GENERALIZED AMALGAMATION AND n-SIMPLICITY
BYUNGHAN KIM, ALEXEI S. KOLESNIKOV, AND AKITO TSUBOI
Abstract. We study generalized amalgamation properties in simple theories. We state the
definition of generalized amalgamation in such a way so that the properties are preserved
under interpreted theories. To deal with the properties, new notions of generalized types
and generalized Morley sequences are introduced.
We consider several sub-hierarchies of simple theories that affect generalized amalgamation; examples are given to show the difference. A notion of an heir base model is introduced
to prove positive amalgamation results about one of the hierarchies.
Introduction
Generalized amalgamation properties were introduced by Shelah in the context of classes
of atomic models [13] and later used by him in the proof of Main Gap theorem in [12]. For
a certain subfamily of simple first order theories, generalized amalgamation for algebraically
closed sets was studied by Hrushovski in [4, 2].
In short, the generalized amalgamation is about being able to embed increasingly complex
systems of algebraically closed sets (boundedly closed sets, models, etc.) into a model of the
theory. One of the simplest such properties (P − (3)-amalgamation) was taken by Kim and
Pillay in [8] as the basis for the Independence Theorem, a characteristic property of simple
theories.
Motivated by [2] and a conjecture of Shelah in [15], Kolesnikov introduced a family of
generalized type-amalgamation properties for an arbitrary simple first order theory in [9].
These type-amalgamation properties generalize the Independence Theorem in the same way
as P − (n)-amalgamation generalizes P − (3)-amalgamation.
This paper is an extension of [9] in several directions. First, it states the definitions of
generalized amalgamation in such a way that they are preserved under interpreted theories.
We believe that the resulting notions will be quite useful in the subsequent study of simple
theories. Second, it exposes an increasing complexity of amalgamation notions themselves
as the dimension increases. For example, for 3-amalgamation (we drop the P − part) it does
not matter whether we amalgamate types or boundedly closed sets; for 4-amalgamation
and above it really does; 3-amalgamation over algebraically closed sets is a consequence of
stationarity for stable theories; 4-amalgamation over algebraically closed sets fails in stable
theories in general. More subtly, the two hierarchies: n-simple theories and K(n)-simple
theories (two reasonable ways to describe when a theory would have amalgamation properties
up to certain “dimension”) coincide for n = 1, 2, but are shown to be different for n ≥ 3.
We here point out that the definition of n-simplicity previously suggested in section 7 of [9]
is correctly modified in this paper. We shall explain why we should take this new definition.
Finally, we begin to address the question of what is “reasonable” to expect when it comes
Byunghan Kim was supported by KOSEF grant R01-2006-000-10638-0.
1
to amalgamation of dimension 4 and above. Two different methods are explored in the last
two sections of the paper. We assume the reader is familiar with the ideas and techniques
developed in [9], as one of the main parts of this paper is to lift those into the context of the
new definitions.
It is worth mentioning that higher-dimensional amalgamation properties were used in [3] to
study the group configuration problem; and, implicitly, in [11] to address Vaught’s conjecture
for a subclass of simple theories.
The paper is organized as follows. In Section 1 we discuss the motivation, state the
definitions of generalized amalgamation notions and establish some basic facts. Another
important definition in Section 1 (and Section 3) is that of a generalized type. In Section 2
we introduce the notions of generalized indiscernible and Morley sequences. We use those to
state the definitions of n-simplicity and K(n)-simplicity. We summarize the results of the
paper at the end of Section 2.
Section 3 is devoted to various results concerning generalized amalgamation and characterizations of n-simplicity. In Section 4 we provide examples, most interesting one is that of
K(3)-simple theory that fails to have 5-amalgamation. The construction can be described
as a variant of Hrushovski construction “with torsion”.
Section 5 contains analysis of K(3)-simple theories. An important notion introduced there
is that of an heir base for a finite Morley sequence. By a result of Lascar and Pillay [10], in
any simple theory for any element a0 there is a model M such that whenever a1 is a nonforking extension of tp(a0 /M ) to M a0 , we have that {a0 , a1 } is a coheir sequence over M .
In stable case, any finite independent sequence over any model has to be a coheir sequence,
but in simple case the answer depends on the strength of amalgamation hypotheses.
Section 6 introduces the notion of model-n-complete amalgamation; a property that holds
in all stable theories and was used in [3].
1. Definitions of generalized amalgamation
1.1. Motivation. A key property of forking in simple theories is the Independence Theorem, proved by Kim and Pillay in [8]. The theorem asserts that, under appropriate conditions, three types p1 (xy), p2 (xz), and p3 (yz) can be simultaneously realized by a triple of
tuples (x, y, and z are strings of variables). The conditions demand that p1 (xy) be realized by ∅-independent {a, b}; and the remaining two types by independent {a, c0 }, {b, c1 }
respectively with lstp(c0 ) = lstp(c1 ). Equivalently, suppose p1 is realized by independent
{a, b}; then we are looking for a common independent solution for the system of two types
{p2 (a; z), p3 (b; z)}. Thus, in the terminology of our paper, the Independence Theorem corresponds to 3-amalgamation property.
The idea behind 4-amalgamation property is to see if we can simultaneously realize four
types p1 (xyz), p2 (xyw), p3 (xzw), and p4 (yzw) by four tuples. Equivalently, suppose p1 is
realized by an independent abc; then we are looking for a common solution to the system of
types {p2 (ab; w), p3 (ac; w), p4 (bc; w)}.
Some of the restrictions that should be imposed on such a system of types are obvious:
we do want independence; and the types p2 , p3 , and p4 must be coherent. For instance,
p2 (ab; w) ¹ a = p3 (ac; w) ¹ a, and so on.
2
The purpose of the discussion below is to demonstrate that the demands made on the
system of types in Definition 1.5 are natural and necessary. The reader who is already
convinced that the definition is quite natural may skip the discussion.
A tetrahedron-free hypergraph (with a ternary edge relation R) gives an example of a
simple theory which clearly fails to have the 4-amalgamation, precisely because one of the
axioms forbids a common solution to R(abw), R(acw) and R(bcw), where R(abc) holds. On
the other hand, we expect a random graph with a binary relation R to have n-amalgamation
properties for all n. Indeed, it is easy to establish the amalgamation properties in the home
sort for Trg . Simplicity is preserved under interpreted theories, so 3-amalgamation does hold
in M eq for M |= Trg (the theory of random graph). However, 4-amalgamation may fail unless
additional compatibility conditions are imposed.
The first two examples show that w must enumerate a boundedly closed set; and that the
“base” elements a, b, c must be boundedly closed. Since in our Trg eliminates hyperimaginaries, we work in M eq and with algebraic closures.
We show that M eq fails 4-amalgamation if a, b, c, w are not closed.
Example 1.1. [w must enumerate a boundedly closed set]
Let a, b, c, d1 and d2 be distinct elements in the home sort of M eq such that tp(ab) =
tp(ac) = tp(bc). Let d be the imaginary d = {d1 , d2 }, where M |= R(a, d1 ) ∧ R(b, d2 ), and
there are no other relations between a, b, d1 , d2 . Let p1 (ab; w) := tp(d/ab); and let p2 (ac; w)
and p3 (bc; w) be the appropriate conjugates of p1 .
It is easy to see that in this case the necessary independence holds, and even p1 ¹ a = p2 ¹ a,
and so on. However, there is no common solution for the system of types. The reason is that d
induces an equivalence relation with two classes among the realizations of tp(a/d) = tp(b/d),
and tp(abd) says that a is not equivalent to b. The system of types expresses that there are
at least three equivalence classes.
To avoid this problem, we must demand that d be boundedly closed.
Example 1.2. [a, b, c must be boundedly closed]
Now let a = {a1 , a2 }, b = {b1 , b2 }, and c = {c1 , c2 } be imaginary elements and d be
an element in the home sort such that {ai , bi , ci | i = 1, 2} and d are all distinct, M |=
R(a2 , b2 ) ∧ R(a2 , c2 ) ∧ R(b2 , c2 ), M |= R(d, a1 ) ∧ R(d, b2 ) and no other relations hold.
Let p1 (ab; w) := tp(d/ab); and let p2 (ac; w) and p3 (bc; w) be the appropriate conjugates of
p1 . It is easy to see here as well that the three types cannot be amalgamated, even though
the independence and coherence of types requirements are met.
We now point out that one should be careful about the order in which the elements of the
bounded closures are listed. This can be illustrated on a version of Example 1.2.
Example 1.3. Let a = {a1 , a2 }, b = {b1 , b2 }, and c = {c1 , c2 } be imaginary elements and
d be an element in the home sort such that {ai , bi , ci | i = 1, 2} and d are all distinct,
M |= R(d, a1 ) ∧ R(d, b1 ) and no other relations hold. Let p1 be the type of the bounded
closure bdd(abd): p1 := tp(aa1 a2 ; bb1 b2 ; d). As in Example 1.2, let p3 be a conjugate to a
type over the bounded closure of bc; but now we carelessly invert the order of b1 , b2 . Namely
p3 := tp(bb2 b1 ; cc1 c2 ; d0 ), where of course M |= R(b2 , d0 ) ∧ R(c1 , d0 ) and no other edges exist.
We see that even though tp(bd) = tp(bd0 ) and tp(bb1 b2 ) = tp(bb2 b1 ), already p1 and p3 are
inconsistent because of the difference in enumerating the bounded closure.
3
We now turn to a more subtle point: the generalized amalgamation should really be
about amalgamating bounded closures (as opposed to amalgamating single boundedly closed
points).
Let us start by noting that the Independence Theorem implies 3-amalgamation for boundedly closed sets. That is, in simple theories we can find an element b such that tp(bdd(a0 b)) =
tp(bdd(a0 b0 )) and tp(bdd(a1 b)) = tp(bdd(a1 b1 )), while holding bdd(a0 a1 ).
We can view 4-amalgamation property as extension for 3-amalgamation in the following
sense. Suppose that we have managed to amalgamate tp(bd) and tp(cd) over tp(bc); and
now want to extend the amalgam to the one over a. Namely, we are trying to amalgamate
tp(d/ab) and tp(d/ac) while preserving the type of abc and the type tp(bcd) of the original
amalgam. Using this to extend 3-amalgamation of boundedly closed sets, we see that bdd(bc)
becomes bdd(abc), and so on. Thus we end up needing to amalgamate four boundedly closed
sets.
The last comment before we state the definitions: while the usual 3-amalgamation implies
3-amalgamation for boundedly closed sets, the situation changes dramatically when we go
to 4-amalgamation. As shown in [3], 4-amalgamation for algebraically closed sets over an
algebraically closed set need not hold even in a stable theory. If one drops the demand
to amalgamate the algebraic closures, then 4-amalgamation (for types) easily follows from
stationarity.
1.2. Generalized amalgamation for boundedly closed sets. In what follows S will be
a partially ordered set with the least element 0S , such that the infimum s ∧ t of any two
elements s, t ∈ S exists, and if S contains an upper bound of s, t ∈ S, then the supremum
s ∨ t exists in S as well.
Typical examples of the partially ordered sets that we will be using are: (1) subsets of
P(n), ordered by inclusion, closed under initial segments; and (2) sets of the
form [Λ]≤n ,
S
where Λ is a set, [Λ]k is the set of all k-element subsets of Λ and [Λ]≤n is k≤n [Λ]k . The
order is again the inclusion. If in addition Λ is a linearly ordered set, then we identify the
set of k-element subsets with the set of all <Λ -increasing k-element sequences from Λ.
Definition 1.4. For a partially ordered set S as above, consider a directed family h{As }s∈S , {πts }s≤t∈S i
of boundedly closed sets. Namely, As is boundedly closed for all s ∈ S, and πts : As → At is
an elementary map for s ≤ t with πts ◦ πut = πus and πss = idAs .
We say that a directed family h{As }s∈S , {πts }s≤t∈S i is an independent system of boundedly
closed sets indexed by S if for all s ∈ S
(1) if u, v ≤ s and t := u ∧ v, then πsu (Au ) ^ πsv (Av );
πst (At )
´
³S
(2) As = bdd t<s πst (At ) .
We omit the maps πts if they are all inclusion maps. We are now ready to state the
definition of n-amalgamation property.
Definition 1.5. Let h{Au }u∈P − (n) , {πvu }u⊂v∈P − (n) i be an independent system of boundedly
closed sets indexed by P − (n) = P(n) \ {n}. We say that a theory T has the n-amalgamation
property if there is a boundedly closed An and elementary maps πnu : Au → An for u ∈
P − (n) such that h{Au }u∈P(n) , {πvu }u⊂v∈P(n) i is an independent system of boundedly closed
sets indexed by P(n).
