Pacific Journal of
Mathematics
THEORIES WITH A FINITE NUMBER OF MODELS IN AN
UNCOUNTABLE POWER ARE CATEGORICAL
A LISTAIR H. L ACHLAN
Vol. 61, No. 2
December 1975
PACIFIC JOURNAL OF MATHEMATICS
Vol. 61, No. 2, 1975
THEORIES WITH A FINITE NUMBER OF
MODELS IN AN UNCOUNTABLE POWER
ARE CATEGORICAL
ALISTAIR H. LACHLAN
In this paper are considered complete, countable, firstorder theories which have a finite number of models in some
uncountable power. It is shown that any such theory is either
(^-categorical or ωi-categorical. This confirms a conjecture of
W. K. Forrest.
Some of the background of this theorem is as follows. Shelah
has shown that every non-ω-stable theory has infinitely many models
in every uncountable power. A proof of this will appear in Shelah
[9]. Thus here we may confine our attention to ω-stable theories.
In [7] Shelah showed that a non-αvcategorical, ω-stable theory has
si | α + 11 models in power fctα Another proof of the same result
was published by Rosen thai [6]. Thus the only uncountable cardinals
tc in which a complete countable theory can have a finite number of,
and at least two, models are the \ξn for 1 < n < ω. The methods
used in this paper come from Baldwin and Lachlan [1] and from
several works by Shelah.
1* Preliminaries* Most of our notation is patterned on [8]. An
additional notation used here is <p(M, a) for the set of solutions of
the formula φ(x, a) in the structure M. Also, if A c | M | by cl(A)
we mean the union of all finite subsets of M definable using parameters from A. We call cl(A) the algebraic closure or simply closure
of A. The reader will require some knowledge of the theory of
strongly minimal sets such as may be found in [1].
Let M be a structure. A formula φ(x, a), a e M, is called strongly
minimal if it has an infinite solution set in M and for every elementary extension M' of M, formula ψ(x, y), and beM', one of the
formulas φ(x, a) Λ ψ(x, b) and φ(x, a) Λ —ι ψ(x, b) has a finite solution
set in ikf'
A formula φ(x, α), ae M, is called co-categorical if its solution
set is infinite and for every n < ω there are at most a finite number
of w-types over Rng a realized by ^-tuples in the solution set of
φ(x, a).
If Th (M) is superstable, φ(x, a) is ^-categorical in M, and
A c Iilf I is finite, then φ(x, a) is also ω-eategorical in W the expansion of M obtained by adjoining names for the members of A.
This follows from the following definability result: Let Ύh(M) be
465
466
ALISTAIR H. LACHLAN
super stable, be M, and A be a subset of \M\ definable without
parameters, then there exists ce A such that if φ(z, y) is any formula
with lh (b) = lh (y) then some equivalence relation ^ on M is definable
from c such that A/ξ? is finite and A Π φ{M, b) is ^-closed. We
shall not prove this here. A stronger theorem in which A need not
be definable will appear in [9]. Besides, if one looks closely at the
proof of our main theorem it is apparent that we can avoid using
the fact that an ω-categorical formula remains ω-categorical when
a finite number of elements are named.
In passing we note that if φ(x, a) is ω-categorical in M and
Th (M) is stable it may not be the case that φ(x, a) is ω-categorical
in every inessential expansion of M. For example, consider the case
in which the language consists of binary relations Ro, Bu
. Let
ω
ω
\M\ = ω[J ω and for i < ω let Rf{a,b) hold just if ae ω and
a(i) = b. Then Th (M) is stable and 3yR0(y, x) is α)-categorical in M
ω
but not in M' obtained by adjoining a name for a member of ω
which has infinite range.
If φ(x, a) is an s.m. formula in M, its dimension in M denoted
dim (φ, (x, a), M) is the cardinality of a minimal set Aaφ(M, a) such
that φ(M, a) c cl (A U Rng a). From the theory developed in [1] the
choice of A makes no difference. Likewise we talk of independent
sets of solutions of φ{x, a), of a basis for φ(x, a) in a particular
model and so on. We adjoin names for the members of a and then
use the definitions already made for formulas without parameters.
We call φ(x, b) a copy of φ(x, a) if 6 realizes the same type as α.
All theories considered are assumed to be complete and countable.
2* Stability and strongly minimal formulas* In this section
are presented some lemmas which will be useful for the proof of
the main theorem. Most of the lemmas have been proved in a more
general context than is necessary for their application in this paper.
This is because of our continuing interest in the theory of stability
in general and strongly minimal formulas in particular.
LEMMA 1. Let A be a subset of a model of a stable theory T.
There exists a model M of T such that Aa\M\
and further:
for
each aeM and formula φ(x, y) there exist ceA
and θ(x, z) such
that \= θ(a, c) and for all be A
{\=φ(a, b)} =*
{\=Vx[θ(x, c)
> φ(x, b)]} .
This was discovered independently by Shelah and the author.
Shelah was the first to formulate it explicitly [8, p. 275]. A proof
THEORIES WITH A FINITE NUMBER OF MODELS
467
of it can be extracted from the second half of the proof of the
model extension theorem in [2].
LEMMA 2. Let Th (M) be ω-stable, N be an elementary extension
of M, ae N, φ(x, a) and ψ(x) be formulas. Further, suppose φ(x, a)
implies ψ(x) and that φ(N, a)f]\M\ = 0 then φ(x, a) has rank less
than that of ψ(x).
This can be proved using the same technique as in the proof
of Lemma 7 of [2]. In the application we make of this lemma M
is the prime model and φ(N, a) Π | M | = 0 is satisfied because φ(x, a)
has no solution realizing an isolated type over 0 .
A formula φ(x) is said to have the Vaught property in a theory
T if there are models M and N of T such that N is a proper
elementary extension of M, φ(M) = φ(N), and | φ{M) | ^ ω.
LEMMA 3. Let T be a superstable theory and φ(x) be a strongly
minimal formula which has the Vaught property. Suppose further
that φ(x) has infinitely many algebraic solutions. Then there exist
arbitrarily large models in which the dimension of φ{x) is finite.
Proof. Since φ(x) has the Vaught property and T is stable
there exists an arbitrarily large model M of T with | φ(M) \ = o).
This is immediate from the extension theorem of [2]. By Theorem
3.1 of [7] if such M is sufficiently large it contains a subset I of
power ω indiscernible over φ{M).
We shall first establish that cl (I) Π φ{M) has finite dimension.
For proof by contradiction suppose the contrary. From Theorem
6.13(A) of [8] it is clear that if ί o c J is finite then cl (70) Π ψ{M)
has finite dimension. Also from the indiscernibility of I over φ{M),
cl (70) Π φ{M) depends only on \I0\. Let e0, elf
be an enumeration
of a basis of cl (I) Π <p(M), and a0, au
be an enumeration of I.
