
G1
=
.
!! G 1. Thus t h e e q u a t i o n xn yn = (xy)"
y i e l d s elements
ho, hl
€ H
such t h a t
g x h g Y h o = g x Y hl Hence
y
have
xz
k
-1 hg y
-1 hl ho € H
=
-1
xpi = g x g
.
Fix
z"
=
g z g
-1
,
From now o n , ie.
G1,
x
1
a contradiction. Go.
Then f o r a l l
So f o r a l l z € Go,
x
!! G o , w e
i t follows t h a t
Hence
Go.
g x z g Thus
5
x'(g)
= g
-1
=(xz)pi=gxg
as required.
@:G x
-1
+
Sym(G)
for all
=pi
.
0
w i l l denote the l e f t regular representation,
x , g c G.
LEEIMA 7
With t h e n o t a t i o n o f l e m m a 2 , s u p p o s e t h a t satisfy: (i) (ii)
"[HI = H . x
c G\H.
pi
€ Aut G
and
g € H
Complete Universal Locally Finite Groups of Large Cardinality
,7 c
s(H)\H~
(iii)
u
(iv)
xn H # g" x H. Then
f
285
can b e chosen so t h a t
PROOF Since
T
assumption x
and
#
H , XH
(g")-'
), t h e r e e x i s t s
p(l'"
and
x'
(g')-'
x'
H
h E H
xf(dP(g)"
h"
# h.la" H in
a r e d i s t i n c t c o s e t s of
as c o s e t r e p r e s e n t a t i v e s .
h
such t h a t
.
G.
By W e choose
Then
=
(g-l
=
(g")-l x" 10'
and
LEMMA
a.
Let
G
be a
ULF
an o u t e r automorphism. 7
C
group of c a r d i n a l i t y Then t h e r e e x i s t s
X > w and l e t
u € S(G)
n €
Aut G
be
such t h a t f o r a l l
:
>\GP
is infinite.
PROOF Let
X = < gs
element occurs
X
I
5
< X > be a l i s t of t h e elements of
times.
Since
G
G
such t h a t every
i s simple, w e may use leuma 6 t o o b t a i n a n
S. THOMAS
286 expression,
G
=
U
E<X
, satisfying
G
c o n d i t i o n s ( a ) t o ( c ) of lemma 5 ,
together with: (d)
TT
(e)
g5 € G
(f)
[G ] = G
5
5 '
5 ' x € G
There e x i s t s
E+1
\G
5
such t h a t
xTTG
#
5
g :
The r e s u l t now follows e a s i l y from lemmas 5 and 7.
x G5
0
DEFINITION.
u € Sym(G), t h e n
If
sup(cr)
=
I g'
{g € G
# g}
.
t h e subgroup c o n s i s t i n g of t h o s e permutations s a t i s f y i n g Alt(G)
Isup(u)
I
< o
is
.
i s t h e subgroup of f i n i t e even permutations.
5 Sym(G,o)
Note t h a t
5 Sym(G)
Sym(G,w)
Sym(G,w)
5 S(G).
Hence i f
i s t h e element given by lemma 8 ,
u
then
is a l o c a l l y f i n i t e group of c a r d i n a l i t y
Thus i f T
x = 8
€ < u , Gp
T €
H
I t i s e a s i l y checked t h a t
0 E Sym(G,w)
Consequently f o r a l l
9
g € G,
Aut H
extends t h e o u t e r automorphism
theorem 11.4.6 of S c o t t [7] s a y s t h a t t h e r e e x i s t s q ( h ) = Y-l h Y
and
then
Z,
Suppose t h a t
is any element, w i t h
X.
for a l l
h E H.
Thus f o r a l l
y
g € G,
n € Aut G.
E Sym(G)
Since
such t h a t
Complete Universal Locally Finite Groups of Large Cardinality
By theorem 10.3.6 of S c o t t [ 7 ] , t h e r e exists But t h i s means t h a t
(y(g)n)-'
u y(g)
TI
g € G
such t h a t
Y
281
a contradiction.
€ H,
@(g)n
=
.
So we have
proved : LEMMA 9.
The o u t e r automorphism
H.
of
Aut G
TI €
cannot b e extended t o a n automorphism
0
The f o l l o w i n g lemma w i l l b e u s e f u l d u r i n g t h e main c o n s t r u c t i o n . LEMMA 10.
Let
XC
G
b e a subgroup of c a r d i n a l i t y
X
.
Then
NH(X) = NG(X).
PROOF.
Suppose t h a t
e
€ Sym(G,w)
and
h € H T C
normalizes
0 €
N(Xp).
[e,
Xp] = 1.
Clearly
n > 1. Then f o r
Thus
4.
[e,
Xp]
T C
N(Xp)
(3-l p(g) 0 € G p
Suppose t h a t p(x) € Xp
8 # 1. L e t
, we
Then
h = 0.7
f o r some
Applying t h e c a n o n i c a l p r o j e c t i o n ,
< Gp, cr >.
p:H + < Gp, cr >, w e s e e t h a t
Xp.
and hence
i f and o n l y i f sup(€)) = {gl,
T € Gp.
[e,
We a l s o o b t a i n
p ( g ) ] = 1. Hence
..., gn}
,
where
have
# 1, a c o n t r a d i c t i o n .
0
THE CONSTRUCTION
To c o n s t r u c t nonisomorphic
ULF
g r o u p s , w e r e q u i r e a w e l l known
combinatorial f a c t . LEMMA 11.
Let such t h a t :
x
be a r e gu la r uncountable c a r d in a l.
Then t h e r e e x i s t s
A 5 '2
288
S. THOMAS
(ii)
If
'I
z
C
A , t h e n 15 < x
I
A
In particular, there exists
+~(5))
'(5) C
"2
is stationary in
.
x
s a t i s f y i n g t h e above c o n d i t i o n s .
We s h a l l b u i l d smooth s t r i c t l y i n c r e a s i n g c h a i n s
, G;
where each r)
ULF
is a
and. t h a t n:G'I
# zE A
+
X.
group of c a r d i n a l i t y
The r e g u l a r i t y of
i s a n isomorphism.
Gz
Suppose t h a t
k
ensures t h a t
CE
+1
< X
i s a c l o s e d unbounded set i n n[Gq]
= Gg
5
X+.
1
= G;
5 < A+
By leuima 11, t h e r e e x i s t s
such t h a t
~ ( 5 .) By p u t t i n g i n " o b s t r u c t i o n s t o isomorphism",
q(5) #
and
[$I
we s h a l l e v e n t u a l l y reach a c o n t r a d i c t i o n .
A
For t h e n e x t few pages, w e w i l l f i x q € i . e . w e write
all
G
=
U G5. 5
and suppress index
,
T)
W e s h a l l now d i s c u s s our s t r a t e g y f o r k i l l i n g
of t h e o u t e r automorphisms of
G.
The next lemma i s a n easy g e n e r a l i z a t i o n
of lemma 3 . 2 of Hickin [Z]. LEMMA 1 2 .
Let
TI
C Aut G
unbounded set (i)
n[Ga]
(ii) n rGa
C
5 X+
a C C:
such that f o r a l l
= Ga.
is an o u t e r automorphism o f
Ga
Shelah [9] has shown t h a t t h e G.C.H.
X z w
.
Then t h e r e e x i s t s a c l o s e d
be a n o u t e r automorphism.
0
*
implies t h a t
Assuming t h a t t h e u n d e r l y i n g set of
Ga
+
h o l d s f o r all
o ( X )
is t h e l i m i t o r d i n a l
La
a s t a n d a r d argument g i v e s :
LEMMA 1 3
(G.C.H.)
There e x i s t s a sequence of f u n c t i o n s f o r any automorphism
n € Aut G
c fa:Ga
+
+z
Gala < h
such t h a t
,
Complete Universal Locally Finite Groups of Large Cardinality
1
{a < A+
COROLLARY
n r G a = fa)
.xi
is stationary in
289
0
(G.C.H.) € Aut G
Let
be an outer automorphism. Then there exists a <
x-k
such that: n[Gal
(i) (ii)
TI
r
= Ga
is an outer automorphism of
Ga = f a
0
Ga*
So we know in advance who the potential trouble makers are.
constructed Ga,
fa
we can check whether the function
is an outer
automorphism. If it is, then lemma 9 provides an extension Ha
fa
cannot be extended to an automorphism of Ha.
n € Aut G
then be constructed so that if induces an automorphism
cp € Aut
H
cp
r
3
n
5
such that
Ga = f a , then
q.)
= 0.
Let H = cyclic group
8
n z 2 Cn.
An, where each A
Then Go
is the direct
is an arbitrary ULF
sum
of A
copies of the
group of cardinality
H.
which contains LEMMA 15. Let
g E Go
be a nonidentity element. Then
Itagal a € Go
is an
x.
=
involution}
PROOF. Let
g have order n 2 2
[
1
1
*<
involution such that h € Go
rf
Ga = fa, a contradiction.
We begin the construction (continuing to suppress the index STEP ONE
Ga
The rest of the chain will
satisfies
with
When we have
such that
z2 z
.
where
Go
< z o > :: < z1 > :: Cn
z 2 z o z 2 = zl.
h yo = g.
contains a subgroup
Let An =
h Putting gi - yi
and
8 < y >
i
i
.
for i z 0,
z2
is an
There exists
X
S. THOMAS
290 h
An
= < g
that
zi
19 [
=
STEP TWO.
< g.>
@
i > o
and
5
is a l i m i t o r d i n a l .
<
=
a
+
Then
is a n i n v o l u t i o n and
= z;f
Ga
u
E
fa
c a n n o t b e e x t e n d e d t o a n automorphism o f < Sym(Ga,o),
(ii)
If
T
(a)
I
CASE 2
,
E < G g , ua>\G:
I T-1p ( g )
there exists
0
Ga.
(i)
(b)
a g a = g i'
be a n e l e m e n t s a t i s f y i n g :
S(Ga)
{ p ( g ) 6 G,"
such
h a s been c o n s t r u c t e d .
i s a n o u t e r automorphism o f
fa
Let
a
u € Go
1.
Assume i n d u c t i v e l y t h a t CASE 1.
i > 0, there e x i s t s
For each
g
STEP THREE.
z y = gi.
1.
Gap
, ua
z
.
then
T
p(a) € GZ
)I
E G:
x
<
such t h a t
[p(a)-'
p(a),
T
TI #
1.
otherwise.
Let
aa = S ( G a )
s a t i s f y ( i i ) above.
(We s l i g h t l y d e l a y t h e p r o o f t h a t
c l a u s e ( i i ) ( b ) can always be s a t i s f i e d . ) Define
Ha
= < Sym(Ga,w),
, ua
GE
~ f ;= cn
a
n
where
= 2
if
q(a) = 0.
3
if
q(a)
=
(Here "wr"
.:G
Then
=
wr
>
H~
1.
denotes the r e s t r i c t e d wreath product.) IKa( = X,
and i t i s e a s i l y checked t h a t
Let
Ka
be t h e b a s e g r o u p of
i s a maximal a b e l i a n
Ka
subgroup of .:G We s h a l l d e f i n e
Ga+l
u
Ga
chain,
Ga+l
=
i c X
l '
t o be t h e u n i o n o f a smooth s t r i c t l y i n c r e a s i n g with
G:
= G i
.
The
GY
are c h o s e n , u s i n g lemmas
4 and 5 , s o t h a t t h e f o l l o w i n g c o n d i t i o n s are s a t i s f i e d .
Complete Universal Locally Finite Groups of Large Cardinality
is a
(1) Ga+l
group.
ULF
(2)
If
Y 5 Ga
(3)
If
x € Gy\G:
,
there exists
a € Go
.
N
G:
By lemma 1 0 ,
"
= N
*a
(Y)
=
u
S(H )
have been d e f i n e d .
5
Y 5 Ga
has c a r d i n a l i t y
(The proof
Suppose i n d u c t i v e l y
To e n s u r e t h a t ( 3 ) h o l d s , w e s l i g h t l y modify
s a t i s f y i n g c o n d i t i o n s ( a ) t o ( c ) o f lemma 5 , and u s e t h i s t o
U
5 < x HE
€
Suppose t h a t
# 1.
x a, x]
[a
(Y).
Ga
We i n d u c t i v e l y c o n s t r u c t a smooth c h a i n
d e f i n e t h e a c t i o n of a s u i t a b l e T~
N
=
-1
such t h a t
satisfying clause (ii)(b).)
t h e c o n s t r u c t i o n i n lemma 5. GY =
Gi
N
h a s been c o n s t r u c t e d .
G;
N a (Y)
t h a t c o n d i t i o n (3) can always b e s a t i s f i e d .
w i l l a l l o w us t o choose that
then
(Y) a n d , by Kegel and W e h r f r i t z [ 3 ] p. 74, Ga Lemma 5 d e a l s w i t h ( 2 ) f o r i > 0.
N
(Y). Ha F i n a l l y w e show
(Y)
,
X
i s a s u b s e t of c a r d i n a l i t y
C l e a r l y (1) p r e s e n t s no d i f f i c u l t y .
X
29 1
T € S(Gy).
Then
Suppose t h a t
lc H g , z5 >\HE
I
H5
c X
and
.
So as we
proceed t h r o u g h t h e c o n s t r u c t i o n , w e may i n d u c t i v e l y d e f i n e a l i s t of t h e "new" elements.
Let
there e x i s t s
u
€ c
h € H
5
Hg, zc >\HC such that
be the ha
cth
element i n t h i s l i s t .
Then
# h.
CLAIM There i s a n i n v o l u t i o n
a € Go\Hc
such t h a t
a h-lhu
a
H
5'
PROOF OF CLAIM If
h-'
h" € Go,
h - l ha € HE\G0
t h e c l a i m f o l l o w s from lemma 15.
Otherwise
f i r s t a p p e a r e d as t h e r e s u l t of one o f t h e f o l l o w i n g t y p e s of
extension: (a)
a r i g h t r e g u l a r e x t e n s i o n , as i n lemma 5.
(b)
a wreath product extension.
(c)
a k i l l i n g e x t e n s i o n , i.e.
f o r some
P
5 a
.
(Throughout t h e rest o f t h i s p a p e r , a n element
x € G\Go
w i l l be s a i d t o
292
S. THOMAS
have type (a), (b) or (c) depending on where it first appeared.) -1 u g = h h , A = A2 5 Go
Then g € N(B)
and so
and B = CA(g).
A.
IBI
It follows that
claim holds.
[A:CA(g)]
A, and the
=
0
We choose H 5+1
that it includes a, and then use lemma 4 to ensure
so
[aua, a] # 1. The construction is completed!
that
5.
Let
THE CONSTRUCTION WORKS
LEMMA 16
A
q €
Let
.
If S
5
Gq
is a soluble subgroup, then
IS1 5
A
.
A, then
IN(H)
I
=
PROOF
5 Gq
By construction, if H Suppose that
S
5 Gq
is a soluble subgroup of cardinality i € o
inductively that for each A+,
cardinality
has cardinality
the
A+.
A.
We shall prove
ith derived subgroup D(i)(S)
has
an obvious contradiction.
i = O By assumption. i = j + l Assume that K = D(')(S)
I [K,K]I
#
A+.
Since
[K,K] a K, we must have A.'
is an abelian group of cardinality abelian group of cardinality contradiction. Hence
A+,
has cardinality
A
.
[K,K] = D(i)(S)
I [K,K]I
Let
I
Then
and suppose that < A
.
Hence K /[K,Kl
[K,K] a K ' [K,KI = A
and K O
also has cardinality
4
K
be an
,a
A+.
The theorem will follow from the next lemma. LEMMA 17 Suppose that q. n[Gz]
= G:.
then:
T €
A
and
n:Gq
-c
GT
is an isomorphism. If
0
Complete Universal Locally Finite Groups of Large Cardinality
293
B e f o r e p r o v i n g t h i s lemma, we s h a l l show how i t c o m p l e t e s t h e proof o f t h e theorem. LEMMA 18 If
r)
9
T
C A
, then
and
Gr)
GT
are nonisomorphic
ULF
groups.
PROOF. Suppose t h a t
n:Gq
i s a n isomorphism.
GT
+
Then t h e r e e x i s t s
k
a
such t h a t :
LEMMA 1 9 (G.C.H.) r) C A ,
If
then
i s a complete
Gr)
ULF
group.
PROOF.
We s u p p r e s s t h e i n d e x phism.
By c o r o l l a r y 1 4 , t h e r e e x i s t s
(ii)TI
r
Gu = f a
TI
€ Aut G
a C X+
i s a n o u t e r automorphism o f
By c o n s t r u c t i o n , = .c
fa
i s a n o u t e r automor-
such that:
+
n[Ga]
= G L = Ka
(iv)
n[K,]
= Ka.
x
.
p
C Aut Ha
By leonua 1 7 , w e have
Ha.
Consider t h e automorphism,
where
cp = p
-1
Bp,
g i v e n by
i s t h e c a n o n i c a l p r o j e c t i o n , and satisfies
Ga'
c a n n o t b e e x t e n d e d t o a n automorphism o f
Sym(Ga,o), G Z , aa >
(iii)
'p
Suppose t h a t
n[GaI = Ga*
(i)
H
q.
cp
e(Ka
x) = Ka n ( x ) .
Ga = f a , a c o n t r a d i c t i o n .
Then
n
S. THOMAS
294
W e r e t u r n t o t h e proof o f lemma 1 7 . LEMMA 20
U <
.
q € A
Let
+ X .
i s a maximal a b e l i a n
Then K:
SL ,3roup
of
G"
each
ir
PROOF.
W e have a l r e a d y n o t e d t h a t Suppose t h a t
g € G'\(G:)+.
i s a m a x i m a l a b e l i a n subgroup o f
K:
IK:I
Since
q € A
For t h e n e x t few p a g e s , we f i x
A,
g (I N(k:).
0
and s u p p r e s s t h e i n d e x
q
.
Let
b e t h e f u n c t i o n d e f i n e d by
a, € K u
a,(g)
where
=
.'):G(
if
= 1
otherwise
i s a generator of
z
g = 1
= za
Cn
a
.
Then
LEMMA 21 Suppose t h a t
x € G\G,
.
p z a
x € Sym(G ,a) f o r some
B
If
c xg
.
Ig
is abelian, then
€ Gu >
x €K
B
or
PROOF. If
x
i s a n e l e m e n t o f t y p e (a), t h e n
< xg
Ig
€ Ga z
i s nonabelian.
we need o n l y c o n s i d e r e l e m e n t s o f t y p e s ( b ) and (c). TYPE ( b ) . Assume t h a t
f i r s t a p p e a r s as
x
x = f.b € K where
f
# 1 and B
c a s e s that simple, sup(f) =
b € Ga
2 a
and
{gl,
..., g
HB
Suppose t h a t
b € Hp\G,,
b
}
f i n i t e l y many e l e m e n t s of
# 1. C o n s i d e r i n g s e p a r a t e l y t h e
w e see that
is i n f i n i t e and hence
[Ga:CG ( b ) ]
a
.
B
[G,,
b]
{ b g ( g € Gal
b e t h e s u p p o r t of t h e f u n c t i o n Ibg
1
g € Gal
# 1. S i n c e Ga
such t h a t :
f.
is
is i n f i n i t e . There a r e only
Let
So
Complete Universal Locally Finite Groups of Large Cardinality (i)
c n} n
1 9 i
tgi
S i m i l a r l y t h e r e are only f i n i t e l y many (ii)
{gi 11
i
n
n}
{gig
g 1 1 5 i C n}
( i i i ) {gig-'b Choose
g € Gu
[xg, x]
=
1.
c
#
{gibg 1 1 5 i 5 n}
11 n
_C
0.
g € Gu
i 5 n}
#
295
such t h a t :
0
{gig 11 C i 5 n }
# 0.
so t h a t c o n d i t i o n s ( i ) t o ( i i i ) f a i l .
Suppose t h a t
Write fg
xg =
. bg
= h.c.
Then fch.bc
x xg =
b fch = h f .
Hence
b h f.cb.
=
xgx
We have
5 s u p ( f C ) U sup(h)
sup(fch)
{gibg 11 d i
=
c
n}
U {gig 11
c
S i n c e c o n d i t i o n ( i i i ) f a i l s , we have a n e q u a l i t y above. support of c a r d i n a l i t y sup(hbf)
2n, =
i
c
n}
.
Hence
b h f
has a
and we must have
b sup(h ) U s u p ( f )
= { g i g b I 1 5 i C n } U {gil l 5 i C n ) .
But, s i n c e c o n d i t i o n s ( i ) and ( i i ) f a i l , we have
a contradiction.
Consequently
b = 1 and
x € K
B'
TYPE (c) Suppose t h a t
where
e # 1
and
f i r s t a p p e a r s as
x
p
1 a
.
Applying t h e c a n o n i c a l p r o j e c t i o n , it f o l l o w s t h a t
S. THOMAS
296 c zg
b
1
g € Ga
is a b e l i a n .
t
# 1. L e t
Hence
z = p(b)
..., 9,)
s u p ( 0 ) = {gl,
f o r some
For any
B'
Suppose t h a t
be t h e support o f t h e permutation
.
8
This should not cause
(We are u s i n g "sup" i n two s e n s e s i n t h i s p a p e r . confusion!)
b € G
g € Ga,
and
Arguing as i n t h e p r e v i o u s case, w e r e a c h a c o n t r a d i c t i o n .
x
e c
=
SP(G
B
,a).
Suppose t h a t
a
x E K~
or
Since
I<
B
Let
x
n:G'
i s a n isomorphism such t h a t
G"
-+
c
S ~ ( G ~ , O f)o r some
I
g € G :
a:
L
>I
=
xglg € G:
.
t
.
x = n(a,) , w e have
Suppose that
L = < z g ( g € G:
z €K'
B
or
5
N(L)
z € H".
B
t
=
K.;
Thus
.
z
€Kt
B '
S i n c e K:
(ii) n ( a a )
.
Then
is a b e l i a n o f z
E Hi.
, is a maximal a b e l i a n subgroup of
So w e have shown:
Suppose t h a t t h e isomorphism n[G;]
x EK
Suppose t h a t
LEMMA 22
(i)
Let
and
a contradiction. nK []:
2 a
Then
7 c xglg € Ga t
K,:
Then i f
.
n[Gz] = G:
X , we must have
X. By lemma 21,
5 H"B
8
B
z € n[K:]\c
cardinality Then
0
b e t h e element p r e v i o u s l y d e f i n e d .
€ K:
Thus
= G :
E K;
. f o r some
2 a.
n:Gq + G"
satisfies:
Complete Universal Locally Finite Groups of Large Cardinality Then
n[K:]
.
=
p
Suppose t h a t
.
a
>
Then
T h i s i n d u c e s a n isomorphism
where
cp = p
-1
0 p'
are t h e c a n o n i c a l p r o j e c t i o n s , and
p, p'
297
etK?,x) =
Ki
n(x).
Hence
IP
satisfies:
cpIG21
(i)
"
Ga.
=
By theorem 11.4.1 of S c o t t [ 7 ] , subgroup o f (ii) Let
.H:
Thus:
cp[Alt(G:)]
X'l
a'
X"
B
i s t h e u n i q u e minimal normal n o n t r i v i a l
Alt(G:)
Alt(G").
=
B
d e n o t e t h e sets o f 3-cycles of
Alt(G:),
respectively.
Alt(G")
B
By theorem 11.4.2 of S c o t t [ 7 ] , w e have:
Let
x
Its o r b i t under c o n j u g a t i o n by e l e m e n t s o f
= (a b c ) € .X:
G:
is
CLAIM If
x , y € :X
[x, yp(g)1
y !t Ox U 0
and
X
# 1.
,
then there e x i s t s
g € G :
such t h a t
PROOF OF CLAIM Suppose t h a t yp'g)
= (a
Pa
Since
B
x = (a b c ) €
-1
x = (a b c) a
ya
-1
a)
and
y = (a
and
y = (a
If
g = a-l a ,
s a t i s f i e s o u r requirements.
> a , t h e r e e x i s t elements
$
B Y).
B
y),
a , b , c € G"\G:.
where
B a, B , Y
then 0
Let € :G
.
Then f o r a l l
298
S. THOMAS
for a l l
(b)
g E :G
,
[ x , ~ ' ( ~ ' 1= 1.
But t h i s c o n t r a d i c t s t h e c l a i m and (i), (iii) above.
W e conclude:
LEMMA 2 3 Suppose t h a t t h e h y p o t h e s i s o f lemma 2 2 h o l d s .
Then:
Thus t o p r o v e lemma 1 7 , it i s enough t o show t h a t
p
? a.
F o r t h e s a k e of c o n t r a d i c t i o n , assume t h a t
n [ < ag ( g E :G Suppose t h a t z
E Sym(GT,w).
P
z
\
E n[Kz]
N(n
Hence, l e t t i n g
2.
Sym(Gc,w)
Then
.
Then, a r g u i n g as i n t h e p r e v i o u s c a s e ,
E H.:
z
P
g C GZ
Sym(G;,w).
w e must h a v e
n ( z ) C Sym(GT,o).
There e x i s t s
5
YL?,I
[.
C o n s i d e r any e l e m e n t that
c x g l g 6 G:
5
n ( a a ) E Sym(GG,w).
for
Hence
f~
Since
> ]
n ( a a ) f Sym(Gi,w)
Then f o r a l l
g E ,:G
[ z , ag ]
Let
such t h a t
n(h) = g
f o r some
h E :G
we o b t a i n
# 1. Suppose
Complete Universal Locally Finite Groups of Large Cardinality
1
[ n ( z ) , n(aa)n(h)
1
=
a contradiction. Hence for all 1 # z € H:
299
, , n(z)
f Sym(G;,o).
We conclude
that :
n
rf[K?, M]:H
(*)
Sym(G;,o)
=
n[K:].
For the sake of concreteness, suppose that :a handled similarly.) Express
Then for each g
€
7
Ga
x = n(aa)
=
1.
(The other case is
as a product of disjoint transpositions
=
We m y inductively define a sequence of elements of GG,
2 = c g51
5
-z
h >
such that: (i)
go
=
1.
':(pus (ii) if i # j c h , then sup(x gi) )n Then
c xglg € 2 z
= 4
.
is an abelian group of cardinality
A.
Consider the
element
8 € Sym(G;,o)
Then (iii)
e gi x = x gl
.
(iv)
(x )
(v)
(xg )i e = xgi
Hence 0
=
and
X.
normalizes
for I c i < h . c xglg € 2 > and
18, n [ K ! ]
1. Consider
,
S. THOMAS
300 Since
IBl
such that
=
A,
N(B) cK:
M. :H
Hence there e x i s t s
n ( z ) = 8 , contradicting ( * ) .
z € (K:
H:)\
:K
This completes the proof of lemma 1 7 .
Complete Universal Locally Finite Groups of Large Cardinality
301
REFERENCES t11 P. Hall, Some constructions for locally finite groups, J. London Math. SOC. 34 (1959), 305-319.
r21
K. Hickin, Complete universal locally finite groups, Trans. her. Math. SOC. 239 (1978), 213-227.
r31 0. H. Kegel and B.A.F. Wehrfritz, Locally finite groups, North-Holland, Amsterdam, 1973.
r41 A. Macintyre, Existentially closed structures and Jensen's Principle Israel J. Math. 25 (1976), 202-210.
0,
A. Macintyre and S. Shelah, Uncountable universal locally finite groups, J. Algebra 43 (1976), 168-175. B. H. Neumann, On amalgams of periodic groups, Proc. Roy. SOC. London, Ser. A. 255 (1960). 477-489. W. R. Scott, Group theory, Prentice-Hall, Englewood Cliffs, N.J., 1964. S. Shelah, Existentially closed models in continuum, preprint, 1977. S. Shelah, Models with second order properties 111. Omitting types for L(Q), Arch. Math. Logik 21 (1980). 1-11.
303
LOGIC COLLOQUIUM '84 J.B. Paris. A.J. Wilkie, and C.M. Wilmers (Editors] 0 Elsevier Science Publishers B. V. (North-Holland], 1986
p-H -CATEGORICAL STRUCTURES 0
Carlo T o f f a l o r i I s t i t u t o Matematico "U.Dini" Universita d e g l i Studi d i Firenze Florence, I t a l y
Let
be a countable, complete 1 s t o r d e r theory with no f i n i t e models; f o r
T
I=
T,
s u b s e t s of
M,
every
M
let
be t h e Boolean a l g e b r a of p a r a m e t r i c a l l y d e f i n a b l e
notice t h a t
A,
n i t e cardinal
B(M)
B(M ) i s atomic and
IB(M)
d e f i n e A-Boolean spectrum of
t y p e s of t h e a l g e b r a s
B(M) f o r M F T ,
T
IMl = A.
I
For every i n f i -
= [MI.
t h e s e t of t h e isomorphism
A problem c l o s e l y l i n k i n g model
t h e o r y and Boolean (meta) a l g e b r a i s t o c l a s s i f y 1 s t o r d e r t h e o r i e s by looking
a t t h e i r Boolean s p e c t r a ; a l a r g e d i s c u s s i o n of t h i s problem a l r e a d y appears i n we only recall t h a t it looks more approachable i f
[MT2],
A =
H0
( i n view of
t h e Ketonen c l a s s i f i c a t i o n of isomorphism t y p e s of countable atomic Boolean algebras theory
[K] ), i n p a r t i c u l a r i f we l i m i t o u r s e l v e s t o T
w-stable t h e o r i e s ( a s a
i s w-stable i f and only i f , f o r every countable
M
= ! T , B(M)
is a
superatomic a l g e b r a , and a system of very simple i n v a r i a n t s f o r superatomic algeb r a s i s provided by t h e Cantor-Bendixson a n a l y s i s ) .
A f i r s t s t e p towards t h e
c l a s s i f i c a t i o n of 1 s t o r d e r t h e o r i e s by t h e i r /j -Boolean s p e c t r a i s t h e i n t r o 0
ducing of p - 8 - c a t e g o r i c i t y LMTlS] : a theory 0
i f its
T
i s s a i d t o be p-
k0-Boolean
4-c a t e g o r i c a l
spectrum c o n t a i n s only one isomorphism type. p-fi- c a t e g o r i c a l 0 t h e o r i e s were s t u d i e d i n a s a t i s f a c t o r y way i n [MTl], t h e aim of t h i s paper i s t o develop t h o s e r e s u l t s , emphasizing an a l g e b r a i c d i r e c t i o n , namely t h e study of p-JI - c a t e g o r i c a l s t r u c t u r e s (a s t r u c t u r e M i s s a i d t o be p - f l - c a t e g o r i c a l i f Th(M) 0 0 i s ) . One could n o t i c e t h a t a l o t of examples of p - 3 - c a t e g o r i c a l s t r u c t u r e s a r e 0 just provided by t h e g e n e r a l t h e o r y , f o r i n s t a n c e & - c a t e g o r i c a l s t r u c t u r e s , H -c& 1 0 t e g o r i c a l s t r u c t u r e s , w-stable s t r u c t u r e s of Morley rank 1 a r e p-h - c a t e g o r i c a l . 0
Some m o r e r e s u l t s , mainly concerning r i n g s , were proved i n LMTl],
it may be u s e f u l
t o r e c a l l t h a t a l g e b r a i c a l l y closed f i e l d s , r e a l closed f i e l d s , d i f f e r e n t i a l l y c l o s e d f i e l d s of c h a r a c t e r i s t i c 0 are p - h o i c a t e g o r i c a l .