4
Definition 1.6. We say that a simple theory T has the n-complete amalgamation property,
or n-CA, if T has the k-amalgamation property for each 3 ≤ k ≤ n.
Remark 1.7. If n-CA fails over arbitrary sets, but does hold for some set B (equivalently,
for sets A∅ of certain kind), then we say that T has n-complete amalgamation over B. For
instance, stable theories may fail to have 4-CA, but they do have n-CA over models for all
n < ω [3].
1.3. Amalgamation for system of types.
Proposition 1.8. Let T be a simple theory and let n ≥ 3. Let S be a non-empty collection
of proper subsets of n = {0, . . . , n − 1}, closed under subsets; let B be a set in the monster
model Cheq . Suppose that, for each u ∈ S, we are given a complete type ru (xu ) over B in
possibly infinitely many variables xu such that
(1) if u ⊆ v, then xu ⊆ xv and ru ⊆ rv ;
(2) for any au |= ru , {a{i} |i ∈ u} is B-independent, where a{i} is a subtuple of au consisting
of realizations corresponding to the variables x{i} ⊂ xu ;
(3) au is as a set bdd(∪i∈u a{i} B), and the map fu : au → xu is a bijection.
Then T has n-complete amalgamation property if and only if there are complete types {ru (xu ) |
u ∈ P(n)} over B such that (1), (2), and (3) hold for all u ∈ P(n).
Proof. We prove a stronger statement by induction on n ≥ 3. Namely, we show that, for
each n,
(∗) the existence of types indexed by the entire P(n) and satisfying (1)–(3) is equivalent
to n-complete amalgamation and
(∗∗) given a non-empty S ⊂ P − (n+1) closed under subsets indexing a system of types that
satisfies (1)–(3), assuming n-complete amalgamation we can extend such a system to
the one indexed by P − (n + 1).
For n = 3, all the statements involved hold outright in any simple theory by Independence
theorem; the details are left to the reader.
For induction hypothesis, suppose both the equivalence (∗) and the statement (∗∗) are
true for some n − 1 ≥ 3 (I.H.). We first show that (∗) holds for n.
(⇒) Assume the left hand side. To show n-CA, assume there given an independent directed
system of boundedly closed sets indexed by P − (n). Let us produce the corresponding system
of types satisfying (1)–(3).
This can be described as a level-by-level construction: at the bottom, enumerate A∅ using
the variables in the set x∅S
, and let r∅ (x∅ ) := tp(A∅ /B). For k-th “level”, i.e., for the kelement u ⊂ n, let xu := v(u xv ∪ yu , where yu is the list of new variables enumerating
´
³S
v
the set Au \
v(u πu (Av ) . With xu enumerating the set Au in the natural way, we let
ru (xu ) := tp(Au /B). Using the definition of a directed system of sets, it is not hard to check
that the properties (1)–(3) hold.
By the left hand side, there is rn (xn ) completing the system of types. Then conversely, with
an |= rn (xn ), the original system of closed sets are completed; the corresponding projections
are πnu := fn−1 ◦ fu .
(⇐) Assume n-CA. By I.H., it suffices to show the case when S = P − (n). For u ∈ P − (n),
let au |= ru . We claim that {au | u ∈ P − (n)} forms an independent system of boundedly
closed sets indexed by P − (n). Indeed, letting πvu := fv−1 ◦ fu , we see that {au | u ∈ P − (n)} is
5
a directed system by the assumption (1); it is an independent system by (2) and each set is
boundedly closed by (3). Thus, there is a boundedly closed set an and the projection maps
πnu , u ∈ P − (n), that commute with the given system of projections. It is straightforward to
check that, reversely, rn = tp(an ) completes the system {ru | u ∈ P − (n)}.
Now we deal with (∗∗). Take S ⊆ P − (n + 1) closed under subsets. Let S 0 := S ∩ P(n).
The system of types {ru | u ∈ S 0 } satisfies (1)–(3), and so can be extended to the system
indexed by P(n) by I.H. and what we proved in (*). For S0 := P(n) ∪ S, we have defined
the types ru for each u ∈ S0 . It is easy to check that the resulting system {ru | u ∈ S0 } still
satisfies (1)–(3).
Iterate this construction: suppose we have enriched
the system indexed by S to a system
S
indexed by Si , i < n + 1, such that Si = S ∪ j≤i P((n + 1) \ {n − j}). Let Si0 := Si ∩
P({0, . . . , n \
− i − 1, . . . , n}). As the system of types {ru | u ∈ Si0 } still satisfies (1)–(3), can
again extend it to the system indexed by the full P({0, . . . , n \
− i − 1, . . . , n}). We have defined the system {ru | u ∈ Si+1 } that satisfies (1)–(3), where Si+1 := P({0, . . . , n \
− i − 1, . . . , n})∪
Si . Finally, Sn+1 = P − (n + 1). This establishes (∗∗).
a
Here’s yet another way to look at n-amalgamation property. We will use it throughout the
paper to avoid cumbersome notation coming from the directed systems of boundedly closed
sets.
In a nutshell, the approach is to view the (n + 1)-amalgamation as amalgamation of n
objects over the base: for instance 3-amalgamation can be treated as amalgamation of two
types while fixing the base; 4-amalgamation as simultaneously realizing three “sides” in a
tetrahedron over the base triangle, and so on.
Definition 1.9. Let {Aw | w ∈ P − (n)} be a directed independent system of boundedly
closed sets (with inclusion maps), Aw ⊂ Cheq . We say that {pw (xw ; Aw ) | w ∈ P − (n)} is a
coherent system of types over {Aw | w ∈ P − (n)} if the following conditions hold:
(1) dom(pw ) = Aw ; and if Cw |= pw , then Cw ⊃ Aw ;
(2) if w ⊆ v, then xw ⊆ xv and pw ⊆ pv ;
(3) for all w ∈ P − (n) the map fw : Cw → xw is a bijection.
Denoting Cw∅ := fw−1 ◦ f∅ (C∅ ), we demand that
(4) Cw is bdd(Aw ∪ Cw∅ ) for all w ∈ P − (n), and Cw∅ ^ Aw .
A∅
The intuition is that given an (n + 1)-directed system {Au | u ∈ P − (n + 1)}, we let, for
w ∈ P − (n), the set Cw to be Aw∪{n} , so in particular C∅ = A{n} . And then we enumerate
the elements of Cw by variables xw in a coherent way. The following proposition shows that
(n + 1)-amalgamation corresponds to strong n-amalgamation described in [9, 4.3 or 7.5]:
Proposition 1.10. Let T be simple and let n ≥ 3. Then T has (n + 1)-complete amalgamation property if and only if for
S every 3 ≤ k ≤ n, for every coherent system of types
over {Aw | w ∈ P − (k)} the union w∈P − (k) pw (xw ; Aw ) is realized (amalgamated) by Ck and
Ck∅ ^ {A{i} |i ∈ k}.
A∅
Proof. (⇒) Suppose T has (n + 1)-CA.
S Take a coherent system of types {pw (xw ; Aw ) | w ∈
P − (n)}. We are showing that p∗ := w∈P − (k) pw (xw ; Aw ) is a consistent type. To do that,
6
we construct an independent directed system of boundedly closed sets {Au | u ∈ P − (n + 1)},
whose amalgam will produce
of the needed type.
³S a realization
´
First, let An := bdd i∈n A{i} . This is possible since all the sets Aw are inside Cheq .
Thus, for u, v ∈ P(n), we have the boundedly closed sets Au , and the projections πvu := idAu .
Now we get the sets and projections for u ∈ P − (n+1) of the form w∪{n}, where w ∈ P − (n).
So let u := w ∪ {n}, let Au := Cw , where Cw |= pw (xw ; Aw ). The corresponding projections,
for v ⊂ u, are: πuv := idAv for v ∈ P − (n), and πuv := fw−1 ◦ fv\{n} for other v ⊂ u. Immediately
from definitions we see that {Au | u ∈ P − (n + 1)} is an independent directed system of
boundedly closed sets. Let An+1 be its amalgam, in which we identify the sets Au with
u
(Au ). It remains to note that the sets Cw , w ∈ P − (n), are the sets
their projections πn+1
S
Aw∪{n} , and so f ∗ := w∈P − (n) fw is a (partial) function from An+1 into the list of variables
S
∗
w∈P − (n) xw , whose domain realizes p .
(⇐) We use induction on n ≥ 3. For n = 3, both statements are true in any simple theory,
so the implication holds. Now we take an independent directed system of boundedly closed
sets {Au | u ∈ P − (n + 1)}, and show that is has am amalgam An+1 . First, by induction
hypothesis, we have n-CA. So we may assume that the part {Aw | w ∈ P(n)} of the directed
system is contained in Cheq .
And now simply build the types pw (xw ; Aw ) for w ∈ P − (n) by induction. For k < n we
construct, for w ∈ [n]k , the variables xw , bijections fw : Cw → xw , and the types pw (xw ; Aw )
such that
(1) Cw = Aw∪{n} ;
u∪{n}
(2) fu = fw ◦ πw∪{n} for u ⊂ w; and
(3) pw (xw ; Aw ) = tp(Cw /Aw ).
For k = 0, let f∅ be a bijection from A{n} onto a list of variables x∅ ; and p∅ (x∅ ; A∅ ) :=
tp(A{n} /A∅ ). Having constructed the types pw (xw ; Aw ) for w ∈ [n]<k , take a k-element
subset w ∈ P − (n). Define
[ u∪{n}
[
fw0 :
πw∪{n} (Au∪{n} ) →
xu
u(w
u(w
in the natural way: for an element a of the union, choose the smallest in size u such that
a = πwu (b), for some b ∈ Au . Let fw0 (a) := fu (b). It is easy to check that the definition of
fw0 does not depend on the choice of u. Now add the variables zw to enumerate Aw∪{n} \
S
S
u∪{n}
0
u(w xu ∪ zw , and let fw ⊃ fw be a bijection
u(w πw∪{n} (Au∪{n} ) if necessary, let xw :=
between Aw∪{n} and xw . It remains to define pw (xw ; Aw ) := tp(Cw /Aw ), where Cw := Aw∪{n} .
−
Thus,
S we get a coherent system of types over {Aw | w ∈ P (n)}. By hypothesis, the
union w∈P − (n) pw (xw ; Aw ) is a type realized by a set Cn . Then there is a natural bijection
S
S
fn := w∈P − (n) fw between Cn and w∈P − (n) xw . We also have Cn∅ ^ An . We take An+1 :=
A∅
w∪{n}
−1
bdd(Cn ) and πn+1 := fn ◦ fw . It is easy that the resulting system of boundedly closed
sets indexed by P(n + 1) is a directed system. To check independence, we need to see if
{i}
{πn+1 (A{i} ) | i < n + 1} is an independent set in An+1 over A∅ . For the first n members this
{n}
{n}
is immediate, and πn+1 (A{n} ) ^ An since πn+1 (A{n} ) = Cn∅ .
a
A∅
7
A significant part of the paper will be in the context where we have n-CA, and are trying
to prove (n + 1)-CA, i.e., the next “level” of amalgamation under additional assumptions.
By Proposition 1.10, it is enough to find a common independent realization for coherent
systems of types over {Aw | w ∈ P − (n)}. By n-CA, we do not loose the generality when
assuming that the domain {Aw | w ∈ P − (n)} for the system of types lies inside Cheq . As a
by-product of Proposition 1.8, we know that these sets can be enumerated in a coherent
S way.
In particular, if the tuple ai enumerates the boundedly closed A{i} , then Aw = bdd( i∈w ai ).
This leads to the following notational agreements.
Notation 1.11. Given a coherent system of types, by the restriction of pu (xu ; Au ) to Av ,
where v ⊂ u, we mean the type pv (xv ; Av ). When convenient, we denote the restriction as
pu dAv .
If w = {i0 , . . . , in−1 }, then we use the symbol ai0 ,...,in−1 for the appropriate enumeration of
Aw . We further agree to drop the subscripts w from the type p and the variable list x. That
is, we write p(x; Aw ) or p(x; ai0 ,...,in−1 ) for pw (xw ; Aw ).
To avoid too many subscripts, it is sometimes convenient to use a, b, and c in the place of
a0 , a1 , and a2 . In this case, we use p(x; abc) for p012 (x012 ; a012 ). These types with simplified
notation are called generalized types. One needs to keep in mind that the list of variables x
is actually different in a generalized type p(x; ab) than in p(x; ac); and both lists are different
than the one in the generalized type p(x; ab)da = p(x; a).
We shall see these issues again in Section 3.