For each n < ω let m(n) be the least number such that {β0, ely
, en}a
cl {a0, au
, am{n)).
It is clear that m(n) is nondecreasing and
unbounded as n increases. Since it will simplify the notation without
materially affecting the argument we shall suppose that m{n) — n.
Let θn(x, aQ, au
, an) be a formula with only a finite number of
solutions one of which is en. Let ψn(x) denote the formula:
#o(eo, x) Λ θfa,
ao,x)Λ--Λ
0n(en, α 0 , alf
, an-l9 x) .
Notice that en+ί, en+2,
all realize the same type over {et\ i ^ n) U
{α^: i^n}. By suitable choice of θn+1 for any sufficiently large finite wczω
\= -i Ix A {θ«+1(em, a0,
, an, x): mew}
.
468
ALISTAIR H. LACHLAN
Thus the formulas ψn(x)Aθn+i(em, α0,
, an9 x) for m = n + 1 , n + 2,
which all have the same degree in the sense of Shelah [8], and the
first of which is φn+ί(x), witness that Deg ψn+ι < Deg ψn. This contradicts the superstability of T by Corollary 6.10 (A) of [8]. Thus
cl (I) Π φ(M) does indeed have finite dimension.
To complete the proof of the theorem extend I to a set /',
indiscernible over φ(M), of any desired cardinality and apply Lemma 1
to cl (/').
LEMMA 4. Let T be a complete theory and φ(x) be a strongly
minimal formula which does not have the Vaught property, then
T is a)^categorical.
Proof. From [1], see proof of Lemma 8, p. 83, the present
lemma is obvious when T is ω-stable. Suppose therefore that T is
not α>-stable. Form an increasing sequence M(Q), ikΓ(l),
of countable models of T such that \S(M(0)) \> ω and such that for all i
<p(M(i +1)) Φ φ(M(i)). Let M = \J {M(i): i < ω}. Let {φt(x9 a*): i<ω)
be an enumeration of all formulas having only x free and parameters
from M, similarly let (ψt(x9 y, b*): i < ω> be an enumeration of all
formulas having at most x9 y free. We construct a sequence of
triples «#*(#, cl), nίy a^'.i < ω) such that for all i the following five
conditions hold:
(i)
(ii)
ni < ni+ι < ω, c* e M{n%), and at e φ(M(ni+ι))
i+1
l
θi+1(x, c ) implies θ^x, c )
ι
1
1
(iii) θi(x, c ) implies one of the formulas φt(x, a ) and —i φt(x, a )
(iv)
_(v)
I S ( I W ) n {p: θt(x,
ci)ep}\>ω
if N *x*y(θt(x, c') Λ φt(x, y, bι) A φ(y)), then θt(x, c*) A ψt(x9
i+1
aif 60 is consistent with >ω members of S(M(nt)) and θi+ι(x9 c )
implies ψt(x9 ai9 6*)The only part of the construction which is not straightforward
is the choice of α4. Suppose the triples prior to {θt(x9 c% nt9 a^ have
all been suitably chosen, and that θt(x9 c*) and nt have been chosen
so that (i)-(iv) are satisfied. Call a formula χ(x, a) good if its
parameters are in M(ni)9 it implies θt(x9 ci)9 and
I S(M(n<))
Π {p: χ ( x ,
d)ep}\>ω.
Notice that if χ(x, a) is good, then there are disjoint good formulas
1
χo(x, α°) and χx(x9 a ) such that χ(x, a) is equivalent to χo(x, a°) V
1
Zi(», α ). Suppose there is a good formula χ(xt a) such that
t= -i lxly(χ(x, a) A fi{x, y9 b*) A φ(y)). Then we can satisfy (v) by
replacing θt(x9 cι) by χ{x, a). The choice of at is immaterial. If for
some good χ(x, a) the solution set of 3x(χ(x, a) A fi(x, V, b*) A φ{y))
THEORIES WITH A FINITE NUMBER OF MODELS
469
is nonempty but finite, then for some b e φ(M(nτ)), X(x, a) A ψi(x, b, δ*)
is good. Thus to satisfy (v) in this case we just take at = 6. Otherwise let ai^φiMiyii + 1)) — φ{M{n$). For each good χ(x, a) we now
have from the strong minimality of φ(x)
N lx(χ(x, a) A fi(x, aif b*) A φ(at)) ,
whence this choice of at will again satisfy (v).
Let a realize the type over M generated by the formulas θt(x, c*).
By the omitting types theorem there is a model M' D | M\ U {a} such
that φ(M') = φ{M). Thus φ{x) does have the Vaught property.
5. Let T be a stable theory for which φ(x) and ψ(x)
are s.m. formulas.
Further suppose that each of these two formulas
has infinitely many algebraic solutions. Then either in every model
of T φ{x) and ψ(x) have the same dimension, or each model M of
T has an elementary extension M' such that ψ(x) has the same
solution set in M! as in M while φ(M') = cl (φ(M) U {b}, M) for some
b in I M'\ — I M\. The former case holds if and only if there is a
formula χ(x, y) such that the following are valid in T:
LEMMA
χ(x, y) — > φ(χ) A φ(y), i<ωy{χ(χ, y) A f(y))
and 3-ωy3xχ(x, y).
Proof. To prove the first part of the lemma we attempt to
construct M'. Let A = \M\ U {&} where b is a solution of φ(x) not
in M. Apply Lemma 1 to A to form M'.
Suppose that ψ(x) has a solution a in | M' \ — \ M\. Taking
τjr(x) A x Φ y0 as <p(x, y) in the statement of Lemma 1 we find a
formula
θ(x, y, z) and
ceM
such t h a t
N θ(a, b, c)
and θ(x, b, c)
has
no solution in M. We can suppose that θ(x, y, z) implies ψ(x) A ψ{y)
Clearly N lxθ(x, b, c), b is not algebraic over \M\9 and φ is s.m.
k
It follows that for some k < ω, |= 3 y -π lxθ(x, y, c).
Similarly,
ι
N 3α? —i 3yθ(x, y, c) for some I < ω. Now all the solutions of θ(x, b, c)
satisfy ψ{x) and none are in M, hence there are at most a finite
number, say m. We may suppose that
μ Vy(lyθ(x, y, c)
> lmxθ(x, y, c)) .
Since cίJlf, θ(a, y, c) cannot have a solution for y in Mf whence
θ(a, y, c) has only a finite number since φ{x) is s.m. Hence we may
suppose that for some n < ω
N Vx(3xθ(x, y, c)
> lmyθ(x, y, c)) .