W e want here t o d e a l with
t h i s problem f o r groups. W e w i l l proceed s t e p by s t e p , s t a r t i n g of course from
C. TOFFALORI
304
Abelian groups, or, more generally, from modules (over a countable ring); by combining some classical model theoretic results about modules (see [ Z ] Ketonen's analysis of countable atomic Boolean algebras [K]
,
)
and
we will show
that every module (hence every Abelian group) is p-h -categorical. 0
The general problem for groups seems to be very difficult; even if we limit ourselves to the w-stable case, the situation keeps confusing, because the problem of classifying w-stable groups is still open and looks essentially intractable. No benefit is got by restrictingourselvesto some comparatively slight generalization of Abelian groups, like nilpotent groups of class 2 ; in fact, w-stable nil-2 groups are a real enigma, as well as all w-stable groups (see [ B C M J ~ ~ ~ [Me] ) ; this difficulty hinders the understanding of p-x -categorical groups, but 0 lets us give an example of an w-stable nil-2 group which is not p-3-categorical. 0
and LMT27; we will use the multiplicaMost of our notation comes from [MTl] -1 ,u) for groups, but we will prefer the additive one tive notation ( - , (+,
-,
0)
while dealing with Abelian groups.
Thanks to G. Cherlin and M. Ziegler for their valuable suggestions.
s
1 - All modules are p-H,-categorical
"
1. We consider here (left) modules over a countable ring R-modules are represented in a 1st order language L(R) and a 1-ary functional symbol for every
rCR.
R
with identity.
containing
+,
-, 0,
We recall some basic facts about
the model theory of modules (the main ones concerning the definable subsets of a module) A
.
formula (of L(R)) y(;)
is said to be an equation if
y(i)
is of the follow-
ing kind
r v + r v 0 0 1 1
+....+
r v n n
=
ro, r l , ..., r
0
-
E R. A formula $(v) is said to be a pp-formula (posin tive primitive formula) if $(v) is of the following kind
where
- -
where y.(v, z ) 1
is an equation for every
i 6 m.
Notice that, if
M
is a module
305
p- No-Categorial Structures
and
$(v,
i)
is a pp-formula, then
+(M,
is a subgroup of
e M,
pp-definable subgroup) an<, for every is a coset of
0)
$01,
either
+(M,
M
a)
=
(called a
0 or $(M, a)
0).
Baur's quantifier elimination procedure implies that every parametrically definable subset X of M
where
m, n. < w
SubgrOUpS of and
M
can be represented in the following way
13
13
1
are pp-definable
(without loss of generality $.,(MI is a subgroup of
a , , a , , €M. We can also say that, if
where
O < m < w , n. < w
and, for every
as above, moreover [$i(~) A
i < m, j < ni, gi(M), $.,(MI
and, for every
:
X #
17
B,
+.(M))
then
i, j, $i(M),
+,1 3,(MI, ai'
a are defined ij
+ij(~)] is infinite.
very important result for our purposes is the Neumann Lemma.
Lemma 1.1
ktn
-
(8. H. Neumann)
subgroups of
G, ao, al,
such that
H / HnHi
G
If
..., a
is an Abelian group, Ho, HI,
, aE
is
H + a -C
..., H
,
H
are
.u
(H + a , ) and there is 1 i finite if and only if i t k , then G,
In fact, the following proposition is a direct consequence of Lemma 1.1 Lemma 1.2
where k
- Let M be a (countable)module,
+(M), +k(M)
are pp-definable subgroups of M, a, ak t M
< m, $k(M) is a subgroup of +(M) and [+(MI
for every subgroup +'(M) of +(MI are infinitely many elements
(bi
: +k(M)]
satisfying [$(M) :
i
E: W )
in
X
:
and, for every
is infinite. Then,
$'(M)]
such that
infinite, there
C. TOFFALORI
306
*
if
*
f o r every i 6 w,
-
Proof __
m = 0 m
-+
then
i #j,
:
<
k
b
m,
- b , $ l
i
either
$'(M); [$'(M)
i s i n f i n i t e or, f o r a l l
$'(M)fl O k ( M ) ]
:
+ b . ) A ( $ k ( ~ +) a k )
($'(M)
=
0
.
W e p r o c e e d by i n d u c t i o n on m.
remember t h a t
m+l : l e t
many e l e m e n t s
= ($(MI + a ) 0 b , (iE w ) i n X
k < m,
is infinite.
: $'(M)]
kym( $ k ( +~ a k ) ,
-
x
and, f o r every
n
[$(M)
either
i # j,
such t h a t , i f
0
[$I
(MI
:
then there e x i s t i n f i n i t e l y
$'(M) n gk(M)]
then
b
i
-
bj
$ ' (M)
is i n f i n i t e o r
+ a ) = 0 for a l l i E w. Choose a maximal s e t I b , : i E w ) k k 1 s a t i s f y i n g t h e s e p r o p e r t i e s , c o n s i d e r $ ( M ) + a and s e t I = { i € w : ($'(M) i b , ) m m 1 n ($,(M) + a m ) = 0 1 ($'(M) + b , )
( $ (M)
.
1 s t case:
[$'(M)
:
$'(M)O $ (M)]
is i n f i n i t e . Therefore
2nd c a s e :
[$'(M)
:
$'(M)n$ (M)]
i s f i n i t e and
I
is coinfinite. [ b
[$'(M)
:
$'(bl)n+m(M)]
i s f i n i t e and
I
i s c o f i n i t e , namely f o r
m
m
works. 3rd case: almost a l l
i Ew
then t h e r e e x i s t
For e v e r y
let
I
=
k
<
m
bcX
0 Again, t h e r e e x i s t k
Therefore
+ am) # 0
($'(M) + b i ) A ($,(M) a
m.0
(=
a ), m
such t h a t
...,
a
m,n-1
.
a
k,O
(=
a
k
),
a
: i e w } s t i l l works.
L e t n = [$'(M)
is f i n i t e
:
...,
i
i
: i#
k,n(k)-1
.
satisfying
I}
: $'(M)n$,(M)],
such t h a t
$'(M)n $ (M)] k : ($'(MI + b ) n ($k(M) + a k ) # 0
[$'(MI
b
(n(k), say),
p- Ho-Categorial Structures T h i s c o n t r a d i c t s Lemma 1.1, s i n c e in
$(M)
$'(M),
...,
$O(M),
307 $,(M)
have i n f i n i t e index
.-
Some l a s t remarks about modules: every module i s s t a b l e , furthermore, i f a module and
Apparently
p
p
S(M)
( t h e d u a l space of
i s axiomatized by
+
-
p U p
,
B(M)
M
define
and determined by
p
+
.
L e t u s t u r n our a t t e n t i o n now t o countable atomic Boolean a l g e b r a s .
2.
be u s e f u l t o recall t h e Ketonen r e s u l t s phism t y p e s of t h e s e a l g e b r a s .
Let B (
I t may
about t h e c l a s s i f i c a t i o n of isomor-
LK]
# ( 0 ) ) b e a countable atomic Boolean
a l g e b r a then we can d e f i n e an i n c r e a s i n g sequence of
is
( I v(B) : v o r d i n a l )
of i d e a l s
i n t h e following way:
B
*
I ( B ) = (01,
*
I (B ) = i d e a l g e n e r a t e d by A t B , 1
*
I (B) =
*
I
0
U
vth
X
v+l
X i s a l i m i t ordinal,
if
I (B)
v
( B ) = preimage of
I ~ ( B ~ I ~ ( B i)n) t h e n a t u r a l homomorphism of
B
onto
B/IV(B).
Let
b
if
E B;
b
CB-rank
#
IU(B)
b =
-
(CB = Cantor-Bendixson);
such t h a t
b E I (B) V
and w e d e f i n e
then w e d e f i n e
o t h e r w i s e , it i s easy t o s e e t h a t t h e l e a s t o r d i n a l
v
i s a successor o r d i n a l , and we s e t i n t h i s c a s e
CB-rank b = min Apparently, i f
v,
f o r every o r d i n a l
v
CB-rank b = v ,
:
b E IV+I(B) then
b
I
1
.
IV(B)
i s a f i n i t e element i n
B/Iv(B),
C. TOFFALORI
308 CB-degree
I (a E
b =
At(B/I
(B))
:
a
5
blIv(B)lI
.
I n g e n e r a l , an obvious c a r d i n a l i t y argument i m p l i e s t h a t t h e r e i s that I (5) = I (B). a cX+l CB-rank of B , hence
Q-rank
If
such
w1
The l e a s t o r d i n a l w i t h t h i s p r o p e r t y i s c a l l e d t h e
CB-rank b + 1 : b E B , CB-rank b <
B = sup
CB-rank B = a ,
<
a
then e i t h e r
B/Ia(B) > ( O ) or B/I
a
-1. ( t h e f r e e countable
( B ) 1. A
Boolean a l g e b r a ) . Finally, i f = min
p E S(B)
( t h e d u a l space of B ) , we d e f i n e
CB-rank b : b E p }
Then, t h e isomorphism t y p e of
a.
can b e determined i n t h e following way:
B
i s superatomic (namely, B / I a ( B ) 2 ( 0 ) ): i n t h i s c a s e ,
B
a = B
o r d i n a l and, i f of
CB-rank p =
.
+
1,
then
B/I
i s given by t h e o r d e r e d p a i r
B
6
(B)
( a , d)
is f i n i t e .
a
is a successor
The isomorphism t y p e
where
a = CB-rank B = CB-rank 1 + 1
is called the
(d
a - t y p e of B
b.
B:
CB-degree of
w e g i v e h e r e , and t h e one of
CB-rank
B [b]
5
B/I
a
( B ) 1. A .
b i n B)
i n t h e superatomic c a s e ) .
b E B
.
such t h a t
Of c o u r s e , i f
B
CB-rank b < B Cb]
f o r every
b 6 B , U(b) = { p
b 6 Ia(B),
then U(b) =
: b E p , CB-rank p =
f o r every
p € S ( B ) (such t h a t CB-rank p =
0; and, i f b l I
-,
t h e Boolean
i s uniform and
In t h i s case, w e define:
cz S(B)
is called the
( a , d)
t h e d e f i n i t i o n o f CB-rank
( w e denote h e r e by
CB-rank B [b*]
a l g e b r a of sub-elements of
*
[MTIJ
i s uniform, namely, f o r every
B
then
t h e ordered p a i r
B,
t h e r e i s a s l i g h t d i f f e r e n c e between
-
}
B
2 (O),
(notice that, i f
( B ) = b ' ( 1 ( B ) , t h e n U(b) = U ( b ' ) ) ; a),
r ( p ) = min { CB-rank B [x]
:
xEp}.
p- N
,-
Categorial Structures
Then we s e t , f o r every
b C B, f ( b l I ( B ) ) B a i s a s t r i c t l y a d d i t i v e f u n c t i o n of B / I
fB
every f
B
w,
(8 < w
IJ E w l , v
of f
)
1
+
B > 0
of
B
sup t r ( p ) : p E U ( b ) (B)
(2A) i n w
(here,
i s given by a
. isn
bilal
1
.
( i f we p u t , f o r
a 1 Moreover, w e d e f i n e t h e derived f u n c t i o n s
i n t h e following way: f o r every
=
when
PI).
IJ = max { v ,
=
309
b E B,
\
b
Ia
=
CB-rank B and by t h e sequence ( f B (1
abridges
b ( I a ( B ) ) . Therefore, t h e isomorphism type B
c. I n g e n e r a l , every countable atomic Boolean a l g e b r a
B
I a)
:
(1( 0 ) )
B < w
).
1
can b e decom-
posed i n one and o n l y one way (up t o isomorphisms) a s a d i r e c t product B 2 B where
>
B
1 CB-rank B
3.
i s superatomic,
B2
of
M
B(M)
B1 = ( 0 )
or
X
CB-rank B
B
1
The i d e a w e w i l l f o l l o w t o prove that every (countable)
i s p-k - c a t e g o r i c a l i s t o show t h a t t h e Ketonen isomorphism i n v a r i a n t s 0
can be obtained from t h e l a t t i c e of pp-definable subgroups of
and hence are preserved by elementary equivalence.
Th(M),
Of course, t h e f i r s t s t e p
# 8;
by r e c a l l i n g t h e
concerns t h e CB-analysis of
B(M).
g e n e r a l decomposition of
( s e e l ) , it i s e a s y t o see t h a t , i n o r d e r t o calcu-
l a t e CB-rank
X
X
and ( i n case)
Let
X € B(M), X
CB-degree X ,
w e may l i m i t o u r s e l v e s t o t h e c a s e
$(M) and $.(MI are pp-definable subgroups of M , 3 n, $ . ( M ) i s a subgroup of $(M) and [$(MI : (M) 3
where j
<
Lemma 1.3
Proof CB-rank X
2
>
2'
(The superatomic c a s e )
module
i s uniform and e i t h e r
1
aj
- For every a
€
M,
CB-rank X
=
CB-rank
1
(x +
and, f o r every is infinite.
a).
I t s u f f i c e s t o show (by i n d u c t i o n on v ) t h a t , f o r every o r d i n a l
2
v
i m p l i e s CB-rank (X + a )
v.-
v,
C. TOFFALORI
310
-
Lemma 1.4
( i ) CB-rank X = CB- rank $(M).
<
( i i ) I f CB-rank X
CB-degree X = CB-degree $(MI.
m,
Proof -
( i ) Let u s show t h a t , f o r every o r d i n a l
only i f
CB-rank $(M)
(
L
cases
(
($(.MI + a )
2
IJ + 1 .
If
Otherwise, t h e r e i s
i s obvious.
L
i f and
v
j
We proceed by i n d u c t i o n on
f
v l i m i t are trivial.
u = 0,
CB-rank
CB-rank X
u.
i s obvious ( s e e Lemma 1 . 3 ) .
+
u,
Let
u = p
CB-rank
( .U
+
1 ,
CB-rank $(MI
($.(MI
i < n 1 ( f o r instance,
< n
the
v;
L
v ; therefore
+
a , ) ) < p, t h e claim 3 j =O) such t h a t
CB-rank ($.(M) + a , ) 2 IJ. By Lemma 1.2, t h e r e e x i s t i n f i n i t e l y many elements 3 3 { b , : i E w l i n X such t h a t , i f i # j, then b . - b . $O(M) and, f o r any 1 3 i f [$O(M) : $O(M) A $,(M)] i s f i n i t e , then j < n,
4
3 f o r a l l i E w. 3 t h e i n d u c t i o n hypothesis i m p l i e s
( $ o ( 1 4 ) + bi) fl ( $ , ( M ) 3
all
i 15 w,
for a l l
i c!
hence
w,
?,
i
-
Lemma 1.5
CB-rank X
p
<
($.(M) + a , ) ) 3 3
(
CB-rank $(M)
+
1
<
m
CB-rank ( 4 ( M ) + b , ) 0 I
As
2
p
. therefore
m,
(see ( i ) ) . -
u
i f and o n l y i f t h e r e i s no i n f i n i t e descending
sequence of p p - d e f i n a b l e subgroups Proof -
+
CB-rank X = CB-rank $(MI = u <
( i i ) Suppose
CB-rank (
+ a,)= 0
($n(M) : n
€
w) such t h a t $ ( M ) 0
=
$(M).
Otherwise, we can d e f i n e a n i n f i n i t e descending sequence of
)
ordinals. (
f
)
CB-rank $(M) =
Let
CB-rank $ ( M )
< =
L
6 + 1,
$(MI, X I fl X CB-rank X
where
2
Oi(M),
=
2 m)
=
0,
;
$..(M) 13
m;
i f a = CB-rank B ( M ) and 6
hence t h e r e e x i s t CB-rank X
1-
> 6,
X1, X2 C B(M)
CB-rank X
> 6
2 -
2
a,
then
such t h a t
X I , X2
(namely CB-rank X
without loss of g e n e r a l i t y , we can assume, f o r
s a t i s f y t h e u s u a l c o n d i t i o n s (and $ . ( M ) 1
1
=
i = 1, 2,
i s a subgroup of
for
p- Ho-Categorial Structures
and, by Lemma 1 . 1 ,
Q2(M)
has i n f i n i t e index i n
i f CB-rank Q ( M ) <
Notice that,
then Q ( M )
m,
31 1
$ ( M I ; i n p a r t i c u l a r Q ( M ) C Q(M).2 f
admits a t most f i n i t e l y many
pp-definable subgroups of f i n i t e i n d e x , otherwise we can d e f i n e a s t r i c t l y dec r e a s i n g i n f i n i t e sequence of pp-definable subgroups
($,(MI
:
i n the
n E w)
following way Qo(M)
*
= Q(PU
Qn+l(M)
that
=
Qn(M)A$(M) where @ ( M )
[Q(M)
i s f i n i t e and Q ( M )
$(M)]
:
i s any pp-definable subgroup of
g $(M).
I n p a r t i c u l a r , t h e i n t e r s e c t i o n o f a l l pp-definable subgroups of
i s f i n i t e i s a pp-definable subgroup of
Q(M)
in
and admits no proper pp-definable subgroup of f i n i t e index.
Q(M)
$(MI,
t h i s subgroup; n o t i c e t h a t , f o r every
pp-definable Subgroup of
LQ ( N )
Q(N),
: Qo(N)]
h a s f i n i t e index
N F M, Q o ( N )
= [$(MI
whose
+(M)
index i n
denote by Q o ( M )
such
Q(M)
: Qo(M)]
Let us is a
and
Q"(N)
has no proper pp-definable subgroup of f i n i t e index.
- If
Lemma 1.6
CB-rank Q ( M ) <
m,
t h e n , f o r every v < w l , CB-rank Q ( M ) L V
if and o n l y if t h e r e i s a pp-definable subgroup Q ' ( M )
has i n f i n i t e index i n Proof ( -+
1
-
(
+
)
CB-rank Q ' ( M )
of Q ( M )
such t h a t Q ' ( M )
2 v.
is t r i v i a l .
By r e p l a c i n g ( i f n e c e s s a r y )
= CB-rank Q o ( M ) ) ,
of f i n i t e index. E B(M)
Q ( M ) and
+ 1
such that
$ ( M ) with
we may assume t h a t Q ( M ) As
QO(M)
( a p p a r e n t l y , CB-rank Q ( M ) =
has no pp-definable p r o p e r subgroup
i n t h e proof of Lemma 1 . 5 , we have that t h e r e e x i s t
X I , X2
@ ( M I , XI
n X2
Without l o s s of g e n e r a l i t y , f o r every
=
0, CB-rank X
i = 1, 2,
X1, X z e
> v , CB-rank X 2 L v
1-
.
C. TOFFALORI
312 where
$i(M), $ , , ( M ) s a t i s f y t h e u s u a l c o n d i t i o n s and 11 Lemma 1.1 i m p l i e s
@(M).
Of
Lemma 1 . 7 Proof -
-
$l(l.l) C ~
#
CB-rank $(M) <
If
then
m,
I t s u f f i c e s t o show t h a t
$i(M)
i s a subgroup
) o r $2(M)CL$(M).-
M
#
CB-degree $(MI =
[a
(M)
$O(M)]
:
The proof i s s i m i l a r
CB-degree $O(M) = 1.
t o t h e one of 1.6.Theorem 1.8
-
If
i s a c o u n t a b l e module and
M
B(M)
i s superatomic, then
M
i s p-,c(-categorical. 0
-
Proof
*
Let N
=
IN I
M,
.
=
We have
CB-rank M <
B ( N ) i s superatomic; f o r ,
-,
CB-rank B ( M ) = CB-rank B ( N ) ; r e c a l l t h a t similarly
CB-rank @(MI
4.
L
v = 0, v
i f and only i f , CB-rank $(N)
l i m i t are trivial.
L w
CB-rank $(M)
$ ' (M)
satisfying
$' (N)
i s a subgroup of
$(v)
,
u
that
Hence CB-rank $(N)
*
w
and
hence we have t o show t h a t CB-rank M =
more g e n e r a l l y , w e w i l l prove t h a t , f o r every pp-formula
and f o r every o r d i n a l
The c a s e s
CB-rank B( M) = CB-rank M + 1
CB-rank B ( N ) = CB-rank N + 1,
CB-rank N ;
=
and, by Lemma 1 . 5 , CB-rank N < - , t o o ;
i f and only i f
[$(MI :
$ 0
(M)
J
If $(M)
w
= i.1
: $'
(N)]
+ 1,
v.
Lemma 1 . 6 i m p l i e s
admits a pp-definable subgroup CB-rank $ ' ( M ) >
i n f i n i t e and
$ ( N ) , [1@(N)
L
u
.
Therefore,
i s i n f i n i t e and CB-rank L$' ( N )
2
2- v.
CB-degree B(M) = CB-degree B ( N )
:
see Lemma 1.7.-
(The uniform c a s e ) F i r s t w e w i l l show:
Theorem 1.9 B(M)
-
For every c o u n t a b l e module
M,
e i t h e r B(M)
i s superatomic o r
i s uniform.
Proof -
Suppose t h a t
CB-rank X = w <
-.
B(M)
i s n o t superatomic, and choose
A s u s u a l l y , we s e t
X E B(M)
such t h a t
1-1.
313
p- Ho-Categorial Structures
where
a
every
i
a
EM, $i(M), $ . , ( M ) ij 17 m an6 j < mi, $ , , ( M )
i’
<
is infinite. i
11 CB-rank B
Therefore
< m ( i = 0,
0
CB-rank M =
*
$
- bk
O0(M)
[M
m ,
if h
v + 1;
:
( ( 4 (M) + a 0
0
-
)
,O
j<m0
[$O(M) :
g0(M)
Consequently,
h
c.
CB-rank B [ M
w,
f
M
-
= f
B(M) i n t h e following way ( s e e 2 ) :
B(M) i s uniform.
of B(M) / I (B(M)
*
f i n a l l y , f o r any
X E B(M), s e t f
= CB-rank
M
6
;
satisfying
M
P u t a =CB-rank B ( M ) ,
which i s defined
p , CB-rank p =
(Xla) = sup
i s determined by a and
- For every
(X
v
m
}.
1 CB-rank a[yJ
r(p) : p E U(X) }
:
Ydp
1,
where
XIIa(B(M)).
W e define as i n 2 t h e derived functions
0,
=
hence, by
in
( 2 A ) i n w1
f o r any p 4 S(M) ( s a t i s f y i n g CB-rank p = =), l e t r ( p ) = min
If
))
w e have t o c o n s i d e r t h e s t r i c t l y
B(M),
*
Proof -
< n,
01
XI
XE B ( M ) , l e t U ( X ) = { p e S(M) : X
Lemma 1.10
i
a
-
f o r any
type of B( M)
+
i s f i n i t e , then f o r a l l
*
abbreviates
(M)
CB-rank ( $ (M) + b ) = CB-rank ( ( $ ( M I + b h ) X ) = v. 0 h 0 v + 1. I t follows t h a t B ( M ) i s uniform.-
b e a countable module such t h a t
a d d i t i v e function
Oj
(bh : h E w)
I\ O i ( M ) ]
i n o r d e r t o g e t t h e isomorphism t y p e of
Xla
J
(M) : + i j ( ~ )
0.
0
In particular, f o r a l l
($
i s i n f i n i t e f o r every
$i(M)]
h E w ( $ ( M I + b h ) n ($i(M) + a i ) =
M
[e,
o ~ ( M ) and
on t h e o t h e r s i d e , t h e r e i s
# k,
i < m, i f
f o r every
L e t now
=
t h e r e e x i s t i n f i n i t e l y many elements
Lemma 1 . 2 , bh
i s a subgroup of
[XI
and, f o r
M
f o r s i m p l i c i t y ) such t h a t
CB-rank $ ( M ) = CB-rank
since
a r e pp-definable subgroups of
f B (6
(fM(Mla)
X e B ( M ) , a € M,
6
=
+
a ) and n o t i c e t h a t
<
BM
B < wl,
r e c a l l t h a t , f o r every p E U(X)
w ) of f M , then t h e isomorphism 1 :
6 < wl).
B
6
f (Xla) = fM((X + a)Ia). M
X E B ( M ) , a C M,
i f and only i f
CB-rank X = p + a
=
C . TOFFALORI
314 = { Y
a : Y E p } F- U ( X
f
+
a ) , and r ( p ) = r ( p
y < B
assumeourclaim t r u e f o r every have
X
X,la 1
3<m
y < B
L e t now on
3
Xla =
every
and
E
j
= f
M I
B ( M ) , X = ($ ( M )
+
a)
Proof -
- For every
F i r s t suppose
B < B
L e t now
Y
((X. +
M
- .u
1
I
( $ , (M)
7
ja
a) l a )
+ a
.)
1
B
B
(X
.-
f b , ( x l a ) = fM($(M)la)
wl,
= 0,
a).
and we show it f o r
i f a n d o n l y i f ( X + a ) la =
< m, f y ( X l a )
$(M) and $ , ( M I ) . 1
Lemma 1.11
+
j
8. + a)
B > 0,
we
I n f a c t , we
Ia
,
where, f o r
(with t h e usual conditions
.
w e p r o c e e d b y i n d u c t i o n on n. n = O
:
t h e claim i s
trivial.
+ 1 : i f X ' = ($(M) + a ) - ." ( $ , ( M ) f a . ) , t h e n f ( X ' l a ) = f M ( $ ( M ) / a ) , 1 1 i
.
4
Therefore
f
(Y
M h
(a)= f ( $ ( M ) l a ) M n
f M ( X l a ) = f M ( X ' l a ) = fM($(M)l a )
13 > 0;
L e t now
p r o v e it f o r
f M ( X ' l a ) . Since
=
.
Y
CXC X', h -
a s above, we suppose o u r c l a i m t r u e f o r e v e r y
it f o l l o w s
y < B
and w e
The problem l i e s i n showing t h a t , i f X = ( $ ( M ) + a ) - , U ( $ , ( M ) + a . ) 15n 3 3 and X ' = ($(M) + a ) ($.(M) + a , ) , t h e n , f o r e v e r y decomposition 13 1 Xila, w e c a n d e f i n e a decomposition X ' la = X ' la such t h a t , Xla = i<m i<m i f o r a l l y < B and i < m, f:(Xila) = f:(X;/a), and c o n v e r s e l y .
6.
.u
.c
F i r s t suppose
x
=
6 xi;
i <m
where
6
u Xila; w i t h n o loss o f g e n e r a l i t y , w e c a n assume i<m moreover, f o r e v e r y i < m, i f xi 0,
t, > 0
Xla
=
and f o r e v e r y
+
h < t . X L
ih
i s contained i n a s u i t a b l e coset
315
p- Ho-Categorial Structures
+ aih (qih(M) a pp-definable subgroup of $(M)) and ( J l , ( M ) + a , ) ih ih ih i s t h e union of f i n i t e l y many smaller c o s e t s corresponding t o pp-definable
$ , (M)
subgroups of i n f i n i t e index i n Q(M)
X,
ih
Notice t h a t Lemma 1.1 i m p l i e s t h a t
Jlih(M).
e q u a l s t h e union of t h o s e c o s e t s
+ a
-
Jlih(M)
+ aih
such that
[ $ ( M ) : + i h ( ~ ) ] i s f i n i t e . of c o u r s e w e can assume ( + ( M I + a ) - ,U ( $ , ( M ) + a , ) # 2 n 1
#
p
such t h a t
(hence [ Q n ( M )
is infinite; l e t
then
bU
v i t y of
hence
Define
-
bv
f i
#
As
[$(M) :
bo,
3 [Q(M)
:
bn(M)A $ih(M)]
$,(M)n + i h ( M )
)
.
is i n f i n i t e , there e x i s t
i s f i n i t e and
: +ih(M)]
..., bq- 1 &
$,(M)]
i
( Q n ( M ) + a n ) n(Jlih(M)
= q i s f i n i t e , while
qih(M) + a
< m, +a, ) # ih
: Qn(M)nJlih(M)]
such t h a t , i f u < v < q , ih By u s i n g t h e i n d u c t i o n h y p o t h e s i s , t h e a d d i t i -
and Lemma 1.10, we g e t , f o r every
y < B,
C. TOFFALORI
316 then
X'
d X' (where X ' fl X ' = 0 i f 1 < s < n ) , and f Y ( X ' l a ) = f y ( X ) a ) l
=
f o r every
Let now X ' la = u i <m such t h a t , f o r e v e r y
X i la,
< m,
i
A s above, we c a n assume
"1
iJ xih
X'
,u
X. la
61<m < 8 , f y ( X , la) = f y ( X ! l a ) ( t h e n , f L ( X 1 a ) l f M (X'Ia)).
y
M
= ,U
i
M
i
X ] and, f o r every
i<m I where e a c h
(ti > O),
= h
we c l a i m t h a t t h e r e i s a decomposition X(a =
Xih
< m,
i
if
X;
+
9,
is contained i n a s u i t a b l e c o s e t
qih(M)
+ aih (bih(M) a p p - d e f i n a b l e subgroup of $(M)) and +aih) - X ' ih i s t h e u n i o n of f i n i t e l y many s m a l l e r c o s e t s c o r r e s p o n d i n g t o p p - d e f i n a b l e sub-
groups of i n f i n i t e index i n and
XYh =
2nd c a s e :
XYh #
M
i < m
For e v e r y
(such t h a t
Xi
# 0)
h < ti, consider
1 s t case:
fy(X:'
qih(M).
ih
(a)= f y ( X ' M
3rd c a s e :
[$(MI
:
X"ih
#
xih; Xih,
b u t [+ih(M)
la)
ih
Xih
$ih(M)]
those cosets
Y
t h e n f y ( X " ( a )= f M ( X i h ( a ) f o r a l l M ih
for a l l
y
is infinite); + a
Jllk(M)
lk
1
i s i n f i n i t e ; again
$n(M)f?+ i h ( M ) ]
< 6
[+ih(M)
and
1 < n, k < t
there exist
:
(use t h e induction hypothesis);
i s f i n i t e (hence,
: $ n ( M ) / ?k i h ( M ) J
by Lemma 1.1
such t h a t such t h a t
y < 6;
$(M)
+
a
qlk(M)]
[$(MI
:
[$(MI
: $
lk
(M)]
e q u a l s t h e union of
is f i n i t e ; therefore i s f i n i t e and
Notice t h a t :
*
$
(M)n q i h ( M ) n q l k ( M ) in
q.
*
$,(MI
[$lk(M)
:
4
n Jiih(M) (M)O
X"
every
lk
such t h a t
b
$lk(M),
b u t f i n i t e index, say
;
qlk(M)]
I t f o l l o w s from Lem.a
in
has i n f i n i t e index i n
is infinite.
1.2 t h a t t h e r e a r e i n f i n i t e l y many e l e m e n t s { b
-
bv
$,(M)A i i h ( M ) f+l lk(M)
n Jllk(M)
if
u # v,
us w 1
and, f o r
+ b U )A X" excludes only f i n i t e l y lk many c o s e t s corresponding t o s o m e p p - d e f i n a b l e subgroups o f i n f i n i t e i n d e x i n
u E
w,
($,(M) A $ih(M)
d
: U
317
p- Ho-Categorial Structures
On(M)n$ih(M)n
qlk(M)
of c o u r s e , from
( i n p a r t i c u l a r , it i s d i s j o i n t from
$,(M)
notice that, for a l l
an).
t
y
<
t
in
$ih(M)
aih
and,
We r e p l a c e
(and f o r a s u i t a b l e element
6
qih(M)
c
+ aih),
and
X:
# 0)
comes o u t e v e n t u a l l y i n t h i s way.
Put
X, if i < m , ih
Xi#
For e v e r y
i
X
i
< m
=
(satisfying
U
h < ti
1
and
h < t
(Xihf?Xik
if
let
X
ih
be t h e s e t which
= Ii f h < k < t . )
X i = O i f x : = O .
Apparently
X = !iJm
Xi
f o r every
i < m
and
Lemma 1.12
-
p
= min
If
(and y < 6.-
L S(M)
{ CB-rank B [$(MI]
Proof -
5
i s obvious.
where, f o r every
i
< m,
:
X . fl X = 1 1
(and CB-rank
1a
0 M
0 if
p
such t h a t
Conversely, l e t
i< j<m),
=
-1,
f y ( X , la) = fy(X' ( a ) M i M i
then
$(MI t a E p )
Y E p;
r(p)
=
.
as above, s e t
Y =
u
i <m
Yi,
C. TOFFALORI
318
$l(M), $ , . ( M ) s a t i s f y t h e u s u a l c o n d i t i o n s f o r e v e r y i < m , j 11 T h e r e f o r e , f o r some i < m , Q i ( M ) + a i € p and, f o r e v e r y i < m , and
CB-rank B [ Y j
2
CB-rank B c Y . 1
Theorem 1.13
-
If
.
ni
CB-rank B [ $ , ( M ) + a , ] = CB-rank B [ $ , ( M ) ]
=
i s a c o u n t a b l e module and
M
<
i s uniform, t h e n
B(M)
M
.is
p-6 - c a t e g o r i c a l . 0
Proof -
Let
N
M,
1NI
$'o;
=
B(N)
i s uniform b e c a u s e of 1 . 8 and 1.9.