2. n-simplicity
2.1. Generalized Morley sequences. Until now, we dealt with independent directed systems of boundedly closed sets indexed by P − (n), for some n. The generalized Morley sequences that we define below are examples of directed systems indexed by sets [Λ]≤n partially
ordered by inclusion, where |Λ| ≥ n + 1, in particular Λ can be infinite. We start with a
simple example of such a system.
Example 2.1. Let I = {ai | i < α} be a sequence in Cheq indexed by an ordinal α. By an
[α]1 -sequence over A we mean the union {bdd(A)} ∪ {bdd(Aai ) | i < α}.
Increasing the exponent, an [α]2 -sequence over A is the union of sets {bdd(A)}, {bdd(Aai ) |
i < α} and {bdd(Aai aj ) | i < j < α}. We hide here that the elements of such sequences are
ordered tuples, so we will need to be careful about the enumeration in the formal definition.
It is easy to see that an [α]2 -sequence is a directed system of boundedly closed sets (all
the projections are identities) indexed by [α]≤2 .
We now formalize the notion.
Definition 2.2. Let hΛ, <Λ i be a linearly ordered set. The set [Λ]≤n is the following union
{∅} ∪ {{i} | i ∈ Λ} ∪ · · · ∪ {s | s ⊂ Λ, |s| = n}.
A set of ordered tuples {as | s ∈ [Λ]≤n } is a [Λ]n -sequence if
S
(1) as is an ordered tuple enumerating the set bdd( t⊂s at ) without repetitions for every
s ∈ [Λ]≤n ;
(2) if s ⊂ t, then as is a subtuple of at .
8
Again, it is immediate that a [Λ]n -sequence is a directed system of boundedly closed sets
indexed by [Λ]≤n .
Now we turn to indiscernibility of [Λ]n -sequences.
Definition 2.3. Let Λ be a linearly ordered set. Let g : Λ → Λ be a partial order-preserving
function. For each n < ω, there is a unique natural extension (partial) ḡ : [Λ]≤n → [Λ]≤n of
the function g: ḡ(hi1 , . . . , ik i) := hg(i1 ), . . . , g(ik )i.
A [Λ]n -sequence I = {as | s ∈ [Λ]≤n } is [Λ]n -indiscernible if for every partial orderpreserving function g : Λ → Λ, for every s1 , . . . , sN ∈ [dom(g)]≤n we have tp(as1 . . . asN ) =
tp(aḡ(s1 ) . . . aḡ(sN ) ).
If n ≥ 1, I = {as | s ∈ [Λ]≤n } is [Λ]n -indiscernible, and in addition the sequence {a{i} |
i ∈ Λ} is independent over a∅ , we say that I is a [Λ]n -Morley sequence.
When the indexing set is clear, we may omit the prefix [Λ]n , and simply say that I is
indiscernible, or a generalized Morley sequence.
Remark 2.4. (1) For 1 ≤ n < ω, an [n]n−1 -Morley sequence is the same as [n]n -Morley
sequence.
(2) If I = {ai | i < ω} is a Morley sequence over A, the sequence of bounded closures
{bdd(A), bdd(Aai ) | i < ω} is not automatically an [ω]1 -Morley sequence. Enumeration of
bounded closures matters.
The assertion (1) can be verified directly by examining the definition. For (2), let us revisit
the examples in Section 1. In the context of the random graph, let di := {ai , bi } be imaginary
and for all i < j < ω we have R(ai , aj ) with no other relations. The sequence {di | i < ω}
is a Morley sequence over the empty set, but if we are not careful about enumerating the
bounded (in this case, algebraic) closure of, say d1 , the resulting sequence of closures will
not be indiscernible. However, for infinite Morley (or simply indiscernible) sequences this
problem is entirely due to unfortunate enumeration:
Proposition 2.5. Suppose Λ is an infinite linearly ordered set, and I = {a{i} | i ∈ Λ}
≤n
is indiscernible over a set A. Fix an arbitrary n
is an
S < ω. For every s ∈ [Λ] there
≤n
enumeration as of the bounded closure bdd(A ∪ i∈s a{i} ) such that {as | s ∈ [Λ] } is a
[Λ]n -indiscernible sequence.
Proof. Let I 0 ⊃ I be a long extension of the sequenceSI indexed by Λ0 ⊃ Λ. There are
boundedly many ways to enumerate the closure bdd(A∪ i∈s a{i} ), so using Morley’s method
we can “extract” an infinite sequence J := {b{i} | i S
< ω} with the same type diagram over
A as I together with enumerations bs of bdd(A ∪ i∈s b{i} ) such that {bs | s ∈ [ω]≤n } is
[ω]n -indiscernible.
Now we can use compactness to extend the sequence {bs | s ∈ [ω]≤n } to the one indexed
by [Λ]≤n . It remains to note that an automorphism f ∈ Aut(C) sending
the sequence
S
{b{i} | i ∈ Λ} to I over A maps the set enumerated by bs to bdd(A ∪ i∈s a{i} ). Letting
as := f (bs ), we are done.
a
Immediately we get a corollary that will be useful later. It states that any finite, but infinitely extensible indiscernible sequence can be made into a generalized indiscernible sequence.
Corollary 2.6. Suppose Λ is a finite linearly ordered set, and I = {a{i} | i ∈ Λ} is indiscernible over a set A. Suppose further that there is an infinite Λ0 ⊃ Λ and an A-indiscernible
I¯ = {a{i} | i ∈ Λ0 } that extends the sequence I.
9
≤n
Let n :=
is an enumeration as of the bounded closure
S |Λ|. For every s ∈ [Λ] there
≤n
bdd(A ∪ i∈s a{i} ) such that {as | s ∈ [Λ] } is a [Λ]n -indiscernible sequence.
A useful way of thinking about [Λ]n -indiscernible sequences is in terms of partial automorphisms of the monster model.
Lemma 2.7. A sequence I = {as | s ∈ [Λ]≤n } is indiscernible if and only if there is a family
{fg | g : Λ → Λ partial order-preserving function} of elementary maps such that
(1) dom(fg ) = {as | s ∈ [dom(g)]≤n },
(2) if g1 ⊂ g2 , then fg1 ⊂ fg2 ,
(3) fg (as ) = aḡ(s) for all s ∈ [dom(g)]≤n .
Proof. The direction (⇐) is straightforward. For (⇒), take an order-preserving function
g. Since I is [Λ]n -indiscernible, using compactness we have that tp(has is∈[dom(g)]≤n ) =
a
tp(haḡ(s) is∈[dom(g)]≤n ). Therefore, there exists an elementary map fg taking as to aḡ(s) .
Definition 2.8. Let I be a [Λ]n -indiscernible sequence. We use the symbol FI for the
associated set of elementary maps {fg | g : Λ → Λ partial order-preserving function} given
by Lemma 2.7. If necessary, we use fg,I for fg to point out that the elementary map is
associated with the sequence I.
2.2. Extension for generalized Morley sequences; n-simplicity. We now define the
notion of extension for generalized indiscernible sequences. The definition of extension combines three different kinds of extension: for the base, length, and dimension.
Definition 2.9. Let Λ be a linearly ordered set and let I = {as | s ∈ [Λ]≤k } be a [Λ]k indiscernible sequence. We say that a [Λ0 ]n -indiscernible sequence J = {bs | s ∈ [Λ0 ]≤n } is an
extension of I if
(1) a∅ is a subtuple of b∅ ;
(2) Λ ⊆ Λ0 ;
(3) k ≤ n; and two main conditions are
S
(4) for all t ∈ [Λ]≤n , the tuple bt enumerates bdd(b∅ ∪ i∈t a{i} ), extending the tuples as
for each s ∈ [t]≤k ;
(5) for any partial order-preserving g : Λ → Λ, for all s ∈ [dom(g)]k the map fg,J extends
the map fg,I .
If Λ = Λ0 , k = n, then we say that J is a base extension of I. If in addition b∅ ^ I, then
a∅
we call J an independent base extension of I.
If a∅ = b∅ and k = n, then we say that J is a length extension of I.
Finally, if only the dimension k increases, J is a dimension extension of I.
To see the role of conditions (4) and (5), let us prove various extension properties for
[Λ]n -indiscernible sequences, where Λ is infinite. In particular, we show existence of [Λ]<ω indiscernible sequences, i.e. [Λ]<ω -indexed sequences that are simultaneously [Λ]n -indiscernible
for all n < ω.
Lemma 2.10. Let Λ be an infinite linearly ordered set and I = {as | s ∈ [Λ]≤k } be [Λ]k indiscernible. For any tuple b∅ ⊃ a∅ enumerating a boundedly closed set, for any Λ0 ⊃ Λ, and
any n ≥ k there is an automorphic image of I that has an extension to a [Λ]n -indiscernible
sequence J = {bs | s ∈ [Λ0 ]≤n }.
10
Proof. We treat the various extension properties separately.
Length extension follows easily by compactness, the same argument as for extension of
usual indiscernible sequences of hyperimaginaries. Clearly, in the case of length extension
we have that the sequence I itself, not just an automorphic image, is extended.
Dimension extension. We go up by one in dimension, n = k + 1. Let I be a [Λ]k indiscernible sequence. We may assume that Λ = ω. We show that, for any n < N < ω,
n
≤n
there is an
tuple at enumerates the set
S [ω] -sequence J = {at | t ∈ [ω] } such that the
bdd(a∅ ∪ i∈t a{i} ), extending the tuples as for each s ∈ [t]≤k , and for any partial orderpreserving map g : ω → ω, | dom(g)| ≤ N , for all s ∈ [dom(g)]k the map fg,J extends the
map fg,I . This is enough by compactness.
Let µ = sup{|as | | s ∈ [ω]k }+|T |. Let λ := (iN (2µ ))+ , and extend I to a [λ]k -indiscernible
0
≤k
n
sequence
S I = {as | s ∈ [λ] }. For t ∈ [λ] , let at enumerate the bounded closure
) in some way. S
Now color each sequence i0 < · · · < iN −1 < λ according to
bdd( i∈t a{i}S
the type tp( t∈[i0 ,...,iN −1 ]n at / s∈[i0 ,...,iN −1 ]≤k as ). More precisely, for each i0 < · · · < iN −1 ,
take an order-preserving function g : ij 7→ j and consider an automorphism
of C that exS
tends the corresponding elementary map fg . The automorphism sends s∈[i0 ,...,iN −1 ]≤k as to
S
n
corresponding bounded closures.
s∈[0,...,N −1]≤k as and all the at , for t ∈ [i0 , . . . , iN −1 ] , to theS
S
µ
There are at most 2 many different types of the form tp(fg ( t∈[i0 ,...,iN −1 ]n at )/ s∈[i,...,N −1]≤k as ).
By Erdös-Rado theorem, there is an infinite monochromatic subset
of λ, which, without loss
S
of generality, includes ω. The resulting enumerations of bdd( i∈t a{i} ), for t ∈ [ω]n , give the
needed [ω]n -sequence J.
Base extension. We may assume that Λ = Λ0 = ω and k = n. We show that for any
N < ω, there is an automorphic image J = {a0s | s ∈ [ω]n } of I such that J has a base
extension to an ([ω]n , N )-indiscernible sequence over b∅ . That is, thereSis an [ω]n -sequence
J 0 = {bs | s ∈ [ω]≤n } such that the tuple bs enumerates the set bdd(b∅ ∪ i∈s a{i} ), extending
the tuple as for each s ∈ [ω]≤n , and for any partial order-preserving map g : ω → ω,
| dom(g)| ≤ N , for all s ∈ [dom(g)]n the map fg,J 0 extends the map fg,J . This suffices by
compactness.
The argument is similar to the one in the dimension extension part. We take µ =
sup{|as | | s ∈ [ω]k } + |b∅ | + |T |, λ := (iN (2µ ))+ , and extend I to a [λ]n -indiscernible
IS0 = {as | s ∈ [λ]≤n }. For s ∈ [λ]≤n , let bs enumerate the bounded closure bdd(b∅ ∪
i∈s
Sa{i} ) in some way.
S Color each sequence i0 < · · · < iN −1 < λ according to the type
tp( s∈[i0 ,...,iN −1 ]n bs / s∈[i0 ,...,iN −1 ]≤n as ). By Erdös-Rado and compactness, we find b0∅ |= tp(b∅ )
such that I has a base extension to an ([ω]n , N )-indiscernible sequence over b0∅ . Conjugating
b0∅ to b∅ , we get the desired J.
a
The main idea behind the n-simplicity analysis is to see whether various extension properties hold for finite generalized Morley sequences. It is important to note that independence,
not just indiscernibility, is now in the picture as well.