From Theorem 3.1 (A) ([8], p. 306) we can find a formula θ(x, y, z')
470
ALISTAIR H. LACHLAN
and parameters c' in φ{M) U ψ(M) such that
N 3fc2/ -i 3a0'(α, y, c') Λ 3'a? -. 3y0'(Bf y, c') A
VxVy(θ'(x, y, c') — ψ(x) Λ 9>(»)) Λ Vy(lxθ'(x, y, c') -* lmxθ'(x, y, c'))
A Vx(lyθ'(x, y, c')
>Tyθ'(x, y, c')) .
To see this we apply the theorem of Shelah just cited taking A to
be φ(M) U ψ(M), φ(x, y) to be θ(x, y, z) where z plays the role of x
and (x, y) the role of y, and p to be the #-type of c.
Since both φ(x) and ψ(x) have infinitely many algebraic solutions
there is a formula τ(z') with a finite nonempty set of solutions any
of which will serve as c'. The parameter less formula 32'(r(δ') Λ
*'(»> y9 5')) defines a correspondence between the solution set of ψ(x)
and the solution set of φ{x). The correspondence is finite-one in both
directions, all but a finite number of solutions of ψ(x) have mates in
φ(x) and vice versa. In a given model of T let X be a basis of ψ(x).
Form Y by choosing for each e in X one of its mates under the correspondence to be a member of Y. It is easily seen that Y is a basis
of φ(x) and that its cardinality is necessarily the same as that of
X. This shows that the lemma is true when ψ(x) has a solution in
There remains only the task of showing that φ(M') is cl (φ(M) U
{6}, M'). This is immediate from the application of Lemma 1 by
which M was formed. We just take φ(x, y) in the statement of the
lemma to be φ{x) Λ x Φ y0The second part of the lemma is immediate from the discussion
above. We can take χ(x, y) to be 3z'(τ(z') Λ θ'(x, y, z')).
Let φ(x, a) and ψ(x, b) be two s.m. formulas in a model M of
n
T. Adjoin names for the members of Rng(α 6) and then names for
suitable members of φ(M, a) and ψ(M, b) in such a way that if
Aa\M\ is the set named, then | A | <; ω, A is prime over Rng (αn&),
and
I cl (A) n φ(M,a) I = ω = I cl (A) Π φ(M, b) \ .
Let M' be the resulting expansion of M, and T be the corresponding
extension of T. We say that φ{x, a) and ψ(x, b) are linked if and
only if in every model of T they have the same dimension. Note
that this definition is independent of the way in which M! is formed.
Clearly, if φ(x, a) and ψ(x, b) are linked then in any model of
T containing a and 6, φ(x, a) and ψ(x, b) have the same dimension
modulo ω. The following may easily be deduced from the last lemma.
Since it is useful we state it explicitly and leave the proof to the
reader:
THEORIES WITH A FINITE NUMBER OF MODELS
471
LEMMA 6. Let φ(x) and ψ(x) be s.m. formulas of a stable theory
T. Let T be an inessential extension of T. If φ(x) and ψ(x) are
linked in T they are linked in T.
LEMMA 7. Let T be stable and φo(%), •••, 9>»-i(&) be s.m. formulas. For each K > co there exists a model M of T of power tc such
that if φ(x, a) is any s.m. formula of M then \ φ(M, a)\ = it if and
only if φ(x, a) is linked to one of φo(x),
, φn-ι(x).
Proof. Without loss of generality we may assume that each
Ψi{x) contains infinitely many algebraic elements. Choose Ao,
, An^
such that for each i < n, At is an independent set of solutions of
(Pi{x) and | At | = fc. We now form a model M of T including
A — Ao U
U -AΛ_! using the same technique which yields Lemma 1.
In fact I M\ = {δ*: i < fc} where b0, blt
are chosen in turn. To
make M a model we arrange that if a formula φ(a9 biQ,
, δίfc_1, x)
is consistent where a 6 A then there exists j < fc such that (= φ(a, 6ίo,
•••, bik_l9 bd). For the rest we simply choose b{ in such a way that
for any formula φ(x, y) there exists a formula θ(x, z) and c e A U
{bj'i j < i) such that |= θ{bu c) and such that for any beA[j {bά\ j < i)
if μ φψi, b) then \= Vx(θ(x, c) -• <p(x, b)). Using the stability and
countability of T the model M can easily be formed according to the
above prescription.
Let φ(x, a) be an s.m. formula in M whose parameters are from
{bj : j < i) where ω < i < tc. Let C = cl (A U {bό: j < i}) then without loss we may suppose that φ(M, a) Π C is infinite. Suppose
I φ(M, a) Π C | > | i |. Then there exists j <n and a formula %(ajf x) =
^(6, ά°, ά1,
, α3', αy, α?) such that be {bji j < i}9 b extends α, ak e Ak
for each k ^ i,
αy G Aj - cl (Rng 6 (J Rng α° U
U Rng a3')
and X(ajf x) has a solution in
φ(M, a) - cl (Rng 6 U Rng α° U
5
U Rng a ) .
The formula χ(y, x) A φ{x, a) A ψAy) defines a correspondence between
the solutions of φ(x, a) and those of φό{x), whence in every elementary
extension of M φ(xy a) and <Pj{x), have the same number of solutions.
From Lemma 5 any model containing two unlinked s.m. formulas has
an elementary extension in which those formulas have different
numbers of solutions. Thus in this case φ(x, a) and ψj{x) are linked.
Suppose I <p(M, a) Π C \ ^ | i \ then it is easy to show by induction
on j that if i <^ j <ί K, b e A U {bk: k < j} and ψ(x, b) has a solution
in φ(x, α), then it has a solution in φ(M, a) Π C. It follows that in
472
ALISTAIR H. LACHLAN
this case φ(M, a) — φ(M, a) Π C which has power <fc. This completes
the proof of the lemma.
1
8. Let φo(x, α°) and φλ(x, a ) be linked s.m. formulas in
1
a model M of a stable T. Let dim (φt(x, a ), M) — nt < co for i < 2.
1
Then for any model N of T containing a° and a
LEMMA
1
dim (φo(x, α°), N) — dim (φ^x, ά ), N) - n0 — nx
provided that one of the dimensions on the left is finite.