Moreover CB-rank B ( N ) = CB-rank B ( M ) s i n c e CB-rank B(M) = =
sup { CB-rank
a
Let
*
B
5
f M ( @ ( Ml a) ) and
.
f N ( $ ( N )l a ) .
p / N b e t h e h e i r of
Let p
p € S(M) in
S(N)
such t h a t (see
s t a b 1 e ) ; t h e r e f o r e @ ( N ) E p l N , CB-rank plN =
=
CB-rank B
[$(N)]).
=
plN E U ( $ ( N ) )
Then, f o r e v e r y
such t h a t
r(p) = r(q)
.
f M ( $ ( M ) l a )L f N ( $ ( N ) l a ) there i s
q E U($(M))
po E U ( $ ( N ) )
> M. and CB-rank p =
recall that
and, f i n a l l y , r ( p ) = r ( p l N )
m,
$(v),
p E U(@(M)),
CB-rank B
there exists
$J(M)]
q
=
=
.
I t s u f f i c e s t o show t h a t , f o r e v e r y
such t h a t
is
Th(M)
r(p) = r(q)
.
For any
p E U($(N)),
p E U($(N)),
define
i n t h e f o l l o w i n g way
p i = { $(N) :
Po g s ( N ) ;
N
$(M) E p
[LP],
( s e e Lemma 1.12, r e c a l l t h a t , f o r e v e r y pp-formula
*
CB-rank $ ( N ) .
=
t o show t h a t , f o r e v e r y pp-formula
0; w i t h o u t l o s s of g e n e r a l i t y we can assume
=
(and s i m i l a r l y
}
I3 B f M ( @ ( M ) l a=) f N ( $ ( N ) l a )
$ ( v ) and B < w l r Suppose
then it suffices
m
$ ( v ) , CB-rank $(M)
and, f o r e v e r y pp-formula
CB-rank B ( M ) ,
=
CB-rank $(M) <
$(M) + 1 : $ ( v ) pp-formula,
f o r CB-rank B ( N ) )
3b E
in fact. l e t
i n p a r t i c u l a r , f o r every
N
such t h a t
$ J ~ ( N )E
i
< n,
p:
(i
$(N) + b E p }
+
< n), $ j ( ~ ) pi, a j
there is
suppose towards a c o n t r a d i c t i o n t h a t
.
b. E N 1
such t h a t
N
(jt < m ) ,
$Ji(N)
+
b, L p; 1
m,
319
p- Ho-Categorial Structures if
iln( q i ( N )
b E
whereas
+ bi),
+ bi E p ,
qi(N)
then
i < n
I
1
$- p
Furthermore
po E U($(N)) and r ( p ) = r ( p n ) .
r ( q ) = '(Po)
= r(p)
B > 0,
L e t now
$(M) l a
Let
sentation
=
w e assume o u r c l a i m t r u e f o r a l l where, f o r every
By u s i n g t h e elementary equivalence oE $(N) l a
Set q
j
y
< m,
N and M ,
p f' OIM'
=
I
( f o r , q+ = { $ ( M I : @ ( N ) E p:
U X .la j < m I
,u
($,(N) + a , + b) = i < m I 3 f o r every i < n , j < m
( $ . ( N ) + bi)
$,.(N) + a j + b
)
0,
.
then q E U($(M)),
.
< 6 and we prove it f o r
6.
X , admits t h e u s u a l repre3
we g e t a s i m i l a r decomposition
Y , l a where, because of Lemma 1.11 and t h e i n d u c t i o n h y p o t h e s i s , j < m 3 f o r every y < B and j < m, =
Theorems 1.8, 1 . 9 , 1.13 imply t h e main r e s u l t o f t h i s s e c t i o n : Theorem 1.14
s
2
-
- Every module i s p-4 - c a t e g o r i c a l . 0
A n i l - 2 w - s t a b l e non-p-h
- c a t e g o r i c a l group
0
N i l p o t e n t groups of c l a s s 2 ( s h o r t l y n i l - 2 groups) p r o v i d e a comparatively s l i g h t g e n e r a l i z a t i o n of t h e c l a s s of Abelian groups; i n f a c t a group s a i d t o be n i l - 2 i f t h e commutator subgroup of Z(G) of G.
G
G
is
i s contained i n t h e c e n t r e
However, while Abelian groups a r e q u i t e understood from a model
t h e o r e t i c p o i n t of view and, i n p a r t i c u l a r , t h e r e i s a complete c l a s s i f i c a t i o n of w-stable Abelian groups, t h e s i t u a t i o n f o r n i l - 2 groups, even i n t h e w-stable c a s e , i s v e r y confusing; Mekler's paper
[Me]
suggests t h a t t h e s t a b i l i t y
c l a s s i f i c a t i o n of n i l - 2 groups i s t h e key p o i n t towards t h e s t a b i l i t y c l a s s i f i c a t i o n f o r groups, r i n g s and, more g e n e r a l l y , a l l s t r u c t u r e s with f i n i t e l y many r e l a t i o n s and o p e r a t i o n s ; i n p a r t i c u l a r , t h e hardness of c l a s s i f y i n g w-stable
C . TOFFALORI
320
groups t r a n s f e r s t o w-stable n i l - 2 groups, however t h e e x i s t e n c e of so many
w-stable n i l - 2 groups j u s t s u g g e s t s t h a t t h e r e may be w-stable n i l - 2 groups which a r e n o t p-,4' - c a t e g o r i c a l ; we propose t h e following example, using M e k l e r ' s 0 arguments [Me]
.
ro ro,
Let b E
{ a E w
=
:a
A(a, b)
tf
I
l(a) = l(b)
{
or
I t i s easy t o s e e t h a t
ro,
E
(r)
B
CB- type
then
i
f o r every
< l(a) } ,
and f o r any a ,
define (
r
# 0, a ( i ) 5 i
ro
+
1, a ( i ) = b ( i ) f o r every i < l ( b )
i s an w-stable n i c e graph, hence, f o r every c o u n t a b l e
r0
i s a superatomic a l g e b r a , b u t
+
1, I),
p-fi - c a t e g o r i c a l , s i n c e 0 i s a c o u n t a b l e model of Th(To) and r $ r 0' r c o n t a i n s a t l e a s t one s t a r , p o s s i b l y ti,
r
= ( 2 , I ) , while, i f
B(To)
CB-type B ( T ) = ( w
is not
as
many s t a r s , whose elements admit i n f i n i t e l y many a d j a c e n t nodes. Consider now t h e n i l - 2 group of exponent
p G
0
= G(T,),
f o r completeness' sake
we remember t h a t
v E.
ro,
u, u'
E
where, f o r every
[ u,
N = (
ro *
Z
3
:
any well o r d e r i n g
0
= Z(Go)
lA(u,
*
u'
ro,
A(u,
>
G
u')).
i s w-stable s i n c e
0
F P
is a basis for
i s a v e c t o r space o v e r
IF
Z
,
0
is interpretable i n
Z x 0
?r
Z y
0
is; give
[u,
Go;
{ Zo v : v E
i n f a c t , set i n
i f and only i f
C
G
u']
:
u,
u' C
r
0'
0'
and
PI
and
Go/z o ;
* r
ro
then:
i s a v e c t o r space over
u'), u < u'
Go/Zo
<,
2 d e n o t e s t h e f r e e n i l - 2 product and
( v ) 2 Z/p,
(x)
0
=
Go/Zo
CG ( y ) 0
r
0
is a basis for
p- Ho-Categorial Structures and i n t h e q u o t i e n t
A(z~xI%, r 1( G 0)
Let
i. i f
W E
excluding
(Go/Zo)/%,
ZoI%
321
71 Zoy and [ x , y ]
zoyI%) i f and o n l y i f Z x 0
0
r 0'
then
v l , v 2 CI
< h l , h2 < p
iii. i f
k
...., hk
< p,
Z v/%
I
=
0
r0
Zov
h
: 0
,
< h < p 1
A ( v l , v 1, 2
then
Z
v1
Zo
and t h e only a d j a c e n t nodes a r e
1'
...,vk
vl,
2,
r o I=
and
E
ro
0
~ 1 %
admits some
~ ~ 1 I% ,
vlI%
Zo
Z
hl
hl Z v 0 1
i v . otherwise,
r 2 ( G0 )
0
...v kh k l %
=
{
hl Zo(vl
...vkhk )t
:
1
< p
0 < t
= (
Z
0
XI%
E
r 1 (G 0)
:
IZoxI%l
w e a n a l y s e t h e d u a l space and, f o r every
is
p - 1, Z
0
over
y
r2
IF -independent and g e n e r a t e s
u r3
that
enlarge
v E r );
{ Zy:y
E
over B
u r3
-as w e l l as [(B U IF
P
1'
r3),
r 3(B
U
IF }
P
XI%
has some
i s isomorphic
B(GO),
f o r every
%
( Zx : 2 ~ 1 %E
r 3 of rl (GI,
( s e e i i i . ) ;d e f i n e now G/Z
G
1
B € G
over
IF
i s a d i r e c t addendum of
2,
define a subset
r 2 (G)
- c l a s s of
t o a subset
i s a b a s i s of
G:
P h a s a graph s t r u c t u r e induced by
r,)]
has no
I
0
r 1 ( G0)
about t h e s t r u c t u r e of
[Me]
by choosing an element
P A(Zxl%, 2 ~ 1 % ) } and
S(Go),
x E G, tp(xlGo).
r e c a l l some r e s u l t s from
2 ~ 1 %when
< h
0
hence t h e CB-type of B ( G ) , 0 by c o n s i d e r i n g G > G 0' G w 1- s a t u r a t e d , I G ( = h 1' L e t us a b b r e v i a t e Z ( G ) by 2 , f i r s t w e
I n o r d e r t o determine t h e isomorphism t y p e of
B
=
r 0'
G
1%;
Z vl% ;
with t h e graph s t r u c t u r e induced by
a d j a c e n t nodes }
of
2
are a d j a c e n t t o a s u i t a b l e v E r O and
a d j a c e n t node. Therefore,
v
0
h2
then
and t h e only a d j a c e n t node i s
to
.
= u
be t h e corresponding graph, we have:
a d j a c e n t nodes; ii. i f
1,
{ Zo
=
(choose
such t h a t
{Zy:yET3}
Zyl% E rZ(G) such t h a t
r2 in
v
B
satisfying
n r3
=
0
(see iv.); n o t i c e t h a t
r 1 (G); a f t e r let
X
noting
be a b a s i s
of i t s coaddendum (which i s an Abelian group of exponent
p,
hence
C. TOFFALORI
322 a vector space over
in other words, relations.
Then
r3
G has
as a set of generators, with the suitable
r0
Enlarge now the well ordering of
(if
lA(yI,y2)
y
yl,
2
IF
6 T3)
B <
x,
is a basis of
)
to a well ordering for
y2
: yl,
, too (and
:
is another set of generators of
G
P
BJX
y2 €
r3 CJ
B , y1 C y 2 ,
2;
Zy
IF
r3U
then
X i i t [yl,
and
P
is a vector space over
G/Z
U X
U B
is a vector space over
Z
*
P
r 0 < r2 - r0 < r3 - r2 <
such that
*
.
F )
y€
r3 U
B }
is a basis of
G/Z).
If
r;
U B' U X'
then there exist a graph isomorphism bijection IGI
of
Q,
=h,,G
B
@r
of
r3
1
'
@
of
G
rot
fixing
$x of X onto X'
onto B', and a bijection
1 defined by setting
=
r;
onto
w -saturated); hence an automorphism
@(X)
(defined in the same way), a
(remember
fixing
G
0
can be
3
(x) if x 6 B
x.
x) if x E
(; :
I
we can classify now
1. x c Go; 2.
x f!
Go,
(x I
{ tp
G )
0
:
x E G }.
in this case, CB-rank tp(x1 but there is
g
€ Go
G )
0
such that
We set for simplicity
= 0.
Z x = Zg;
first suppose that x
can be expressed as a product of commutators of elements of
-
what elements of
G
* Let x
yo]
= g
[v,
G
0
G,
*
g
-1
the problem is
are needed in this decomposition.
....
[v,
yn]
where
v
€
ro,
n r2
w,
ZyoI%,
... , Zynj?, €
3 23
p- NO-Categorial Structures
r 2 (GI - r2(G0) 2 ~ ~ Ir2(Go)), 1 %
where
a r e placed i n t h e same star.
I%, ..., Z y ' 1%
Zy;
an automorphims
ro U
$r
all
5
i
n
C r (G) and 2
0
of
G
}
to
r3
...,y
yo,
isomorphism
r3
of
t o get
fixing
p
I%, ...,
tp(zyo
=
r;
. .. , Zy;l%)
and mapping
Go
ro U {
and
onto
(Zybls,
y,!,
fixing
..., y,: 1 ro
then t h e r e is
,k p ,
x
i nt o x'
to
r;;
and mapping
(enlarge
extend t h e into
y.
1
y:
1
for
I t i s easy t o deduce t h a t CB-rank t p ( x ) G ) = 1.
$).
0
More g e n e r a l l y , i f t h e decomposition of classes in
Let
notice t h at , i f
r 2 (G) - r 2 ( G0)
x
only needs elements of
G
whose
are placed i n t h e same star, then CB-rank
t p ( x IG ) = 1. 0 L e t now
x = g
where
v E
II
i t n
ro,
Zy.,
13
jtni
E r2(G) - T2(Go)
1%
Zy.
in
1%
for of
Zy:
,
13
G
1%
II
i t n
n
j t n .
CB-rank t p ( x
Go
1 G0 )
Cv,
n
and
i
<
i
j 5ni,
5
n
and
,
1 5 n.
Y;~I I=
and mapping = n4 1 ,
5
l i e on t h e same s t a r f o r a l l
E TZ(G),
fixing
i
l i e on d i f f e r e n t stars i f
Zyiol%, Zyl01%,
x ' = g
f o r every
x
then t h e r e i s an automorphims
p,
into
x'
(proceed as above).
i n f a c t notice t h a t , i f
Moreover,
$
3 24
C. TOFFALORI v €
where
. ..,
I
27;
ro,
%)
adjacent to
...,
0 < h 0 belong t o
1
Zv
hn
<
IJ
E
n+ 1
in
n
different stars,
More g e n e r a l l y , i f t h e d e c o m p o s i t i o n of such t h a t
1%
Zy'
P
1 i s t h e power of a maximal s e t
are
A
*
IF - i n d e p e n d e n t , P
L e t now and
B), into
,
a l s o n e e d s some c l a s s e s
of e l e m e n t s
CB-rank t p ( x l G ) 0
where v €
ro,
261%
5
8
B',
x
n
1
+ n2
into
x'
are
A
r 2 (G) - r 2 ( G 0 ) , (Zy' : Z y ' g Y ) ,
. [v,
rl(G),
a']
(6' like
B
and mapping
.
CB-rank t p ( x l G ) = w : t h e key remark t o show t h i s c l a i m i s t h a t , 0 then 26, A ' a r e IF -independent i f and o n l y i f Z B ' , A ' are. P
[v,f3]=b,B0],
,
have no a d j a c e n t nodes i n Zy..
( i5 n,
13 (m+l)w + n
- if
j
+
5
1,
ni)
ZBO,
B;]
n!i
13
B;,
ZBO1%,
...,
0
and
..., "mI%
26,' A ' are F - i n d e p e n d e n t ; P I n t h i s c a s e , CB-rank t p ( x l G ) = 0
j t nn i
p o s i t i o n , t h e n t h e r e i s a n automorphism
Bh, y . . i n t o
ro;
ET
because:
[vht
x ' = 9 m!h
rl(G),
a r e a s above.
...,v m + l
where v o ,
i s a n i n i t i a l segment of t h e w e l l o r d e r i n g f i x e d i n
-
and Z y ' ,
such t h a t
Zy'
A'
hence
I G0 ) 5 n. Zy' 1 %
h a s no a d j a c e n t c l a s s e s i n
yij]
=
r 2 (GI
a r e F -independent. Notice t h a t , i f x ' = g P t h e n t h e r e e x i s t s a n automorphims $ o f G f i x i n g Go
Z
Furthermore, if
B]
x = g [v,
then
Y
n,
whose c l a s s e s i n
G
i s t h e number o f t h e stars i n v o l v e d i n
n
5
t h e n CB-rank t p ( x
i s a d ja c e n t b u t does n o t belong t o
IF - i n d e p e n d e n t , and
n2
x
i
.
= zy' ( w i t h o u t l o s s of g e n e r a l i t y , Zyi = Zy;) 1 ~ ( i ) I n g e n e r a l , i f t h e decomposition o f x needs e l e m e n t s o f
a r e p!.aced
Zyjla,
such t h a t , f o r e v e r y
S
zy,
r 2 ( G ) - r 2 ( G0 )
similarly
a r e p a i r w i s e d i s t i n c t , and n e i t h e r e q u a l n o r
then there e x i s t s
%,
. . . , 2 . ~ ~ 1 %(and
p, Zyol%,
r2 (G) ,
and hence
y!., 11
t h e decomposition o f
x
x
[v $ of
into
~ + y~i j ,] G
fixing
h a s a s i m i l a r decomG
0
x':
is e s s e n t i a l l y unique, s i n c e ,
if
and mapping
325
p- No-Categorial Structures
z6hl,,, E
and
belong t o
h a s no a d j a c e n t nodes
T1(G)
2 ~ ~ 1 % for
every
5
h
ro,
5
-and s i m i l a r l y f o r
m+l
without l o s s of g e n e r a l i t y , w E
(h
m ) , Zy.
,I% ( i5
13
n, j
5
ni)
a r e p a i r w i s e d i s t i n c t and n e i t h e r equal nor a d j a c e n t t o
T2(G),
..., w
wo,
I n g e n e r a l , i f t h e decomposition of
- some c l a s s g Z6ls
such t h a t
,I%-,
x
then,
Zy! 11
(provided some f a c t o r s
ZBh = 26;
0 equal o r a d j a c e n t t o
ZS;ll%,
a r e neglected)
with
Zw
and Z y . . = Z y ! , 11 11
.
needs has
281%
no a d j a c e n t nodes and
26, A '
are
IF -independent,
P
- sane c l a s s e s
Zy
r 2 ( G ) , and Z y ' ,
' 1%
i s a d j a c e n t b u t does n o t belong t o
and
T2(G),
m i s t h e power of a maximal set
of elements
Y
26
such t h a t
(26 : 26 € Y),
a r e IF -independent, P
A'
-
1%
A a r e IF -independent, P
i n a d d i t i o n t o some c l a s s e s i n
-
Zy '
such t h a t
n i s t h e sum of t h e number of involved stars i n i" ( G ) - r (G and t h e p o w e r 2 2 0 Y of elements Zy' such t h a t (Zy' : Z y ' C Y), A a r e
of a maximal set IF -independent,
P
then
CB-rank t p ( x
F i n a l l y , if
I
G
Z x = Zg
0
5m
+ n.
w
f o r some
but
g € Go,
g
X .
-1
cannot be expressed a s
IF -independent commutators, then t h e s e c o n d i t i o n s i s o l a t e P t p ( x l G ) and it i s easy t o deduce from t h e previous remarks t h a t CB-rank G 2 tp(x G 1 = w
a product of
I
.
0
3 . For every
g d Go, Zx # Zg;
of t h e class
Zx.
*
There i s exist tp ( x
*
go E Go
v E
I G0) h
= w
h € { 1, 2
+
go E Go
There i s
...,
ro,
c p,
same s t a r .
such t h a t
and
1,
t h e problem i s now t o measure t h e complexity
Z (x
-1 go
..., p-1
)
1%
E T2(G)
such t h a t
[- r Zx
=
(G )
ho Zv g
0
,
otherwise t h e r e
]:
then,
CB-rank
a s w e l l a s i n t h e following more g e n e r a l case. such t h a t
zyol%,
Zx =
Z
g
h~
o y o ...
..., z y n l % E r 2 ( G ) 1-
hn where n E w, 0 < h 0' 'n r 2 ( G o ) 1 a r e placed i n t h e
C. TOFFALORI
3 26
*
More generally, suppose that there is
n, ni E w,
where
go E G
0
such that
< hij < p , Z y ,. I % E T2(G) - r2(Go)
0
for every
11
i, j,
1 % are placed in the same star for every i 5 n, but ZYi0/% ZYini Zyio(s, Zylol% belong to different stars if i < 1 5 n. Therefore . . . I
- if x ' E G and Zx' $ of
G
has a similar decomposition, then there is an automorphism
fixing G
0
and mapping
x
into
x';
- a standard compactness argument implies that CB-rank tp(x I G (recall that, if T2(G),
to
-and similarly
are pairwise distinct and
h 0 ZY0
h
... ",Y
=
if and only if there is A
. . ., Zym
Zyo I%,
0
h
"'b o
S
m+ 1
such that Z y ' A
Zy',
1%
Zy' 1 % - belong
< P, then
such that, for all
similar argument works if the decomposition of
Zy' 1 %
- - . ,hm
w2 + n + 1
=
h m
0
h
< ho,
0
.. .,
Zy'bls,
)
i s m,
Zy, = Z y ' 1 o(i)).
also involves some classes
Zx
is adjacent but does not belong to
T2(G)
and
are IF -independent.
P
* Let finally x
6. G
such that
Zx, A ' are IF -independent, notice that P and it is easy to deduce from the
this condition determines tp ( x I Go), previous remarks that Therefore CB-type B(G of
Th(T
0
)
and
T
P
0
To
CB-rank tp(x = (w2 + w
1 G01
+ 1,
= w2 + w
l),
.
while, if
T
is a countable model
(so that CB-type B(T) = ( w + 1 , l ) ) ,
then
G(T)
is elementarily equivalent to
Go, but, by proceeding as above, we can see that
CB-rank B(G(T)) > w2 + w + 1.
Hence, Th(GO)
is w-stable but not
p-,k -catego-
rical. Theorem 2.1
- There are (w-stable) non p-h -categorical nil-2 groups. 0
0
p- Ho-Categorial Structures REFERENCES
[BCM]
[K]
[LP]
Baur-C-. Cherlin-A. M a c i n t y r e , T o t a l l y c a t e g o r i c a l g r o u p s and r i n g s , J. Algebra 57 ( 1 9 7 9 ) 407-440
W.
J . Ketonen,
The s t r u c t u r e o f c o u n t a b l e Boolean a l g e b r a s , Ann. Math.
108 (1978) 41-89
D. Lascar-B. P o i z a t , 44 ( 1 9 7 9 ) 330-350
An i n t r o d u c t i o n t o f o r k i n g , J. Symbolic Logic
Marc ja-C. T o f f a l o r i , On pseudo-h - c a t e g o r i c a l t h e o r i e s , Z e i t s c h r . 0 f . Math. Logik, t o a p p e a r
[MTl]
A.
[MTZ]
A.
(1,
Marc ja-C. T o f f a l o r i , On Cantor-Bendixson s p e c t r a c o n t a i n i n g 1 ) . I - Logic Colloquium 83, 11 - J. Symbolic Logic, t o appear
Mekler, S t a b i l i t y o f n i l p o t e n t g r o u p s of c l a s s 2 and prime exponent, J. Symbolic Logic 46 (1981) 781-788
[Me]
A.
[Z]
M.
Z i e g l e r , Model t h e o r y o f modules, Annals of Pure and Applied Logic 26 ( 1 9 8 4 ) , 149-213
3 21
LOGIC COLLOQUIUM '84 J.6. Paris. A.J. Wilkie, and G.M. Wilmers (Editors) 0 Elsevier Science Publishers 6. V. (North-Holland), 1986
3 29
ON SENTENCES LNTERPRETRBLE I N SYSTEpls OF ARITHMETIC
A.J.
wilkie
Department of Mathematics, The U n i v e r s i t y , nanchester, U13 ~ P L , UK
.
W e g i v e a c h a r a c t e r i z a t i o n of t h e n l s e n t e n c e s t h a t are i n t e r p r e t a b l e i n n2 e x t e n s i o n s of bounded arithmtic. 1. INTRODUCXION
L = ( o , l , + , . , < ) be t h e u s u a l language of a r i t h m e t i c . Recall t h a t a formula of L is c a l l e d bounded (or A. or co or no) i f a l l its q u a n t i f iers occur i n c o n t e x t SZ < t . . . , v j i < t . . . , where t i s a term o f L not c o n t a i n i n g the v a r i a b l e s t. The h i e r a r c h i e s of En and nn formulas a r e defined as u s u a l . W e denote by IA,, the scheme o f i n d u c t i o n f o r bounded formulas ( w i t h parameters) together w i t h a s u i t a b l e base t h e o r y (see e . g . [ 3 ] ) . N o t e that [Ao is a n, t h o e r y and hence preserved t o i n i t i a l segments ( c l o s e d under.) of models of I&. I f u is a n, s e n t e n c e of L t r u e i n t h e s t a n d a r d model IN, s a y u is vii ~ib*(ii,?) Where u' is A ~ t, h e n T~ denotes t h e t h e o r y In, + Vlay VG<x 3C < y u'(iJ,3). Our aim i n t h i s paper is t o d e f i n e a n a t u r a l map u 5 , t a k i n g t r u e n, Sentences t o t r u e n, Sentences, having the p r o p e r t y t h a t €or any II, s e n t e n c e A, i f g 1 A t h e n To + A is interp r e t a b l e i n To and, p r o v i d i n g o satisfies a c e r t a i n smoothness c o n d i t i o n , conversely. I n t e r p r e t a b i l i t y is being used here i n t h e s t r o n g sense, namely i f S , T are any t h e o r i e s of L and T ~ I At h e~ n we s a y t h a t S is i n t e r p r e t e d i n T by Q, where @ is a formula of L c o n t a i n i n g j u s t one f r e e v a r i a b l e , i f Let
--
,
1.1
T 1
1.2
T b VX.Y ( ( M Y )
1.3
T 1
1.4
T t A@, f o r each sentence A
@(O)
A X
VX,Y ( ( @ ( X ) A
< Y)
MY))
---
@(X)),
-- (O(X+l)A@(X+Y)A@(X.Y))), 6
S where
be r e l a t i v i z i n g a l l q u a n t i f i e r s i n
A
and
A@
d e n o t e s the sentence obtained
to
0.
Thus 'S is i n t e r p r e t e d i n T by @ ' m a n s t h a t i n any model M d e f i n e s an i n i t i a l segment of M, which is also a s u b s t r u c t u r e of s a t i s f y i n g S (and c l e a r l y also I&).
of M.
T
d
.Ws h a l l also i n v e s t i g a t e the c l o s u r e p r o p e r t i e s o f d e f i n a b l e i n i t i a l segments of models of Tu i n t e r n of t h e n a t u r a l 'growth f u n c t i o n ' a s s o c i a t e d w i t h u. Concerning t h i s we should mention the paper [ 2 ] where t h e following r e s u l t w a s proved i n answer t o a q u e s t i o n o f R. so1ovay:P r o p o s i t i o n 1 (Paris-Dimitracopoulos) M be a nonstandard countable model o f Peano arithmetic (PA). Then there is an i n i t i a l segment I of M (closed under . ) such that if J is any i n i t i a l segment of I d e f i n a b l e i n t h e s t r u c t u r e ( M , I ) ( i . e . a unary predicate symbol i n t e r p r e t i n g I is added t o L) t h e n J is not c l o s e d under exponentiation.
Let
I A o + exp is n o t i n t e r p r e t a b l e i n In,, where exp is t h e n, s e n t e n c e Vx,y 9 z z = xy , and ' Z = xy* is the n a t u r a l A . formula d e f i n i n g t h e graph of e x p o n e n t i a t i o n (see [ l ] ) . I t t u r n s o u t , however. t h a t if we t a k e u t o be vx 3 y y = x + 1 then 5 is ( e q u i v a l e n t i n IA,, t o ) exp, so a corollary t o o u r main theorem i s
This r e s u l t shows t h a t
A. J . WI LKlE
330 Theorem 2 F o r any n, s e n t e n c e IA,, + e x p tA.
A,
IA,
+
A
is i n t e r p r e t a b l e i n
IA,
if
and o n l y
if
(Theorem 2 w a s stated w i t h o u t p r o o f in [ 3 ] where it w a s used t o o b t a i n c o n s i s t e n c y r e s u l t s for c e r t a i n s y s t e m s of a r i t h m e t i c . I t 1s a l s o referred t o i n [ 4 ] where similar problems a r e i n v e s t i g a t e d b y p r o o f - t h e o r e t i c methods. The t e c h n i q u e s used i n t h i s p a p e r are m o d e l - t h e o r e t i c . ) O U T main r e s u l t n o t i c e t h a t s i n c e a n, s e n t e n c e is a s s e r t i n g t h a t a c e r t a i n f u n c t i o n ( w i t h A,, g r a p h ) i s t o t a l , t h e problem of d e t e r m i n i n g t h e map CT E is o n e of f i n d i n g an o p e r a t i o n , w i t h s u i t a b l e p r o p e r t i e s , which maps total f u n c t i o n s t o t o t a l f u n c t i o n s . The n e x t s e c t i o n i n v e s t i g a t e s s u c h o p e r a t i o n s .
To r e t u r n t o
--
-2.
UNIFORM FUNLTIONS AND OPERATIONS ON THEM
W e call a function
-+ I N
f:IN
2.1
x < y -+ f ( x ) < € ( y ) ,
2.2
f(0) = 1
u c i f o ~i f i t sa t i sfies for a l l
f(x) >
d
IN,
and
C l e a r l y 2 . 1 and 2 . 2 m p l y , f o r a l l 2.3
x,y
x c: I N ,
X.
now d e f i n e t h e o p e r a t i o n s ( i t e r a t i o n ) and * ( i t e r a t e d i n v e r s e ) on uniform f u n c t i o n s by t h e r e c u r s i o n s : -
We
2.4
(1)
i
2.5
(1)
f*(O)
(0.X)
=
= 1
x ,
(11) i ( Y t 1 . X )
,
(11) f ( y )
< x
= f(i(Y,X)).
< f(Yt1)
-+
f*(X)
=
l+€'(Y).
Then f * : CN -t I N is c l e a r l y non-decreasing ('and w e l l d e f i n e d by 2 . 1 , 2 . 2 and 2 . 3 ) and f : IN2 -+ I N is s t r i c t l y i n c r e a s i n g i n b o t h a r g u m e n t s . W e now d e f i n e €1: I N 2 -+ I N by:2 6
i
fl(y,X) =
(f*(Y),X).
C l e a r l y t h e d i a g o n a l of f l , x F-+ f l ( x , x ) , is uniform so we may repeat t h e c o n s t r u c t i o n on t h i s f u n c t i o n . To make the n o t a t i o n mre c o n c i s e , however, we f i r s t make t h e c o n v e n t i o n t h a t i f h : IN^ -+ IN is any b i n a r y f u n c t i o n t h e n h ( x ) d e n o t e s h ( x , x ) . Thus i f t h e d i a g o n a l o f h s a f i s f i e s 2 . 1 and 2.2 f h e n h and h* a r e w e l l - d e f i n e d - a n d , f o r example, h satisfies h ( y + l , x ) = h(h(y,x)) = h(h(y.x). h ( y , x ) ) . The b i n a r y f u n c t i o n s 2.7
fn+l(Y,X)
fl,f2,
fn ( f ; ( y ) ,
..., f n , ... are
now d e f i n e d by 2 . 6 and for
X).