One of the main results of the first part of the paper is the following theorem.
Theorem 2.11. Let T be simple, n ≥ 1. The following conditions are equivalent:
(1) T has (n + 2)-complete amalgamation property;
(2) for each 1 ≤ k ≤ n, for every [k + 1]k -Morley sequence I and every boundedly closed
b0∅ , if the subsequence I 0 := {as | s ∈ [k]k } of I has an independent base extension
to a [k]k -Morley sequence J 0 = {b0s | s ∈ [k]k } over b0∅ , then I has an independent
11
base extension to a [k + 1]k -Morley sequence J = {bs | s ∈ [k + 1]k } over b∅ , where
tp(b0k ) = tp(bk ).
(3) for each 1 ≤ k ≤ n, every [k + 1]k -Morley sequence I can be length-extended to a
[k + 2]k+1 -Morley sequence.
Definition 2.12. Let n ≥ 1. We say that T is n-simple if T satisfies the condition 2, and
therefore each of the equivalent conditions, in Theorem 2.11.
Until we prove Theorem 2.11, we take the condition 2 as the definition of n-simplicity. As
we pointed out earlier, this replaces the old definition in [9, 7].
Remark 2.13. Condition (2) is visibly more complicated than (3); and there is a good
reason for this. To simply demand for any [k]k -Morley sequence to have a base extension
is too much to ask in a simple theory for k ≥ 3. We illustrate this point on a very simple
example: in a model of the theory of a random graph, take a Morley (over the empty set)
sequence {a0 , a1 , a2 }. Let b be such that R(a0 , b) ∧ ¬R(a1 , b). We certainly cannot find
b0 |= tp(b/a0 a1 ) such that {a0 , a1 , a2 } are b0 -Morley. To be able to find the base extension,
we need to demand that {a0 , a1 } are b-Morley to begin with. After we take the bounded
closures into account, the condition (2) becomes what it is.
We saw in Lemma 2.10 that given an infinite [ω]k -Morley sequence I and any possible
extension of the base, we can move the sequence (or equivalently the base) so that the image
becomes [ω]k -Morley over the larger base. A natural question to ask is can we do it while
fixing an initial segment of the sequence I. The resulting hierarchy of properties (originally
introduced in Kolesnikov’s paper [9]) is what we call K(n)-simplicity in this paper. It turns
out that the two hierarchies (n- and K(n)-simplicity) stay together for n = 1, 2, but diverge
at n = 3.
Definition 2.14. Let T be a simple theory, n ≥ 1. We say that T is K(n)-simple if
for each 1 ≤ k ≤ n, for every [ω]k -Morley sequence I and every boundedly closed b∅ , if
the subsequence I 0 := {as | s ∈ [k]k } has an independent base extension to a [k]k -Morley
sequence J 0 = {b0s | s ∈ [k]k } over b0∅ , then I has an independent base extension to a
[ω]k -Morley sequence J = {bs | s ∈ [ω]k } over b∅ , where tp(b0k ) = tp(bk ).
To see what the definition states, let us go through the cases n = 1, 2 in the definition
while removing the bounded closures. For n = 1, we have: if I = {ai | i < ω} is a Morley
sequence over A (remember A = a∅ ) and B ⊃ A with B ^ a0 , then there is B 0 |= tp(B/Aa0 )
A
such that I is B 0 -Morley. Certainly, this property holds in every simple theory. We are not
gaining anything with the additional demands on bounded closure.
In the case n = 2, a simplified version of the definition says that given an A-Morley
sequence I and B ⊃ A with B ^ a0 a1 , assuming that {a0 , a1 } is a B-Morley sequence, we
A
can find B 0 |= tp(B/Aa0 a1 ) such that I is B 0 -Morley. In this case, the stronger indiscernibility
demand is necessary to make things work in C(h)eq .
Immediately from definitions, a theory T is 1-simple if and only if it is K(1)-simple if and
only if T is simple. For larger values of n, here is the behavior:
Theorem 2.15. I. The following theories exist.
12
(1) For each 2 ≤ n < ω, there is a theory which is (n − 1)-simple and K(n − 1)-simple,
but neither n-simple, nor K(n)-simple.
(2) There is a theory both n-simple and K(n)-simple for all n < ω.
II. Let T be simple. The following are equivalent.
(1) T is K(2)-simple;
(2) T has 4-amalgamation property;
(3) T is 2-simple.
III. If a theory is n-simple, then it is K(n)-simple. There is a theory which is K(n)-simple
for all n, but fails to be 3-simple.
Even though K(n)-simplicity is not equivalent to (n + 2)-amalgamation, the former family
of properties seems to generalize smoothly the definition of non-dividing. For example, if
n = 2, then K(2)-simplicity provides a stronger consequence of independence of B from a0 a1
over A than non-dividing. Moreover originally, K(1)-simplicity (not 1-simplicity) is the key
property showing the Independence Theorem (=3-amalgamation) in [8]. So it is of interest
to study the family of K(n)-simple theories for n ≥ 3. A first step is a proof of amalgamation
for “big” types. This is the content of Theorem 5.2 in Section 5.
3. Amalgamation results
This section contains various consequences of n-amalgamation property, involving amalgamation over more complicated systems than the ones indexed by P − (n). This is used in
the proofs of Theorems 2.11 and 2.15.
We start by introducing a notion of the coherent system of types indexed by a partially
ordered set S. This generalizes Definition 1.9 in a natural way.
Definition 3.1. Let hS, <S i be a partially ordered set with the least element 0, in which
any two elements have the infimum and, provided there is an upper bound, supremum.
Let {As | s ∈ S} be a directed independent system of boundedly closed sets, As ⊂ Cheq .
We say that {ps (xs ; As ) | s ∈ S} is a coherent system of types over {As | s ∈ S} if the
following conditions hold:
(1) dom(ps ) = As ; and if Cs |= ps , then Cs ⊃ As ;
(2) if s <S t, then xs ⊆ xt and ps ⊆ pt ;
(3) for all s ∈ S the map fs : Cs → xs is a bijection;
(4) for Cs0 := fs−1 ◦f0 (C0 ), we demand that Cs is bdd(As ∪Cs0 ) for all s ∈ S, and Cs0 ^ As .
A0
Given such a coherent system of types, by the restriction of ps (xs ; As ) to At , where t <S s,
we mean the type pt (xt ; At ). The reverse relation is an extension. When convenient, we
denote the restriction as ps dAt , and the extension pt |As . Note that due to (4), it follows that
xs∧t = xs ∩ xt . In a coherent system of types, any two types ps (xs ; As ) and pt (xt ; At ) are
compatible in the sense that ps dAs∧t = pt dAs∧t .
As in 1.11, we can give the similar notational convention by calling the types in the
system generalized types. Let us say more on our terminology generalized types. When we
call tp(d/A) a generalized type over A, we mean the usual type tp(bdd(dA)/ bdd(A)) over
bdd(A). When we deal with generalized types we shall make it explicit. Otherwise types are
the usual ones. Indeed the terminology of generalized types (and the coherent system of
types) will more or less exclusively used in the following context.
13
Definition 3.2. Let A = {aj |j ∈ J} be B-independent. Let {Ai |i ∈ I} be a certain
collection of nonempty subsets of B ∪ A with B ⊆ Aj and B ∪ A = ∪j Aj . We say types
pi (x) ∈ S(Ai ) are compatible over B if each pi does not fork over B and for i, j, pi dAi ∩ Aj =
pj dAi ∩ Aj . (Hence pi dB = pj dB). We say these B-compatible types pi (x) are (generically or
independently) amalgamated if there is q(x) ∈ S(BA) nonforking over B such that ∪i pi ⊆ q
(i.e. q ` pi ).
For generalized types, we can define similar notions with extra cares. Namely, if each pi (x)
is a generalized type tp(di /Ai ) (which stands for tp(bdd(di Ai )/ bdd(Ai ))), then pi (x) indeed
is pi (xi ; bdd(Ai )) with the variable xi for bdd(di Ai ) containing x for bdd(di B). The generalized types pi are B-compatible if di ^ Ai and pi dAi ∩ Aj := tp(bdd(di (Ai ∩ Aj ))/ bdd(Ai ∩
B
Aj )) = pj dAi ∩ Aj . But a priori, the variable for bdd(di (Ai ∩ Aj )) must be xi ∩ xj . In
these manners, the B-compatible generalized types pi induce a canonical coherent system of types over the family of intersections of the sets Ai with the least element B and
the greatest element A. Again, given an intersection, say Ai ∩ Aj ∩ Ak , the variable for
it comes from the same combination, say xi ∩ xj ∩ xk (note that, due to independence,
bdd(Ai ∩ Aj ∩ Ak ) = bdd(Ai ) ∩ bdd(Aj ) ∩ bdd(Ak )). Now we say these pi (x) are generically
amalgamated if ∪i pi is realized by some d (for real bdd(dBA)) such that d ^ A.
B
Having these notational conventions, we can further simplify our description 1.10 of generalized amalgamation as follows: T has (n + 1)-amalgamation over B iff for B-independent
A = {a1 , ..., an } and any B-compatible generalized types pi (x) over Ai where Ai = BAr{ai }
(i = 1, .., n), p1 ∪ ... ∪ pn is generically amalgamated.
So (n + 1)-amalgamation is the property on a particular n-cover of an independent parameter set. The following “free amalgamation” theorem asserts that it gives more general
property for n generalized types over any n-cover of the parameter set.
3.1. Free amalgamation theorem.
Theorem 3.3. Suppose that T is a simple theory having (n + 1)-complete amalgamation.
Assume that sets B, A, Ai and B-compatible generalized types pi (x) are given as in 3.2 with
I = {1, .., n}. Then the types pi can be amalgamated.
Proof. We prove this by induction on n ≥ 2. When n = 2, the statement follows from the
Independence Theorem. Assume the theorem holds for n − 1. For simplicity assume B = ∅.
Let Ai1 ...ik denote Ai1 ∩ · · · ∩ Aik . For k < n, let
S
Ck = {Av | v ⊆ I, |v| = n − k}.
Note that then C0 = A1...n ⊆ ... ⊆ Cn−1 = A.
Claim 1. There is a generalized type q1 over C1 that is compatible with each type pi ,
i = 1, . . . , n.
S
Proof of Claim 1. Note that since A1...î..n ∩ j∈{1,...,î,..n} A1...ĵ...i...n = A1...n , we have that
{A1...î..n |1S≤ i ≤ n} is A1...n -independent. Moreover by compatibility of pi , we get that the
types pi d j∈{1,...,î,..n} A1...ĵ...i...n are again A1...n -compatible. Then by (n + 1)-amalgamation
S
of the types pi d j∈{1,...,î,..n} A1...ĵ...i...n over A1...n , we get the desired q1 . Hence Claim 1 is
proved. ¤
14
We shall now show that for k = 1, . . . , n − 1, there is a generalized type qk over Ck such
that
qk−1 ⊆ qk
qk is compatible with each type pi , i = 1, . . . , n.
(∗)
Note that the n − 1 case finishes the proof of the theorem, and Claim 1 gives (∗) for k = 1.
Hence assume that (∗) holds for k − 1 (k > 1) and we have qk−1 ⊇ qk−2
S · · · ⊇ q1 . We are
proving (∗) for k. For this, let Vk = {v ⊆ I| |v| = n − k}, and let AV := v∈V Av for V ⊆ Vk .
By induction on |Vk |, we show that there is a generalized type qV over bdd(Ck−1 ∪ AV )
such that
qV is compatible with each pi and extends qk−1 .
(∗∗)
For V = ∅, we are given a type qk−1 over Ck−1 . So it suffices to show the following claim
which provides the induction step for (∗∗), and, as AVk = Ck , completes the proof of (∗) for
k.
Claim 2. Given a subset V of Vk , and a generalized type qV satisfying (∗∗) and w ∈ Vk \ V ,
there exists a type qw (or written qV ∪{w} ) over bdd(Ck−1 ∪ AV ∪{w} ) satisfying (∗∗) and
extending qV .
Proof of Claim 2. Let w = {i1 , . . . , in−k }. Now consider n − k + 1 many types: qV and
pil dDl where Dl = Ail ∩ Ck−1 AV ∪{w} for l = 1, .., n − k. We first check that they are
compatible: For 1 ≤ l 6= m ≤ n − k, Dl ∩ Dm = Ail im . Hence by compatibility of pi ,
pil dDl ∩ Dm = pil dAil im = pim dDl ∩ Dm . Moreover Ck−1 AV ∩ Dl = Ail ∩ Ck−1 AV . Therefore
by (**), qV dCk−1 AV ∩ Dl = qV dAil ∩ Ck−1 AV = pil dCk−1 AV ∩ Dl . Thus, the types are indeed
compatible.