Proof. Suppose the hypotheses are satisfied. For i < 2 choose
j
i
b such that Rng b is an independent set of solutions of Φi(x, a*),
<n
and <Pi(x, a*) has infinitely many solutions in cl(3 &*). Further
1
on
make these choices so that b^b is prime over a a\ Within this
1
restriction let b° and b both be of maximal length. Let mt = lh (&*)
for i < 2. Let T be formed by adjoining names for the members
1
1
of α°, a , 6°, and 6 . Let N be an arbitrary model of T containing
1
α° and a and N' be an expansion of N to a model of T. Let c°
1
1
and c correspond to 6° and 6 respectively under the expansion of
on
0
N to N'. Let c d be a basis for φo(x, a ) in N. We claim that d
is a basis for φo(x, CL°) in N'. If not, then there exist j < co and a
2
suc
formula ^(^°> ^\ ^ » Ψ> %) h that
μ ψ(a\ a\ c\ c\ ώ0,
m
, ds) A ^ zf(a\
a\ c°, c\ 4
, di-i, «)
where d0,
, d^ are distinct members of d. Abbreviate ^(α°, a\ c°,
1
c , %) Λ At** Φofa, a?) to χ(^). From Theorem 3.1 (A) [8, p. 306] the
relation on 9>0W α°) defined by χ(^) can be defined using only parameters from φo(N, a0). Further since there is a model N° of T
0
1
such that 3°, a\ c°, ^eiSΓ and φQ(N°, α^cclίRngα^c^c ), it follows
that there is a formula equivalent to χ(x) containing only parameters
from φo(N, a°) which are algebraic over αonc°. But then we should
have
N θ(d0,
, dj) A l%nzθ(d0,
, dd.lf z)
for some n < ω and θ(x) containing only ά° and c° as parameters.
This contradicts the choice of d. We have shown that
dim (φo(x, α°), N) = dim (φo(x, α°), N') + m0
provided that the lefthand side is finite. From Lemma 5
dim (φo(x, 3°), N') = dim (φfa a1), N').
It follows that dim (φo(x, a1), N) is finite, and hence in exactly the
same way as for φo(x, α°)
THEORIES WITH A FINITE NUMBER OF MODELS
473
dim (φfa, a1), N) = dim fafa a1), N') + m,.
This is enough to prove the lemma.
For the next lemma we are indebted to Shelah who pointed it
out to the author in the course of conversation.
LEMMA 9. Let φ(x) be a formula without parameters, Mo and
M1 be models of an co-stable theory, and F: φ(M0) —> φ{M^ be an
elementary onto map. There is a model M and elementary embeddings Fo, Fx of Mo, M\ into M respectively such that
F0(φ(M0)) = F^Md)
= Ψ{M) .
Proof. We may suppose that F is the identity on φ(M0), that
N is an elementary submodel of both Mo and M1 prime over φ(M0),
and that (| Mo | - | N\) Π (| M, | - | N\) = 0 . We define a theory T*
for the language obtained by adjoining names for all the members
of I Λf01 U I Λfi I as follows. Let θ(x, y) be a formula of T and a be
a tuple in \MX\ — \N\ such that lh(x) = lh(a). From Shelah [8]
Theorem 3.1 (A) there is a formula ψθ,z(y) with parameters from N
such that N θ(a, b) <-• ψθ,ι(b) for all b in N. Let T* have as nonlogical axioms all sentences θ(a, b) where ae \Mι\ — \N\, beMQ, and
N ψθΓa(b). I t is easy to see that T* is consistent, complete, and
extends Th(M0, | Mo |) U Th(Ml9 \ M, |).
A prime model ikf* of Γ* exists because T is ω-stable. The
reduct of ikP to the language of T is the desired model M. One
has to verify that if χ is a formula of T such that 3z(χ(α, 6, z) Λ
φ(z)) is a theorem of T*, where ά e l Λ f J — |iV| and δeilί 0 , then
χ(α, 6, c) is a theorem of T* for some c e φ(N). Using the theorem
of Shelah quoted above let %0(y) and χx{y, z) be formulas with
parameters from N such that for all b'f cf in N (= 3^(Z(^> δ', ^) Λ
φ(z)) ~ XQ{V) and μ X(a, b', c') A φ{c') — XJb', c'). Clearly in N we
have N Vy^zX^y, z) <— X0(y)).
From the definition of Γ* since
3^(Z(α, 6, a;) Λ 9>(^)) is a theorem, we have 1= X0(b) in ilί0 whence there
exists c in 2>(ikf0) such that f= Z^δ, c) in Λf0. From the definition of
T* again we have %(α, b, c) A φ(c) a theorem of T* as required.
Let φo(x, α°) and ^^cc, ά1) be two s.m. formulas in a model M
of T. We say that <£>0(#, ά°) and ^(α;, α1) are linked by the pair
(X(xQ, XU y°, y1), m> if the following formulas are valid in M
X(x0, xl9 α°, a1)
> φo(xo, α°) Λ φ&u a1), ^^x^x,,
and
l*»xyxJi^X(xOf
x» a\ a1)] .
χlf α°, ά1) ,
474
ALISTAIR H. LACHLAN
Let <p(x, a) be an s.m. formula in a model Λf of T we say that
all copies of φ(x, a) are uniformly related if there exists a pair
<Z, m), such that for any model N of Γ, and linked copies φ(x, ά°),
9>(a?f α1) of <£>(#, α) in iV are linked by (X, m>.
10. Let φ(x, a) be an s.m. formula in a model of an
ω-stable theory T. Suppose that all copies of φ{x, a) are uniformly
0
1
related. If φ(x, a ) and φ(x, a ) are two linked copies of φ{x, a) in
a model of T, then they have the same dimension.
LEMMA
Proof. For proof by contradiction suppose If is a model of T
1
in which φ(x, ά°) and φ(x, a ) are linked copies of φ(x, a) with different
dimensions. Notice that both must have finite dimensions: if both
were infinite then we should have
1
1
dim (φ(x, α°), M) = ( φ(M, α°) | = | φ(M, a ) | = dim (φ(x, ά ), M),
while if one were finite and the other infinite we would have an
infinite indiscernible set contained in the closure of a finite set which
would contradict the superstability of T. Without loss of generality
suppose that dim (φ(x, α°), M) = 0 and that dim (φ(x, ά1), M) — 1.
Using Lemma 9 we can form a model which we shall also call
1
M in which there are ω pairwise linked copies φ(x, ά°), φ(x, α ),
of φ(x, a) such that dim (φ(x, a1), M) — i and such that for any i and
on in
n J
i(] i+1C]
n
ί+i
j, α α
a ' and ά a . -. α
realize the same type. Without
3
loss we shall suppose that for any i, j φ(M, a*) Π φ(M, a ') = 0 and
1
ύ
that φ(M, a ) and φ(M, a ) are linked by (X, m).