These f u n c t i o n s h a v e some p l e a s a n t p r o p e r t i e s which are c r u c i a l for t h e s e q u e l and which w e now e s t a b l i s h .
fo 2.8
fA(fn(x))
2.9
€A
2.10 fn
t h e f o l l o w i n g hold f o r a l l
= f
= 1
+
n.x,y,z
fA(x).
( i n ( Y , X ) ) = y + fA(X). (Y+Z.X) = f n ( y , f n ( z , x ) ) ~~
f n ( f n ( y , x ) , x ) = f,(y.f,(x,x)).
2.11 n
1
2.12
(1)
fn+l(x)
2.13
in+l(Y,x)
-$
=
fn(x),
( i i ) f n ( x ) b fntl
i n ((Zy-l)-fk ( x ) , ~ ) .
(x.0).
E
IN:-
n a 1.
On Sentences Interpretable in Systems of Arithmetic
PrOof
2.8 is imnediate from the d e f i n i t i o n of €A and d e f i n i t i o n of f n and 2.8 b y i n d u c t i o n o n y . i n d u c t i o n on y . FOK 2.11 we have
fn(fn(Y,x),x)
=
i n - 1 (€;-1(fn(y,X)),X)
=
f n - 1 (fn-l(fn-l(fi-l(y),X)),X)
*
= in-1 ( f f ; - l ( Y )
+
= in-l(fi-l(Y
in-l( f;-l(x),x))
) r
For
3
follows f r o m the is also clear b y
fi-l(X),X)
(by 2.7).
(by 2.7).
( b y 2.9).
( b y 2.10). as r e q u i r e d .
we h a v e
2.12 (i)
fntl(X) = fn
2.10
(by 2.7).
-
= fn(Y,fn(X,X))
2.9
331
(by 2.7).
(fi(X),X)
in (1,~).
= fn(x).
as r e q u i r e d .
2 . 1 2 ( i i ) we u s e i n d u c t i o n on x. F o r x = 0 the r e s u l t is clear s i n c e both s i d e s are 1. Suppose f n ( y ) 6 x < f n ( y + l ) and t h a t t h e r e s u l t is t r u e for y. Then we h a v e For
f n t l (x,o)
=
2,
=
i n (l+fi(y).O)
( b y d e f i n i t i o n of
f;),
=
fn (in(fA(y),O))
( b y d e f i n i t i o n of
in),
(~:(X),O)
(by 2.7),
=
fn (fntl(Y.0))
(by 2.7),
6
fn(fn(y))
( b y o u r i n d u c t i v e assumption
fn(x) ( b y the s u p p o s i t i o n o n y , x ) , W e establish 2.13 b y i n d u c t i o n o n y . Firstly, 6
fntl(0.X)
= x =
fn(0,X)
=
in ( ( 2 ' - l ) . f ~ ( x ) , x ) ,
Now suppose 2.13 is t r u e
for y .
fntl(Yt1.x) = fntl(fn+l(y,x)) =
( b y d e f i n i t i o n of
fn+l),
( b y d e f i n i t i o n of
fn),
( b y d e f i n i t i o n of
= in(fi(in((2Y-l)'fA(X),X)). =
i n ( ( 2y-1)*€;(
=
X)+fA(
(X).
€,+I),
( b y convention )
*..
= fn (fn(fntl(Y,x)), fntl ( Y ~ x ) )
= in(Zy'fA
as r e q u i r e d .
as r e q u i r e d . Then w e h a v e
i n + l ( Y, x ) 8 inti( Y, x ) )
.
),
( b y 2.7)
f n + l ( y . x ) ) (by i n d u c t i v e hyp.
X), in+l(Y,X) )
( b y 2 . 9 1,
i n ( ( 2 Y - l ) . f ~ ( x ) , x ) ) ( b y i n d u c t i v e hyp.
i n ((2Y+1-1).f~(x),X)
( b y 2.10),
),
),
as r e q u i r e d . 0
W e now d e f i n e , 2.14
Fn(x)
=
To relate t h e d e f i n e d by
2.15
for
n a 1,
Fn's
( i )e,(x) = x.
F,(x)
Fn: IN
-+
IN
by
in ( f * ( x ) , O ) . to
I we i n t r o d u c e t h e f u n c t i o n s en,e;
( i i )e n t l ( x ) =
E !% For a l l n, x E I N , n > 1, 2.17 F n ( f ( x ) ) = f n ( F n ( x ) ) . 2.18
the f u n c t i o n
= i(e;(f*(x)),o).
we have
: IN
--4
IN
332
A.J. WILKIE
~$n+x) =
2.19
write
(we p r e f e r t o
i(e,(f*(x))-l,o).
~ i ~ ) ( xf o) r
~1 ( n . x ) h e r e . ) 2.20 KKZf
or
Fin)(X)
<
Fn+l(x)
for
X 3 2.
we have
2.17
Fn(f(X)) = in(f*(f(X)),O)
( b y 2.14).
= &.,(l+f*(x),O)
(by 2.8),
in),
= f n ( i n ( f * ( x ) , O ) ) (by d e f i n i t i o n of =fn( Fn( x ) )
, as
(by 2.14)
bp
required.
For 2.18 notice t h a t 2 . 1 3 w i t h x = 0 ( r e c a l l f i ( 0 ) = 1) w e have, f o r a l l n , y E IN, €,+I ( y , O ) - = fn(2y-1,0). Hence by i n d u c t i o n o n n, f o r a l l n , y e IN, f n ( y , O ) = f ( e , ( y ) , O ) , so 2.18 follows upon s e t t l n g The case n = 1 is y = f*( x ) h e r e . we prove 2.19 also by i n d u c t i o n on n . clear from 2.13 ( w i t h n=O) and 2 . 1 4 . For n b 1 w e have
~ $ n + l )( x ) = F ~ ( F ~ ( ~ ) ( x ) ) , = kl(f*(Fl(n)(X)),O) =
(by 2.14).
= kl(en(f*(x)),O)
(by 2.9
= i(en+l(f*(x))-l,o) For 2.20 f i r s t
For
= eei+l(x)
e;+z(x)
since
( b y 2.13,
w e note t h a t f o r a l l n , x
e,+l ( x ) > e n ( x ) . then
( b y i n d u c t i v e hyp. ),
kl(f*(i(e,(f*(x))-i,o)),o)
e i ( x ) = 2x-1 >
-
1
>
f"(O)=l),
E IN,
x
21+e~(x)-
2.20 now f o l l o w s from 2 . 1 8 and 2 . 1 9 s i n c e
as r e q u i r e d .
2.15),
x > 2
with
= e,(x)
and l f
1 = 2.en+l(x)
x > 2
w e have e,+l(x)
> e,(x)
-1 > e n + l ( x ) .
implies f * ( x ) > 2 . 0
An example.
2 . 2 1 suppose
x + 1.
f(x) =
Then it is easy t o check t h a t f l ( y , X ) = y + x + 1 (so f l ( x ) 2.19, t h a t f o r
all
n b 1
2.22
F,(x)
= e,
( x + l ) , so
2.23
~ ( l n ) ~ =)
en(x+l)-l.
+
+
f"(x) = x 1, f ( y , x ) = y x and hence t h a t + 1 ) . W e t h e r e f o r e have, u s i n g 2.18 and
= 2x
:-
rl(x) =
2x+1 -1,
and
I t is rather t e d i o u s t o compute f, exactly for n > 2 ( i n t e r n of mre f a m i l l a r f u n c t i o n s ) b u t it can be e a s i l y e s t i m a t e d u s i n g t h e f u n c t i o n s wn d e f i n e d by:2.24
(Thus Then it is easy t o see t h a t wn(y) = e , ( x + l ) f o r a l l x , n , y c IN such t h a t e & x ) < y < e, ( x + l ) and hence ( b y 2 . 1 7 and 2 . 2 2 ) w e have f o r a l l n , x E
IN:-
2.25
Wn(X)
3. THE MAP
<
f n ( X ) < Wn('+(X))-
0
-+ 5
we now r e t u r n t o say u
=
AND INTERPRFPATIONS IN
the
TU
s i t u a t i o n of section 1 and l e t
viB3 u'(a,3)
where
u'
is
u
A ~ ,Define t h e
be a true
n, sentence,
a, formulas
f'(x.y).
On Sentences Interpretable in Systems of Arithmetic
f(X.Y)
by:-
3.1
f'(X,y)
3.2
f(X,y)
-
Vs 3
<
3q < y
X
< y (f'(X,z)
2
(5.9)
0'
y =
A
A VW
< y
+
1t
Xz
- Vs
<
*<
X
W
U'(s,q),
2 ) .
C l e a r l y Tu 1 Vx 31 y f ( x , y ) and it is harmless t o write f ( x ) = y f o r f ( x , y ) and t o regard f as a term o f L when working i n To ( s i n c e i n To one can prove i n d u c t i o n for bounded formulas even if f is allowed t o o c c u r i n t h e terms bounding q u a n t i f i e r s ) . Notice also t h a t 2 . 1 and 2 . 2 (and 2 . 3 ) are provable from indeed w e have :T,,, 3.3
T,, 1 Vx ( f ( x )
> xz
t 1).
W e now want t o r e p r e s e n t i and f * by A. formulas i n Tu. T h i s can be done q u i t e n a t u r a l l y u s i n g t h e fact t h a t IA, admits A,-definable f u n c t i o n s ( x ) ~ ( = " t h e ( y + l ) s t nwnber o f t h e sequence coded by and 1x1 ( - " t h e l e n g t h of t h e sequence coded by x") s a t i s f y i n g for some s u i t a b l e polynomial p, :3.4
I A .
3.5
I A,,
V
U
3 X <
VX,W,U
A
(X')w = U
(1x1
p(U)
(1x1 =
1
A
W
s
1 A (X),
= U),
+ 3x' < x . p ( u )
Vy <
W
((X')y
(
Ix'I
and = w+1 A
(X)y)).
of t h e r a p i d growth of f given by 3.3 (which, t o g e t h e r with 3 . 4 and 3.5, i m p l i e s a y l y n o m i a l bound, s a y p ' ( x , y , z ) on t h e code f o r t h e sequence x,f(l,x),f(2,x) f ( y , x ) = z ) i t is e a s y t o show that if we d e f i n e ' f ( x , y ) = z ' by t h e A,, formula:-
NOW because
,...,
t h e n , i n any model M of Tu f d e f i n e s a partial f u n c t i o n w i t h t h e p r o p e r t i e s t h a t for a l l x B M t h e set Ix = ( y t M: MFZIz z = f ( y L x ) ) is an i n i t i a l f ( y , x ) is total and segment of M c l o s e d under +1 and t h e f u n c t i o n y i n c r e a s i n g on I, w i t h range c o f i n a l i n M. F u r t h e r , 2 . 4 i s s a t i s f i e d f o r a l l x 6 M and y t I,.
--
W e may s i m i l a r l y d e f i n e f* (which w i l l i n fact be provably t o t a l i n To), t h e f n ' s and t h e F n ' s (which w i l l be p a r t i a l ) by n a t u r a l 4 formulas and because
o n l y simple i n d u c t i o n s were used, it is clear t h a t a l l the i d e n t i t i e s and i n e q u a l i t i e s e s t a b l i s h e d i n s e c t i o n 2 for I N w i l l hold any model o f To whenever t h e f u n c t i o n s under c o n s i d e r a t i o n are d e f i n e d . I n f a c t i f one s i d e of such an i d e n t i t y (or t h e " g r e a t e r s i d e " of an i n e q u a l i t y ) is d e f i n e d then so is t h e other and t h e i d e n t i t y (or i n e q u a l i t y ) h o l d s . (For example, f o r 2.11 one can prove, f o r each n t I N w i t h n > 1, i n T,, t h e sentence V
X,y,U,V ((fn(Y,X)=U
A
fn(U,X)=V)
-+
32
(Z=fn(X,X)
A V
= fn(Y,Z))).)
We can now i n t r o d u c e o u r main d e f i n i t i o n s r 3.7
For each n t I N , (Recall f, = f.)
3.8
8
denotes t h e
Theorem 5 . For each n
d
IN,
n, un
un d e n o t e s t h e sentence
n,
Vfly y
-
sentence
Vlay y
-
fn(x).
Pl(x).
is i n t e r p r e t a b l e i n To by a nn+l
formula.
333
A.J. WI LK IE
334 As it s t a n d s
is nn+3, b u t it is c l e a r l y e q u i v a l e n t i n T, to @n(X) A VY ( @ n ( Y )-+ (32 ( Z = f n + l ( Y , X ) ) A VZ (Z=fn+l(Y,X) -+ @ n ( Z ) ) ) ) , which c a n o b v i o u s l y be p u t i n t o nn+z form.
n o t e first t h a t we h a v e @,,(O) ( b y 1.1 for en). t h r o u g h o u t t h i s p r o o f , ) Suppose h ( y ) . Then b y 1 . 4 for On we may choose z s u c h that h ( Z ) A 2 = f n ( Y ) . But b y 2 . 1 2 ( i i ) there is z ' ( z such t h a t z ' = f n + l ( y . O ) and, f u r t h e r , On(z' ) h o l d s ( b y 1 . 2 for h) as r e q u i r e d .
TO
show 1.1 holds f o r
(we a r e working i n
T,
Now 1 . 2 for follows e a s i l y from t h e c o r r e s p o n d i n g p r o p e r t y of On and t h e fact that fn+l is i n c r e a s i n g i n its second argument, and 1 . 3 c l e a r l y follows froin 1.2 and 1 . 4 ( s i n c e f n ( x ) > f,(x) > x2+1, b y z . l z ( i ) ) . Hence it r e m a i n s t o show that i f @ n + l ( ~ ) t, h e n f o r some z , z = f n + l ( x ) A o n + 1 ( z ) . ( N o t e t h a t the formula ' 2 = f n + L ( x ) ' is & so t h a t we may d r o p t h e r e l a t i v i z a t i o n . ) Now an+1(x) i m p l i e s @,(x) and h e n c e ( u s i n g 3.9 w i t h 'y-' ) we a t least h a v e a z such that z = f n + l ( x , x ) A o n ( z ) . Now suppose @,,(y). Then ( b y 3.9) for some z o , zo = f n + l ( y , x ) A h ( z o ) . NOW u s i n g 3.9 a g a i n ( w i t h 'y=z,') t h e r e is a 21 s u c h t h a t 21 = f n + l ( z o , x ) A O n ( z l ) . But b y 2 . 1 1 fn+1(Zo,X) = f n + l ( f n + l ( Y , X ) t X )
f n + l ( y , f n + l ( x , x ) ) = f n + l ( y , z ) . Thus w e h a v e shown t h a t O n ( z ) A VY (rn,(y) 92, ( z , = f n +(lY , z ) A on(z,)) , i . e . z = f n + l ( x , x ) , as r e q u i r e d . =
--
where
h+l(z).
0
For 3.10
and
MkTu -1
F,
(M)
n =
E
(a
IN, n E M:
1,
~
w e now d e f i n e : -
3 y y= ~ ~ ( a ) ) .
The f o l l o w i n g lemM f o l l o w s from 2.17 and we l e a v e t h e e a s y details t o the
reader. Lemna 6 ... suppose M ~ T , . Then for a l l n E IN, n > 1, F;'(M) is a n i n i t i a l segment of M ( p o s s i b l y w i t h a g r e a t e s t e l e m e n t ) . F u r t h e r , Mbun ( i . e . f n is t o t a l i n M ) if and o n l y i f P;'(M)kT, ( i . e . f i s t o t a l i n F;'(M)). 0
we a r e now r e a d y t o p r o v e t h e main r e s u l t of t h i s s e c t i o n .
Th%.r_en! 2
L e t A be a i n To.
n,
sentence such t h a t
1
A.
Then
Tu
+
A
EE?.?.
suppose A = VP B (%), where B ( P ) is Ao, and t h a t e a s y compactness argument ( a n d 3 . 8 ) i t f o l l o w s t h a t : 3.11
I A,
1 vw,y
( ~ 1 " ()w )
= y
--$
w f < w B (2)).
for some
is i n t e r p r e t a b l e
Now by a n
1 A. n
E
IN.
Define ~ ( w 0 ) 3 y ( F ~ + , ( w ) = y A @ n + t ( y ) ) , where is t h e formula ( g i v e n by theorem 5 ) i n t e r p r e t i n g an+l i n Tu. L e t M be any model o f T, and let J = (a 6 M : Mp$(a)) and I = ( a 6 M : M b t ~ ~ + ~ ( a ) )Then . I is a n i n i t i a l segment of M s a t i s f y i n g un+l and h e n c e c l e a r l y IkTu. F u r t h e r J = F i t l ( I ) and so by lemma 6 ( w i t h I4 = I ), J is a n i w t i a l segment of I ( a n d hence of M ) s u c h t h a t J&. we are g o i n g t o show t h a t JCA, so l e t b c J . W e must show JFVP < b B ( f ) for which i t s u f f i c e s t o show t h a t ItVa < b B ( f ) . Now working i n I we h a v e f o r some c E I , c = Fn+,(b), and hence b y 2 . 2 0 , f o r some c ' E I , C ' = (b). S i n c e c e r t a i n l y I R A o , t h e r e q u i r e d c o n c l u s i o n f o l l o w s from 3.11. Thus w e h a v e shown t h a t Ta t A is i n t e r p r e t e d i n T, b y W.
Fin)
0
The a i m of t h e rest of t h i s paper is t o p r o v e a c o n v e r s e of theorem 7 and to show t h a t theorem 5 c a n n o t be s i g n i f i c a n t l y unproved, for which we r e q u i r e a t h e o r y of games p l a y e d i n models o f I A o t exp.
On Sentences Interpretable in Systems of Arithmetic
335
4 . CONSTRUCCING I N I T I A L SEGHEN.PS BY GAMES
we denote by LT t h e r e l a t i o n a l language of arithmetic ( i . e . LI is the same as L except t h a t +, . a r e t e r n a r y r e l a t i o n symbols) and by L r ( J ) t h e language o b t a i n e d by adding t o Lr the unary r e l a t i o n symbol J . W e f i x f o r t h e rest o f t h i s s e c t i o n a nonstandard c o u n t a b l e model, M s a y , of I A. + exp and a nonstandard B E M, and w e denote by L ~ ( J , B ) the language obtained by adding t o L r ( J ) a c o n s t a n t symbol for each d E M,d < B. W e assume t h a t Lr(J,B) is m e 1 numbered i n M i n some s t a n d a r d way. The classes 3,. V n (for n E I N ) of L r ( J ) formulas a r e d e f i n e d as follows:-
a
4.1 So = V, = t h e class of q u a n t i f i e r - f r e e formulas of L r ( J ) t h a t begin w i t h at mst one negation symbol. (This i n e s s e n t i a l r e s t r i c t i o n is f o r t e c h n i c a l reasons explained b e l o w . ) 4.2
3ntl
4.3
Vn+L = {QZO
= {X%0
: 0 E
Vn).
0
3n).
:
E
For s E M, 3,(s) ( v n ( s ) ) d e n o t e s t h e (M-coded) set of a l l 3, ( v , ) formulas having W e 1 number < 8 . Thus i f 8 is nonstandard S n ( s ) c e r t a i n l y c o n t a i n s a l l s t a n d a r d ?In formulas. We a l s o w r i t e a n ( s , B ) f o r t h e (M-coded) set of a l l formulas o f L r ( J , B ) o b t a i n e d by s u b s t i t u t i n g c o n s t a n t s o f L r ( J , B ) f o r (some o f ) the v a r i a b l e s of an 3n(s)formula. S i m i l a r l y f o r V n ( s , B ) . If 0 E 3 n ( s , B ) u V n ( s , B ) , 0* d e n o t e s t h e prenex negation of 0 obtained by i n t e r changing V ' s and 3's i n 0 and removing t h e i n i t i a l negation symbol frcm t h e q u a n t i f i e r - f r e e m a t r i x of 0 if t h e r e is one, or adding one i f t h e r e is n o t . C l e a r l y 4.1 e n s u r e s t h a t 0** = 0. L e t u s suppose now t h a t nonstandard
are given s a t i s f y i n g t h e following:4.4
a < b
4.5
q
and
for all n
en(b)< B
a,b,t
d
E IN,
M
and an
M-coded f u n c t i o n
g
and
i s s t r i c t l y i n c r e a s i n g on its domain, which i n c l u d e s a l l x E M w i t h x 6 b, & x , a ) is d e f i n e d f o r a l l x E M w i t h x 6 Z t , and & Z t , a ) = b .
W e are now ready t o describe the game r ( n , s , a ) , where n E IN and s, a (and we s h a l l always assume that 8 k m, where m E I N i s chosen so that 3 J m . B ) Z 0 ) , which i s "played i n M by two p l a y e r s , I and 11. Let H be t h e set of s e n t e n c e s i n 3n( s , B ) and suppose t h a t f o r some i < a the
E
M
p l a y e r s have c o n s t r u c t e d a (M-coded) sequence, O L , ~ L , . . . , O z i - l , O L l , of s e n t e n c e s from H, called a play of l e n g t h i. The game i s continued according t o one of t h e following t h r e e r u l e s : 4.6
I
chooses any
@zi+z = *:i+l. 4.7 I chooses some
E
j
< i,
H
where
and $I does not begin w i t h an choose c < B and set O z i + z 4.8
I
and
V
@,itl
chooses some
j
and
3, = V
11
a,
must set e i t h e r
or
=
is n o n - t r i v i a l l y o f t h e form 32 V ( 2 )
and sets
(E).
O,i+z
@,itl= 0,j.
Player
I1
< i, where e2j is n o n - t r i v i a l l y of t h e form
does n o t begin with an V , and some F < B . = 0 (t) and 11 must set 02i+z = V ( E ) .
PLayer
I
must then V k ; )
then sets
The game i s f i n i s h e d after
a such moves when a p l a y of l e n g t h a , s a y Ol,Oz,..., Ora-i,Ora w i l l have been c o n s t r u c t e d . W e t h e n d e c l a r e p l a y e r I t h e winner i f a t least one of t h e following f i v e c o n d i t i o n s hold ( I 1 wins o t h e r wise):-
there is no t r u t h assignment, v, t o t h e s e n t e n c e s J ( E ) ( f o r c < B ) such that v ( G Z i ) = True for a l l i < a w i t h 0,i E 3 , ( s , B ) , where, f o r V an atomic sentence of H , V ( V ) = True i f and o n l y i f ( M , C ) C < ~ w ,or 4.9
A.J. WILKIE
336
4.11
o,, ozl
4.12
Q,,
4.13
for
4.10
-
is
o,:
is
~ ( 9 )f o r some
IS
J(Q)
f o r some
i
C,d < B for some
with i,j
<
<
a,
OL
6 a,
01
< a, or c < g ( d ) , Ozl
€of Some
s010(3
J(C)
i,]
1
a,
is
J(4)
and
a2j
16
it is e a s y t o show t h a t t h e r e a r e A, formulas ( i n v o l v i n g some s u i t a b l y l a r g e parameter, s a y e,,(B)) d e f i n i n g the sets of p l a y s , winning p l a y s for I and winning p l a y s f o r 11, and t h e s e sets w i l l hence be M-coded. F u r t h e r , s t r a t e g i e s ( f o r I o r 11) are slmply M-coded f u n c t i o n s (mapping p l a y s --+ H) having t h e obvious ( A , ) p r o p e r t i e s , and because t h e game is M-finite, an easy induction i n M shows t h a t e i t h e r p l a y e r I or p l a y e r I1 has a winning s t r a t e g y . We, of course, support p l a y e r I1 so o u r next am is t o f i n d a l a r g e a (as a f u n c t i o n of n and 8 ) f o r which I1 has a winning s t r a t e g y for r(n.s.a). NOW
cemna
8.
Player I1 has a winning strategy for t h e game
r ( 0 , s . a )providing
< [t/s],
a
?Lp_f. Let H be t h e s e t of s e n t e n c e s i n 3 , ( s , B ) . Observe t h a t i f o c H t h e n c l e a r l y (assuming a reasonable W e 1 numbering) there i s a n M - c o d e d set c ( o ) = (c,,..,cs), where c i < c = + <~ B for i = l,..,s-l, s u c h t h a t t h e t r u t h value of 0 is n a t u r a l l y determined i n M whenever t r u t h v a l u e s for J(G,),..,J(~~) a r e given.
s is a s t r a t e g y for I f o r r ( o , s , a ) . I t is s u f f i c i e n t t o show how 11 can d e f e a t t h i s s t r a t e g y , so suppose t h a t @ , , Q z , . . . , ~ , i - l , @ t l is a p l a y of l e n g t h i ( < a ) of r ( o , s , a ) i n which I h a s been using S. we suppose f u r t h e r that t h e r e a r e a i , b i E M s u c h t h a t t h e following t h r e e (A,) conditions hold:-
NOW suppose t h a t
4.14 4.15
4.16
ozj
a < &2t-is , a i ) < b i < b. i U c ( o 2 , ) n ( a i , b i ) = @.
j=1
If J ( E ) is given t h e v a l u e t r u e f o r g e t s the value t r u e for each j 6 1.
c4ai
and false for
c k b i , then
Notice that by 4.5 these c o n d i t i o n s are s a t i s f i e d for i=O i f w e d e f i n e a, = a and b, = b. Suppose t h a t S now dictates t h a t I p l a y s t h e sentence E fl. ( o n l y r u l e 4.6 is r e l e v a n t h e r e . ) s a y c(oZi+,) = ( c l , . .,cs) i n i n c r e a s i n g o r d e r and let u s suppose t h a t a i 4 cl < cs 4 b i ( o t h e r w i s e choose a convex subsequence of c l , ...,cs maximal w i t h t h i s p r o p e r t y ) and d e f i n e c, = a i and c t 1 = b i . I now claim t h a t f o r some j < s w e have 6(2t-(8+1)s, c,) < c,+,. For suppose n o t . Then for a l l j 6 s we would have
.
; j ( ~ t - ( i + l ) s,+c,) j > cjtl.
POK
t h i s is clear f o r
j = 0, and i f t r u e
for
some j < s, t h e n c ~ < +$(zt-(i+1)s, ~ c ~ +< ~: ( z)t - ( i + r ) s , g(pt-(i+l)s+j,,,)) < 3 ( z t - ( i + l ) s + j + l , c,) ( c f . 2 . 1 0 ) , so is t r u e f o r j + l . But now ( s e t t i n g j = s )
we have c o n t r a d i c t e d 4.14,
and my claim is proved.
Now d e f i n e j, to be t h e least j < s such that < ( 2 t - ( i + l ) s , c,)< c,+, and set = cj,, and b i t , = cj,+,. C l e a r l y 4.14 and 4.15 hold ( f o r i = i + i ) , and and 11 may choose + z r + r e i t h e r @ri+r or @:i+l aa d i c t a t e d by 4.16 ( f o r i = i + 1 ) .
n o t i c e t h a t if a < [:I, t h e n 2 t - a ~ 3 2 so that c e r t a i n l y ba > g ( ; h ) ( b y 4.14 w i t h i = a ) . Thus t h i s c o n s t r u c t i o n can be continued f o r a steps and c l e a r l y t h e r e s u l t i n g p l a y o f l e n g t h a s a t i s f i e s none of 4.9-4.13, So I1
NOW
has defeated
s as r e q u i r e d .
0
9 -
Suppose
n
E
IN, s, a
Q
M\(O), and t h a t p l a y e r
I1
h a s a wining s t r a t e g y f o r
On Sentences Interpretable in Systems of Arithmetic
337
t h e game r ( n , s , a ) . Then p l a y e r I1 h a s a winning s t r a t e g y for t h e game r ( n + l , s , O ) f o r any D 6 H w i t h D < [ l o g , a ] . m t
.
be the set o f s e n t e n c e s i n a n + , ( s , B ) , and l e t U be a winning s t r a t e g y f o r I1 f o r r ( n , s , a ) . W e f i r s t i n t r o d u c e some n o t a t i o n c o n c e r n i n g t h i s game. If i < a, t h e n P ( i ) d e n o t e s the ( H - c o d e d ) s e t o f p l a y s of l e n g t h 1 o f P ( i ) then E(p) r ( n , s , a ) i n which 11 u s e s t h e s t r a t e g y U and i f p c is Phe s e t o f s e n t e n c e s d e n o t e s t h e set of moves made by I1 i n p , i . e . E ( p ) o c c u r i n g w i t h e v e n subscript i n p . I f p c P ( i ) , p' 6 P ( ] ) and 1 < 7 t h e n w e w r i t e p < p' i f t h e s e q u e n c e p is an i n i t i a l segment o f t h e sequence p ' . Let
ti
NOW suppose t h a t
i s a n y s t r a t e g y f o r I f o r r ( n t + , s , D ) . As i n lemna 8 I1 c a n defeat S , so suppose that +l,+z,...,@~l-i, i s a p l a y of l e n g t h i ( < D ) o f t h e game r ( n + l , s , p ) i n which I has been u s i n g S . W e suppose f u r t h e r t h a t t h e r e is some k, c M and pi 6 P(k,) s u c h t h a t t h e f o l l o w i n g f o u r (A,) c o n d i t i o n s h o l d : S
i t i s s u f f i c i e n t t o show how
k, < a For a l l
4.19
If
4.20
w(?)
(k+t+. .+
.
4.17 4.18
,
< i,
3
+).
i f Qz3
then
3, ( s , B )
6
OZ3
6
E(pi).
3 6 i and Qz1 is n o n - t r i v i a l l y o f t h e formdoes n o t b e g i n w i t h a n 3) t h e n for some c
I f - ] < i and e2] is n o n - t r i v i a l l y of t h e form V ( x ) does n o t b e g i n w i t h a n V ) and C < B, t h e , and p c P ( k i + h ) w i t h pi < p , with h 6% 6 E(P).
Now suppose t h a t S d i c t a t e s t h a t I now p l a y s +,,+2 so t h a t 4 . 1 7 - 4 . 2 0 are preserved.
@ z l + l .W e must d e t e r m i n e
I is i n v o k i n g r u l e 4 . 6 . @,it16 H and I1 must set e i t h e r @,i+2 = @,i+lor Ozi+2 = +,t+l. NOW let u s suppose f o r t h e m n t t h a t a ~ , ~ + +f 3 ( s , ~so ) t h a t we may where V ( z ) - 6 3,(s,B) c l e a r l y assume t h a t Q,i++ i s o f t h e form 32 V($) ( ~ ( x ) n o t b e g i n n i n g w i t h a n 3 ) and where q u a n t i f i c a t i o n ' 3 x ' is n o t c < B and p c P ( k i + h l ) w i t h v a c u o u s , Now i f there_ is some h i < p i < p s u c h t h a t w(r) c E ( p ) , t h e n we set k i + + = k i + h i , p i t i = P and +,it, = @,i++. C l e a r l y 4 . 1 7 , 4 . 1 8 and 4 . 1 9 are preserve. To see t h a t 4 . 2 0 is too, suppose j < i + 1, a ,, is of t h e form VxV'(x) ( n o n - t r i v i a l l y ) ,
Then
%,
h
<%++, p'
Then c l e a r l y
and
P (ki+++ h ) , pi++ < p'
c
j
<
i
and, s e t t i n g
h'
the
=
hi
V'(5)'
+
h,
6
E (0')
w e have
and p i < p ' , which v i o l a t e s 4 . 2 0 . If t h e r e are no h i , p r o p e r t i e s t h e n we may c l e a r l y set k i + l = k i p P i + i = P i NOW for
t h e case t h a t @,l+l we h a v e : -
i < D < [log2a] 4.21
4 6%
and hence
6
h' <
c, p and
>,
f o r Some p'
2
< B.
c P(ki t h ' )
w i t h t h e above @ r i + r = @:i+r.
let us f i r s t observe t h a t since
3,(s,B),
k, < a-4.