Then by the induction hypothesis of the theorem for n − k + 1(< n), there is qw , an
amalgam of qV and pil dDl . It is compatible with pil since qw ∩ pil = pil dDl . So is with other
pj for j ∈
/ w: Note that Aj ∩ Ck−1 AV ∪{w} = Aj ∩ Ck−1 AV . Thus qw , an extension of qV over
Ck−1 AV ∪{w} already contains pj dAj ∩ CAV ∪{w} = pj dAj ∩ CAV ⊆ qV .
This finishes the proof of Claim 2 and the proof of the theorem.
a
3.2. Amalgamation results. The reader is probably convinced by now that results under
the notions of strong (n−1)-amalgamation in terms of types, and strong n-simplicity in terms
of Morley sequences in [9] can be smoothly lifted into the new context of n-amalgamation with
generalized types, and n-simplicity with generalized Morley sequences, since after changing
the set of definitions appropriately into the new context, the same ideas lead to the corresponding results.
The following lemma is stated in the context of definitions of n-amalgamation in this
paper. This is the first result of [9] that we lift to the new context, so we provide a detailed
proof. In particular, we do not use the full power of our notational agreement 1.11 and the
remark below Definition 3.2.
Lemma 3.4. Fix n ≥ 2 and let N ≥ n be given. Let {Aw | w ∈ [N ]≤n−1 } be a directed
system of boundedly closed sets, and let {pw (xw ; Aw ) | w ∈ [N ]≤n−1 } be a coherent system of
types over the directed system. If T has (n + 1)-amalgamation property, then the system of
typesScan be independently amalgamated. That is, there is a type pN (xN ; An ), where AN =
bdd( w∈[N ]n−1 Aw ), such that {pw (xw ; Aw ) | w ∈ [N ]≤n−1 } ∪ {pN (xN ; AN )} is a coherent
system of types.
Proof. We prove the statement by a stronger induction hypothesis. Namely we prove for any
N = {0, ..., N − 1} ≥ n,
15
(1) the coherent system of types over {Aw | w ∈ [N ]≤n−1 } can be independently amalgamated by a type pN (xN ; AN );
(2) in addition, the type pN (xN ; AN ) is an independent amalgam of the following coherent
system of types
(∗) qs , where for every s ∈ [{N − n + 1, . . . , N − 1}]n−2 , the type qs is an amalgam
of
{pw (xw ; Aw ) | w ∈ [{0, . . . , N − n} ∪s]n−1 };
|
{z
}
N −n+1 elements
(∗∗) the type p{iN −n+1 ,...,iN −1 } .
The first statement for the base case N = n follows from (n + 1)-amalgamation property
by Proposition 1.10. The second statement is immediate: given an (n − 2)-element subset
s of {1, . . . , n − 1}, the type qs is simply the type pw (xw ; Aw ) in the original system, where
w = {0} ∪ s.
Suppose the statement is true for some N ≥ n, and assume we are given a coherent system
{pw (xw ; Aw ) | w ∈ [N +1]n−1 }. For each t ⊂ {N −n+2, . . . , N } of size n−2, consider the N element set wt := {0, . . . , N − n + 1} ∪t. Using the induction hypothesis, we get n − 1 types
|
{z
}
N −n+2 elements
qt := pwt (xwt ; Awt ). Together with the type pw0 (xw0 ; Aw0 ) (w0 = {N −n+2, . . . , N }) we obtain
a coherent system of n types, which has an independent amalgam by Proposition 1.10. a
For the remainder of the section, we give the proofs of the new amalgamation results only,
and for the parallel results coming from old results in [9], we provide precise references.
We now turn to amalgamation “over Morley sequences”. In the simplest set-up, suppose we
have a 3-element Morley sequence {a0 , a1 , a2 } that we want to extend to a 4-element sequence,
using 4-amalgamation. The idea is quite clear: take the type p01 (x; a0 a1 ) := tp(a2 /a0 a1 ),
conjugate to a0 a2 and a1 a2 , and amalgamate the resulting three types p01 , p02 , and p12 over
the base a0 a1 a2 . Looking a bit ahead, we then can use Lemma 3.4 to extend the sequence
further, getting an infinite Morley sequence that extends {a0 , a1 , a2 }.
In the full setting, we are dealing with the generalized Morley sequences, and need to
account for the bounded closures. Therefore, the entire system of types must be conjugated.
This motivates the following definition.
Definition 3.5. We say that a system of boundedly closed sets {Bv | v ∈ [Λ0 ]≤n } is isomorphic to {Au | u ∈ [Λ]≤n } if there is an order-preserving bijection g : Λ → Λ0 and
a corresponding elementary map fg : {Au | u ∈ [Λ]≤n } → {Bv | v ∈ [Λ0 ]≤n } such that
fg (Au ) = Bg(u) for all u ∈ [Λ]≤n .
Let {pu (xu ; Au ) | u ∈ [Λ]≤n } be a coherent system of types over {Au | u ∈ [Λ]≤n }, and let
{Bv | v ∈ [Λ0 ]≤n } be an isomorphic system of boundedly closed sets via an elementary map
fg , where g : Λ → Λ0 is an order preserving bijection.
By conjugate system to {pu (xu ; Au ) | u ∈ [Λ]≤n } over {Bv | v ∈ [Λ0 ]≤n } we mean the
system of types {qv (xv ; Bv ) | v ∈ [Λ0 ]≤n } such that
(1) if v = g(u) for some u ∈ [Λ]≤n , and v ⊂ Λ ∩ Λ0 (in particular, if v = ∅), then we
already have the variables xv , and qv is obtained from pu by replacing each parameter
a ∈ Au by fg (a) ∈ Bu ;
(2) assuming xv0 have been defined for v 0 ( v, let xv be the list of variables that extends
xv0 , v 0 ( v, and contains new variables for each variable in the list xu \ ∪xu0 . As
16
before, qv is obtained from pu by replacing each parameter a ∈ Au by fg (a) ∈ Bu ,
and the variable list xu by the list xv .
Immediately from the definitions, we get the following.
Proposition 3.6. Let I = {as | s ∈ [ω]≤n } be a generalized Morley sequence.
1. If X, Y ⊂ ω have the same cardinality, then IX := {as | s ∈ [X]≤n } is isomorphic to
IY .
2. If {pu (xu ; au ) | u ∈ [n]≤n } is a coherent system of types (on the first n elements of
the sequence), and for every v ∈ [ω]n the system {pw (xw ; aw ) | w ⊆ v} is conjugate to
{pu (xu ; au ) | u ∈ [n]≤n }, then the entire system {pw (xw ; aw ) | w ∈ [ω]≤n } is a coherent
system of types over the sequence I.
Remark 3.7.
(1) Suppose that I = {as | s ∈ [Λ]≤n } is a generalized Morley sequence.
Let J = {bs | s ∈ [Λ]≤n } be an independent base extension of I to a base b∅ . For each
s ∈ [Λ]≤n , let ps (xs ; as ) be the type tp(bs /as ). It is easy to see that {ps | s ∈ [Λ]≤n }
form a coherent system of types over I. We refer to such a system as the system of
types generated by the extension J over I.
(2) We also take simplified notational convention for generalized Morley sequences. When
we call I = hA; a0 , ..., an−1 i (where the pure sequence ha0 , .., an−1 i is A-Morley) an
[n]k -Morley sequence, we indeed mean I to be
{bdd(A ∪ {ai | i ∈ s})| s ⊆ n, |s| ≤ k}.
We can also say ha0 , ..., an−1 i is [n]k -Morley over A. When A = ∅ (k = n resp.), we
may omit A ([n]n resp.).
We now provide the main amalgamation results from [9], stated in the context of this
paper. Firstly, the following is a direct consequence of Lemma 3.4. We also refer the reader
to Proposition 4.6 in [9].
Proposition 3.8. Suppose that T has (n + 2)-amalgamation. Take a linear order Λ with
Λ ≥ n (even more n = {0, ..., n − 1} ⊆ Λ), and let I := {as | s ∈ [Λ]≤n } be a generalized
Morley sequence.
Let J := {bs | s ∈ [n]n } be an independent base extension of the initial segment {as | s ∈
[n]n } of I to the base b∅ . Let {ps | s ∈ [n]≤n } be the coherent system of types generated by the
extension J over {as | s ∈ [n]n }. For an arbitrary u ∈ [Λ]n , let pu (xu ; au ) be the conjugate
system. Then the system of types {p(xu , au ) | u ∈ [Λ]n } can be generically amalgamated.
Corollary 3.9. Suppose T is a simple theory with (n + 2)-complete amalgamation. Then T
is n-simple and K(n)-simple.
The following proposition is a re-statement of Theorem 5.4 in [9] in the context of the
present paper.
Proposition 3.10. Suppose T is simple, n ≥ 1. Then the following conditions are equivalent:
(1) T has (n + 2)-complete amalgamation;
(2) for all 1 ≤ k ≤ n, every [k + 1]k+1 -Morley sequence can be length extended to an
infinite [ω]k+1 -Morley sequence over A;
(3) for all 1 ≤ k ≤ n, every [k+1]k -Morley sequence can be length extended to a [k+2]k+1 Morley sequence.
17
Using the length extension terminology, the equivalence (1) ⇔ (3) can be restated as
follows:
Corollary 3.11. Let T be a simple theory, n ≥ 1. Then (n + 2)-complete amalgamation is
equivalent to length extension property for [k + 1]k -Morley sequences, 1 ≤ k ≤ n.
We finish the subsection with re-statement of Theorem 6.5 in [9].
Proposition 3.12. If a theory T is K(2)-simple, then T has 4-amalgamation.
3.3. Proofs of Theorems 2.11, 2.15. As promised, we now provide with the proof of
Theorem 2.11.
Proof of Theorem 2.11. The equivalence (1) ⇔ (3) is given by Proposition 3.10. The implication (1) ⇒ (2) follows from Proposition 3.8.
For (2) ⇒ (3), take a [k + 1]k+1 -Morley sequence I = {as | s ∈ [k + 1]k+1 }, and let
0
I := {as | s ∈ [k]k } be the initial segment of the sequence I. Letting b0∅ := a{k} , we note
that the original sequence I can be viewed as a base extension J 0 of I 0 to the base a{k} .
Applying (2), we let b∅ be such that I has an independent base extension to the sequence
J = {bs | s ∈ [k + 1]k+1 } and the generalized type of bk is the same as the generalized type
of b0k .
Now for t ⊂ {0, . . . , k, k + 1}, k + 1 ∈ t, we let at := bt\{k+1} (in particular, a{k+1} := b∅ ).
Thus, together with the original {as | s ∈ [k+1]k+1 }, we get a system of boundedly closed sets
indexed by [k + 2]k+1 . Then by construction, the system is a [k + 2]k+1 -Morley sequence. a
We note that above proof, Corollary 3.9, and Proposition 3.12 clearly establish parts II
and III(1) of Theorem 2.15. The rest examples of 2.15 will be constructed in the next section.
4. Examples
All the examples presented in this section are simple theories with elimination of hyperimaginaries, weak elimination of imaginaries, trivial forking, and trivial algebraic closure in
the home sort. Thus, we will be working in Ceq and with the algebraic closure in the place
of bounded closure. We begin the section with a further simplification.
Lemma 4.1. Let T be simple with weak elimination of imaginaries. Suppose that T has namalgamation property for algebraically closed sets in the home sort for some n ≥ 4. Then
T has n-amalgamation property for algebraically closed sets in Ceq .
Remark 4.2. It is helpful to keep in mind that the theorem fails without weak elimination
of imaginaries; namely it is not possible to replace dcleq by acleq in the proof below.
Proof. Let {Au | u ∈ P − (n)} be a directed system of boundedly closed sets, Au ⊂ Mueq . Let
A0u := Au ∩ Mu= . By weak elimination of imaginaries, Au = dcleq (A0u ). By our hypothesis,
there is an algebraically closed A0n completing the directed system {A0u | u ∈ P − (n)}. Now
it only remains to check that dcleq (A0n ) completes the original system {Au | u ∈ P − (n)}.
By weak elimination of imaginaries, dcleq (A0n ) = acleq (A0n ), so An is in fact algebraically
closed. Now fix u ∈ P − (n). Given the elementary map πnu : A0u → A0n , we need to produce
an extension π̄nu : Au → An . This amounts to having tp(dcleq (A0u ) = tp(dcleq (πnu (A0u )), which
is immediate.
a
18
Remark 4.3. In all the examples we present in this section, acl= (A) = A for all A ⊂ C.