1
J
If ceφ(M, a ), c'eφ(M9 a ), and i < j we say that c and c' are
f
3
X-related written cXc if t= X(c, c\ a\ a '). Likewise if c, c' are se1
3
quences in φ(M, a ) and φ(M9 a ') respectively and i < j we say c is
X-r elated to c', written cXc'f if corresponding members of the two
9
sequences are Z-related. Let c, c be sequences we write ctcc' if there
3
exist i, j and sequences e\ P independent in φ{M, α^), φ{M9 a ) respec3
ι
3
tively such that i < j , cXe\ c'Xe , and e Xe '. Our immediate goal is
to prove:
PROPOSITION. Let π(x0,
, xn_u y0,
, yn^) be a formula containing at most parameters from α° and the free variables indicated
such that, if cnc' is an independent sequence of length 2n in φ(M, α°)
with lh c = lh c' = n, then t= π(c, c'). Then c, c' can be found in
φ(M, aQ) such that \= π(c, c') and cκcf.
Towards establishing the proposition we make two observations.
Firstly, if i < j and e* is independent in φ(My a1) then there exists
THEORIES WITH A FINITE NUMBER OF MODELS
475
e3' in φ(M, a3) such that eιXe3 and any such e3' is independent. The
reason for this is that aίf)a3' realizes the same type as ά?{λa3~\ If
we were to adjoin in turn basis elements βjj,
, βi_i to φ(M9 a0)
there would simultaneously be generated in φ(Mf a3''*) elements
el~*9 , β£i*i %-related to < •• ,βi»1, respectively. Further e*~* =
(ei~\
, βίii> is independent because no term is in the algebraic
closure of those preceding it. Secondly, notice that there exists
k < ω such that for every i at most k members of φ(Mf a*) have no
%-predecessors in φ(M, a0).
To prove the Proposition consider a fixed formula π(x, y) satisfying
the hypothesis for which the conclusion fails. Let Iaω be infinite
and m < ω be given. From the remarks in the last paragraph we
can deduce the following. There exists iel, an infinite set / c / ,
an independent m-tuple e(m) in φ(M9 a*), and an m-tuple c(m) in
φ(M, α°) such that for every j e J there exists an independent m-tuple
e3'(m) in φ(M, a3'), and an m-tuple c3\m) in φ(Mf a0) such that
e(m)Xe3(m) & c(m)Xe(m) & co\m)le3\m) .
Further c(m) may be chosen such that if c is any w-tuple of distinct
members of c(m), and d is an w-tuple in φ(x, a0) independent over
Rng c, then N ττ(c, d). To see this, in the first instance choose c(m)
and e(m) of length much greater than m. There is a formula π'(x)
equivalent to
3^o3-ω2/i
**ωV*-Mx,
y) & <P(Vo, α°) &
& φ(yn_lf a0))
0
for arguments in φ(M, a ) because in any family of finite subsets of
s.m. sets which are uniformly definable there is a member of greatest
cardinality. Our aim can be achieved by thinning c(m) to an m-tuple
such that for any w-tuple c of distinct members of c(m) we have
N τc'(c). Since we are supposing that the conclusion of the Proposition fails for π(x, y) if c is an w-tuple of distinct members of
c(m) and cr is the corresponding ^-tuple of members of c'(m) then
N -i π(c,
c).
Using the compactness theorem we can find Mω ^ M, sequences
(c(m):m<ω), (cω(m):m<ω), (i(m):m<ω), (e(m):m<ω), (eω(m):m<
ω
ω
ω), and a e M such that the following conditions are satisfied.
ω
ω
Firstly a realizes the same type as a. For each m, c(m) and c (m)
ω
are m-tuples in φ(M, α°) and φ{M , α°) respectively, i(m) < ω, e(m) is
an independent m-tuple in φ(M, aUm)) and eω(m) is an independent
m-tuple in φ(Mω, aω). Also, if c is any w-tuple of distinct members
ω
of c(m) then \= π'{c), and if c is the corresponding w-tuple of
ω
members of c (m) then N ~i π(c, c'). Finally,
e(m)Xeω(m) & c(m)Xe(m) & cω(m)Xeω(m)
476
ALISTAIR H. LACHLAN
ω
where Z-relatedness is extended to the copy φ(%, a ) of φ(x, a) in
the obvious way.
ω
ω
For each m let d (m) be a subsequence of c (m) chosen such that
ω
ω
ω
d (m) is independent over a in φ(M, α°), and such that c (m) is in
ω
n
ωΠ
cl (Rng (d (m) a a0)).
ω
Now lh(d (m)) <n
ω
because if c
is a subse-
ω
quence of c (m) of length n and c is the corresponding subsequence
of c(m) then N —i π(c, cω) and N 7r'(e) whence cω is not independent.
Notice that eω(m) is in cl (cω(m)nαωnα°) and recall that eω(m) is an
independent sequence of length m in φ(Mω, aω). Let dω be a sequence
in φ{Mω, α°) which is independent over aω and which has maximal
ω
ω
ΰ>n ωn
length < n. Comparing d and d (m) we see that cl (d α α°) ΓΊ
ω
ω
ωΓ] ωf) 0
ω
ω
<p(M , a ) has dimension at least m. Hence cl (d a a ) Π <p(ikf , α )
has infinite dimension. This is inconsistent with T being superstable.
This completes the proof of the Proposition.
From the Proposition by compactness we can obtain M* ^ M a*
and a" in M! realizing the same type as α, and an independent
n
sequence (ci:i<ω) (c'j:i<ω)
in φ(M',a°) with the following property. There exist sequences <e : i < ω) and (eί' i < ω) in ^(.M, α')
and φ(M, a") respectively such that <c€: i < ω> is %-related to
<β : i < α>>, <e : i < ώ) is Z-related to <β": i < ω> and <c : ΐ < ω> is
Z-related to <β": i<ω).
Again the notion of Z-relatedness is extended
to the present context in the obvious way. Let α° and <cέ: i < ω)
be fixed. We describe the relationship between ά', a" and <(<?': i < a))
by saying that α' n α" permits <c : i < ω>.
Given two copies of φ(x, a) and a solution c of one there are at
most a finite number of c' in the second copy to which c is X-related.
Also for all but a finite number of c' in the second copy there are
only a finite number of c in the first which are Z-related to c'. Thus
n
ω
if I CI ^ λ, b b' permits at most 2 + λ sequences in C. Choose λ
ω
ω
+
such that λ > λ ^ 2 and M to be λ -saturated. Choose C S φ(M, 6°)
such that I C\ = λ and C is independent over <c*: ί < ω>. There are
λω sequences {cj: i < co) in C which have no repetitions and which
n
thus make <cέ: i < ω> <c : i <ω) independent. Each one is permitted
n
by some 6 P and at most λ are permitted by the same δ n δ\ Hence
over (Rng 6°) \J {ct:i < ω}\J C there are at least Xω types realized by
2^-tuples δ n δ' in Mr. This contradicts the super stability of T and
completes the proof of the lemma.