Thus f o r some @,i+, 6 (@,i++, @:i+%), t h e r e is E E ( l + k i ) such t h a t p i + L = < p i , eZi++,a,=+, > and w e h e n c e set k i + + = 1 + k i . The f a c t that 4.17-4.20 are p r e s e r v e d now f o l l o w s e a s i l y f r o m 4 . 2 1 .
- -
case
2. I is i n v o k i n g r u l e 4 . 7 . Then €or some j < i, @+i+L = +,j = ( w n o t b e g i n n i n g w i t h a n 3). where '3x' is non-vacuous. By 4 . 1 9 , w ( G ) 6 E ( p i ) f o r gome c < B and = UI ('Z) so t h a t 4 . 1 7 hence we may set k i t l = k i , p i + + = p i and @,itz 4.;9 a r e - t r i v i a l l y preserved and t h e o n l y way 4 . 2 0 might n o t be is if w(c) = Vy w' ( S , y ) ( V ' n o t b e g i n n i n g w i t h a n V ) and for some d < B,
h <
;&
ki + h <
+
,
p E
a
so, b y 4 . 8 ,
P(ki
h)
with
pi 6
we had
p ();,>
v*(<,s)*
c E (p).
is a p l a y of l e n g t h
However, k i + h +I
A.J. WI LKI E
338 (Ca)
of
for
I
g
e
s
r ( n , s , a ) i n which r I h a s o b v i o u s l y used ( b y 4.10) - a c o n t r a d i c t i o n I
is i n v o k i n g r u l e
S b u t which is w i n n i n g
4.8.
Then f o r some- 3 < i, OZj i s V&(;) ( u n o t b e g i n n i n g w i t h a n v ) and is u(_c) for some < B. we are f o r c e d t o set oZit, = u(c). NOW suppose I invoXes_ r u l e 4.6 i n t h e game r ( n , s , a ) - f o l l o w i n g t h e p l a y p i t o form
a
33
V(e) = u' (;,GI ( n o n - t r A v i a l l y , 3 ) and we h a v e , f o r s m d < B ,
where
u*(E,G)
does n o t b e g i n w i t h a n
< p i , V ( ~ ) , u ( ~ ) , u ( ~ ) , u ' (6 ~P,(~2 t)k>i ) . ( P l a y e r I has invoked r u l e 4 . 7 h e r e and I1 h a s used U, which i s p e r m i s s i b l e b y 4 . 2 1 . ) W e now l e t p i t l be t h i s p l a y and set k i t l = 2 t k i . Now b y 4 . 2 1 , 4.17 i s p r e s e r v e d . A l s o o u r c o n s t r u c t i o n g u a r a n t e e s t h a t 4.18 and 4.19 are p r e s e r v e d and t h e same h o l d s f o r 4 . 2 0 s i n c e i f j c is1 and eZj is o f the form s t a t e d i n 4 . 2 0 t h e n n e c e s s a r i l y 3 6 1. Further, p c
h Cz<+l
P(kith'1
where
and
p E
h'
= 2 t
P(kl+lth) h
and
C 2 t%+.
we h a v e now shown how t o c o n s t r u c t a p l a y
pitl
<
4
<:
<
p
c l e a r l y imply pi <
and
p
(by 4.21)
= @l,oz,. . . , ~ ~ ~ ~ , 4 ,of , - ,t h e game r ( n t 1 , s . p ) i n which I u s e s S , and a p l a y pp of r ( n , s , a ) i n which I1 u s e s U, s a t i s f y i n g 4'17-4.20 ( f o r i = D ) . S i n c e p~ is a win f o r 11 i t c l e a r l y f o l l o w s that 4 is a win €or 11, so I1 has d e f e a t e d S as r e q u i r e d .
0
lo
C-OLOLl*.KY
suppose n c I N , s, a 6 M, a n o n s t a n d a r d , and e, ( a ) I1 h a s a winning s t r a t e g y for t h e game r ( n , s , a-2).
C
[ -tI .
Then player
Proof n . For n = 0 the r e s u l t follows from lenuna 8 . s u p p o s e S i n c e S-tl ( a ) =%- ( Z a - l ) , p l a y e r I1 h a s a winning s t r a t e g y f o r t h e game r ( n , s , Za-3) ( b y t h e i n d u c t i v e h y p o t h e s i s ) . SO b y lemna 9, 11 h a s a winning s t r a t e g y for r ( n t 1 , s . p ) where P = [1og,(Za-3)]-1 = a-2 ( s i n c e a is n o n s t a n d a r d ) , as r e q u i r e d . By i n d u c t i o n on
eG,L (a) < &.
0
10 t o c o n s t r u c t i n i t i a l segments of M which are c l o s e d under g and i n which d e f i n a b l e i n i t i a l segments are q u i t e l o n g . we first d e f i n e ( B , K ) where K i s a n i n i t i a l segment of &n lo t c o n t a i n i n g B, to be t h e L r ( J ) - S t N C t U r e w i t h d-in (c B M: n\.c < B) where L r - r e l a t i o n s are i n h e r e t e d from M and J i s i n t e r p r e t e d a s K.
W e now u s e c o r o l l a r y
Theorem 11 Suppose n e I N \ ( 0 ) and i n i t i a l segment K of M
p E M U N satisfy such t h a t : -
i s closed u n d e r the f u n c t i o n
g (and
e, ( 0 )< t . g
Then there i s a n
is t o t a l on K ) , and
4.22
K
4.23
a
4.24
i f 4 ( x ) i s any ( s t a n d a r d ) 3, o r Vn f o r m u l a of L r ( J ) ( p o s s i b l y i n v o l v i n g parameters f r o m ( B , K ) ) s u c h t h a t ( B , K ) H ( O ) and P (B,KbVX ( O ( X ) ( X t l ) ) , then ( B , K ) k @ ( 2 ' In') for some m t I N \ ( O )
6
K, b f K,
and
--*
LrEf s i n c e p n + 1 we can clearly f i n d a n o n s t a n d a r d s e M s u c h t h a t e, ( [ / r ] + 2 ) < [t/s]. By c o r o l l a r y 10, p l a y e r I1 h a s a w i n n i n g s t r a t e g y , u say, for the game r ( n , s , a ) where a = [ p / z l . Let el,@*, .,%. ( n c IN)
..
..
On Sentences Interpretable in Systems of Arithmetic
339
be an i n f i n i t e l y r e p e t i t i v e ennumeration of 3 , ( s . B ) (recall t h a t M is c o u n t a b l e ) . W e c o n s t r u c t p l a y s po 6 pI <...C pr C... of r(n,s,a) s a t i s f y i n g for each r c 1N:t h e r e is
4.25
nr
6 IN \
(0)
such that
pr
6 P(ir)
1 ir C a(l---),
for some
nr
where we are using here t h e same n o t a t i o n as i n the proof o f lemna 9 ( i n p a r t i c u l a r , P ( i ) d e n o t e s the set of p l a y s of r ( n , s , a ) i n which I1 u s e s U). suppose pr has been c o n s t r u c t e d for some r 6 I N ( p o i s the empty p l a y ) . case.&
r t 1 = 3q. is a s e n t e n c e (set = pr, nr+, = nr if n o t ) t h e n f o r some "4 6 (4q,e$) w e have
eq
I f eq i s a sentence ( n o n - t r i v i a l l y ) of t h e form a&(;) ( 0 not beginning-with an 3), and -qtq 6 E ( p r ) ( s e t pr++ = pr, n r t I = nr i f n o t ) t h e n for some c < B, <pr, eq, w ( ~ ) >6 P ( l + i r ) and w e l e t pr+I be t h i s p l a y and set nr+, = 1 + nr.
r
case_l
+
1 = 3q t 2 .
Suppose qtq = % ( x ) c o n t a i n s j u s t one free v a r i a b l e and t h a t e i t h e r ( i ) both % ( g ) and $(4)* a r e i n E(pr) or ( i i )b o t h %(Q)* and %(g) are i n E(pr),
where 6 = 2' / r n e l ( s e t pr+I = pr, nr+, = nr i f n o t ) . I f ( i )h o l d s we c o n s i d e r t h e p l a y p r t I = <pr,xr,x;, ,xy,x;> of r ( n . 6 . a ) . where y = [&I, d e f i n e d
.. .
as follows. Each x i is of t h e form % ( g i ) for some c i < 6 and I i s invoking r u l e 4.6, and x: is 11's response as d l c t a t e d by U. F u r t h e r , c I is 2 Y - I and for i = r , . ? . , ~ , c i is ci-s t 2y-l if xi-I = and c i is ci-l
-
2Y-i
xi--L= X Z - ~ .
if
Then c l e a r l y p r t r hard t o see that:4.26
f o r some
(and
ci <
6 6
6 P(
<
i.j
i r t I )where
xi
Y,
2Cp/5nrI
is
irtI = i,
+
qtq(si) and
xi
is
)
%(sj)*
and and
i t is not c j = ci+l
).
A l s o , if ( i i )above h o l d s we could o b t a i n pr+l would hold with t h e a s t e r i s k interchanged.
T h i s completes t h e c o n s t r u c t i o n of
=rcyN
y 6 a (1-
pr+l.
.
K = (c < B : J(c)
6 T) Since no pr satisfies either 4 . 9 or 4.10 it is e a s y t o check ( u s i n g cases 1 and 2 of she c o n s t r u c t i o n ) t h a t for-any ( s t a n d a r d ) formula $ ( x ) 6 8, u W n and c < B , we have ( B , K ) ~ ( c ) i f and o n l y i f @ ( c )6 T . S i n c e no or satisfies any of 4.11-4.13, t h i s i m p l i e s 4 . 2 2 and 4 . 2 3 . F u r t h e r , 4.24 is c l e a r l y now implied by 4.26 ( f o r a s u i t a b l e q ) .
Let
T
E (pr)
and d e f i n e
i n a similar way and 4.26
0
5-EZW!S ON.. N O N : I ~ ~ W ~ Z L I T y Our aim i n t h i s f i n a l s e c t i o n is t o u s e theorem 11 t o show that theorems 5 and 7 are n e a r l y best p o s s i b l e . Let u s f i r s t f i x a nonstandard model, M say, o f are t o t a l Fl,Fz,. T,- t exp, so t h a t c l e a r l y a l l t h e f u n c t i o n s f * , f , f I , . i n M and s a t i s f y everywhere a l l t h e p r o p e r t i e s e s t a b l i s h e d i n s e c t i o n 2 . Let n 6 IN \ ( 0 ) and a E n \ IN. we d e f i n e t , b , g . and D as follows:-
..
5.1
t
=
€*(a).
b = F(2t.a). is t o t a l i n n.)
5.2
..
( i t is e a s y t o show t h a t t h e f u n c t i o n
f f ( x c M: Ht=x C b ) .
5.3
g =
5.4
e, ( 0 )< t C e; ( p + r ) .
XI-+ ? ( 2 f w ( x ) , x )
A. J . WI LKIE
340
W e may f u r t h e r suppose t h a t there is some B E M s u c h t h a t e,(b) < B for f o r a l l m E I N (by taking an elementary extension o f M i f necessary).
Now 4 . 4 , 4 . 5 and t h e h y p o t h e s e s o f theorem 11 are s a t i s f i e d so t h e r e i s a n i n i t i a l segment K o f M s a t i s f y i n g 4.22-4.24 and i n p a r t i c u l a r K B u . Now u s i n g a r e s u l t o f [l] (namely t h a t To f o r m u l a s o f L c a n be w r i t t e n ( p r o v a b l y i n IA,texp) i n t h e form 3p < e- ( x ) W , w h e r e W i s q u a n t i f i e r - f r e e x :Z), o f and m E I N ) i t i s e a s y t o show t h a t formany Cn ( o r n n ) formula, L t h e r e is a n 3, ( o r V n ) f o r m u l a , x * ( x ) , of L r ( J ) p o s s l b l y i n v o l v i n g parameters f r o m ( B , K ) ) such t h a t : -
5.5
for all
e
< B, (B,K)kx*(?)
i f and o n l y l f e E K
and
Kbx(B).
W e are now r e a d y t o p r o v e : -
msoE%!!Ls For n > 2, un+,
is n o t i n t e r p r e t a b l e i n
P_roof suppose @ ( x ) is a Define the f o r m u l a s
Cn W(x)
or
nn
and
5.6
W(X)
U
32 ( 2 =
Pn+,(X)
5.7
T)(X)
CJ
32 ( 2 =
F(X.0)
A
by a
T , ,
f o r m u l a of
n ( x ) by:-
L
En
nn
or a
interpreting
un+l
formula. in
Tu.
O(2)).
A W(2)).
Notice t h a t i f $ ( x ) is En t h e n so are W(X) and q ( x ) . Also b y a n argument smilar t o the one used i n t h e proof o f t h e o r e m 5 ( c f . 3 . 9 ) . i f O ( x ) is nn t h e n so are W(x) and n ( x ) ( s i n c e n 2 ) .
+
C o n s i d e r the f o r m u l a s @ * ( x ) , W*(x) and q * ( x ) ( C f . 5 . 5 ) . By o u r s u p p o s i t i o n ( w e a l s o u s e 5 . 5 w i t h o u t mention h e r e ) & ( x ) d e f i n e s i n ( B , K ) an i n i t i a l Hence b y 5 . 6 . 3 . 1 0 and lema segment o f K c l o s e d u n d e r the f u n c t i o n f n t 1 . 6 , w * ( x ) d e f i n e s i n ( B , K ) an i n i t i a l segment o f K c l o s e d under f . I t now c l e a r l y f o l l o w s from 5 . 7 t h a t q * ( x ) d e f i n e s i n ( B , K ) a n i n i t i a l segment o f K ( c o n t a i n i n g 0 a n d ) c l o s e d under + 1. Hence by 4.24 there i s m 6 I N \(O) s u c h t h a t :5.8
(B,K)kq,;s!,
5.9
6 = 2r”/m’.
where
NOW by 5 . 7 We o b t a i n (B,K)kV*(k(€i,O)), and b y 5 . 6 ( B , K ) ( r @ x ( F n + l ( ? L 6 , 0 ) ) ) . C l e a r l y we h a v e a c o n t r a d i c t i o n ( b y 4 . 2 3 ) i f w e c a n show b < F n t l ( f ( b . o ) )
by 2 . 9 . € * ( ? ( 6 , 0 ) ) = 6+1 and h e n c e b y 2 . 1 8 Fn+, ( i ( 6 , O ) ) =-?(<+,(6t;),O). By 5 . 9 we c e r t a i n l y h a v e e;( 6 + 1 ) + 2 e;( p t l ) and so b y 5. 4, Fritz( f ( 6 . 0 ) ) + f ( 2 2 t - 1 , o ) = i(Zt,;(Zt-l,O)) ( b y 2 . 1 0 ) . NOW it 18 e a s y t o shw t h a t f o r a l l X d M, > x ( b y i n d u c t i o n on x ) , so b y 2 . 1 8 ( f o r n =l), 5 . 1 and 5 . 2 we o b t a i n F n + , ( ? ( 6 , 0 ) ) > i’(Zt,a) = b, as r e q u i r e d .
NOW
m)
0 Of c o u r s e there is s t i l l a gap between theorems 5 and 12 and w e l e a v e t h i s as a n open problem. However, t h e y o b v i o u s l y imply:-
Theorem 13 The t h e o r y (un : n E I N ) is n o t i n t e r p r e t a b l e i n Tu. I n p a r t i c u l a r ( c f . the example at the end of s e c t i o n 2 ) the t h e o r y (wx3y y = w,(x) : n E I N ) i s n o t i n t e r p r e t a b l e i n I A o ( a l t h o u g h e v e r y f i n i t e subset i s ) . 0
( S t r i c t l y s p e a k i n g , t h e f u n c t i o n f o f 2.21 d o e s n o t s a t i s f y 3 . 3 , b u t t h i s Fl,Fz, have c o n d i t i o n w a s o n l y t o g u a r a n t e e t h a t the f u n c t i o n s f,,f,, A. g r a p h s and t h i s i s c e r t a i n l y the case for t h i s p a z t i c u l a r f . )
...,
...
34 1
On Sentences Interpretable in Systems of Arithmetic
we s h o u l d remark here t h a t J . B . paris has m o d i f i e d t h e games used i n s e c t i o n 4 t o allow any p a r a m e t e r s from M t o o c c u r i n the s e n t e n c e s p l a y e d (and n o t j u s t those less than some f i x e d b o u n d ) p r o v i d e d MCPA. I n t h i s case the games are M - i n f i n i t e b u t the strategies for player I1 s t i l l e x i s t as M - d e f i n a b l e f u n c t i o n s . Using a v e r s i o n of corollary 11, Paris t h e n o b t a i n s the f o l l o w i n g c o n s i d e r a b l e improvement of p r o p o s i t i o n 1:Proposition 14
MW A be n o n s t a n d a r d and c o u n t a b l e . Then there is a n i n i t i a l segment ( c l o s e d u n d e r . ) I o f M s u c h t h a t i f J is a n y i n i t i a l segment of I d e f i n a b l e i n ( M , I ) t h e n J is n o t s i m u l t a n e o u s l y c l o s e d u n d e r a l l the f u n c t i o n s (un: n e I N ) . Let
Let u s now t u r n t o theorem 7 . U n f o r t u n a t e l y i f the f u u n c t i o n f associated w i t h has v e r y erratic growth it may be t h e case t h a t no fixed number o f i t e r a t i o n s of P, d o m i n a t e s 2x ( e v e n though for f t h e slowest growing of all uniform f u n c t i o n s , namely fix) = x t 1, w e h a v e F , ( x ) = 2x+1-1). Hence T,- may n o t imply I A ~+ e x p so w e h a v e no hope of p r o v i n g t h e c o n v e r s e of theorem 7 i n g e n e r a l (see theorem 16 b e l o w ) . However, w e d o have the f o l l o w i n g r e s u l t : -
u
Theorem 15
Suppose A is a Then T,- + e x p 1
sentence of
Ill A.
L
such that
T ,
+
A
is i n t e r p r e t a b l e i n
Tu.
P E f
+ A i n To. Let @ " ( x ) be the n a t u r a l t r a n s l a t i o n f o r m u l a and choose n E I N and a n 3, formula o f L ~ ( J ) , +(x) say, which i s l o g i c a l l y e q u i v a l e n t t o J ( X ) A +m*(x)J. NOW s u p p o s e A = V j2 A ' ( % ) ( A ' ( i t ) a .A f o r m u l a ) a n d , for c o n t r a d i c t i o n , t h a t M i s a c o u n t a b l e n o n s t a n d a r d model of q t exp, a d M, and M W % < a A'(%). Now i t is e a s y t o s h o w t h a t x k-+ 1 ( e n + l ( f * ( x ) ) , x ) is a t o t a l f u n c t i o n i n M ( i t i s dominated b y F$"+')(x)) so we may d e f i n e : Let
of
@'(x) interpret
@ ' ( x ) i n t o an
Tu
Lr
-
5.10
t
5.11
b =
5.12
g
=
en ( f * ( a ) ) ,
?
= f
(Zt,a),
r
(x
€
and
M: q x < b ) .
may f u r t h e r suppose ( b y t a k i n g a n e l e m e n t a r y e x t e n s i o n o f M i f n e c e s s a r y ) t h a t there i s a B e M s u c h t h a t B > e i ( b ) f o r a l l m E IN, so t h a t b y theorem 11 there i s a n i n i t i a l segment K o f M s a t i s f y i n g 4 . 2 2 - 4 . 2 4 w i t h P = f " ( a ) - l . S i n c e K F T ~ it f o l l o w s t h a t 0 d e f i n e s i n ( B , K ) an i n i t i a l segment s a t i s f y i n g A and closed u n d e r f . I n p a r t i c u l a r : We
5.13
( 8 , K *-@(
a).
As i n t h e proof of theorem 1 2 , we d e f i n e
q(x)
Q
32 ( z =
h
f(x,o)
A
@ ' ( z ) ) and
c o n c l u d e , b y 4 . 2 4 and 5 . 5 , t h a t ( B , K ) F q s l * ( 6 ) , where 6 = z[@'nI, and P = f* (aL-1. c l e a r l y t h i s implies ( ~ , ~ ) b q ( f * ( a )and ) hence ( B , K ) ) = @ ( f ( f * ( a ) . O ) ) . However, b y i n d u c t i o n on x one can e a s i l y show that for a l l x e M , i ? ( f " ( x ) , o ) > x , and h e n c e ( B , K ) k @ ( a ) c o n t r a d i c t i n g 5 . 1 3 as r e q u i r e d . 0
we c a n now state o u r main r e s u l t , which f o l l o w s i m n e d i a t e l y f r o m theorems 7 and 1 4 and 2 . 2 2 .
Theorem 16 s u p p o s e T,- t.. exp. Then for a n y n, s e n t e n c e A, T~ + A is i n t e r p r e t a b l e i n if and Only i f % 1 A. I n p a r t i c u l a r , IAo t A is i n t e r p r e t a b l e i n I& if and o n l y i f 14 + exp 1 A.
342
A.J. W I L K I E
[l] H. Gaifman and C. Dimitracopoulos, Fragments of Peano's Arithmetic and the MRDP theorem, in Logic and Algorithmic Monographie No.30 de L'Enseignment Mathematique, Ceneve 1982, pp. 187-206. 121 J.B. Paris and C. Dimitracopoulos, A note on the Undefinabrlity of cuts, J . of symbolic Logic, voi.48, ~ 0 . 3 ,sept.1983, pp.564-569. [ 3 ] J.B. Paris and A.J. Wilkie, On the scheme of Induction for Bounded Arithmetic Formulas, submitted to the Annals of Pure and Applied Lcqic.
[4] Pave1 Pudlak, Cuts, Consistency Statements and Interpretations, J. Of SymbOllC Logic, VO1.50, N 0 . 2 , June 1985, pp.423-441.
LOGIC COLLOQUIUU '84 J.E. Paris, A.J. Wilkie. and C.M. Wilmers (Editors) @ Elsevier Science Publishers E. K (North-Holland), 1986
343
ON THE MODEL THEORY OF EXPONENTIAL FIELDS (SURVEY)
Helmut Woiter Humboldt Universitat zu Berlin 1086 Berlin Unter den Linden 6-8
INTRODUCTION The p r e s e n t paper i s an extension of t h e l e c t u r e "Some r e s u l t s about exponential fields" which I gave a t the Conference "Table Ronde de Logique" (Paris, 15/16 Octobre 1983). I t p r o v i d e s a survey on some r e s u l t s concerning the theory of exponential f i e l d s . The investigations of this theory a r e motivated b y A . TARSKI'S decidability problem concerning the f i e l d of r e a l numbers with t h e additional exponential f u n c t i o n e x . In r e c e n t y e a r s a l o t of people have been concerned with exponential fields and r i n g s and have obtained many i n t e r e s t i n g r e s u l t s (see e.g. C R I , C M I , C W i 3 , C D r l l , C H R I , C D W 1 3 , C D W 2 3 , C D a I , C W o l I ) , b u t TARSKl's problem i s s t i l l open and a solution i s n o t in sight a t the moment. A complete solution seems t o b e v e r y d i f f i c u l t . i n general, t h e r e a r e several methods t o solve a decidability problem b u t i n this special case attemps were made i n two directions O n the one hand there a r e attemps t o p r o v e the elimination of quantifiers This method has the advantage that one has t o investigate only the standard model and hence one can use all the tools of analysis b u t i t has the disadvantage that one has t o know almost a l l dtrout this model. On the other hand i t has been t r i e d t o approximate the theory of the standard model b y suitable axiom systems i n order t o find perhaps a complete system by means of model theoretic tools as f o r instance: model completeness prime model and others In both cases great difficulties have t o b e overcome and as long as we do not know whether the theory i s decidable we should not ignore that i t could be undecidable. Even independent of T A R S K I s decidability problem the class of exponential fields i s a very i n t e r e s t i n g s u b j e c t of investigation. Only the interplay of analytical and algebraic means yields fundamental r e sults where the algebraic methods often have to be developed f i r s t . Of c o u r s e , i t i s impossible t o give h e r e a complete survey on all r e s u l t s concerning the theory of exponential f i e l d s . We want t o r e s t r i c t ourselves mainly t o t h e r e s u l t s obtained by our r e s e a r c h group in Berlin. We investigated d i f f e r e n t classes of exponential fields w i t h t h e i n t e n t i o n to obtain more information on such s t r u c t u r e s and classes and their theories i n order t o give perhaps a c o n t r i b u t i o n t o the solution of TARSKl's decidability problem Definition. If F is from F into F , E ( x + y ) = E(x)E(y) c a s e E i s said
a f i e l d of c h a r a c t e r i s t i c o and E a unary function then (F,E) i s said t o b e an exponential f i e i d i f f o r all x , y e F and i f E(o) = 1 , E(1) 1. In this t o b e an exponential function on F .
We could also r e g a r d exponential fields of p o s i t i v e c h a r a c t e r i s t i c , b u t then E ( x ) = 1 f o r all x and this case i s n o t i n t e r e s t i n g . In the following l e t L b e a language f o r exponential fields, i . e . L contains, symbols
H.WOLTER
344 + ,-
,.,
symbol
-1 E
f o r the usual f i e l d operations a n d an additional u n a r y f u n c t i o n f o r an exponential f u n c t i o n . F u r t h e r l e t Eax b e the s e t of
axioms E ( x + y ) = E ( x ) E ( y ) , E ( o ) # 1 , E ( 1 ) k 1 a n d l e t EF b e an V-axiom system f o r fields of c h a r a c t e r i s t i c o augmented b y E a x . Then EF determines the theory of exponential f i e l d s . The most important models of EF are (R,e) a n d (C,e), where R a n d C a r e t h e fields of r e a l a n d complex numbers, r e s p e c t i v e l y , a n d e i s the usual exponential f u n c t i o n in these f i e l d s . I n the following Q denotes the f i e l d of r a t i o n a l numbers, Z the s e t of integers, F an a r b i t r a r y f i e l d of c h a r a c t e r i s t i c 0 , F = (F,E) an exponential f i e l d a n d , u n l e s s s t a t e d otherwise, m , n , k , l , i , j denote n a t u r a l num-
fl
b e r s . i c a n also b e , the a c t u a l meaning c o n t e x t . If F i s an o r d e r e d f i e l d a n d a c F , value of a. a is said t o b e infinitesimal i f p o s i t i v e r a t i o n a l number, a n d a i s f i n i t e i f Notions a n d denotations n o t specially explained usual.
of i w i l l b e c l e a r from the then I a I i s t h e absolute I a I i s smaller than e v e r y I a I > q f o r some q € 0 . in this p a p e r a r e u s e d as
N o w our aim i s t o give a c o n t r i b u t i o n t o finding a r e c u r s i v e a n d complete axiom system of Th(R,e) p r o v i d e d t h a t s u c h a system e x i s t s . S o we t r y t o approximate t h i s theory b y a p p r o p r i a t e a n d n a t u r a l axioms.
2 , UNORDERED EXPONENTIAL FIELDS F i r s t of all we want t o p r o v i d e some simple well-known
facts.
1 . I n (F,E) E i s n o t uniquely determined by F a n d EF. Indeed, if i s an additive f u n c t i o n from F i n t o F a n d E ( f ( 1 ) ) f 1 , then T ( x ) = =E(f(x)) i s an exponential f u n c t i o n on F , t o o .
2 . (C,e) i s s t r o n g l y undecidable, model i s undecidable.
i.e.
The field of r a t i o n a l s i s definable in Y ( x ) := ] y
3 k
z(E(y) =
E(z) = 1 A L
e v e r y t h e o r y h a v i n g (C,e) a s a
(C,e)
p
f
b y t h e formula
o h x = y x).
G.
In f a c t , (C,e) ey = 1 i f f y = 2 q T i , where q E Z and i = Since Q i s s t r o n g l y undecidable ( s e e e . g . C S h I ) , we h a v e t h e claim a n d , m o r e o v e r , we obtain
3.
EF
i s undecidable.
The n e x t lemma implies t h a t the r a n g e of the exponential f u n c t i o n in e v e r y EF-existentially complete model i s the whole f i e l d , e x c e p t i n g 0. Lemma -1 . (i)
C DW1 I
If a c F
and
F * = (F*,E*)
F * I= (ii)
If
F
F
3
a
0, then t h e r e e x i s t s an e x t e n s i o n
of
F
such that
F * C EF a n d
x ( ~ * ( x )= a ) .
is
c 3~
EF-existentially
complete,
( E W= a ) f o r ail
a € F, a
then
P
0.
Similar as f o r (C,e), t h e r e e x i s t s a formula in L r a t i o n a l s in e v e r y EF-existentially complete model. Theorem 2 .
C DW 1 1
defining the f i e l d of
On the Model Theory of Exponential Fields L e t F b e E F - e x i s t e n t i a l l y complete. Then, f o r a l l a$ Q i f f F k 3 x ( E ( x ) = l h E ( a x ) = 2 ) := P ( x ) . Hence 7 Y ( x ) d e f i n e s ( C , e ) , we o b t a i n Corollary 3 .
(C, e)
C
Q
in
F . Since T?' ( x )
345 a € F,
does n o t define
complete.
By compactness arguments a n d t h e s t r o n g u n d e c i d a b i l i t y of t a i n from t h e a b o v e theorem:
(i) (ii)
Q we f i n a l l y ob-
C DW1 1
EF
i s n o t companionable
Every existentially
Theorem 5 . ~-
in
DW1 1
i s n o t EF-existentially
Corollary 4 .
Q
C DVd1
(and hence
h a s n o model completion).
EF
complete e x p o n e n t i a l f i e l d i s s t r o n g l y
undecidable.
I
( R , e ) h a s n o e x i s t e n t i a l c l o s u r e , i . e . t h e r e i s n o E F - e x i s t e n t i a l l y comp l e t e e x t e n s i o n of (R,e) t h a t c a n b e embedded i n e v e r y s u c h e x t e n s i o n .
Our r e s u l t s show t h a t the t h e o r y of EF i s r a t h e r complicated a n d s i n c e EF has models with q u i t e d i f f e r e n t p r o p e r t i e s , EF i s n o t a g o o d a p p r o x i mation of T h ( R , e ) . T h e r e f o r e , in the n e x t c h a p t e r we r e s t r i c t o u r s e l v e s t o more s p e c i a l c l a s s e s o f s u c h f i e l d s , namely t o o r d e r e d exponential f i e l d s . N e v e r t h e l e s s , (C, e ) i s a v e r y i n t e r e s t i n g b u t d i f f i c u l t s u b j e c t of i n v e s t i g a t i o n . S e v e r a l attemps h a v e b e e n made t o g e t a s u r v e y on t h e d i s t r i b u t i o n of the z e r o s of s u c h f u n c t i o n s i n (C , e ) d e f i n e d b y terms in L , b u t s t i l l without s u c c e s s . InLHR] C . W . HENSON a n d L.A. RUBEL p r o v e d , b y means of v e r y c o m p l i c a t e d a n a l y t i c a l t o o l s (NEVANLINNA-Theory), t h e following c o n j e c t u r e of S . SCHANUEL: If L' i s t h e language of e x p o n e n t i a l f i e l d s with p a r a m e t e r s from C b u t without d i v i s i o n a n d f i s a f u n c t i o n from Cn into C defined b f i s n o w h e r e equal t o o , then f has the form e , term of L ' a n d i f where g i s a definable f u n c t i o n , t o o . Using analogous methods H . KATZBERG p r o v e d t h e f o l l o w i n g r e s u l t f o r f u n c t i o n s f,g on C d e f i n e d b y terms of L'.
4 "
Theorem 6 . C ~(i)
K 1
L e t f: C
+C .
p(x)eg('), (ii)
L e t f: Cn + C , f ( x ) = e9(x) zeros
.
f ( x ) has f i n i t e l y many z e r o s i f f i t h a s t h e form
where
p
i s a polynomial o n
n > 1.
C
and
f o r some d e f i n a b l e g ) o r
f(x)
g
i s definable.