Thus, the generalized types, generalized Morley sequences, and amalgamation of coherent
systems of generalized types over algebraically closed sets simplify to just types, Morley
sequences, and amalgamation of coherent types, all in home sort.
4.1. Bi-partite tetrahedron-free hypergraphs. Let k ≥ 3. The theories Tk we introduce
below were studied in [9]. Fix k ≥ 3, let Lk := {P, R, S}, where P is an unary predicate, S
and R are k-ary predicates. Let Tk be the model completion of the following set of sentences
in Lk :
“R ⊂ P k ;”
“R is symmetric irreflexive (with respect to all permutations);”
“S ⊂ P k−1 × ¬P ” (we use the notation x̄ S y, x̄ is understood to be a tuple in P k−1 );
“S is symmetric
· irreflexive in the first k − 1 variables;”
¸
W
(5) ∀x1 . . . xk , y R(x1 , . . . , xk ) → w⊂{1,...,k} ¬(x̄w S y) , where for w = {i1 , . . . , ik−1 } we
(1)
(2)
(3)
(4)
put x̄w := xi1 . . . xik−1 .
|w|=k−1
It was shown in [9] that the theories Tk are SU -rank 1, ω-categorical, and simple unstable.
Moreover, for every k ≥ 3 the theory Tk is K(k − 2)-simple, but fails to be K(k − 1)-simple.
Visibly, Tk fails to have (k + 1)-amalgamation, so it is not (k − 1)-simple. It is not hard to
check that k-complete amalgamation holds in Tk (this is essentially the content of the proof
of Proposition 2.6 in [9]). Thus Tk is (k − 2)-simple.
Finally, the theory Trg of the generic (binary) graph is an example of an ω-simple(= nsimple for all n) and K(ω)-simple(= K(n)-simple for all n) unstable theory.
4.2. Tetrahedron-free hypergraphs. Fix k ≥ 3, and let the language L := {R} consist of
a single k-ary predicate. Let T F Hk be the model completion of the following set of sentences
in L:
(1) “R is symmetric
W irreflexive (with respect to all permutations);”
(2) ∀x0 . . . xk , y w⊂{0,...,k} ¬R(xi1 . . . xik−1 ), where w = {i1 , . . . , ik−1 }.
|w|=k
Similarly to the examples in the previous subsections, it can be shown that the theories
T F Hk are SU -rank 1, ω-categorical, and simple unstable. Also for every k ≥ 3 the theory
T F Hk is both (k−2)-simple and K(k−2)-simple, but is neither (k−1)-, nor K(k−1)-simple.
4.3. K(n − 1)-simple theories without (n + 1)-amalgamation; even n. We fix an even
number n ≥ 4 and let R an n-ary relation symbol. We consider symmetric and irreflexive Rstructures. For such an R-structure A, let no(A) denote the number (modulo 2) of n-element
subsets of A satisfying R.
Definition 4.4. Let (∗) be the following condition on an R-structure A:
(∗) If A0 is an (n + 1)-element subset of A, then no(A0 ) = 0.
Let K be the class of all finite (symmetric and irreflexive) R-structures having (∗).
Lemma 4.5. (Main lemma) Let A be an R-structure with n + 2 elements. If A ∈
/ K, then
there are at least two (n + 1)-element subsets A0 , A1 ⊂ A with no(Ai ) 6= 0 (i = 0, 1).
19
Proof.
Let Ai (i = 1, ..., n + 2) be an enumeration of (n + 1)-element subsets of A. Then
P
i=1,..,n+1 no(Ai ) = 2 ∗ no(A). So if one of Ai satisfy no(Ai ) = 1, then there must be another
Aj with no(Ai ) = 1.
a
Lemma 4.6. K has the amalgamation property (in the sense of generic structures).
Proof. Let Aa, Ab ∈ K. Let B = Aab be the free amalgam of Aa and Ab over A. So
RB = RAa ∪RAb . We show that B can be expanded to an R-structure with (∗), by preserving
R on Aa and Ab. We prove this by induction on |A|. Let A = {a1 , ..., am , am+1 }. By the
induction hypothesis, we may assume that B0 = {a1 , ..., am , a, b} = B r {am+1 } already has
the property (∗). For each X0 ⊂ {a1 , ..., am−1 } with |X0 | = n − 3, we can determine whether
R(X0 ∪ {am+1 , a, b}) or not so that no(X0 ∪ {am , am+1 , a, b}) = 0 holds. Now we show that
B has the property (∗). Assume otherwise. Then there is an (n + 1)-element subset Y ⊂ B
with no(Y ) = 1. Y is contained neither in Aa nor in Ab. Y is not contained in B0 by the
induction hypothesis. Y does not have am , by our construction. So Y must have the form
Y0 ∪ {am+1 , a, b} for some Y0 ⊂ {a1 , ..., am−1 } with |Y0 | = n − 2. Let us consider the set
Y ∗ = Y ∪ {am }. Y ∗ does not have the property (∗), because no(Y ) = 1. By the lemma
above, there must be an (n + 1)-element set Y 0 ⊂ Y ∗ , Y 0 6= Y with no(Y 0 ) = 1. But this is
impossible.
a
By the last lemma, K has a generic structure M . Let Un be the theory of M . By a
back-and-forth argument, we can easily show that Un is ℵ0 -categorical and with elimination
of quantifiers. It is also clear that acl(A) = A holds in any model of Un . We work in a big
model M of Un .
Lemma 4.7. Un has weak elimination of imaginaries. (n ≥ 4).
Proof. We show for n = 4. The cases of other n can be proved similarly.
Claim 4.8. Let X = ABCD be an R-structure such that
• ABC, ABD, ACD ∈ K,
• A, B, C, D are mutually disjoint.
Then there is an R-structure Y ∈ K such that
• the domain of Y is ABCD,
• Y |ABC = X|ABC, Y |ABD = X|ABD and Y |ACD = X|ACD,
where Y |ABC denotes the substructure of X with the universe ABC.
First we show the claim by assuming B = {b}, C = {c}, D = {d}. Let A = {a} ∪ {ai }i≤m .
Let the universe of Y be the set ABCD. We define RY . For a 4-element subset F of Y ,
(1) if F ⊂ ABD, F ⊂ ABD or F ⊂ ACD then Y |= R(F ) if and only if X |= R(F ),
(2) if F = {ai , b, c, d} then Y |= R(F ) if and only if no(F a) = 1 in X (so no(F a) = 0 in
Y ).
Now we prove that Y belongs to K. If this is not the case, then Y has a five element
subset F 0 with no(F 0 ) = 1. By our construction (properties 1 and 2), F 0 must have the form
F 0 = {ai , aj , b, c, d}. Let us consider 6 element subset F 0 a. F 0 a does not belong to K, so by
the main lemma, there is a five element set F 00 ⊂ F 0 a with no(F 00 ) = 1, F 00 6= F 0 . We must
have F 00 = {a, ai , b, c, d} or F 00 = {a, aj , b, c, d}. In either case, we have a contradiction with
property 2. Now we prove the claim by induction on |B|+|C|+|D|. So let D = D0 ∪{d}. By
20
the induction hypothesis, there is Y0 with the universe ABCD such that Y0 |ABCD0 ∈ K,
Y0 |ABC = X|ABC, Y0 |ABD = X|ABD and Y |ACD = X|ACD. Let A∗ = AD0 . We now
consider four sets A∗ , B, C and {d}. Since |B|+|C|+|{d}| < |B|+|C|+|D|, by the induction
hypothesis again, there is an R-structure Y ∈ K with the universe A∗ BCd(= ABCD) such
that Y |A∗ BC = Y0 |A∗ BC, Y |A∗ Bd = Y0 |A∗ Bd and Y |A∗ Cd = Y0 |A∗ Cd. For this Y we
have Y |ABC = X|ABC, Y |ABD = X|ABD and Y |ACD = X|ACD. This finishes the
proof of Claim 4.8.
Claim 4.9. We work in M. Suppose that two formulas ϕ(x̄, ā) and ψ(x̄, b̄) define the same
set A ⊂ Mk . Then A is definable over ā ∩ b̄.
By the ℵ0 -categoricity and the elimination of quantifiers, we may assume that ϕ(x̄, ā) and
ψ(x̄, b̄) are quantifier free formulas, and that they define a quantifier free complete type p(x̄)
¯ ā) then d¯ ∩ (ā ∪ b̄) = ∅, since algebraic closure is
over ā ∩ b̄. We can also assume that if ϕ(d,
trivial in M. We show that p(x̄) is equivalent to ϕ(x̄, ā). Suppose otherwise. Then there is
a quantifier free formula ϕ0 (x̄, ā) such that
ϕ0 (x̄, ā) ` p(x̄) ∪ {¬ϕ(x̄, ā)}
Let d¯0 ∈ M realize ϕ0 (x̄, ā) and d¯00 ∈ M realize ψ(x̄, b̄). Identifying d¯0 and d¯00 , we have an
¯ ā) ∧ ψ(d,
¯ b̄). X may not be a structure in K.
R-structure X = āb̄d¯ such that X |= ϕ0 (d,
¯
However, by applying claim A to X with A = ā ∩ b̄, B = ā − b̄, C = b̄ − ā and D = d,
¯ X|b̄d¯ = Y |b̄d,
¯ X|āb̄ = Y |āb̄. By the
we get an R-structure Y ∈ K such that X|ād¯ = Y |ād,
∗
∗ ∼
¯
¯
genericity, we can find āb̄d ⊂ M such that āb̄d =āb̄ Y . In M, we have ¬ϕ(d¯∗ , ā) ∧ ψ(d¯∗ , b̄),
contradicting the equivalence of ϕ and ψ. So we conclude that p(x̄) is equivalent to ϕ(x̄, ā).
We have proved Claim 4.9.
Let āE be an imaginary element, where E(x̄, ȳ) is a ∅-definable equivalence relation. By
reordering x̄ and ȳ if necessary, we can choose ā0 , b̄0 , b̄1 with the following properties:
• E(b̄i ā0 , ā) (i = 0, 1)
• ā0 ∈ acl(āE ) and b̄1 ∩ b̄2 = ∅.
Then, by claim 4.9, we have E(x̄, ā) is equivalent to a formula with the parameter ā0 . So we
have āE ∈ dcl(ā0 ) and ā0 ∈ acl(āE ).
a
Proposition 4.10. Un is K(n − 1)-simple, (and simple).
Proof. By the last lemma, we can work in M instead of working in Meq . Let I = ha0i ...aki :
i < ωi be infinite indiscernible over C. Let J = ha0i ...aki : i < n − 1i be the first n − 1
terms of I. We assume that J is BC-indiscernible. We need to find B 0 ⊂ M, B 0 ∼
=JC B
0
0
such that I is B C-indiscernible. It is possible to find such B (outside M) so that I is
B 0 C-indiscernible in the structure IB 0 C. So, by the genericity of M, it suffices to show that
the structure ICB 0 has the property (∗). Suppose otherwise and choose (n + 1)-element set
A = {a0 , ..., an−1 , e} ⊂ ICB 0 with no(A) = 1. We show that A cannot be a counterexample.
If A ⊂ IC then it is a substructure of M (so it satisfies (∗)). So we may assume that
e ∈ B 0 . If one of the ai ’s belongs to C, then there is A0 ⊂ JBC ⊂ M with A0 ∼
= A, so
k(i)
A cannot be a counterexample. Hence we can assume ai = ai (i = 0, ..., n − 1). Let
m = |{k(i) : i = 0, ..., n − 1}|. We proceed by induction on m.
Case m = 1. We may assume the common value of k(i) is 0. If no({a00 , ..., a0n−1 }) =
1, then by the indiscernibility, we would have no({a00 , ..., a0n }) = n + 1 = 1 mod 2. So
21
no({a00 , ..., a0n−1 }) = 0. If R(a00 , ..., a0n−2 , e) holds, then by the indiscernibility of J over e,
we have R(X, e) holds for any (n − 1)-element subset X of {0, ..., n − 1}. So we must have
no({a0 , ..., an−1 , e}) = n = 0 mod 2. A contradiction.