3* The main result.
We prove:
Let T be countable, ω-stable, and neither ω-categorical nor ωrcategorical. Then T has ^ ω models in every infinite
power.
THEOREM.
THEORIES WITH A FINITE NUMBER OF MODELS
477
Proof. As in [3] we work within a highly saturated model M
of T. Let φ(x, a) be an s.m. formula. Note that it may be impossible
to choose a in the prime model. The proof splits into two cases
according as φ(x, a) is α>-categorical or not. In the former case we
modify the proof given in [3]. In the latter case we can distinguish
enough different models by examining the dimensions of various copies
of φ(x, a).
Until further notice let φ(x, a) be ω-categorical. Let a realize
an Z-type pQ. Since T is not ω-categorical assume without loss of
generality that T has infinitely many 1-types. Choose ψ(x) of least
rank and degree in the sense of Morley such that ψ(x) belongs to
infinitely many distinct 1-types. By an observation of Shelah ψ(x)
can be chosen such that: there is a formula ε(x, y) defining an equivalence relation on ψ(M), φ(x) is equivalent to lyε(x, y), if 1= ^(α) then
ε(x, a) has rank a and degree 1 where a denotes the rank of ψ(x),
and the equivalence classes are indistinguishable in the sense that
for any formula X(x)
N VxVy(lz(Φ, z) A X(z)) A e(y, y)
Fix an infinite power tc and n < w.
> Zw(e(y, w) A X(w))) .
We now describe a model
M(fc, ri) of T of power K. Let a(/c, ri) = (aQ(fc, n),
po
, α^/c, n)) realize
Without loss of generality we may suppose that ψ{xo)epo.
I(/c, ri) = {bQ(/c, ri), , 6%-i(/c, n)}
be
such
that for
Let
all ael(fc, ri),
\= ε(α, ao(tc, n)) and α realizes a type of rank a over (/(/:, n) — {a}) U
Rng ά(/c, n). Further let J(fc, n) = {c^/r, ri): i < it] be a set of power
tc such that for any a e J(/c, n), t= φ(a, a(/c, ri)) and the type realized
by a over Rng a(/c,ri)U I(tc, ri) U (J(κ, ri) — {a}) is nonalgebraic. M(tc, ri)
is to be a model prime over Rng a(fc, n) U I(fc, ri) U J(κ, ri).
Observe that I(tc, n) is a subset of the solution set of ε(x, ao(/cf n)),
a formula of rank a and degree 1, whence I(/c, n) is indiscernible
over Rng ά(/c, n). To see that in fact I(/c, ri) can be suitably chosen
note that we can find b^tc, ri) such that for each i < ω |= ε(bi(ιc, ri),
ao(fc)) and b^tc, ri) realizes a type of rank a over {bj(/c, n): j < i) U
U Rnga(/r, ri). Because for each ί there is only one such type containing ε(x, ao(/c, n)) the sequence <&*(£, ri): i < ω) is indiscernible over
Rng α(/c, ri). Since the theory we are dealing with is stable, the set
{bi(/c,ri):ί< ώ) is indiscernible. Similarly we see that J(tc, ri) can be
found, that J(/c,ri)is indiscernible over I(ιc,ri)U Rng (ic, n) and that the
type over Rng a(tc, ri) realized by a sequence of distinct members of
I(/c,ri)U J{ιc, ri) depends only upon which members of the sequence fall
into I(tc, ri) and which into J(κ, ri). The following should be evident:
PROPOSITION
1. If m > n then the type realized by
478
ALISTAIR H. LACHLAN
a(/c, m)n<6i(/c, m): i < n)n(Ci(fc, m): i < fc)
is the same as that realized by
a{ιc, ri)n(bi(fc, n):i < /^>n<ci(/c, ri): i < fc) .
We also have:
2. If pe Sm(I(/c, ri) u J(κf ri) U Rng α(/c, ri)) is isolated
so is p\ (I(tc, ri) U Rng a(fc, ri)).
PROPOSITION
To see this let p be isolated over I(fc, ri) U Rnα a(fc, ri) U Jo where
Jo = {c09 ••*>ci-i} ΐ s a finite subset of J. Now ^(α;, a(yc, ^)) is s.m.
and ω-categorical, and as observed earlier φ(x, a(ιc, n)) remains cocategorical if names for a finite number of elements are added to
the language. It follows immediately that <c0, •• , ^ _i> realizes an
isolated i-type over l(tc, n) U Rng a(/c, n) which suffices.
We shall now show that if m > n then M(/c, m) and M(tc, n) are
not isomorphic. For proof by contradiction suppose that M(ιc, m)
is isomorphic to M(κ, n) and that m > n. Then some extension
in Sι+n+1(I(/c, n) U Rng a(κ, n) U J(tc, n)) of the type q+ realized by
a(fc, m)n(bQ(/c, m), •••, bn(/c, m)> over 0 is isolated. By Proposition 2
some extension q' of q+ in Sι+n+1(I(/c, n) U Rng α(/r, n)) is isolated.
Let π(x, y, z) be a formula such that
π(x, y, a(/c, ri), bo(/cf n),
, h.^/c, ri))
generates q* where y stands for xi+n.
Notice from Proposition 1 that
q, the type realized by a(κ, nY{bQ{tc, ri), , bn^(fc9 n)) is included in
q+ and hence in q'. _Let lh(a^) = lh(W) = i, «M6°) = iM^"1) = n, αonδ°
realize g, and t= π(a\ b\ 61, ά°, 6°). Without loss assume that |= ε(αj, «2);
otherwise we may replace π(x, y, z) by a similar formula which does
have this additional property. For example, if ψ(x) has degree 3,
π(x, y, z) may be replaced by
l&ly*l&ly\π(x9
yf z1) A π(z\ y\ z°) A π(z°, y0, z)) .
One should bear in mind here the indistinguishability of the ε-equivalence classes.
The following claim will enable us to complete the first half of
the proof of the theorem:
3. There exist α°, 6°, b\ a\ b\ b\
such that
iCi
i
i
for all i < ω, a b* realizes qflh(a ) = lflh(b ) — n, and
ί+1
ί+
b , b \ a\ V) and
j
for 0 < % < ω, ¥ realizes a type of rank a over {b : 0 <
Φ %}.