C ( a n d hence
f ( x ) has n o z e r o s in
has i n f i n i t e l y many
3 . ORDERED EXPONENTIAL FIELDS Next we a r e g o i n g t o s t u d y some p a r t s of t h e u n i v e r s a l t h e o r y o f t h e o r d e r e d f i e l d o f r e a l numbers w i t h e x p o n e n t i a t i o n . L e t OF b e an V-axiom system for o r d e r e d fields and
T =
OFU E
ax
u [ ( l + l / n ) n 4 ~ ( 1 6) ( l + l / n ) " + l : n >
01.
Since t h e statement Vx 7 OVy(E(y) = 1 + l / x + E ( x y ) c E ( 1 ) ) i s t r u e i n (R,e) T i s weaker b u t n o t in a l l non-Archimedean T-models, t h e V-theory of than Thy(R,e). Hence we r e g a r d t h e b e t t e r approximation
H. WOLTER
346 OEF' = OFU E a x u
f
1
E(x)
+
.
x\
The following theorem, which c a n b e p r o v e d b y s t a n d a r d a r g u m e n t s , shows t h a t the t h e o r y of o r d e r e d e x p o n e n t i a l f i e l d s OEF' i s sufficiently strong t o c h a r a c t e r i z e t h e e x p o n e n t i a l f u n c t i o n uniquely i n t h e s t a n d a r d model (R,e). Theorem 7 . ~If
(F,E)
DWl 1
[:
k
OEF',
t h e n i t h o l d s in (F,E)
that
(i)
z 1 + x, E i s s t r i c t l y monotonously i n E ( x ) b 0 , x f. 0 - E ( x ) c r e a s i n g a n d t a k e s a r b i t r a r i l y small a n d l a r g e p o s i t i v e v a l u e s ( b u t n o t n e c e s s a r i l y all p o s i t i v e v a l u e s ) .
(ii)
x
(iii)
E
>OA
E(y) = 1
+
l/x
--+ E ( x y ) <
E ( l ) < E((x+l)y).
E'(x) = E ( x ) .
is differentiable and
H e r e , t h e d e r i v a t i v e i s d e f i n e d b y means of t h e E - d - t e c h n i q u e . F o r p r o v i n g t h e n e x t r e s u l t s some s p e c i a l a l g e b r a i c t o o l s a r e n e c e s s a r y , i n p a r t i c u l a r we n e e d s o - c a l l e d p a r t i a l e x p o n e n t i a l f i e l d s . These a r e o r d e r e d f i e l d s with a p a r t i a l e x p o n e n t i a l f u n c t i o n . Suitable e x t e n s i o n s of the f i e l d s a n d t h e corresponding exponential functions finally yield Theorem 8 . C DW1 1 ~(i)
0EF'-existentially
(ii)
I n e v e r y 0EF'-existentially complete model the statement E has the i n t e r V x > 0 3 y ( E ( y ) = x ) i s t r u e , i . e . i n s u c h models mediate v a l u e p r o p e r t y ( f o r s h o r t : I n t ( E ) ) .
OEF'
complete models a r e r e a l c l o s e d f i e l d s .
i s n o t s u f f i c i e n t l y s t r o n g to p r o v e the V-theory
of
(K e)
Theorem 9 . C D W l 1 ___OEF'F
Vx
> O(E(x)
2
1
On the o t h e r h a n d , OEF' INow we r e g a r d a s t r o n g e r
Ek(x) =
xi/i!
and
+
x
+
2
x /2).
2
Vx > l / n ( E ( x ) 2 1 + x + x / 2 ) for all n b 0 . V-axiom system OEF. F o r t h i s l e t
OEF = OEF'U
{ E(x)
2 Ek(x): k odd,
k
*
31
.
i'k
Similar a s a b o v e , OEF-existentially complete models a r e r e a l c l o s e d f i e l d s . F u r t h e r m o r e , i n s u c h models t h e i n t e r m e d i a t e v a l u e p r o p e r t y i s t r u e f o r all terms without i t e r a t e d e x p o n e n t i a l f u n c t i o n . I t i s an open q u e s t i o n whether t h i s p r o p e r t y i s t r u e f o r all terms a n d i t i s a l s o open whether OEF p r o v e s Thv(R, e l . Remark: One c a n p r o v e t h a t Th(0EF) = Th(0EF' U E k ( x ) : f o r a r b i t r a r y f i x e d n > O a n d a l l o d d k 2 35 m(lx I C l / n +E(x) 1.e. i t s u f f i c e s t o approximate t h e e x p o n e n t i a l f u n c t i o n only in some a r b i t r a r i l y small s t a n d a r d n e i g h b o u r h o o d in o r d e r t o g e t t h e same t h e o r y .
,
On the Model Theory of Exponential Fields
-
4 . ON THE STRUCTURE OF EXPONENTIAL FIELDS NEW EXPONENTIAL FUNCTIONS
In t h e following l e t finded. field
F
and let
{
OEFl = OEFU
F u r t h e r l e t Fo =
I a € F:
Int(E)i O\
a
F 1 = [ a € F: a > 1 \
347
A METHOD FOR CONSTRUCTING
OEF'I b e similarly de-
and let
b e the p o s i t i v e p a r t of t h e o r d e r e d
.
We now want t o s p l i t F o a n d F 1 i n t o s o - c a l l e d additive and rnultiplicat i v e Archimedean c l a s s e s , r e s p e c t i v e l y . Definition. (i)
L e t a,b € F o . a
(ii) Let
5
b iff na
a , b c F1, a a b iff
*
b and nb
a
f o r some
n.
an 3 b a n d bn 3 a f o r some n .
Obviously, Y a n d == a r e equivalence r e l a t i o n s on F o a n d F 1 , r e s p e c t i v e l y . The c o r r e s p o n d i n g s e t s of equivalence c l a s s e s a r e denoted by
?'.
F"O a n d I f F i s Archimedean o r d e r e d , then t h e r e e x i s t exactly one a d d i t i v e a n d one multiplicative Archimedean c l a s s . In a r b i t r a r y o r d e r e d fields t h e Archimedean c l a s s e s a r e segments of F , induces o r d e r i n g s in Definition. (i) (ii)
s
Let
S,S'E
(i)
Po
a c b (in F ) a n d
S , S ' E pl.
and
af s
a-b.
Y
(F,E)
and hence t h e o r d e r i n g i n F
b y the following
S < S' i f t h e r e a r e elements a c b (in F ) and 7 a x b .
infinite
(ii)
F1
and
< s ' i f t h e r e a r e elements
Theorem 10. C Wo2 ~Let
To
and b Q s ' s u c h t h a t
a€ S
and
b E S' s u c h that
1
b e a non-Archimedean exponential f i e l d s u c h t h a t f o r e v e r y
a>0
t h e r e e x i s t s an i n f i n i t e
b
with
b2 6 a.
Then we have
F o i s densely o r d e r e d without f i r s t and l a s t element. 0 F o r all a , b € F : a - b iff E(a)* E(b).
(iii)
If (F,E) b OEF'I, l a s t element.
(iv)
E
then
maps e v e r y c l a s s
Int(E)
is satisfied,
F1 s e
then
i s densely o r d e r e d without f i r s t and
Fo E
i n t o some c l a s s maps
s onto
I f moreover 1 S (and Fo onto F 1. SC?'.
Hence, an exponential f u n c t i o n i s an o r d e r - p r e s e r v i n g map of t h e additive Archimedean c l a s s e s i n t o t h e multiplicative ones a n d i f Int(E) i s s a t i s f i e d , then t h e map i s o n t o . Of c o u r s e , we c o u l d e x t e n d the r e l a t i o n s , ., and
*
t o F and F o , r e s p e c t i v e l y , a n d modify the above r e s u l t s . Theorem 10 g i v e s a l i t t l e hint how t o answer the q u e s t i o n in which o r d e r e d fields exponential f u n c t i o n s a r e definable. Now we want t o i n v e s t i g a t e t h e problem how well OEF d e s c r i b e s the exponential f u n c t i o n in a given model, o r in o t h e r w o r d s , how g r e a t the d i f f e r e n c e between two s u c h f u n c t i o n s can b e in t h e same f i e l d . F i r s t we a r e going t o show t h a t in Archimedean o r d e r e d OEF-models E i s uniquely determined. F o r
H.WOLTER
348 L2
this purpose let
b e t h e language
a second exponential f u n c t i o n a n d the axiom system OEF i s defined analogously.
augmented b y a symbol
L
E* for
OEF2 = OEF(E)UOEF(E*) b e the union of
formulated with
E
and
E l , r e s p e c t i v e l y , OEF21
Theorem 11. C DW 2 3 _____ Let
(F,E,E*)
(i)
If a e F
b e a model of and
E(a) - € * ( a ) (ii)
If
F
a
OEF2.
is finite,
then
E(a),E*(a)
are finite and
i s infinitesimal.
i s Archimedean,
then
Now we r e g a r d an a r b i t r a r y model
E = E*.
(F,E,E*)
o f OEF2
a n d i n v e s t i g a t e the
r e l a t i o n s between E a n d E * . Theorem 7 implies t h a t E,E* are t i a b l e , s t r i c t l y monotonously i n c r e a s i n g (hence i n j e c t i v e ) , a n d t h a t only p o s i t i v e b u t a r b i t r a r i l y small a n d l a r g e v a l u e s . M o r e o v e r , l e t all p o s i t i v e values i n F . Then, f o r e v e r y a P F t h e r e i s e x a c t l y b € F such that h from F i n t o
Proposition 12.
E * ( a ) = E ( b ) . Defjning F such that E ( a )
h(a) = b - a E(a + h(a)).
diffsrenE ,E t a k e E take one
we obtain a f u n c t i o n
IW 0 2 7
(i)
h i s a d d i t i v e , d i f f e r e n t i a b l e a n d the d e r i v a t i v e of e v e r y where.
(ii)
OEF
b V x ( 0 ~ xl I
1 -+E(x)<
Ek(x)
+
xk+l)
is
h
f o r all
o
k
odd.
Of c o u r s e , i f F i s non-Archimedean, then h n e e d n o t b e c o n s t a n t . Now l e t h b e an a r b i t r a r y additiye map from F i n t o f a n d E an I f E (a) = E ( a + h ( a ) ) a n d E (a) 2 E k ( a ) f o r exponential f u n c t i o n on F . all a 6 F a n d all k odd, then E * i s an exponential f u n c t i o n on F in t h e sense of OEF, t o o . The n e x t r e s u l t i s an e x t e n s i o n of Theorem 6 from
LDW2 3 .
Theorem 13. C Wo2 ~If
f u n c t i o n h: (i) (ii)
7
(F,E) k OEFI, If If
a a
then
(F,E*) k OEF
F -+ F s u c h t h a t i s infinitesimal, i s finite,
then
i f f t h e r e e x i s t s an a d d i t i v e
E*(a) = E(a then
h(a)
I h(a) I
+
h ( a ) ) f o r all
< I al"
f o r all
a€ F
and
n.
i s infinitesimal.
(iii)
L e t a b e i n f i n i t e a n d a 7 0 . I f h ( a ) f 0, then h ( a ) c a n b e a r b i t r a r y . If h(a) < 0 , t h e n a + h ( a ) has t o b e p o s i t i v e , i n f i n i t e a n d h(a) 2 -a + n - L n ( a ) f o r all n, where Ln i s the i n v e r s e f u n c t i o n of E .
(iv)
L e t a b e infinitel a n d a < 0. I f h ( x ) < 0 , then h ( x ) c a n b e arb i t r a r y . I f h ( a ) 2 0 , then a + h ( a ) has t o b e n e g a t i v e , i n f i n i t e and h(a) -a - n - L n ( - a ) f o r all n .
This theorem gives a good s u r v e y on the exponential f u n c t i o n s E * definableby E a n d c e r t a i n a d d i t i v e f u n c t i o n s h on F . The n e x t r e s u l t s sho*w t h a t in all non-Archimedean models (F,E) C OEFI exponential f u n c t i o n s E without a n d w i t h t h e p r o p e r t y
I n t ( E * ) , r e s p e c t i v e l y , a r e definable b y means
On the Model Theory of Exponential Fields
349
of E a n d h . F o r t h i s , l e t ( F , E ) C OEFl a n d F b e non-Archimedean. I f we r e g a r d t h e a d d i t i v e g r o u p of F a s a Q-vector s p a c e w i t h a b a s i s B , t h e n we a r e able t o d e f i n e (b means of Theorem 13) a t l e a s t c a r d ( F ) h: B 9 B u {Or with t h e d e s i r e d p r o p e r t i e s . Hence, different functions these functions h y i e l d c a r d ( F ) d i f f e r e n t e x p o n e n t i a l f u n c t i o n s on the same f i e l d F (with additional p r o p e r t i e s ) . Theorem 1 4 . C W02 3 ~(i)
For all infinite
a,b€ F
E * ( x ) = E(x
form
+
0 c a
with
t h e r e i s an
b
4
b
(F,E*)
h(x)) such that
E * of t h e
OEF a n d
S i s the segment r a n g e ( E * ) r l S = 6, where (xcF: x g a or ac x < b or x s b ]
.
(ii)
If E * ( x ) = E(x + h ( x ) ) and ( F , E * ) k OEF t h e r e e x i s t s an n s u c h t b a t h(. .h(b). i t e r a t e d n - t i m e s , t h e n Int(E 1.
(iii)
I f h: F -F
.
all
i
is additive,
and h(b) = 0
Bo = { b . : i
for b € B
-
and for every bC B = 0 , where h i s
. .)
-
C b] 5
Bo, then
B,
h ( b i ) = bi+l
for
Int(E*).
The t e c h n i q u e u s e d h e r e y i e l d s some f u r t h e r i n t e r e s t i n g r e s u l t s ( s e e a l s o LDW 2 I). Theorem -15. (i).
-
In e v e r y OEF21-existentially complete model t h e r a t i o n a l s c a n b e d e f i n e d b y t h e formula
y (ii)
C W02 3
( x ) := Vy(E(y) = E * ( y )
OEF21
E(xy) = E*(xy)).
i s n o t companionable.
(iii)
OEF21
i s undecidable.
(iv)
The t h e o r i e s o f e a c h OEF21-existentially
complete model a n d o f
a l l s u c h models a r e u n d e c i d a b l e .
OEF 2I , i t i s unknown whether TheoI f we u s e s t r o n g e r axiom systems t h a n r e m 15 r e m a i n s t r u e , b e c a u s e i n t h e s t r o n g e r c a s e t h e r e i s p e r h a p s n o t e n o u g h freedom t o d e f i n e a s u i t a b l e f u n c t i o n
E*.
In c o n n e c t i o n with d e c i d a b i l i t y i n v e s t i g a t i o n s o f a t h e o r y T the existence o f a p r i m e model of T ( i n t h e s e n s e of A . ROBINSON) c a n b e h e l p f u l . But w i t h r e s p e c t t o OEFl t h e s o l u t i o n of t h i s p r o b l e m seems to b e v e r y d i f f i c u l t a s we w i l l s e e from the n e x t t h e o r e m , which c a n b e p r o v e d b y means of t h e same t e c h n i q u e as a b o v e . Theorem 6. _ _ _ _1_
C
W02
I
L e t e b e t h e u s u a l e x p o n e n t i a l f u n c t i o n on R and e = e(.. .e(l)...), where e i s i t e r a t e d n-times. If l , e l , n o r a t i o n a l multiples o f n o p r i m e model.
e
and if
Under t h e s u p p o s i t i o n s o f Theorem
e n+l
i s algebraic,
16 o n e o b t a i n s t h a t
and the corresponding constant term
En+l
en+l
.. .,en-l
are
t h e n OEFl has i s algebraic
can b e transcendental
in some
H. WOLTER
350
s u i t a b l e model ( F , E * ) , a n d t h i s c o n t r a d i c t s t h e e x i s t e n c e of a prime model. In p a r t i c u l a r , t h e e x i s t e n c e o f s u c h a model c a n depend o n t h e u n s o l v e d q u e s t i o n whether ee i s t r a n s c e n d e n t a l . By s u c h r e s u l t s t h e i n v e s t i g a t i o n s of e x p o n e n t i a l f i e l d s c a n become extremely d i f f i c u l t . Further important questions a r e those concerning extensions of exponential f i e l d s . We know almost n o t h i n g a b o u t s u c h problems ( s e e a l s o C Ge I ) . I f for instance (F,E) i s an exponential field a n d F t h e r e a l clo-sure of F_ , t h e n i t i s u n c l e a r whether t h e r e i s an e x p o n e n t i a l f u n c t i o n E 2 E o n F . If F i s the p e r j e c t c l o s u r e of F , t h e n , of c o u r s e , t h e r e i s a c o r r e s ponding extension E of E with s u f f i c i e n t l y good p r o p e r t i e s . Great d i f f i c u l t i e s a l s o h a v e t o b e overcome i f we a r e given a n e x p o n e n t i a l f i e l d (F,E) where F i s r e a l c l o s e d a n d we want t o e x t e n d E t o the a l g e b r a i c c l o s u r e of F such that E h a s a p p r o p r i a t e p r o p e r t i e s . With r e s p e c t t o t h i s p r o b l e m H . KATZBERG o b t a i n e d some p a r t i a l r e s u l t s .
5 . SOME ANALYTICAL PROPERTIES OF EXPONENTIAL FIELDS Now we do n o t r e g a r d V-axiom s y s t e m s any l o n g e r b e c a u s e we n e e d s t r o n g e r axioms i f we want t o i n v e s t i g a t e mor,e i n t e r e s t i n g a n a l y t i c a l p r o p e r t i e s of e x p o n e n t i a l f i e l d s . F o r t h i s l e t OEF = OEFw { l n t e r p e d i a t e v a l u e p r o p e r t y f o r terms u { ROLLE's Theorem f o r t e r m s \ . OEF i s f o r m u l a t e d f o r all terms t in L with r e s p e c t t o a f i x e d v a r i a b l e in s u c h i n t e r v a l s where
\
*
xL
E ( can be t h e t e r m s a r e d e f i n e d . By OEF* the inequalities E(x) p r o v e d from t h e o t h e r axioms i f k i s odd and k 3. It a n open p r o b l e m whether t h e terms h a v e t h e i n t e r m e d i a t e v a l u e p r o p e r t y i n OEFImodels.
*
By means of WILKIE's a n d RICHARDSON'S r e s u l t s DAHN was able t o p r o v e
C Da I
Theorem 1 7 . ___If
F C OEF, F
from
5
F , then
F * I= OEF*
Diagram(F1 L, OEF*
and i f
3 y Vx(x
F* I-
t ( x ) , s ( x ) a r e terms with p a r a m e t e r s
y -7t(x)
y.
*
s ( x ) ) := 'p
iff
This theorem f i n a l l y implies
c
Theorem 1 8 . ___If
F,F*
k
Da j
OEF*,
p a r a m e t e r s from
F
C
F,
The l a s t r e s u l t i s a little problem i s s t i l l o p e n .
F*
and
then F k
v(x)
3
i s a q u a n t i f i e r f r e e formula with
x Ip(x)
iff F*
3
x y(x).
h i n t t h a t OEF* c o u l d b e model complete b u t t h e
Now we come t o t h e "Problem o f t h e l a s t r o o t " , which goes b a c k t o A . MACINTYRE ( s e e C D r l 7 ) . I t i s i n d u c e d b y the f o l l o w i n g q u e s t i o n : L e t p ( x ) b e a n o n - z e r o e x p o n e n t i a l polynomial o v e r R . I s t h e r e a n i n t e l l i gible f u n c t i o n depending only on the r e a l p a r a m e t e r s of p which b o u n d s the a b s o l u t e v a l u e s of t h e r e a l r o o t s of p ? The n e x t theorem a n s w e r s t h i s q u e s t i o n p o s i t i v e l y n o t only f o r e x p o n e n t i a l
On the Model Theory of Exponential Fields
35 1
polynomials in the s t a n d a r d model but also f o r all n o n - z e r o exponential terms w i t h one variable in a l l OEF*-models. L e t F != OEF* a n d t(x,S) a term with p a r a m e t e r s H from F a n d the complexity k . Further let s . b e a c o n s t a n t term with the parameters B and the complexity 6 k + l
=c
and let s terms.
be
, where the sum i s t a k e n o v e r the f i n i t e s e t of all these
s;
(It would s u f f i c e t o r e g a r d
I siI , b u t
s =
Isi[
Finally, l e t D b e the c o r r e s p o n d i n g f i n i t e p a r t I s i >0: F C si > O \ u { si = 0: F C si = O \ of Diagram(F1
i s no term). and T = 0EF.u
D.
Then we have Theorem 19. C Wol ~-
I
There i s a computable c o n s t a n t term that (i) (ii)
F k
3 y V x
--
z y(t(x,S)
T
I- Vx(x > t *
T
c Vx(t(x,a) = o
Here "computable"
t(x,?i)
?
t*
depending only on
S such
c ) , then
c' c ) . 1x1 L t * ) .
means t h a t t h e r e e x i s t s a computable n a t u r a l number
.
.
n
s u c h that t * = E(. . E ( s ) . . ) , where E i s i t e r a t e d n-times. This theorem was g e n e r a l i z e d b y J. GEHNE a n d P . GORING ( s e e C Ge 3 , C Go I ) . They extended the language L b y the i n v e r s e of the exponential function and b y a r a t h e r b i g c l a s s of algebraic functions a n d obtained the analogous r e s u l t f o r the e x t e n d e d language L " .
In o r d e r t o p r o v e Theorem 19 we u s e d some Lemmata a n d techniques from DAHN's p a p e r C Da 1 where h e approximated exponential terms in non-Archimedean OEF*-models b y a p p r o p r i a t e d s e r i e s a n d obtained i n t e r e s t i n g r e s u l t s about the limit behaviour of exponential terms. Now we want to p r e s e n t the most important theorem of this p a p e r (which i s n o t n e c e s s a r y t o p r o v e Theorem 1 9 ) . Theorem 20. ____Let
C
Da 1
F, t(x,a)
F b lirn
b e as in the p r e v i o u s theorem a n d
t(x,S) = b,
then t h e r e i s a c o n s t a n t term
b E F . If
t * (with the same
XJQ
p a r a m e t e r 3 and the same number of i t e r a t i o n steps of that F C t = b .
E
as
t ) such
This implies t h a t the l i m i t of a term t belongs already t o the exponential subfield g e n e r a t e d b y the p a r a m t e r s from t . The l a t t e r theorem (and the theorems 1 7 , 18) as well as the techniques developed by DAHN in o r d e r t o g e t t h i s r e s u l t s were extended b y DAHN, GEHNE a n d GORING t o terms from the language L" ( s e e C DG 3 , C Ge 3 , C GOJ ) . By means of Theorem 2 0 one finally obtains the c o n v e r s e of a theorem of RICHARDSON ( s e e C R 1 ) . F o r t h i s , l e t M b e the smallgst s e t of functions a n d t f o r functions t , s € M containing 1 , x a n d l e t i t b g c l o s e d u n d e r + , ( o b v i o u s l y , the o p e r a t i o n t c a n b e e x p r e s s e d in L"). F u r t h e r l e t D b e t h e smallest s e t of c o n s t a n t s s u c h t h a t 16: D a n d i f a , b e D , then
.
H.WOLTER
352 a+b, a.b,
ea,
a-’
t D
Theorem 21. ____[ R
(i)
C
(ii)
1
DG 7
a € D,
then t h e r e a r e f , g E M such that a = l i m f/g x -+If f , g € M a n d f / g i s b o u n d e d (by a r a t i o n a l n u m b e r ) , t h e n If
lirn f / g 6 D U {O\ x 3 d l
.
( i i ) g e n e r a l i z e s a theorem o f v a n den DRIES,
who p r o v e d it f o r f u n c t i o n s
4 2 2 x ( f r o m o n e p o i n t o n ) . He o b t a i n e d a l o t o f o t h e r i n t e r e s t i n g r e s u l t s which aim a t t h e elimination of q u a n t i f i e r s f o r the t h e o r y of e x p o n e n t i a l fields ( s e e C Dr2 3 , C D r 31 1. Finally we want t o r e m a r k t h a t f o r t h e i n v e s t i g a t i o n of o u r t h e o r y i t was e s s e n t i a l t o h a v e a s u i t a b l e d e s c r i p t i o n of t h e e x p o n e n t i a l f u n c t i o n , i . e . we needed an a p p r o p r i a t e f u n c t i o n a l a n d d i f f e r e n t i a l equation a n d a s u f f i c i e n t l y good approximation of E by rational terms. F o r other functions satisfying these conditions corresponding r e s u l t s could b e possible b u t t h e r e a r e only few s u c h n i c e f u n c t i o n s . Arc tan,gens f o r i n s t a n c e h a s s u f f i c i e n t l y g o o d p r o p e r t i e s a n d h e n c e , a s P. PATZOLD p r o v e d in L P 1, many of t h e r e s u l t s p r e s e n t e d h e r e c a n b e c h a n g e d f o r the c o r r e s p o n d i n g model c l a s s .
REFERENCES
C Da 1
C
DG
Dahn, B . I . , The l i m i t b e h a v i o u r of e x p o n e n t i a l t e r m s , t o appear i n F u n d . Math.
I
Dahn, B .I. Preprint.
a n d P. Goring,
Notes o n exponential-logarithmic
terms,
CDrl 1
Van den D r i e s , L . , E x p o n e n t i a l r i n g s , exponential functions, Preprint.
C Dr2 1
Van den D r i e s , L . , Analytic Hardy f i e l d s a n d e x p o n e n t i a l c u r v e s in t h e r e a l p l a n e , P r e p r i n t .
C
Dr3
CDWl
e x p o n e n t i a l polynomials a n d
I
Van den D r i e s , L . , Bounding t h e r a t e of g r o w t h of s o l u t i o n s of a l g e b r a i c d i f f e r e n t i a l e q u a t i o n s a n d e x p o n e n t i a l e q u a t i o n s i n Hardy fields, Preprint.
J
Dahn, B . l . a n d H. Wolter, O n the t h e o r y of exponential f i e l d s , ( 1 9 8 3 ) , 465 - 4 8 0 .
ZML 29
C DW2 3
Dahn, 6.1. a n d H . Wolter, O r d e r e d f i e l d s with s e v e r a l e x p o n e n t i a l f u n c t i o n s , t o appear in ZML.
C Ge 1
Gehne, J . , u b e r Losungsmengen g e w i s s e r exponentiell-algebraischer Gleichungen, D i s s . A (Humboldt-Universitat).
On the Model Theory of Exponential Fields
C
353
Goring, P. , Uber Eigenschaften exponentiell-logarithmischer Diss , A (Humboldt-Universitat).
GO 3
Terme,
C HR 1
Henson, C . W . a n d L . A . Rubel, Some applications of Nevanlinna t h e o r y t o mathematical logic: Identities of exponential f u n c t i o n s , Preprint.
C K I
K a t z b e r g , H . , Complex exponential terms with only finitely many z e r o s , Seminarbericht N r . 49, Humboldt-Universitat z u Berlin, Sektion Mathematik.
CM3
Macintyre,
[ P I
Patzold, P . , Preprint.
CRI
Richardson, D . , Solution of the i d e n t i t y problem f o r integral exponential f u n c t i o n s , ZML 15 (19691, 333-340.
C Sh 1
Shoenfield, J .R. Company, 1967.
C
Wi 1
CWOl
A.
Wilkie, A . J . ,
1
,
The laws of exponentiation,
Preprint.
Uber Korper u n d Terme m i t arcustangens-Funktionen,
,
Mathematical l o g i c , Addison-Wesley
On t h e exponential f i e l d s ,
Publishing
Preprint.
Wolter, H . , O n the problem of t h e l a s t r o o t f o r exponential terms, t o appear in ZML. Wolter, H . , Some r e m a r k s on exponential functions in o r d e r e d fields, P r e p r i n t .
LOGIC COLLOQUIUM '84
355
J.B. Paris, A.J. Wilkie, and G.M. Wilmers (Editors) 0 Elsevier Science Publishers B.! l (North-Holland).1986
BOUNDED ARITHMETIC FORMULAS AND
TURING MACHINES OF CONSTANT ALTERNATION
Alan Woods* University of Malaya Kuala Lumpur, Malaysia
A formula i n the f i r s t o r d e r language f o r the n a t u r a l numbers N with
i s bounded a r i t h m e t i c (or A ) i f a l l i t s q u a n t i f i e r s a r e of t h e forms
=,<,+,*,O,l
3YS,
0
VYIX
*
I t i s e a s i l y seen t h a t every A
formula $(x) i n one f r e e v a r i a b l e is
e q u i v a l e n t t o a formula i n "prenex normal form":
+
+
where $(x,y) i s q u a n t i f i e r f r e e , 3 y i . < x denotes a block of q u a n t i f i e r s
3 yil$ +
x 3yi2s x
+ +
y = yIy2
... yk, +
... 3 yim< x ,
Q is 3or
\d
according a s k i s odd o r even, and
e t c . . The c l a s s i c a l h i e r a r c h y of a r i t h m e t i c f o r n u l a s with
unbounded q u a n t i f i c a t i o n suggests immediately the important question: BOUNDED ARITHMETIC HIERARCHY PROBLEM:
Can k i n (0.1) be fixed independently of $ ?
I n o t h e r words, i s t h e r e a proper A h i e r a r c h y according t o the number of a l t e r n a t i o n s of bounded q u a n t i f i e r s needed t o d e f i n e a p r e d i c a t e ? Although i t has a l r e a d y been considered by many authors, among them Harrow [ l o ] , Lipton [ 161,
Wilkie [ 261, P a r i s and Dimitracopoulos [ 231 ,[61,
t h i s question remains open.
A
s o l u t i o n , e i t h e r way, would have many i n t e r e s t i n g consequences. For example i n Harrow [ l o ] i t i s shown t h a t i f the A hierarchy does n o t 2 c o l l a p s e , then DSPACE(n) (or e q u i v a l e n t l y the Grzegorczyk c l a s s E, ) contains a
s e t of n a t u r a l numbers which cannot be defined by any A
forrmla.
This would
answer a long-standing q u e s t i o n of Parikh [ 241.
On the o t h e r hand suppose t h a t M i s a nonstandard model of Peano Arithmetic and i d e n t i f y t r e a t i n g +,-
a
E
M with {0,1,2,
the theory of ( an,+, - , I ) growing) A
*
..., a-1).
I t i s known (see,e.g.[23])
a s r e l a t i o n s on a, the theory of t h e s t r u c t u r e ( a , + , . , i ) for a l l n
E
N.
-
that determines
But i s t h e r e a nondecreasing (slowly
d e f i n a b l e f u n c t i o n g(x) with g(n)+
as n+-
through N,
Current Address: Dept. of Mathematics. Yale University, Box 2155 Yale S t a t i o n , New Haven, Connecticut 06520, U.S.A.
A. WOODS
356 (e.g.,
g(x) = [ l o g 2 x l , [ l o g 2 log2 x 1 , e t c ) such t h a t f o r a l l a
( a,+,..
I)
determines the theory of ( a g ( a ) , + , . , s ) ?
Lessan [IS] i t can be shown (see [231) t h a t i f the A
E
M,
t h e theory of
By adapting methods of hierarchy collapses there
can be no such g ( x ) . The s p e c i a l c a s e k=I of the A
h i e r a r c h y q u e s t i o n i s of p a r t i c u l a r i n t e r e s t .
I t can be reformulated i n the following way.
Call a s e t X
C
N
bounded Diophantine
i f X i s defined by a formula $(x) of the form
-+ where F(x,y) i s a polynomial w i t h i n t e g e r c o e f f i c i e n t s .
Davis, Matijasevi;,
and
Robinson [ 5 ] ask: BOUNDED DIOPHANTINE QWSTION: Does X being bounded Diophantine imply t h a t i t s
complement Xc i s a l s o bounded Diophantine? Closely r e l a t e d questions have been considered by Manders and Adleman [ 1 7 ] , Hodgson and Kent [ 1 I J,
and Borger [ 3
1.