Case m = m0 + 1. Since other cases are similar, we may assume that |{k(i) : i =
0, ..., n − 2}| = m0 and k(n − 1) ∈
/ {k(i) : i = 0, ..., n − 2}. Now we consider the (n + 2)k(n−2)
∗
element set A = A ∪ {an−1 }. A∗ does not have the property (∗), since no(A) = 1. By
the lemma above, we must have another (n + 1)-element set (counterexample) A0 ⊂ A∗
k(n−2)
k(n−1)
with no(A0 ) = 1. Since A 6= A0 , an−1 must belong to A0 . If an−1 also belongs to
k(i )
k(in−3 ) k(n−2) k(n−1)
A0 , then A0 has the form {ai0 0 , ..., ain−3
, an−1 , an−1 , e}. So by the indiscernibility,
k(i )
k(i
)
k(n−2)
k(n−1)
{a0 0 , ..., an−3n−3 , an−2 , an−2 , e} ⊂ JCB 0 is also a counterexample. This is impossible,
k(0)
k(n−2) k(n−2)
since JCB 0 ∼
= JCB ⊂ M. So A0 must be the set {a0 , ..., an−2 , an−1 , e}. For this A0 ,
the m-value is m − 1. So A0 cannot be a counterexample, by the induction hypothesis.
By a similar argument as above, it follows that if a ∈
/ A then tp(a/A) does not divide over
∅. Thus, Un is simple.
a
Again a similar argument shows that Un is K(ω)-simple. The theory Un is supersimple of
SU -rank 1 and eliminates hyperimaginaries.
Proposition 4.11. Un does not have (n + 1)-amalgamation.
Proof. For i = 0, ..., n, let pi (x0 , ..., xi−1 , xi+1 , ..., xn ) be the complete type generated by the
formula R(x0 , ..., xi−1 , xi+1 , ..., xn ). Suppose that a0 , ..., an ∈ M is a common solution of the
(n+1)-types p0 ,...,pn . Then no({a0 , ...., an }) = n+1 = 1, since n is even. So {a0 , ..., an } ∈
/ K,
a contradiction.
a
The proof of Theorem 2.15 is now completed. We question whether there are similar
examples for odd n ≥ 5.
5. Heir base notion and amalgamation results for K(3)-simple theories
In this section, T is a complete simple theory with elimination of hyperimaginaries. We
work in Ceq , and our bounded closure is acl = acleq .
5.1. Main results. As we have seen under K(3)-simplicity, T has 4-amalgamation, but
need not have 5-amalgamation. Still, 5-amalgamation does hold for certain systems of types.
This is what we are going to argue. We state our main theorems for the cases on generalized
Morley sequences having acl(∅)-base parameters. Of course, it is only for convenience. We
also use notational convention 3.7.
Definition 5.1. We say that [4]4 -Morley ha, b, c, di is extensible to an infinite generalized
Morley sequence or infinitely extensible if there is an [ω]4 -Morley sequence whose initial
segment is the sequence ha, b, c, di.
Theorem 5.2. Assume that T is K(3)-simple. Suppose that ha, b, c, di is a [4]4 -Morley
sequence that is infinitely extensible. Let [3]3 -Morley I = hb, c, di be a segment of ha, b, c, di.
Suppose that each of ap I, aq I, ar I, and as I is [4]4 -Morley and infinitely extensible. Let pI
denote the corresponding generalized type of ap over I, and similarly for qI , rI , and sI . Let
pabc be the conjugate of pI to the [3]3 -Morley sequence ha, b, ci, and the meaning of qabd or
racd is similar. Then pabc , qabd , racd , sI can be generically amalgamated over ha, b, c, di.
22
The above theorem will be deduced from the following two.
Theorem 5.3. (T K(3)-simple.) Suppose that ha, b, c, di is a [4]4 -Morley sequence that is
infinitely extensible. Let I = hb, c, di be a segment of ha, b, c, di, and let ap I, aq I be infinitely
extensible [4]4 -Morley sequences. Then for the generalized types pI and qI as above, we have
that pabc , qabd , qacd , qI are generically amalgamable.
Theorem 5.4. (T K(3)-simple.) Suppose that ha, b, c, di is an infinitely extensible [4]4 Morley sequence with the segment I = hb, c, di. Now, let ap I, aq I be infinitely extensible
[4]4 -Morley sequences. Then pabc , pabd , qacd , qI are generically amalgamable.
5.2. Notion of an heir base. The notion of an heir base was implicitly introduced by
Lascar and Pillay in [10]; our current goal is to generalize that notion in a “multidimensional”
way.
Fact 5.5 (Lascar-Pillay). Given a, A, there is a model M containing A such that a ^ M
A
and for any B ⊇ M , if a ^ B, then tp(a/B) is an heir of tp(a/M ).
M
In particular, if a1 |= tp(a/M ) and a ^ a1 , then tp(a1 /M a) is a coheir extension of
M
tp(a/M ). One of the motivating questions for introducing the notion of an heir base is:
given Morley I over A, can we find a model M containing A such that I is a coheir sequence? Not necessarily.
Example 5.6. In the theory of a ternary tetrahedron-free graph T F H3 , a sequence I =
{a0 , a1 , a2 } such that |= R(a0 , a1 , a2 ) is ∅-Morley, but it fails to be a coheir sequence over any
model. Indeed, if such a model M existed, then we would get b ∈ M with |= R(a0 , a1 , b).
Since I is M -Morley, R(a0 , a2 , b) and R(a1 , a2 , b) also hold. Thus, we have an R-tetrahedron.
But it is possible to find such a model under generalized amalgamation for generalized
Morley sequences, in the form of an heir base. We give the precise definition.
Definition 5.7. Suppose that [n]n -Morley sequence I := hA; a0 , ..., an−1 i is given. We say a
model M (^ I) containing A, an heir base of I (over A), if
A
(1) I has a base extension hM ; a0 , ..., an−1 i over M and
(2) for all k < n and any B ⊇ M , whenever the initial segment J = hM ; a0 , . . . , ak i has
a base extension over B and J ^ B, then tp(J/B) is an heir of tp(J/M ).
M
Remark 5.8. By (2), if M is an heir base of I = hA; a0 , ..., an−1 i, then the pure sequence
ha0 , ..., an−1 i is a coheir sequence over M , i.e. the (usual) type tp(ai+1 /M a0 ...ai ) is an
extension of tp(ai /M a0 ...ai−1 ) and a coheir of tp(ai+1 /M ) = tp(a0 /M ). With some an ,
if hM ; a0 , ..., an i is generalized Morley, then again by (2), the pure ha0 , ..., an i is a coheir
sequence as well.
Lemma 5.9. Assume T has (n + 1)-CA. Let I = hA; a0 , ..., an−1 i be a [n]n -Morley sequence.
Then for any C0 (⊇ A), we have C ≡A C0 such that C ^ I and I has a base extension over
A
acl(C).
23
Proof. This is an easy consequence of n-CA. We can clearly assume that A, C0 are algebraically closed and C0 ^ a0 . There is C1 such that C1 a1 ≡A C0 a0 . Then by amalgaA
mation, we have independent C01 |= tp(Ck / acl(Aak )) for k = 0, 1. Then there are independent Cij such that acl(Cij ai aj A) ≡A acl(C01 a0 a1 A) for i < j < n. Now by iteration
with k-amalgamation for k ≤ n, we have {Cs |s ∈ [n]n−1 } such that for s, s0 ∈ [n]n−1 ,
acl(Cs as A) ≡A acl(Cs0 as0 A). Then finally by (n + 1)-amalgamation, we have the desired set
C.
a
Now we show the existence of an heir base. The proof mimics that of Fact 5.5, but here
we take advantage of generalized amalgamation. Recall that for A ⊆ M and a (possibly
infinite), ClA (a/M ) = {ϕ(x, y) ∈ L(A)| ϕ(x, m) ∈ tp(a/M ) for some m ∈ M }. (x is a
subtuple of the variable for a.)
Proposition 5.10. T has (n + 1)-CA. Let I = hA; a0 , ..., an−1 i be a [n]n -Morley sequence.
Then there is a model M ^ I containing A such that M is an heir base of I.
A
Proof. We assume A = acl(A). We first find a model satisfying 5.7 (1) and (2) with k = n−1.
Consider the following set of models
U = {N | A ⊆ N, I ^ N, hN ; a0 , ..., an−1 i is [n]n -Morley extending I.},
A
which is ordered as follows: N1 < N2 if N1 ≺ N2 , hN2 ; a0 , ..., an−1 i is a base extension of
hN1 ; a0 , ..., an−1 i, and ClA (I/N1 ) ( ClA (I/N2 ). Recall that I as a set is bdd(Aa0 ...an−1 ),
so we let xI be the variable for it. Now, by Lemma 5.9, U is nonempty. Then by Zorn’s
lemma, there is a maximal element M0 . Thus with 5.9 we have that, for any C containing
M0 such that I ^ C and hM0 ; a0 , ..., an−1 i has a base extension over C, if ϕ(a, d) with a ⊆ I,
A
ϕ(x; y) ∈ L(A), x ⊆ xI corresponding variable for a, and d ∈ C, then ϕ(a, m) for some
m ∈ M0 . As we consider the formula over A (not over M0 ), M0 is not yet the desired
model. Hence we iterate this argument to obtain an elementary chain {Mi | i ∈ ω} such
that A ⊆ Mi ≺ Mi+1 , Mi+1 I is an extension of Mi I, I ^ Mi and for any N Â Mi+1 with
A
N I being a base extension of Mi+1 I and I ^ N , then ClMi (Mi I/Mi+1 ) = ClMi (Mi I/N ).
Mi
S
Therefore if we set M = Mi , then it is the desired model.
Now the similar proof of 5.9 using (n + 1)-CA easily shows that M satisfies 5.7(2) for an
arbitrary k < n as well.
a
Now the following important lemma says the notion of coheir also behaves well in the
context of generalized Morley sequences.
Lemma 5.11. Suppose that I = ha0 , ..., an−1 i is [n]k -Morley over M and as a pure sequence
a coheir sequence over M . With additional ai (i ≥ n), let a pure sequence I = hai |i < ωi
be a coheir sequence over M (such a sequence exists). Then there is an extended map from
acl(IM ) to the set of variables for it, so that we have M I, a generalized [ω]k -Morley sequence
over M , forming a length extension of I.
Proof. Let r(x0 , ..., xk−1 ; y{0,...,k−1} ) over M be the generalized type realized by ha0 , ..., ak−1 i
where the variable xi is for the pure tuple ai and y{0,...,k−1} is for the other elements in
24
acl(M a0 ...ak−1 ). Hence we can further assume that for any c0 ...ck−1 ; d |= ψ(x0 , ..., xk−1 ; y) ∈
r, d ∈ acl(M c0 ...ck−1 ). We shall show that
S
{r(ai (i ∈ s), an ; ys∪{n} )| s ⊆ n, |s| = k − 1}
is consistent. Then it says ha0 , ..., an i is a length extension. By iteration we obtain the
lemma. Now suppose not. Then by compactness, there is ϕ ∈ r such that consistent
ϕ(ai (i ∈ s1 ), an ; y 0 ) ∧ ... ∧ ϕ(ai (i ∈ sm ), an ; y m ) implies ¬ϕ(ai (i ∈ sm+1 ), an ; y m+1 ). Due to
coheirness, there is u ∈ M such that
∃y 0 ...y m (ϕ(ai (i ∈ s1 ), u; y 0 ) ∧ ... ∧ ϕ(ai (i ∈ sm ), u; y m )).
Hence there is ui ∈ acl(M a0 ...an−1 ) such that ϕ(ai (i ∈ s1 ), u; u0 ) ∧ ... ∧ ϕ(ai (i ∈ sm ), u; um ),
but again by compactness we can assume there is no um+1 ∈ acl(M a0 ...an−1 ) holding ϕ(ai (i ∈
sm+1 ), u; um+1 ). This violates that ha0 , ..., an−1 i is generalized Morley over M .
a
5.3. Comments about the rest of the section. We finish this section with proofs of
main results. This is probably the most technical part of the paper. Not to complicate the
presentation further, we adopt the following rules:
(1) All types, except coheir extensions or mentioned otherwise, are generalized types. So
for example tp(e/M abc) denotes a generalized type over the system of algebraically closed
sets indexed by P(3) with the root M and the maximal element acl(M abc). Aside from some
obviously necessary care, one can deal with generalized types in much the same way as one
deals with the usual types.
(2) All Morley sequences are generalized Morley, and we use the notational convention 3.7.
But when we talk about a coheir sequence or extension, then the sequence/extension is a
pure sequence/type.
(3) The notion of conjugation is in the sense of generalized types. For instance, we write
a0 b0 c1 ≡eM a0 b0 c2 to express that there is an elementary map f that fixes acl(eM ) pointwise
and sends acl(a0 b0 c1 eM ) to acl(a0 b0 c2 eM ) and respects closures of subsequences, e.g., we
have f (acl(a0 c1 eM )) = acl(a0 c2 eM ).
(4) Due to 5.11 and possibly its more general forms, for generalized types, coheir extensions
are freely taken and their nice properties are preserved.