PROPOSITION
(i)
N π(ai+\
(ii)
j < o) A j
THEORIES WITH A FINITE NUMBER OF MODELS
+1
479
i+1
Notice from (i) that al and a\ are ε-equivalent since \=π(a ,
ίn
ί+in ί+in
ί+1
&') and α δ< realizes q. Further since α δ <δ >
+
+1
ΐ+1
1
2
realizes q , αo and δ are ε-equivalent. Thus from (i) all of δ , δ ,
are ε-equivalent to a°0. Now ε(x, al) has rank a and degree 1 whence
(ii) may be restated by saying that <δ*: 0 < i < ω) satisfies a certain
ω-type: Hence by the compactness theorem it is sufficient for the
proof of the proposition to show that any finite initial segment of
the desired sequence can be found. Suppose we wish to find α°, δ°,
1
1
1
2
j
δ, a , δ,δ,
, ά\ b then we proceed in reverse order. First choose
5
5
+
3
1
a\ b f b realizing q then choose a '~\ δ'"" such that |= π{a\ V, b\
j
1
a ~\ δ'" ). Notice from the choice of π that 3zπ(x, y, z)eq+. Now
choose δ5*"1 to realize the unique type over Rng (a5~inbά~ι) U ψ) which
1
3 1
+
has rank a and contains ε(x, a^ ). Then ά'^b'^ζb '' )
realizes q .
2
2
H
H
%
We can choose α'~ , δ^ such that N π ( S J , V~\ a'~\ 9~ ). Let
J 2
i 2n i 2
δ '~ realize the unique type over Rng (α ~ δ ~ ) U {bjf δ^J which has
2
i 2n i 2n
rank a and contains e(x, at )- Again α " δ " <δ i ~ 2 > realizes q+, thus
we may continue in the same manner. Certainly (i) will be satisfied
fe
and for 0 < ί ^ i, δ* realizes a type over {δ : i < k ^ j} which has
1
2
i
rank a. Further δ , δ , •• , δ are all ε-equivalent. By the same
argument used to show the existence of /(/c, n), for 0 < i ^ i, b*
realizes a type of rank a over {bk: k ^ j A k Φ i}. This suffices to
prove the proposition.
1
2
Of course, from (i) all of δ , δ ,
are solutions of ψ(x) and
moreover are ε-equivalent. From (ii) we get that {¥:0 <i < o)} is
ί+in ί+1Π
i+1
indiscernible. It is clear that the type realized by
a b (b )
in
over Rng(<z ^) is isolated. From this it follows easily that for all
i+1
on
i, b realizes an isolated type over Rng(α δ°). Recall that ψ(x)
was chosen to have least possible rank and then least degree subject
to there being infinitely many 1-types containing ψ(x). Since the
ί+1
type of δ over 0 has rank a it cannot be isolated. From Lemma
2 it follows that δ ί + 1 realizes a type of rank <a over Rng(αonδ°).
Now choose δί for each i, ω ^ i < 2ω, such that t= ε(δ*, δ1) and
ί
on
δ realizes a type of rank a over Rng (ά δ°) U ψ\ 0 < j < i). Using
once again the argument by which we showed I(n, ft) exists, we see
that {bj: 0 < j < 2ω} is indiscernible. But αonδ° determines a partition
of {bj: 0 < j < 2ω} into two infinite sets, since δω, δω+1,
are all the
members of {bj: 0<j<2ω}Rng(<z°, δ°) realizing a type of rank a over.
Thus there are 2ω (I + w)-types over the set {b5: 0<j<2ώ) of power
ω. This contradicts the ω-stability of T. Hence M(/c, m) and M(/c, n)
are not isomorphic. This completes the proof of the theorem in the
case where T has an ω-categorical s.m. formula.
For the rest of the proof suppose φ(x, a) is an s.m. formula and
that in any model M containing a, cl (Rng a) Π φ(M, a) is infinite.
Let a realize p0. Suppose first of all that:
480
ALISTAIR H. LACHLAN
(A) For each n < ω there exists a model M of T and copies
φ(x, α0),
, φ(x, α71"1) of φ(x, a) in M no pair of which are linked.
With an arbitrary model M of T we associate a natural number
i(M) as follows. Let EM be the equivalence relation on B = {a: a e M
M 1
1
and a realizes p0} defined by: ά°E a if φ{x, α°) and φ(x, α ) are linked.
Recall that if two copies of φ{x, a) are linked then they have the
same dimension in M modulo ω. Let i(M) be the number of equivaM
lence classes C under E such that for ae C φ{x, a) has dimension
equal to ||Λf||. If B = 0 let i(M) = 0. From Lemma 7, for each
ic > ω and each n, 1 ^ n < ω, there is a model Λf of power /c such
that ί(M) = w. When K — ω the proof of Lemma 7 can be modified
to show that there is a model M of power ω with i(M) = %.
Alternatively, from [3] Γ has at least y^0 countable models. In either
case we have the desired conclusion. Henceforth we assume that (A)
fails.
0
1
Let φ(x, a ) and φ(x, a ) be two copies of φ (x, a) in a model M
0
1
of T. Recall that <p(x, a ) and φ(x, a ) are linked by the pair
(X(x\ xu y°, ψ), m> if the following formulas are valid
X(xOf xlf a\ a1) -+ φ(xOt α°) Λ φ(xl9 a1) ,
3>w%%(α?0, a?i, ά\ a1)
We shall now show that because (A) fails we have:
1
(B) There exists a pair (X(x0, xίf y°, y ), m) such that if <p(x, α°)
1
and φ{x9 α ) are linked copies of φ{x, a) in a model of T, then they
1
are linked by (X(xQ, xl9 y°, y ), m>. That is, all copies of φ(x, a) are
uniformly related.
Let ((Xif mi): i < (o) be an enumeration of all pairs which might
link two copies of φ(x, a). Notice that it is sufficient to prove that
there exists j < ω such that any two linked copies are linked by
(Xtf m*> for some i ^ j . For proof by contradiction suppose no such
j exists. Then in any (^-saturated model we can find a sequence
(φ(xf a*):i < ω) of copies of φ(x, a) such that for each i, <p(x, α°)
and φ(x, α*) are linked but not by any pair (Xk, mk) with k ^ i.
Observe that for each i and j there exists i' such that if k ;>
V φ(x, a1) and φ(x, ak) are not linked by any pair with subscript ^j.
By the compactness theorem (A) holds. This contradiction completes
the proof of (B).