The main new r e s u l t of the p r e s e n t paper i s a weak h i e r a r c h y theorem f o r A
formulas.
THEOREM 0.1 a A
There i s “0 f i x e d k such that every A.
formula @ ( x ) i s e q u i v a l e n t t o
formula of the form
+ where $(x,y) i s q u a n t i f i e r f r e e .
In o t h e r words, f o r every k
E
N there i s a A
formula which i s n o t
formula c o n t a i n i n g only k q u a n t i f i e r s . This improves on + e a r l i e r work of Alex Wilkie [ 2 6 ] which covered the case where $ ( x , y ) i s allowed
e q u i v a l e n t t o any A
t o c o n t a i n a t most a c o n s t a n t number (independent of $ ) of occurrences of 5
.
Note t h a t an attempt t o go from k blocks of q u a n t i f i e r s t o k q u a n t i f i e r s v i a t h e c l a s s i c a l coding a r w n t r u n s i n t o t h e d i f f i c u l t y t h a t xr code numbers a r e r e q u i r e d i n order t o a s s i g n a d i f f e r e n t code t o each ordered r - t u p l e
y1,yZ,
Yr <
X.
There i s a somewhat s i m i l a r problem i n proving theorem 0 . 1 (and Wilkie’s theorem) i n t h a t f o r each r can take values
5 xr,
terms of f i x e d s i z e .
E
+
N , t h e r e a r e formulas $(x,y) c o n t a i n i n g terms which
and w e wish
t o d i a g o n a l i s e over these using a formula with
This is done by using a space-time-alternation
f o r Turing machines of c o n s t a n t a l t e r n a t i p n .
t o introduce such machines, they do make t h e technique e a s i e r t o understand.
Whether the method is
trade-off
While i t i s c e r t a i n l y n o t necessary
-
and the important i s s u e s
-
s u i t a b l e f o r a t t a c k i n g the f u l l A
hierarchy problem i s n o t y e t c l e a r , b u t i t does seem t o be a good way of s e p a r a t i n g the f u l l h i e r a r c h y from i t s
levels.
Bounded Arithmetic Formulas and Turing Machines of Constant Attention
351
Some p r e l i m i n a r i e s . A l l logarithms l o g x = log x i n t h i s paper a r e t o base 2. 2
r e l a t i o n s on N w i l l be c a l l e d A The c l a s s A
Subsets of N and
i f they can be defined by A
formulas.
of a l l such s e t s can be c h a r a c t e r i s e d i n s e v e r a l o t h e r ways.
Let us adopt the convention, f o r our purposes harmless, t h a t {O,I)* denotes the set of a l l nonempty f i n i t e s t r i n g s over the alphabet I 0 , I J which a r e e i t h e r 0 o r
begin with I , so t h a t sets S C N can be i d e n t i f i e d with languages (i.e.,subsets)
L o 10, I I* by i d e n t i f y i n g each x of 0 ' s and 1's.
E
S w i t h i t s binary r e p r e s e n t a t i o n a s a s t r i n g
I t t u r n s o u t t h a t t h i s correspondence i d e n t i f i e s t h e
sets
A
with the languages L F [ O , 1 I* accepted i n l i n e a r t i m e by Turing machines'of constant alternation. A l t e r n a t i n g Turing machines were introduced i n t o computational complexity Since the idea of a Turing
theory by Chandra, Kozen and Stockmeyer ( s e e [ 4 1 ) .
machine of c o n s t a n t a l t e r n a t i o n deserves t o be more widely known amongst logicians. we give a d e t a i l e d d e s c r i p t i o n of them and some of t h e i r p r o p e r t i e s .
I.
E
TURING MACHINES OF CONSTANT ALTERNATION.
, Turing machines.
A C - TM M i s an o f f l i n e n o n d e t e r m i n i s t i c Turing machine. M has a read only I i n p u t tape, on which i t is given x E { O , l ) * bounded on each s i d e by a marker
symbol 1 , and a f i n i t e number m of r e a d l w r i t e work tapes. w i l l r e p l a c e x by x , # x2#
... #\
where x.
E
IO.l}
*
(Occasionally w e
.) M i s c h a r a c t e r i s e d by i t s
f i n i t e work tape alphabet A, f i n i t e s e t Q of s t a t e symbols, i n i t i a l s t a t e q accepting s t a t e qa, and i t s program P , so w e w r i t e M=
a bold 'old where a
E
not
{O,l,#};
bold,
bnew 'new
DoDl
bnew E Am; qold.qnew
E
.
I' P is a
(1.1)
m
"*
Q ; Di
E
{L
i' R i' 0 ) f o r i = O , I
....,m,
a r e the symbols old c u r r e n t l y scanned by t h e i n p u t and work tape heads, r e s p e c t i v e l y , and qold is
and
a l l D.=O.
The i n s t r u c t i o n ( 1 . 1 )
d i c t a t e s t h a t i f a,b
M's c u r r e n t state, then M should r e p l a c e bold by bnew. qold by qnew. and move the head on tape i one symbol l e f t , r i g h t , o,r n o t a t a l l , according as D. i s Li,Ri,
or 0
(Note that a t l e a s t one head moves.) An instantaneous d e s c r i p t i o n (ID) of M c o n s i s t s of the f i n i t e s t r i n g s on
t h e tapes, t h e head p o s i t i o n s ( r e l a t i v e t o the tape c o n t e n t s ) and t h e c u r r e n t
state.
An ID is a "snapshot" of M a t an i n s t a n t .
has blank work tapes, state q
I
, and
The i n i t i a l ID denoted by IDl(x)
the i n p u t head on the l e f t most symbol of x .
A. WOODS
358
A computation C o f a Z - Ri M on i n p u t x i s a sequence ( p o s s i b l y i n f i n i t e ) I o f i n s t a n t a n e o u s d e s c r i p t i o n s I D (x) = I D , I D 2 , such t h a t : I I
...
(i) (ii)
IDi+l
i s o b t a i n e d from I D . by performing some i n s t r u c t i o n i n P .
I f C ends i n I D . w i t h a,boldsqold as scanned symbols and c u r r e n t s t a t e , t h e n no i n s t r u c t i o n i n P begins w i t h
a b old qold
.
(M must c o n t i n u e i f i t can.)
C is an a c c e p t i n g computation i f
(i) (ii)
C i s f i n i t e , and
the f i n a l I D i n C has c u r r e n t s t a t e q
a'
There may be many computations C on i n p u t x s i n c e more than one i n s t r u c t i o n ( I f t h e r e i s o n l y one, we s a y the i n s t r u c t i o n i s
may begin w i t h a bold qOld.
d e t e r m i n i s t i c . ) The computations C form the branches of a r o o t e d t r e e an I D a t each node and I D (x) a s the r o o t . I
(ID'
(x) with
T
M is a s u c c e s s o r of I D i n T"(x)
i f I D ' can be o b t a i n e d from I D by performing a s i n g l e i n s t r u c t i o n from P . ) i n s t r u c t i o n s i n P are d e t e r m i n i s t i c
-
so T (x) h a s only one branch
a DRI.
M
-
If all
then M i s c a l l e d
Zk and rlr Turing machines. <
(i)
..., Qk>
P,A,ql.qa,Q,,Q2.
Q = Q, U Q2U
. .. U Q k
i s a d e s c r i p t i o n of a 1 - RI k
M if
i s a p a r t i t i o n of the s e t of s t a t e s Q of a .T
1
-
Ri
=
(ii)
For each i n s t r u c t i o n a b old qold bnew qnew Do D I then qnew We write
E
Q. =
m
i n p.
if 9
Qi
old
3,
i f i i s odd.
Vi
i f i i s even. k
-
'IM i s e x a c t l y t h e same e x c e p t t h a t w e t a k e
3 = uJi
a r e c a l l e d e x i s t e n t i a l s t a t e s . The s t a t e s i are called universal s t a t e s .
The s t a t e s q
E W = uvi
D
QiU Q i + l .
A d e s c r i p t i o n of a n
q
* * *
E
1
A computation C of a Zk
o r n - 'I'M k
M on i n p u t x i s any s u b t r e e of T-(x)
M
Bounded Arithmetic Formulas and Turing Machines of Constant Attention
359
such t h a t (I)
(11)
ID,(x)
E
C.
For a l l I D (a)
C w i t h a t l e a s t one s u c c e s s o r i n T - ( x ) , M
i f I D has an e x i s t e n t i a l c u r r e n t s t a t e q ID'
(b)
E
E
3 then
e x a c t l y one successor
of I D i s i n C,
i f I D has a universal c u r r e n t s t a t e q E v t h e n a l l successors of I D a r e i n C.
C i s an a c c e p t i n g computation of M on i n p u t x i f a l l computations of t h e L
I
-
TM
M accepts x i f there
I n t u i t i v e l y a n a l t e r n a t i n g RI
C of M on i n p u t x .
M works i n t h e f o l l o w i n g way.
s c a n n i n g symbols a bold on i t s t a p e s i n s t a t e q old. M chooses
branches i n C a r e a c c e p t i n g
. i s some a c c e p t i n g computation
M
Suppose M i s
I f qOld i s e x i s t e n t i a l ,
o f t h e i n s t r u c t i o n s i n P beginning w i t h a bo l d qold to P e r f o m -
(We s a y M "guesses" t h e move. o r "branches e x i s t e n t i a l l y " . )
I f qold is
u n i v e r s a l . M "branches u n i v e r s a l l y " by spawning enough c o p i e s of i t s e l f t o t r y
a l l instructions
i n P b e g i n n i n g w i t h a b old qOld.
communicating w i t h each o t h e r finds that
- acceptance
These c l o n e s have %way
of
is achieved i f an e x t e r n a l observer
t h e machines produced have h a l t e d i n acceptance.
The language
$
LM =
C {O,Il* a c c e p t e d by M i s d e f i n e d by
{
x : x i s a c c e p t e d by M
L e t T ( n ) , S ( n ) denote f u n c t i o n s of n. i n p u t x o f l e n g t h n.
-
11 .
Consider a computation C o f M on an
(This means x h a s n symbols so n % l o g x.)
C u s e s t i m e T(n) i f a l l branches i n C have l e n g t h 5 max ( T ( n ) , n ) + I .
(The l e n g t h of a branch i s t h e number of I D ' S
max { T ( n ) , n ) C uses
by any
steps
(i.e.,
on i t . )
Thus M p e r f o m s a t most
i n s t r u c t i o n s ) a l o n g each branch of C .
space S(n) i f a l o n g any branch i n
C,
t h e most d i s t a n t c e l l s v i s i t e d
tape head a r e n o more then S ( n ) c e l l s a p a r t . M works i n t i m e T(n) and ( o r ) space S(n) i f f o r a l l s u f f i c i e n t l y l a r g e n
and a l l s t r i n g s x
E
41 of
l e n g t h n , t h e r e i s an a c c e p t i n g computation C on i n p u t
x which uses t i m e T(n) and ( o r ) space S(n) ( r e s p e c t i v e l y ) . M i s a Lk-
RI which works i n t i m e T ( n ) )
Lk-
TIME ( T ( n ) ) = ($:
Lk-
SPACE ( S ( n ) ) = (LM: M i s a L - TM which works i n space S ( n ) l k
A. WOODS
360
TIME (T(n)) * SPACE ( S ( n ) ) =
Lk-
{ LM: M i s a I
k
-
TM which works i n time T(n) and space S ( n ) l
S i m i l a r d e f i n i t i o n s of rk- TIME ( T ( n ) ) , DTIME (T(n)), e t c . are assumed f o r
P
k
-
TM's and D T M ' s .
A function T(n) i s f u l l y time c o n s t r u c t i b l e i f T(n) 3 n and there i s some
DTM
M which, when given a s i n p u t a s t r i n g of n symbols I , h a l t s a f t e r e x a c t l y
T(n) s t e p s .
A l l nonconstant polynomials with nonnegative i n t e g e r c o e f f i c i e n t s
are fully t i m e constructible. Remarks : (i)
We advocate the use of the etc.
(ii)
*
n o t a t i o n t o denote simultaneous resource
S t r i c t l y speaking w e should perhaps w r i t e Ik*TIME(T(n))
bounds i n general.
s i n c e a l t e r n a t i o n can be regarded a s a resource-cf.
Provided T(n) i s s u f f i c i e n t l y honest (e.g., a language L i s accepted by some C complement Lc = { x E { O , l l * = x
k
-
remark ( i i i ) .
i f T(n) i s f u l l y t i m e c o n s t r u c t i b l e )
TM i n t i m e T(n) i f and only i f i t s
d L) i s accepted by some
vk-
'IM
i n time T(n).
..
An (iii) W e do n o t r e a l l y have t o introduce t h e p a r t i t i o n Q=Q U Q 6 .U Qk. 1 2 a1 t e r n a t i v e approach i s simply t o d e f i n e a1 t e r n a t i n g Turing machines
,q , , t/ > with computations C given by (I) and (11) above. l a There i s a n a t u r a l p a r t i t i o n of each branch i n T- (x) i n t o segments M W e could according t o whether t h e c u r r e n t s t a t e of t h e I D i s i n o r v
3
M = < P,A,q
say M is a I
k
-
3
TM i f t h e i n i t i a l s t a t e q l ~ 3 a n df o r every x
a c c e p t i n g computation C on i n p u t x
.
E
LM t h e r e i s some
which has a t most k segments.The c l a s s e s
I: - TIME (T(n)) obtained u s i n g t h i s a l t e r n a t i v e d e f i n i t i o n a r e t h e same a s k those defined above. This approach a l s o allows bounding t h e number of
a l t e r n a t i o n s by a non-constant
f u n c t i o n A(n)- s e e [ 4 1 and [221.
Mainly f o r convenience i n s t a t i n g theorems, we observe t h a t a l i n e a r "speed up" theorem of Book and Greibach [2] f o r 11- TM's PROPOSITION 1 . 1
L =
41 f o r
Let
T(n) 3 n, k 3 1 and c 3 I .
some I: - TM k
generalises: If L
E
Ek-
TIME ( c T(n)) then
M with only 2 work t a p e s ( p l u s t h e i n p u t t a p e ) which works
i n t i m e T(n). Note t h a t t h e s i z e of M ' s tape a l p h a b e t depends on c.
t i m e c o n s t r u c t i b l e t h e analogous p r o p o s i t i o n f o r
-
- TM's k
TI
I f T(n) i s f u l l y
a l s o holds.
k TM's ( r e s p e c t i v e l y ) is c h a r a c t e r i s e d
I n p a r t i c u l a r l i n e a r time on Ck and P by
u
I: - TIME (cn) = Ekk
u
Pk- TIME (en) =
c31
cs 1
TI
k
TIME (n)
-
TIME (n)
Bounded Arithmetic Formulas and Turing Machines of Constant Attention An obvious q u e s t i o n is whether t h e c l a s s e s C
form a proper h i e r a r c h y , and i n p a r t i c u l a r :
k
-
361
TIME (T(n)), rk- TIME (T(n))
Is t h e r e some k such that
LINEAR TIHE HIERARCHY PROBLEN:
u
mEN
C
- TIME(n)
m
= Lk-
TIME(n) ?
Apparently a l l t h a t is known i n t h i s d i r e c t i o n i s t h e theorem of Paul, Pippenger, Szemeredi and T r o t t e r 1211 t h a t d e t e r m i n i s t i c l i n e a r t i m e
(J DTmE(cn)(- DTIME ((I+E)n) f o r any E>O. s e e [ 1 3 J ) i s p r o p e r l y contained i n c2 1 The l i n e a r t i m e h i e r a r c h y problem i s , of course, a l i n e a r analogue
L I - TIME(n).
of t h e S t o c h e y e r polynomial t i m e h i e r a r c h y problem ( s e e e.g.
[41) which presumably
g e n e r a l i s e s t h e P = NP q u e s t i o n . 2.
CONNECTIONS WITH THE A- HIERARCHY PROBLEM. Although t h e connection between A
0
sets and languages i n the l i n e a r t i m e
h i e r a r c h y has been f o l k l a w f o r a considerable t i m e now, the l i t e r a t u r e on t h e s u b j e c t takes a r a t h e r c i r c u i t o u s path.
Bennett [ I ] proved t h a t the A
are i d e n t i c a l w i t h the c l a s s R U D of rudimentary languages.
sets
W e w i l l n o t bother
w i t h the usual d e f i n i t i o n of these except t o s a y t h a t i t i s immediate t h a t
RU D c
,& Ck-
TIME(n). Wrathall [27] defined a l i n e a r time h i e r a r c h y by means of
o r a c l e s , and showed t h a t i t s union i s i d e n t i c a l t o R U D . between h i e r a r c h i e s defined using o r a c l e s
c o n s t a n t a l t e r n a t i o n (see [ 4 1 ) , it follows t h a t A identical.
Modulo the equivalence
and those defined using machines of and
u
kcN
Ek- TIME(n)
are
lhis w a s noted e x p l i c i t l y by Lipton [I61 and o t h e r authors.
The equivalence of t h e A
0
and l i n e a r time hierarchy problems can be obtained
d i r e c t l y v i a a s e r i e s of lemmas which w e now sketch.
(Bennett [ I ] ) The r e l a t i o n z = xy can be d e f i n e d by a A
PROPOSITION 2.1
Besides Bennett s e v e r a l o t h e r writers have described A
formula.
d e f i n i t i o n s of the
graph of exponentiation, among them Quincey, P a r i s , Dimitracopoulos, Gaifman, and Pudlak ( s e e e.g.
[61,[81,[251) so we omit t h e proof.
LEMMA 2 . 2
There is a A
w(x,y,i,k)
++
formula w(x,y,i,k)
such t h a t f o r a l l i
t h e r e are e x a c t l y y occurrences of the d i g i t i i n the k-ary expansion of x.
*
2
+ 1 s h where Suppose x , y , i s a t i s f y t h e r i g h t hand s i d e . Then y f 2 [log kl h is the l e a s t i n t e g e r such t h a t h 2 [ l o g X I + 1. For j f h2 l e t z. denote t h e J number of occurrences of i i n t h e f i r s t j d i g i t s of the k-ary expansion of x
Proof:
(or i n x i f x has less than j d i g i t s ) .
A. WOODS
362 The sequence z h,z2h,.. .,z
c o n s i s t s of h numbers each having a t most h2 Therefore i t can be coded by a number z w i t h h ( [ l o g h ] + 1 )
[ l o g h] + I binary d i g i t s . binary d i g i t s .
Note t h a t
z 6 2 h([log h l + I )
".,
'jh+lSzjh+2S A
3h
x
,C
32
LEMMA 2.3
f
x (h,i,j,k,w,x,y,z
f
d + d x Vy, f x
+ ... Qyk f
where $ i s a q u a n t i f i e r f r e e b equivalent to a A
where JI
have t h e r e l a t i o n s h i p j u s t d e s c r i b e d ) .
Consider t h e formula
jgl
+
x such t h a t w codes
which holds i f and only i f
xt/j 6 h j v
f
f
Using p r o p o s i t i o n 2.1 it i s e a s i l y seen t h e r e i s a
(j+l)h'
formula u(x.y,i,k)
x ( u n l e s s x i s very small ) .
f
h there is a n m b e r w
S i m i l a r l y , f o r each j
d
x
JI (x,;)
formula and d E N i s a c o n s t a n t .
This formula i s
formula of the form
3?, i x v ? ,
x
... QYk+
f
x
$+(x,?)
is quantifier free.
+
Proof : Let -
Y have d v a r i a b l e s V l , V 2 ,
...,Vd
+
(say) f o r each v a r i a b l e v i n y.
-+
$+ i s obtained by s u b s t i t u t i n g f o r each v i n $ ( x , y ) t h e corresponding term
v I x d- I
d-2
+ v2x
d-2 vlxd-I + v2x
+
+
... + V d- 1x
... + vd
f
x
+ Vd and appending t h e a d d i t i o n a l conjuncts
.
d
There i s a c o n s t a n t c E N such t h a t i f L E Z
THEOREM 2.4
formula $(x,;) f o r which -+ L = { x E N : 3;,f xVy2 6 x Q;k+c 6 x
some q u a n t i f i e r f r e e A
...
Proof: Suppose M -
=
works i n time n, and t h a t L = a b
$.
o l d 'old
..., Qk>
is a E
k
-
k
-
TIME(n) then t h e r e is
JI (X
,31.
'IM w i t h m work t a p e s which
Each i n s t r u c t i o n b
new 'new
DODl
Dm
i n P i s a s t r i n g of l e n g t h 3m + 4 o v e r the f i n i t e a l p h a b e t A+ =
A U {o, I , # I u Q, u
...u Q,
u I L ~ , R ~ ,,LR~~ ,
..., L,.R,I
.
where we may assume t h a t L . and R. do n o t occur i n A or the Q . ' s . 3 Now c o n s i d e r any x E LM of l e n g t h n and l e t C b e an accepting computation which uses time n. instructions. (3m
+
Any branch of t h e tree C corresponds t o sequence of a t most n
P l a c i n g these i n s t r u c t i o n s end-to-end produces a word of l e n g t h
4)n over t h e h elenmnt(say) a l p h a b e t A+.
By i d e n t i f y i n g A+ w i t h
...,h-1)
(and adding 1 a t the l e f t ) v e , o b t a i n t h e base h r e p e r e s e n t a t i o n of a number y which
Bounded Arithmetic Formulas and Turing Machines of Constant Attention
363
(for n large) s a t i s f i e s
f o r some c o n s t a n t d E N, s i n c e h and m a r e c o n s t a n t s depending only on M (not on x ) . Thus x i s accepted by M i f and only i f
s
3 y l d xdWy2 where
if
e(x,;)
(i) f o r i
x
d
3y3 s x
d
... Q
Y s~ x
d
says that = 1.2,
..., k ,
each y - codes a sequence of i n s t r u c t i o n s f r w P with
Qi
'old
and ( i i ) the sequence of i n s t r u c t i o n s coded by y l,yz, computation of
then ( i i i )
e(x,;)
( the
t h i s computation of
Z
1
-
..., yk
determines a
'IM corresponding t o M ) ,
ends i n acceptance.
By Lemma 2.3 i t s u f f i c e s t o show
+
e ( x , y ) can be taken t o be a A
formula.
With the h e l p of p r o p o s i t i o n 2.1 p a r t s ( i ) and ( i i i ) a r e e a s i l y t r a n s l a t e d i n t o
A
formulas.
Observe that t o v e r i f y
( i i ) (given t h a t ( i ) h o l d s ) i t is only
necessary to check t h a t : (a)
qold f o r the f i r s t i n s t r u c t i o n is the i n i t i a l s t a t e q l , f o r a l l l a t e r
i s the same as q f o r the preceding old new of the l a s t i n s t r u c t i o n is the a c c e p t i n g s t a t e q
instructions q 'new (b)
i n s t r u c t i o n , and
.
For every tape c e l l the c o r r e c t symbol appears i n the a b
p o r t i o n of the old i n s t r u c t i o n s performed a t the i n s t a n t s when that c e l l i s v i s i t e d by the head.
In t h e case of an i n p u t tape c e l l the c o r r e c t symbol i s the a p p r o p r i a t e d i g i t For a work tape c e l l i t appears i n the b p a r t of the new i n s t r u c t i o n performed on the previous v i s i t , i t t h e r e was one, and i s blank
of x ( o r #). otherwise.
To check c e l l i on tape j i t s u f f i c e s t o check t h e i n s t r u c t i o n s a t each
p a i r of consecutive head v i s i t s . coded by y l ,
..., yk
end-to-end,
I f w i s the word formed by p l a c i n g the i n s t r u c t i o n s then t h e i n s t r u c t i o n s a t which c e l l i on tape j
i s v i s i t e d can be located i n w by means of the subwords w (wo i s a p r e f i x of w) and t h e number of occurrences of R
j
such t h a t w = w w in w
o
0 1
exceeds t h e number
of occurrences of L . by e x a c t l y i , J By p r o p o s i t i o n 2.1 and lemma 2.2 ( f o r "computing" the number of occurrences of R . and L i n w ) the words w,w and t h e i r decoding can be described by means J j o + formulas. Thus e ( x , y ) may be defined by a .A formula and c l e a r l y the number
of A
of b l o c k s of q u a n t i f i e r s i n 0 can be
bounded independently of M.
364
A. WOODS
+
REMARKS. ( i ) "he c o n s t r u c t i o n of the formula on an argument f o r E (ii)
I
The d e f i n i t i o n s of
E
E
k
xl# x # 2
-
i n the proof above i s based
TM's shown t o the a u t h o r by Jeff P a r i s .
TM's and Z
Ek-
i n place of x i n p u t s of the form x I # x ff 2 Then i f L
O(x.y)
-
TIME(n) t h e r e i s some A.
... #xm
E L
*(x1,x2
k
-
TIME(n) can
be g e n e r a l i s e d by u s i n g
... #xm
where x 1,x2,
forrmla
$(x,.
,...,xm) .
...,xm)
..., xm
E
{0,1)
*
.
+ such t h a t f o r a l l x,
This can be proved by a
s t r a i g h t - f o r w a r d (but tedious) modification of the arguments above. I n o r d e r te prove a "converse" of theorem 2.4 w e need t o show t h e r e is some formula
f i x e d k such t h a t f o r each q u a n t i f i e r f r e e A
[XI# x ff 2
... #xm :
m) } i s i n
$(xI,x2,...,x
t h e language
$&
E
k
-
TIME(n).
This is a s p e c i a l case of r e s u l t s i n s e c t i o n 5 and could b e postponed u n t i l then. given by Lipton and Pippenger (see [161) and
D i f f e r e n t proofs have a l s o been
.
( e s s e n t i a l l y ) by Bennett [ I ]
However i t may be of some i n t e r e s t t h a t w e can
take k = 2. LEMMA 2.5
free A
...hm:$(xI,x2. ...,xm) I
L e t L = {x ff x ff
1
fornula.
2
where
+ $(x) i s a q u a n t i f i e r
Then L
E
r\
E2- TIME(n)
TIME(n).
rr2-
Discussion : C l e a r l y + and t e s t i n g of
s
can be performed d e t e r m i n i s t i c a l l y by
Turing machines i n l i n e a r time u s i n g e s s e n t i a l l y the primary school algorithms. How quickly m u l t i p l i c a t i o n of two n d i g i t numbers can be done by m u l t i t a p e d e t e r m i n i s t i c ( o r apparently even n l o r E ) TM's i s an open question. 1
Proof of l e e 2.5: Throughout lower case l e t t e r s ck denote easy t o compute constants.
Consider a E
2
-
TM M which works as follows.
+
Given x of l e n g t h
n
,
M f i r s t guesses t h e values of a l l the terms ( i n c l u d i n g subterms) which appear i n $
.
Operating d e t e r m i n i s t i c a l l y M reduces the problem t o one of
number of m u l t i p l i c a t i o n s .
s t o r e d as binary s t r i n g s of l e n g t h L e t B be t h e power
Then
h . y = E yi B1 i=O
where 0 s yi, zi,wi Yi.Zi'Wi
,
z
< B
.
checking a f i n i t e
For each of these M must check t h a t values w,y.z $
cln
satisfy w = y.z.
of 2 s a t i s f y i n g n h =
.
E z. B1 and w
i=o
<
h =
B < 2n. and l e t .
h =
[:Li
B]
(say) + 1.
E w. B1
i=o
I
Since B i s a power of 2, t h e b i n a r y r e p r e s e n t a t i o n s of
can b e read o f f from those of y,z,w. h i I f y.z = w then w = Z ( E zi-j i-0 j = o
Y.) B1 J
so t h e r e are " c a r r i e s " C . such that
Bounded Arithmetic Formulas and Turing Machines of Constant Attention
i
...,h ,
i = 0.1,
for
w.
i
c.
! I:
j=o
Zi-jsyj
zi-j +
y . + Ciql
1
y.
j=o i - j
ci-l]
and Ch = 0 = C e l
=
from the
.
c a l c u l a t i o n of wi-l
to
The remainder i s w. and the q u o t i e n t C . i s
and then d i v i d i n g by B.
J
(mod B)
J
(wi may be found by adding the c a r r y Ci-l Z z
365
1
c a r r i e d over t o t h e c a l c u l a t i o n of w ~ + ~ . )
M guesses t h e value of t h e c a r r i e s C o , C I , . . . , C h - I . C,
::
2 i B (provided B 6 2 ) .
By induction on i,
Therefore t h e space (and time) r e q u i r e d t o s t o r e the
f o r a l l primes p :: c 3 log n. j By the prime number theorem (or i t s weak v e r s i o n s - s e e p r o p o s i t i o n 4.3 below) these M a l s o guesses the r e s i d u e s of each y. and z
r e s i d u e s take
t o t a l space
f
c n t o s t o r e . (This i s reasonable 4
e s s e n t i a l l y the same information as y and z.)
-
they c o n t a i n
The small primes p can, of course,
be generated i n n e g l i g i b l e time. M does t h i s f o r a l l the m u l t i p l i c a t i o n s i t has t o check.
t o check
w
= y.z
Then i n order
(and s i m i l a r l y a l l the o t h e r m u l t i p l i c a t i o n s i n $ ) M branches
u n i v e r s a l l y over: (i)
a l l yi,z
j
and then, f o r each of these, over a l l p t o check that t h e guessed
r e s i d u e s (mod p) are c o r r e c t , and (ii)
a l l i and then, for each of these, over a l l p t o check
i Z
j =O
Yi-j
z j
+
ci-l
CiB + wi
(mod p)
using t h e guessed r e s i d u e s of y i , z j To check ( i ) . the r e s i d u e
(mod p ) .
(mod p) of
S
= Z
Y . Z J = Y
j=o 3
(mod p ) .
These take time
r i g h t hand s i d e of (2.1)
+ 2 ( Y
I
+
s c5s l o g p
(Y,
2 f
+
... ) ) may b e c o q u t e d by c5
c6 l o g n l o g log n
can be found (mod p) i n t i m e < < n.
8
:: n.
additions
S i m i l a r l y the
To complete t h e
checking of a branch of type ( i i ) , M uses a t r i v i a l m u l t i p l i c a t i o n algorithm t o compute t h e l e f t hand s i d e (mod p ) of (2.1)
I f M a c c e p t s then
i n time
A. WOODS
366
i I: j=O Yi-j
zj
+
Ci-]
:CiB
(mod P)
wi
+
(2.2)
where P i s t h e product of a l l primes p < c 3 log n. But i f c enough, then by the prime number theorem ( o r p r o p o s i t i o n
3
was chosen l a r g e
4.3) P i s l a r g e r than
both s i d e s of (2.2) so
i
yi-j
Z
j =O
= C . B + w.
Ci-]
+
j
.
Since t h i s holds f o r a l l i, i t follows t h a t w = y.z. L
TIME(n). S i m i l a r l y , s i n c e
X2-
E
Hence L
E
TI
THEOREM 2.6
2
$(x) be a A.
Let
z I and
5
X \ J ; ~L
x
E
TIME(n).
E2-
formula of t h e form -t +. QY, L x +(x.Y)
...
$ is quantifier free.
x : $(x)l Proof:
Thus (by p r o p o s i t i o n 1 . 1 )
is q u a n t i f i e r f r e e . Lc
- TIME ( n ) .
3;, where k
-+(XI+
E
I:
k+ I
-
(2.3)
Then
TIME(n).
To t e s t , given x, whether (2.3) holds, a l i n e a r t i m e
M first
Zk+l-'IM
+ branches e x i s t e n t i a l l y t o guess the binary d i g i t s of y I . then u n i v e r s a l l y over a l l
possibilities a X
2
-
+
Finally M t e s t s that
f o r y2, e t c . .
IM i f k i s odd, and a
'II
2
-
+
+ ( x , y ) holds by s i m u l a t i n g
TM i f k i s even.
Theorems 2.4 and 2.6 together show and l i n e a r t i m e h i e r a r c h i e s a r e e q u i v a l e n t .
u
ksN
X - TIME(n) k
= A
and the f u l l A
3. SPACE-TIME-ALTERNATION TRADE-OFF. Theorem 3.2 below gives a way of reducing the product of the t i m e and space used by a computation a t t h e expense of i n c r e a s i n g the space bound and the number of a l t e r n a t i o n s .
3,,v2,3,, ...,
Consider a X - TM M = < P,A,q ,q , Qk > which w i l l be k l a fixed throughout t h i s s e c t i o n . Given two ID'S of M (and t h e r e f o r e of the
corresponding X
I
obtained from I D
-
TM
i) we say t h e r e i s a i preforming a f i n i t e
by I t i m e and space used by
of
i.
2
if ID
2
can be
number of i n s t r u c t i o n s from P.
The
p a t h s i s defined analogously to t h a t used by computations
The idea f o r t h e following lemma goes back t o Nepomnyaschii [ 191.
LEMMA 3.1 Z2m+l-
p a t h from I D I S I D
TM
Suppose T(n) i s f u l l y time c o n s t r u c t i b l e and m
E
N.
There i s a T (n) and I D log n 1' ID2
M l m such t h a t given an i n t e g e r f s a t i s f y i n g I d f d-
( s u i t a b l y coded and i n c l u d i n g x) :
Bounded Arithmetic Formulas and Turing Machines of Constant Attention M l m a c c e p t s i f and only i f t h e r e i s a path from I D ,
(i)
,
space T ( n ) / f and t i m e T(n)fm (ii)
M'
to I D
2
367
which uses
works i n time T(n),
m
where n i s the l e n g t h of x. Proof:
- TM M I m which works 2m+ I Since T(n) i s c o n s t r u c t i b l e , a clock counting t o T(n) and
By p r o p o s i t i o n 1.1 i t s u f f i c e s t o c o n s t r u c t a Z
i n t i m e c T(n). m
marking o f f space T(n) can be included by using e x t r a tapes.
m = 0.
guesses s =[ T ( n ) / f
MIO
i n > T(n)-f b u t
f
1.
T(n) s t e p s . ) If I D
s t e p s s t a r t i n g with I D I . accepts. Induction s t e p . Suppose M'
m
(s can be checked by marking o f f s c e l l s , f times,
then simulates
M'
2
f o r ( a t most) T(n)
i s reached without exceeding work space s
exists.
We c o n s t r u c t M'
,
MIo
m+l. This I:2(m+l)+l-
f i r s t guesses (and checks) s = [ T ( n ) / f ] . Then t o check t h e r e i s a s u i t a b l e path from I D I t o I D 2,
guesses a sequence of ID1 = I D '
0
...,
,IDI1,
f +1 key I D ' S of
IDIf
= ID2
-
Each of these involves work tape space d s i n M. I D V i o t h e r than x (which
s
-
s i n MIel.(
M's
(3.1)
The a d d i t i o n a l information i n
a l r e a d y has ) can t h e r e f o r e be s t o r e d i n space
MI,+,
i n p u t head p o s i t i o n r e q u i r e s space roughly log n E s s i n c e
f ,< T ( n ) / l o g n ) . Thus w r i t i n g these I D ' S branches u n i v e r s a l l y over i = 0,l. MIrn
:
...,
takes time f s
$
T(n).
f-1 and simulates the
to t e s t whether t h e r e i s a path from I D '
on I D ' i . I D ' i + l
time T(n)fm and space T ( n ) / f .
i
M' m + l then Z2m+L-
t o ID'i+l
using
C l e a r l y i t i s p o s s i b l e t o choose t h e sequence
(3.1) s o that t h i s w i l l be the c a s e f o r a l l i, i f and only i f there is a path from ID
1
to I D
REMARK: n2m+l-
which uses t i m e T(n)fel
2
and space T ( n ) / f .
Since T(n) i s f u l l y t i m e c o n s t r u c t i b l e , i t follows t h a t there i s a
M"m
TM
is "0 p a t h
which works i n t i m e T(n) and a c c e p t s f.ID1,1D2
from I D
THEOREM 3.2
1
to I D
2
i f and only i f there
which uses space T(n)/f and time T(n)frn
.
Let T(n) be f u l l y time c o n s t r u c t i b l e , 1 5 f ( n ) 5 T(n) and k,m log n '
E
N.
Then t h e r e e x i s t s K such t h a t Ck-
Proof:
TIME (T(n) f ( n ) m ) * SPACE ( T ( n ) / f ( n ) ) 5 ZK- TIME (T(n)).
W e take K = 2k + 2m + 1.
(A more c a r e f u l argument gives K = 2(k+m) .)
We w i l l assume f ( n ) i s i n t e g e r valued. T(n) by 2 T(n) and use speedup.)
( I f n o t r e p l a c e f ( n ) by
[ f ( n ) ] + 1,
Suppose C i s an accepting computation of t h e
Zk- TM M on an i n p u t x of l e n g t h n, and t h a t a l l branches i n C use space T ( n ) / f ( n )
A. WOODS
368 and t i m e T(n) f ( d m .
A l l ID'S
of M can be c l a s s i f i e d as being
o r Qk according t o whether t h e c u r r e n t or q
E
Qk.
similarly.
state q s a t i s f i e s q
E
ll,v2,33,...,
...,
31,q Ev2.q €j3,
This d i v i d e s t h e branches of C i n t o segments which can b e
classified
An I D ( o r more p r e c i s e l y a node) i n the t r e e C i s Q . minimal i f i t i s
t h e f i r s t I D i n Q . o n t h e path t o the I D from I D (x) o r ( f o r convenience) i f i t 1 is a halting I D i n Q i-1' Consider a Z - M M which, working i n t i m e c T ( n ) , f i r s t quesses f and s, k 1 checks s = [ T ( n ) / f ] , and then s i m u l a t e s the 3,,v2,j,,, p a r t s of M ' s
...
computation C a s follows. An
32i+lsegment.
I D of C. M
1
"1 a l r e a d y has
Unless i = 0,
C a l l t h i s ID2i+1.
i n s t o r a g e an
(For i = 0, take ID1 = ID1(x), the i n i t i a l I D of M.)
simulates M by guessing ID2i+2-
t h e v 2 i + 2 minimal I D foIlowing I D
This i s v e r i f i e d by branching u n i v e r s a l l y (except i f ID2i+2
Along one branch, M
1 M
Z2m+l-
T(n)fm by simulating the
RI
i n C.
i n 3 2 i + l a n d checks
and I D ' which uses space T(n)/f and time
MIrn
from lemna 3.1.
Along the o t h e r branch
continues t o branch u n i v e r s a l l y t o simlate segments of C i n
ID2i+2
2i+l
i s a h a l t i n g ID).
guesses an immediate predecessor ID'of ID2i+2
t h e r e i s a p a t h i n T-(x) between ID2i+l
5
32i+lminimal
v2i+2.
(If
i s a h a l t i n g I D t h i s i s omitted.)
t/2i+2 segments. (a)
M branches u n i v e r s a l l y over 1 a l l p o s s i b i l i t i e s f o r 32i+3 minimal I D ' S
(b)
a l l p o s s i b i l i t i e s f o r an I D i n
,
( a t most)
which use work space s
v2i+2
which uses work space s , and which
by t h e performance of some i n s t r u c t i o n from P y i e l d s an I D which uses space e x a c t l y s + 1, and
a l l p o s s i b l i t i e s f o r an I D i n
(c)
n o t ha1 t i n g .
v2i+2
guesses whether o r n o t t h e r e i s a path 1 I f it which uses time T(n)fm and space T(n)/f.
Branches of type ( a ) . For each ID2i+3 in
\d2i+2from
ID2i+2
guesses "no" then ''yes"
then M
1
to ID2i+3
5 verifies
which uses work space a t most s and i s
M
t h i s by s i r m l a t i n g the
branches u n i v e r s a l l y (except i f ID2i+3
- TM MIrn. I f i t guesses 2m+l i s a h a l t i n g ID). Along one
TI
branch i t checks t h e e x i s t e n c e of the p a t h by s i n u l a t i n g the Z the o t h e r branch (which is n o t used i f ID2i+3 a segment i n
32i+3.
Branches of type ( b ) .
M
1
2m+l
i s a h a l t i n g ID) M
1
-
TM
MIrn.
Along
s i m u l a t e s M on
checks t h e r e i s "0 p a t h t o the r e l e v a n t I D from I D
2i+2
Bounded Arithmetic Formulas and Turing Machines of Constant Attention which uses only t i m e T(n)? Branches of type ( c ) .
t/2i+2 which
n
M
uses t i m e
1
369
-
TM M" and space T(n)/f by s i m u l a t i n g the T 2m+l m checks chere i s "0 path t o the non-halting I D i n
e x a c t l y T(n)fm and space T(n)/f by simulating a
- TM which i s a minor v a r i a n t of M"
2m+l
m'
Correctness. C l e a r l y i f M a c c e p t s x then M1 can a l s o accept by guessing f W e onst a l s o check t h a t if M
a c c e p t s x than M w i l l a l s o accept.
1
t/2i+2 segments,
that could a r i s e i s i n t h e simulation of ID2i+3
.
-
f(n).
The only d i f f i c u l t y
where a path t o
some
on a non-accepting branch might be overlooked by M1 i f f i s guessed too
l a r g e (because them t h e s i m u l a t o r used i n the
type (a) branches may allow too
l i t t l e space) o r i f f i s guessed t o small (because t h e l i t t l e time).
However i f t h e whole computation
branches of type (b) ensure space
of M
1
simulator may allow too
i s accepting, then the
s i s s u f f i c i e n t , and those of type (c) ensure
t h a t enough t i m e i s a v a i l a b l e .
v
and M" Ignoring t h e s i m u l a t i o n of M ' w e s e e t h a t each p a i r 3, of m m' segments of M ' s computation c o n t r i b u t e s 4 segments 3,t/ t o t h e branches of
],v,
M 's computation.
1
Taking i n t o account M '
a d d i t i o n a l segments.
Thus
5
m
i s indeed a E
and M"
m
c o n t r i b u t e s a t most 2m + 1
2k+2m+l-
-
REMARK The condition f ( n ) < T ( n ) / l o g n can be removed a t the expense of i n c r e a s i n g K. To see this n o t e t h a t without loss of g e n e r a l i t y we may suppose f ( n ) ,< T(n) Let
E
k
-
.
f l ( n ) = f(n)m'(m+l)
S T ( n ) / l o g n.
Then
TIME (T(n)f(n)m)* SPACE ( T ( n ) / f ( n ) ) C Ek-
TIME ( T ( n ) f l ( n )
m+ 1
)*SPACE ( T ( n ) / f l ( n ) ) .
Some s p e c i a l cases of theorem 3.2 can be found i n the l i t e r a t u r e .
[ I 4 1 states (modulo COROLLARY 3.3
u
1
kS1 k
-
Kannan
TIME(n) = A o ) :
L e t c and E be c o n s t a n t s with 0 < E < 1. Then
1
k
-
TIME (nC)* SPACE (nl-€)
c
A ~ .
TM's t h i s w a s proved e a r l i e r by Nepomnyashcii [ 1 9 ] . Since t h e r e i s For E 1 no p o i n t i n a L - IM r e p e a t i n g an I D , any C - IM which works i n space log n a l s o 1 I c works (simultaneously) i n space l o g n and t i m e n f o r some c , s i n c e t h e r e a r e only that many d i f f e r e n t I D ' S .
COROLLARY 3.4 ( N e p o w y a s h c i i [ 2 0 ] ) El-SPACE
The space-time-alternation COROLLARY 3.5 and f ( n ) Ek-
-+
( l o g n ) 5 A.
t r a d e - o f f approach t o h i e r a r c h y problems.
L e t k,m b e p o s i t i v e i n t e g e r s , T(n) be f u l l y t i m e c o n s t r u c t i b l e ,
as n
+
-.
Then
TIMe(T(n)f(n)m)*SPACE(T(n)/f(n))
4
u
K21
C K- TIME(T(n)).
A. WOODS
370 = f ( n ) d(m+').
- TM U which works i n time k T(n)fl(n)mcl and space T ( n ) / f l ( n ) , and i s a u n i v e r s a l machine f o r the c l a s s
Proof:
L e t f,(n)
Y = Z By theorem 3 . 2 , L
Z
E
U
k
- TIME K
-
There i s a C
( T ( n ) f ( d m ) * SPACE ( T ( n ) / f ( n ) ) .
TIME(T(n)) and can thus be used t o d i a g o n a l i s e over Y.
-
I n p a r t i c u l a r i f T(n) is f u l l y t i m e c o n s t r u c t i b l e and f o r each k t h e r e
is some K,m and some f u n c t i o n f ( n ) Zk-
+
TIME (T(n)) C Zf
such t h a t
TIME (T(n)f(n)m)* SPACE ( T ( n ) / f ( n ) )
then the Ck- TIME (T(n))hierarchy i s s t r i c t .
x
E
then t h e r e is a A
X
Ao.
by C
1
-
Let C
z
k
-
Similarly i f
TIME (n f ( n ) m ) * SPACE ( n / f ( n ) )
formula which i s u n i v e r s a l f o r ( a s u p e r s e t ) of X
TIME (T(n))denote t h e c l a s s of a l l languages L which a r e accepted
1 1 TM's w i t h only 1 work tape.
Nepomnyashcii [20]has observed t h a t , done with
c a r e , t h e method of Hopcroft and Ullman [ I 2 1 shows t h a t f o r 0 <
zlwhere 0 <
<
E'
and consequently
TIME^
(n
2-E
)
E
< 1,and some C ,
c El- TIME (nC)* SPACE (nl-€')
4 , and t h a t consequently f o r some k, C - TIMEl(n2-E) c C - TIME (n) s A . 1 k
This has t h e c u r i o u s consequence: CORALLARY 3 . 6
I f A O = El- TIME (n) then
VE
>
o
(
zl- TIME^
(n
2-E
)
c zl-
TIME ( n ) ) .
I t i s of course known t h a t any n o n d e t e r m i n i s t i c m u l t i t a p e
TM which works 2
i n l i n e a r t i m e can b e simulated by a 1 tape Cl- TM which works i n t i m e n
(see e.g.
[I311 and that t h i s q u a d r a t i c i n c r e a s e i s c l o s e t o optimal f o r c e r t a i n
languages. (See Mass 1181 .) However a converse i n t h e form of t h e n o n l i n e a r speed up suggested by c o r o l l a r y 3 . 6 seems somewhat implausible, so t h i s may r e p r e s e n t
a p o s s i b l e approach towards proving El- TIME (n) # nl- TIME (n). 4.
A
WEAK .A
HIERARCHY THEOREM
For p >, 1 odd. and x an i n t e g e r , l e t 1x1
LEMMA 4.1
Let k
E
N.
There i s a language
that f o r any q u a n t i f i e r f r e e A that
'4 t
(x0# xl#
P
4, E
denote the i n t e g e r s a t i s f y i n g :
DSPACE (log n ) w i t h the p r o p e r t y
formula e ( x ,xl,...,\)
... #\
He,,,
E
\
++
e(xo,xl
there e x i s t s e
,...,\))
e
E
N such
Bounded Arithmetic Formulas and Turing Machines of Constant Attention Discussion. +(xO,x1,...,\)
Wilkie's h i e r a r c h y theorem [261 i s based on the
371
s p e c i a l case i n which
i s of the form
F(x , x l , . . . , \ ) = 0 (4.1) + where F(x) i s a polynomial with i n t e g e r c o e f f i c i e n t s . One considers a D I M M which, + given x ,x l,...,\ and a Godel number f o r F, computes IF(x) I f o r a s u f f i c i e n t P number :f s m a l l primes p. I f t h e product P = n p i s so l a r g e t h a t IF(;)/< P P then
):(F
+
=
0
++
F(;)
: 0 (mod P)
++
vp(F(:)
I
0 (mod p ) ) .
I n our case +(x) can be assumed t o be a boolean combination of a f i n i t e number of polynomial i n e q u a l i t i e s : F(xo,x l,...,xk)
z 0
(4.2)
+ We can s t i l l c a l c u l a t e the numbers IF(x)
.
I The problem i s how t o use the + P r e p r e s e n t a t i o n of a number X (= F(x), say) by i t s r e s i d u e d i g i t s 1x1 t o determine P whether o r n o t X 5 0 while s t i l l keeping t h e space requirement s u b l i n e a r so t h a t the trade-off
theorem a p p l i e s .
The d i f f i c u l t y of determining t h e s i g n of X from i t s r e s i d u e d i g i t s i s one
of two fundamental problems which p r a c t i c a l computation.
have l i m i t e d t h e use of modular a r i t h m e t i c i n
(The o t h e r i s t h e r e l a t e d problem of d e t e c t i n g overflows.)
These problems were s t u d i e d i n the 196O's'in connection with the design of a r i t h m e t i c u n i t s f o r computers.
The algorithm described below is based on a s i g n
d e t e c t i o n scheme given by Eastman [ 7 1 who d e t a i l s i t s p r a c t i c a l implementation, complete with i n s t r u c t i o n s as t o which w i r e s t o feed through which cores t o b u i l d the arithmetic unit.
141,
I f Y i s an i n t e g e r r e l a t i v e l y prime t o p, l e t satisfying Y Z :X (mod p ) ,
If Y
- 2p < Z
f
denote t h e i n t e g e r Z
.
Y .Y are r e l a t i v e l y prime t o p then 1' 2
so i t makes sense t o w r i t e
X
Consider a system of p o s i t i v e , odd, pairwise r e l a t i v e l y prime, moduli
P~,P~,...,P,.
Let
A. WOODS
312
i
(mod Pi)
(4.3)
'j This is j u s t the f a m i l i a r formula f o r s o l v i n g "Chinese remainder" problems
-
it
can be v e r i f i e d by e v a l u a t i n g both s i d e s (mod p.) f o r a l l j . J (4.3) s a y s t h e r e i s some i n t e g e r t such t h a t
x
i
= t Pi
-j& P i j
.
P2
J
We denote t h i s i n t e g e r t by Remark. I t would n o t m a t t e r f o r our purposes, -
b u t i t i s easy t o check t h a t
c a n c e l l i n g a common f a c t o r of X and P. does n o t e f f e c t the value of
were only X number
interested i n
,Y
prime power moduli we could d e f i n e [$]for
> 0, by taking pl,
...,pr
t o be
I[till.
I f we
any r a t i o n a l
the p r i m e powers i n the decomposition
of Y.
I f i i s small, then [$]is
a good i n t e g e r approximation to
po was d e l i b e r a t e l y l e f t o u t of the
products P . so we can use
by means of
1x1
/Lo P. -j=1 E p j
Definition.
LEMMA 4.2
then
Suppose p
?P.
1 I
> i ( 2 p i + 1). I f
'ij
pj
1
P,
p -i {XI< 0P. 2
1
. 1x1
The modulus PO
to calculate
Bounded Arithmetic Formulas and Turing Machines of Constant Attention
Since
[q
approximates
X p.
$ , the sign
t o within
pol
of
1
t h e s i g n of
-, X
( i ) and ( i i )
.
pi
313
and t h e r e f o r e of X, except i f
<
Ti
.
This proves
I n t h e exceptional c a s e
Therefore
1x1
<
i P . = i p. Pi-l
that
1x1
< (-
po-i 2
Po- i
7. i t follows
and s i n c e , by assumption, i p .
) 'i-1
which e s t a b l i s h e s ( i i i ) . The s i g n t e s t . Suppose we a r e given po > i (2pi
+
1x1 , 1x1 Po
1 ) f o r a l l i , and
p1
,..., 1x1
,,...,pr
t h a t i t i s known i n i t i a l l y t h a t I X I <
Apply t h e lewa w i t h i = r t o test the s i g n of X. ( i i i ) ensures t h a t ( X I <
and po,p
Pr
Pr-l
so
Either the
....1 ( a s
a r i t h m e t i c modulo the p . ' s ,
1x1
....I
PO
.
The important p o i n t i s
computation
2
P
r
.
t e s t succeeds o r
f a r as necessary) we
e i t h e r d i s c o v e r t h e s i g n of X o r deduce (when i = 1) t h a t I X I <
1x1
P -r
2-
w e can apply t h e lemma a g a i n with i=r-1.
2 Continuing i n t h i s way through i = r,r-l,r-2. the s i g n of X i s given by
such t h a t
of t h e numbers
"0 , i n which
2 t h a t by using
case
BYpo
from
involves s t o r i n g ( a p a r t from the i n p u t ) only a c o n s t a n t Pr number of i n t e g e r s whose magnitude i s r e s s than p2 Therefore the space used PO'
)XIpl,
XI
i s l e s s than c l o g p
.
.
The e x i s t e n c e of s u i t a b l e moduli f o r proving
lemma 4.1 is guaranteed
by weak v e r s i o n s of t h e prime number theorem (see e.g.
[ 91) .
W e write f ( n ) % ! h(n)
A. WOODS
314
i f t h e r e a r e c o n s t a n t s c1 > 0,c2 such that
clh(n) < f ( n ) < c2h(n) f o r a l l
s u f f i c i e n t l y l a r g e n. PRDPOSITION 4 . 3
~ ( n )%
(Chebychev)
.
log n
Also,
The number n(n) of prime number
6 n satisfies
I: log p % n, t h a t i s n P = 2h ( n ) psn psn p prime p prime
where h(n) % n
.
Proof of lemma 4.1: M which i s given input x # x # 0 1
Consider a DTM
G d e l number f o r some q u a n t i f i e r f r e e f o r n u l a
... #$#'$'
where '$'is
6 i n some s u f f i c i e n t l y easy t o
decode Gadel numbering, with each $ having i n f i n i t e l y many code numbers. be designed s o t h a t f o r i n p u t s of l e n g t h n i f and only i f JI (xo,xl, '$')
*
choosing e = '$'
VZ
...,x).
c
*'
Since the c o n s t a n t c
(J
... #x#eJI
*
++
*
... # \ # r $ '
x # xl#
M accepts
l a r g e enough w i l l ensure n > c
(M a c c e p t s xo# x l #
M f i r s t breaks
>
M will
depends only on $ (not on
and
therefore
*&)
up i n t o a boolean combination of a f i n i t e number of
polynomial i n e q u a l i t i e s of the form F ( = ) > 0
.
These r e q u i r e n e g l i g i b l e s t o r a g e
space and a r e t e s t e d s e q u e n t i a l l y , so only the space required f o r a s i n g l e t e s t M generates an odd prime p
need be considered.
n2 ( t h e e x i s t e n c e of p
is
< P2 < .-.<
guaranteed by p r o p o s i t i o n 4.3) and uses the d i s t i n c t odd primes p
P,
(some easy t o compute approximation of ) c n l o g log n. Only a fixed 1 number of t h e p . ' s a r e s t o r e d a t any one time, but they can of course be generated l e s s than
whenever necessary.
i n sequence using space c 2 log n (mod pi) M can
generate IF(xO,xl,...,
By p r o p o s i t i o n 5.3, r %
n lo
5' IPi
lo n 18, ng
a
S i m i l a r l y , using a r i t h m e t i c
i n space c 3 log n whenever required. so r(2pr+ 1)
4 (
n lo
lo
fog
n ) (n log log n)
n2 i t follows t h a t f o r n s u f f i c i e n t l y l a r g e , po > r ( 2 p r + 1) + so the f i r s t c o n d i t i o n of lemna 4.2 i s s a t i s f i e d . Since F(x) i s a polynomial
A s po
IF(:))
p,-r 2
some c o n s t a n t cF which depends on F. log log n Therefore f o r n > c
d ZCFn f o r
,< (max (:))CF
But by p r o p o s i t i o n 4.3,
p
2
2'"
r
.
* '
so the i n i t i a l c o n d i t i o n f o r applying the s i g n t e s t i s s a t i s f i e d .
compute
Thus M can
t h e t r u t h value of $(x) u s i n g only space c log n, which can be reduced
t o log n by the standard tape compression argument ( s e e e.g. THEOREM 4.4 every A
,
Let k
E
N.
There i s a A
[131 ) .
formula e k ( x , e ) with the property t h a t , f o r
formula @ ( x ) with a t most k q u a n t i f i e r s , t h e r e i s some e
@
L
N such t h a t
.
Bounded Arithmetic Formulas and Turing Machines of Constant Attention
Proof: By
375
i n c r e a s i n g k ( i f necessary) i t s u f f i c e s t o consider the case where
$(x) i s of the form
3 y1
... QY,
,c x V y 2 5 x
6 x Jl(x.;)
w i t h J, q u a n t i f i e r f r e e . Let
4, be
the language i n lemma 4.1 and l e t
8(x0,x1
,...,%,e)
be the A
formula corresponding t o Lk which e x i s t s by Nepomnyashcii's theorem ( c o r o l l a r y 3.4 p l u s remark ( i i ) a f t e r theorem 2.4).
Then f o r every q u a n t i f i e r f r e e A
formula
e
,...,yk )
Jl(x,yI,y2
vxv; Take
0 (x.e)
k
to be3y1
-...
t h e r e i s some ( ecx,;,
:x v y 2
5 x
N
E
6
&.e* Qy,
0
such t h a t ))
.
x B(x,;,e)
$
Of course theorem 4.4 g e n e r a l i s e s t o t h e case where x i s replaced by a sequence of v a r i a b l e s x1,x2, x m We a r e now ready t o prove t h e e x i s t e n c e o f a A. formula $(x) which
.... .
cannot be defined u s i n g
only k
quantifiers.
Proof of theorem 0.1: Let Then
Bk(x,e) be as i n theorem 4.4 and l e t
$(x) i s n o t e q u i v a l e n t t o any A
For by theorem 4.4 there e x i s t s e 6(e )
6
so the t r u t h v a l u e s of
6
formula
$(x) and
-
c N such t h a t v x ( $(x)
Bk(e6,e6)
++
$(XI be the
A
formula-Bk(x,x).
$(x) with a most k q u a n t i f i e r s .
i9(e6)
++
B (x,e6 )). Therefore
k
,
6(x) d i f f e r a t x = e
6'
5. A CONJECTURF.. The a u t h o r b e l i e v e s t h a t by an e x t e n s i o n of the methods i n t h i s paper i t should be p o s s i b l e t o prove: CONJECTURE: There i s a A. quantifier free A
formula
B(x,e) with the property t h a t Dr every -+ $ ( x , y ) , there i s some e L N such t h a t 6
formula
v x ( e(x.e6)
- 3;
6 x
~l(x,;)).
From t h i s i t would follow t h a t f o r each k t h e r e i s which i s n o t e q u i v a l e n t t o any 3y1 with
s
XVY2
6 quantifier free.
,c x
a A
formula
$(x)
formula of the form
...VYZk :x 3;2k+l ,< x
6(x.;)
Since an a r b i t r a r i l y l a r g e f i n a l block of e x i s t e n t i a l
q u a n t i f i e r s is allowed, a proof would show the e x i s t e n c e of a bounded Diophantine
set whose complement i s n o t bounded Diophantine.
A. WOODS
316
Acknowledgements: The a u t h o r would l i k e t o thank P r o f e s s o r s Paris and Wilkie f o r p o i n t i n g o u t t h a t theorem 0.1 w a s open, and P r o f e s s o r John Crossley and t h e l o g i c group a t Monash f o r making a v a i l a b l e the f a c i l i t i e s which made i n i t i a l work on t h i s research p o s s i b l e . REFERENCES
.
Bennett,J.H.,
On S p e c t r a , Ph.D. D i s s e r t a t i o n , P r i n c e t o n ( l 9 6 2 ) .
Book, R.V., and Greibach, S.A., Theory, 4(1970) 97-111.
Quasi-realtime languages, Math. Systems
Borger, E., Note on bounded Diophantine r e p r e s e n t a t i o n of s u b r e c u r s i v e s e t s , i n Recursion Theory ( I t h a c a , New York 19821, Proc. T h i r t e e n t h Summer Research I n s t i t u t e h e r . Math SOC.. Chandra. A.K., Kozen, D.C.. Mach. 28(1981) 114-133.
and Stockmeyer. L . J . ,
A l t e r n a t i o n , J.Assoc. Comp.
Davis, M. ,Matijasevir,Y., and Robinson, J., H i l b e r t ' s t e n t h problem. Diophantine equations: p o s i t i v e a s p e c t s of a n e g a t i v e s o l u t i o n . A m e r . Math. SOC. Proc. of Symposia i n Pure Math. 28(1976) 323-378. Dimitracopoulos, C . , M a t i j a s e v i r ' s Theorem and t h e s i s , University of Manchester (1980).
Fragments of Arithmetic, Ph.D
Eastman, W.L., Sign determination i n a modular number system, i n Proceedings of Harvard Symposium on D i g i t a l Computers and t h e i r Application, 3-6 April 1961. Harvard University P r e s s (1962) 136-162. Gaifman, H., and Dimitracopoulos, C . , Fragments of Peano's a r i t h m e t i c and the MRDP theorem, i n Logic and Algorithmic, Monographie No.30, L'Enseignement Mathematique. Hardy, G.H., and Wright, E.M., Oxford University P r e s s . Harrow, K., 102-117.
An I n t r o d u c t i o n t o the Theory of Numbers,
The bounded a r i t h m e t i c h i e r a r c y , Information and Control 36(1978)
Hodgson. B.R., and Kent, C.F., A normal form f o r a r i t h m e t i c a l r e p r e s e n t a t i o n of NP-sets. J.Computer System S c i 27(1983) 378-388. Hopcroft, J.E., and Ullman, J . D . , R e l a t i o n s between t i m e and tape complexities. J.Assoc. Comput. Mach. 15(1968) 414-427. ' , I n t r o d u c t i o n t o Automata Theory, Languages, and Computation. Addison-Wesley (1979).
Kannan, R., Towards s e p a r a t i n g n o n d e t e r m i n i s t i c time from d e t e r m i n i s t i c time, IEEE 22nd Annual Symposium on Foundations of Computer Science (1981) 235-243. Lessan, H.,
Models of Arithmetic, Ph.D.
t h e s i s , U n i v e r s i t y of Manchester (1978).
Lipton, R.J., Model t h e o r e t i c a s p e c t s of computational complexity, IEEE 1 9 t h Annual Symposim on Foundations of Computer Science (1978).
Bounded Arithmetic Formulas and Turing Machines of Constant Attention
311
Manders, K.L., and Adleman, L., NP-complete decision problems for binary quadratics, J.Computer Systems Sci. 15(1978) 168-184. Mass, W., Quadratic lower bounds for deterministic and nondeterministic onetape Turing machines, Proc. of 16th Annual ACM Symp. on Theory of Computing (1984) 401-408. Nepowyashcii, V.A., Rudimentary interpretation of two-tape Turing Computation, Kibernetika (1970) No.2, 29-35. Translated in Cybernetics (1972) 43-50. Nepomnjascii, V.A., Rudimentary predicates and Turing calculations, Dokl. Akad. Nauk SSSR 195(1970). Translated in Soviet Math. Dokl. ll(1970) 1462-1465. Paul, W.J., Pippenger, N.,Szemeredi. E., and Trotter, W., On determinism versus nondeterminism and related problems,IEEE 24th Annual Symposium on Foundations of Computer Science (1983), 429-438. Paul, W.J., Prauss, E.J., and Reischuk, R.. (1980) 243-255.
On alternation, Acta Inform. 14
Paris, J.B., and Dimitracopoulos, C., Truth definitions for A
formulae, in
Logic and Algorithmic (Zuric 1980). Enseign. Math. 30(1982) 317-329. Parikh, R., Existence and feasibility in arithmetic, J.Symbolic Logic 36 (1971) 494-508. Pudlak, P.. A definition of exponentiation by a bounded arithmetical formula. Commentationes Mathematicae Universitatis Caroline, 24(1983) 667-671. Wilkie, A.J., Applications of complexity theory to L - definability problems in arithmetic, in Model Theory of Algebra and Arithmetic (Proceedings, Karpacz, Poland 1979), Lecture Notes in Math. 834(1980) 363-369. Wrathall,C., Rudimentary predicates and relative computation, SIAM J.Comput. 7(1978) 194-209.