We have an s.m. formula φ(x, a) which has infinitely many
algebraic solutions. If φ{x, a) does not have the Vaught property
then by Lemma 4 T has an inessential extension which is ωΓcategorical. However, any theory which has an cyΓcategorical inessential
extension is itself αvcategorical; one can see this from the remark
at the end of § 2 in [1]. Thus we may suppose that φ(x, a) has the
THEORIES WITH A FINITE NUMBER OF MODELS
481
Vaught property. Let /c > ω be given. From Lemma 3 T has a
model M of power K containing α such that φ(x, a) has finite dimension
in M. Define j(M) to be the greatest n < ω such that some copy of
φ(x, a) in M has dimension n and to be undefined if there are no
copies of finite dimension. Prom Lemma 10 any linked copies of
φ(x, a) in M have the same dimension, and since (A) fails the number
of pairwise unlinked copies of φ(x, a) in a model is finite. Thus
j(M) is well defined whenever there is some copy with finite dimension. Let {e0,
, em} be a set of solutions of φ(x, a) independent over
I AT I, and let M' be prime over | M\ (J {eQ9
, em}. It is easy to see
that φ(x, a) has finite dimension > m in M'. Hence j(M') > m and
I MΊ = tc. From this it is clear that there are infinitely many nonisomorphic models of power K. This completes the proof of the
theorem.
4* Concluding remarks* As mentioned in the introduction our
theorem together with a result of Shelah confirms the conjecture of
W. K. Forrest that any countable theory having a finite number of
models, but more than one, in an uncountable power is ω-categorical.
Forrest also conjectured that such theories have only a finite number
of models in each power it, fc$0 < & < M« We have verified this. A
question which is not so precise concerns the structure of such theories.
In some sense every known example can be seen as a combination
of a finite number of theories categorical in every infinite power.
Can this idea be formalized and be demonstrated for all such theories?
REFERENCES
1. J. T. Baldwin and A. H. Lachlan, On strongly minimal sets, J. Symbolic Logic,
36 (1971), 79-96.
2. A. H. Lachlan, A property of stable theories, Fund. Math., 7 7 (1972), 9-20.
3.
, On the number of countable models of a countable superstable theory,
Logic Methodology and the Philosophy of Science IV, North Holland 1973, 45-56.
4.
, Two conjectures regarding the stability of ω-categorical theories, Fund.
Math., 8 1 (1974), 133-145.
5. M. Morley, Categoricity in power, Trans. Amer. Math. Soc, 114 (1965), 514-518.
6. J. Rosenthal, A new proof of a theorem of Shelah, Proc. of the 1971 Symp. in
Berkeley in honour of Tarski, published by Association for Symbolic Logic.
7. S. Shelah, Finite diagrams stable in power, Annals of Math. Logic, 2 (1970), 69118.
8.
, Stability, the f.c.p., and superstability, model, theoretic properties of
formulas in first-order theory, Annals of Math. Logic, 3 (1971), 271-362.
9. S. Shelah, forthcoming book on stability in first-order model theory.
Received January 16, 1975.
SIMON FRASER UNIVERSITY
PACIFIC JOURNAL OF MATHEMATICS
EDITORS
(Managing Editor)
University of California
Los Angeles, California 90024
RICHARD ARENS
J. DUGUNDJI
Department of Mathematics
University of Southern California
Los Angeles, California 90007
R. A. BEAUMONT
D. GILBARG AND J. MILGRAM
University of Washington
Seattle, Washington 98105
Stanford University
Stanford, California 94305
ASSOCIATE EDITORS
E. F. BECKENBACH
B. H. NEUMANN
P. WOLF
K. YOSHIDA
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Copyright © 1975 by Pacific Journal of Mathematics
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Pacific Journal of Mathematics
Vol. 61, No. 2
December, 1975
Graham Donald Allen, Francis Joseph Narcowich and James Patrick Williams, An
operator version of a theorem of Kolmogorov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Joel Hilary Anderson and Ciprian Foias, Properties which normal operators share with
normal derivations and related operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Constantin Gelu Apostol and Norberto Salinas, Nilpotent approximations and
quasinilpotent operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
James M. Briggs, Jr., Finitely generated ideals in regular F-algebras . . . . . . . . . . . . . . . .
Frank Benjamin Cannonito and Ronald Wallace Gatterdam, The word problem and
power problem in 1-relator groups are primitive recursive . . . . . . . . . . . . . . . . . . . . . .
Clifton Earle Corzatt, Permutation polynomials over the rational numbers . . . . . . . . . . . .
L. S. Dube, An inversion of the S2 transform for generalized functions . . . . . . . . . . . . . . .
William Richard Emerson, Averaging strongly subadditive set functions in unimodular
amenable groups. I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Barry J. Gardner, Semi-simple radical classes of algebras and attainability of
identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Irving Leonard Glicksberg, Removable discontinuities of A-holomorphic functions . . . .
Fred Halpern, Transfer theorems for topological structures . . . . . . . . . . . . . . . . . . . . . . . . . .
H. B. Hamilton, T. E. Nordahl and Takayuki Tamura, Commutative cancellative
semigroups without idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Melvin Hochster, An obstruction to lifting cyclic modules . . . . . . . . . . . . . . . . . . . . . . . . . . .
Alistair H. Lachlan, Theories with a finite number of models in an uncountable power
are categorical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kjeld Laursen, Continuity of linear maps from C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . .
Tsai Sheng Liu, Oscillation of even order differential equations with deviating
arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jorge Martinez, Doubling chains, singular elements and hyper-Z l-groups . . . . . . . . . . . .
Mehdi Radjabalipour and Heydar Radjavi, On the geometry of numerical ranges . . . . . .
Thomas I. Seidman, The solution of singular equations, I. Linear equations in Hilbert
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
R. James Tomkins, Properties of martingale-like sequences . . . . . . . . . . . . . . . . . . . . . . . . .
Alfons Van Daele, A Radon Nikodým theorem for weights on von Neumann
algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kenneth S. Williams, On Euler’s criterion for quintic nonresidues . . . . . . . . . . . . . . . . . . .
Manfred Wischnewsky, On linear representations of affine groups. I . . . . . . . . . . . . . . . . .
Scott Andrew Wolpert, Noncompleteness of the Weil-Petersson metric for Teichmüller
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Volker Wrobel, Some generalizations of Schauder’s theorem in locally convex
spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Birge Huisgen-Zimmermann, Endomorphism rings of self-generators . . . . . . . . . . . . . . . .
Kelly Denis McKennon, Corrections to: “Multipliers of type ( p, p)”; “Multipliers of
type ( p, p) and multipliers of the group L p -algebras”; “Multipliers and the
group L p -algebras” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Andrew M. W. Glass, W. Charles (Wilbur) Holland Jr. and Stephen H. McCleary,
Correction to: “a ∗ -closures to completely distributive lattice-ordered
groups” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Zvi Arad and George Isaac Glauberman, Correction to: “A characteristic subgroup of
a group of odd order” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Roger W. Barnard and John Lawson Lewis, Correction to: “Subordination theorems
for some classes of starlike functions” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
David Westreich, Corrections to: “Bifurcation of operator equations with unbounded
linearized part” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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