Lemma 5.12. Assume T has 4-amalgamation. Suppose that I = ha0i a1i a2i a3i |i < 2i be
generalized Morley and there is e such that a00 a10 a21 ≡e a00 a10 a22 , a10 a22 a32 ≡e a11 a22 a32 .
Then there is an element e0 ≡a00 a11 a22 a10 a21 a32 e such that generalized e0 I is a base extension
of I.
Proof. Let X be the set {aij : i < 4, j < 3}. For subsets A0 , A1 of X, or types p, q, we write
A0 ∼
= A1 or p ∼
= q if they are conjugates. For A ⊆ X and A0 , A1 ⊆ A, we say {A0 , A1 }
is a critical pair in A if A0 ∼
= A1 and they are maximal among such (i.e. there is no pair
{A00 , A01 } with Ai ( A0i ⊂ A (i = 0, 1) and A00 ∼
= A01 ). Also say e0 ≡A e is good for A if
for any critical pair {A0 , A1 } in A, A0 ≡e0 A1 (assigned by ∼
=). Thus e itself is good for
A0 = {a00 a11 a22 a10 a21 a32 }, as {a00 a10 a21 , a00 a10 a22 } and {a10 a22 a32 , a11 a22 a32 } are all critical
pairs there. Now for A ⊆ X and a ∈ X \ A, we call {A0 , A1 } a critical pair in Aa, the first
kind, if a ∈ A0 and a ∈
/ A1 (or vice versa). Otherwise it will be called the second kind. Now
the following claim finishes the proof.
25
Claim. For any subset A of X containing A0 and a ∈ X \ A, if eA is good for A, then there
is e0 ≡A eA good for Aa.
Proof of Claim. To simplify the notation, we put Aj = {aij : aij ∈ A} (j = 0, 1, 2). For
B ⊆ Aj and C ⊆ Ak , let B u C denote the set {aij ∈ B : aik ∈ C}. So B u C is a subset of
B. B u C and C u B are different unless j = k.
Case 1. a = aj0 for some j.
There are (at most) two first kind critical pairs in Aa. They are
• {B0 , B1 } = {((A0 a) u A1 )A2 , (A1 u (A0 a))A2 } and
• {C0 , C1 } = {(A0 a) u A1 )(A1 u A2 ), (A1 u (A0 a))(A2 u A1 )}.
We put r = p|B1 and s = p|C1 , where p = tp(eA /A). Let us consider the following three
types:
p(x), rB0 (x), sC0 (x).
Since eA is good, any two of the above are compatible. So, by 4-amalgamation (possibly
free, 3.3), there is a common generic extension q ∗ of p(x), rB0 , sC0 , and e∗ |= q ∗ is good for
critical pairs of the first kind. But we have to consider the second kind pairs too. The only
second kind critical pair is the following:
{D0 , D1 } = {(A0 a)(A1 u A2 ), (A0 a)(A2 u A1 )}.
Now let q = q ∗ |D0 . Clearly sC0 = q ∗ |C0 and D0 ⊃ C0 . So we have q ⊃ sC0 . We subclaim
that qD1 and rB0 are compatible: Notice that D = [(A0 a)(A2 u A1 )] ∩ [((A0 a) u A1 )A2 ] =
((A0 a) u A1 )(A2 u A1 ). Clearly we have the following:
(1)
qD |D ∼
= q|((A0 a) u A1 )(A1 u A2 ) ∼
= q ∗ |((A0 a) u A1 )(A1 u A2 ).
1
∗
Since e is good for the first kind pairs, we have
(2) q ∗ |((A0 a) u A1 )(A1 u A2 ) ∼
= q ∗ |(A1 u (A0 a))(A2 u A1 ) ∼
= q ∗ |((A0 a) u A1 )(A2 u A1 ).
The last type is exactly rB0 |D. Combining (1)(2), we have qD1 |D ∼
= rB0 |D.
By the subclaim and by the fact that eA is good for A, we can find an amalgam q ∗∗ of the
three types: qD1 , rB0 , p. Now we consider the following types: q ∗∗ |((A0 a)A2 ), q, p. By the
definition of q ∗∗ , q ∗∗ |((A0 a)A2 ) and p are compatible. Also, by the definition of q and q ∗ , q
and p are compatible. Finally, for (A0 a)A2 ∩ dom(q) = A0 a, we have
q ∗∗ |(A0 a) = qD1 |(A0 a) = q|(A0 a).
So we can amalgamate q ∗∗ |((A0 a)A2 ), q and p for getting p∗ over Aa. Clearly we have
p∗ ⊃ q ⊃ sC0 ; p∗ ⊃ q ∗∗ |((A0 a)A2 ) ⊃ q ∗∗ |B0 = rB0 ; p∗ ⊃ qD1 ; p∗ ⊃ p. So e0 |= p∗ is a good for
Aa. The case where a = aj2 for some j can be treated similarly, so the remaining case is:
Case 2. a = aj1 for some j.
In this case critical pairs in Aa are
• {B1 , C1 } = {((A1 a) u A0 )A2 , (A0 u (A1 a))A2 }
• {B2 , C2 } = {A0 ((A1 a) u A2 ), A0 (A2 u (A1 a))} and
• {B3 , C3 } = {(A0 u (A1 a))((A1 a) u A2 ), ((A1 a) u A0 )(A2 u (A1 a)}.
26
The first two pairs are of the first kind, and the last one is of the second kind. Let r = p|C1
and s = p|C2 . Let p∗ be a generic amalgam of rB1 , sB2 and p. Let q(x) = p∗ |C3 . Since C3 ⊂
B1 , we have q ⊂ rB1 . Moreover qB3 ⊂ sB2 : qB3 ∼
=q∼
= rB1 |C3 ∼
= r|(A0 u (A1 a))(A2 u (A1 a)),
∼
∼
∼
p|(A
u
(A
a))(A
u
(A
a))
s|(A
u
(A
a))(A
u
(A
a))
=
=
= sB2 |B3 . So e0 |= p∗ is good for
0
1
2
1
0
1
2
1
Aa.
a
Now we assume, for the rest of the section, T is K(3)-simple (so having 4-amalgamation).
Let us make some comments about the general structure of the proofs. The common ingredient is the construction of what we call the “projected system of types”. Suppose we need
to amalgamate four types pabc , qabd , racd and sbcd over abcd. To this end, we arrange the
elements a, b, c, and d on the “diagonal” of the sequence Ki that we are constructing. The
initial step is to get the elements b0 , c1 , and d2 such that, firstly, for some e
e |= pabc ∪ qabd2 ∪ rac1 d2 ∪ sb0 c1 d2 .
Intuitively, we are taking types over the three-element subsets of the diagonal, and “projecting” them down to the first three elements in the sequence Ki . Secondly, since the aim
is for the sequence K012 = hK0 , K1 , K2 i to be Morley over e, a priori abcb0 c1 d2 need to be
as Morley as possible. Namely, as similarly to 5.12, we want ab0 c1 ≡e ab0 c, b0 cd2 ≡e bcd2 .
However, finding such elements holding two conditions requires more than 4-amalgamation.
We can only do this, under 4-amalgamation, by the help of an heir base M for ha, b, ci.
...
d
c
p
q
e
d2
b
r
c1
s
a
b0
K0
Having constructed b0 , c1 , and d2 , it is not hard to “fill in” the remaining elements (by using 5.12 for example) in Ki , i = 0, . . . , 3, so that K012 is M e-Morley and K0123 = hK0 , ..., K3 i
is M -Morley. If the sequence K0123 can be infinitely extended, then a single application of
K(3)-simplicity finishes the proof: we would have e0 |= tp(e/K012 ) such that the infinite
sequence is Morley over e0 . Then it follows that e0 would satisfy the four types over abcd. A
priori, there is no way to guarantee that the sequence K0123 is infinitely extensible. It was
pointed out in [9] that one can manage amalgamation with less; this is the role of properties
L4 in that paper. A critical use of the heir base notion in our argument below guarantees
that L4 holds for the sequence K0123 .
5.4. Proof of Theorem 5.3. Suppose that generalized Morley sequences hea0 b1 c2 i, heq a0 b1 c2 i,
and ha0 b1 c2 d3 i, all infinitely extensible, are given.
From 5.10, we have Me (^ Ie) an heir base of I = ha0 b1 c2 i, then as eI is infinitely extensible, due to K(3)-simplicity, we can assume eI is generalized Me -Morley. Then from 5.8,
27
pure sequence eI is a coheir sequence over Me . Similarly there is M 0 (≡I Me ) such that
Id3 is M 0 -Morley and a coheir (pure) sequence over M 0 . Then by amalgamation we have
M |= tp(Me /eI) ∪ tp(M 0 /Id3 ) so that
both eI and Id3 are coheir sequences over M .(†)
We can further assume that (by moving eq if necessary), eq I is M -Morley and a coheir
sequence. Now we write
pa0 b1 c2 = tp(e/M a0 b1 c2 ), qa0 b1 c2 = tp(eq /M a0 b1 c2 ).
Claim 1. There are independent b0 , c1 , d2 independent from M a0 b1 c2 such that
(1) a0 b0 c1 ≡eM a0 b0 c2 , b0 c2 d2 ≡eM b1 c2 d2 , and a0 b1 c2 ≡M a0 b1 d2 ≡M a0 c1 d2 ≡M b0 c1 d2 ;
(2) e |= pa0 b1 c2 ∪ qa0 b1 d2 ∪ qa0 c1 d2 ∪ qb0 c1 d2 ;
(3) both usual types tp(c2 d2 /M b1 c1 a0 b0 ) and tp(b1 c1 /M a0 b0 ) are coheir extensions of
tp(a0 b0 /M ).
Proof of Claim 1. We first find b0 . Due to (†), we have b0 such that the sequence ha0 b0 b1 c2 i
is a generalized sequence over M and an M -coheir sequence as a usual sequence (*). Then
find independent
b0 |= tp(b1 /M c2 e) ∪ tp(b0 /M a0 b1 c2 ) (**).
Then from (*), both usual tp(b1 /M a0 b0 ), tp(c2 /M a0 b0 b1 ) are coheirs. Now we shall find c1 .
From (*)(**) and 5.11, we have c0 such that a0 b0 b1 c0 c2 is generalized M -Morley and as a pure
sequence, a coheir sequence. Choose independent
c1 |= tp(c2 /M a0 b0 e) ∪ tp(c0 /M a0 b0 b1 c2 ).
Then a0 b0 b1 c1 c2 is M -coheir sequence (‡). Also, a0 b0 c1 ≡M e a0 b0 c2 , and with (†)(**),
a0 b1 ≡M e a0 c2 (≡M e a0 c1 ) ≡M e b1 c2 ≡M e b0 c2 ≡M e b0 c1 (***). Next we shall find d2 . Note
that from (***), eb0 c1 ≡M ea0 b1 . Thus from (†), eb0 c1 is an M -coheir sequence. Now there
clearly is d0 such that eb0 c1 d0 ≡M eq a0 b1 c2 (††). Also since eb0 c1 d0 is infinitely extensible,
we can assume, from K(3)-simplicity, that eb0 c1 d0 is an M -coheir sequence. Then we have
the final d2 such that tp(d2 /M a0 b1 c2 b0 c1 e) is a coheir extension of tp(d0 /M b0 c1 e) (?). Then,
from (‡), (3) follows. We subclaim (1) and (2) hold with d2 as well.
Subclaim There is an extended map from A = acl(M a0 b0 b1 c1 c2 d2 e) to the set of variables
for A so that (1)(2) hold with the arrangement of A.
Proof of Subclaim. The proof will be similar to that of 5.11, so we only sketch it. From
(‡) above, the first formula in (1) follows. Note that due to (***)(††) and the properties
of the coheir extension (?), the second and the rest of (1), and (2) do hold as pure types
and sequences. Indeed, by the similar observation as in the proof of 5.11, they too hold as
generalized types and sequences. Namely, it can be easily seen that
{ϕ(eb0 c2 d2 , zeb0 c2 d2 ) ↔ ϕ(eb1 c2 d2 , zeb1 c2 d2 )| ϕ ∈ L(M )}
∪ q(ea0 b1 d2 ; yea0 b1 d2 ) ∪ q(ea0 c1 d2 ; yea0 c1 d2 ) ∪ q(eb0 c1 d2 ; yeb0 c1 d2 ),
is consistent, where q(x0 x1 x2 x3 ; y0123 ) is the (generalized) type realized by eq I, and ϕ(x0 x1 x2 x3 ; z0123 )
is a formula such that z0123 ∈ acl(M x0 ...x3 ), with possible overlaps of y’s and z’s. We have
proved Subclaim and Claim 1. ¤
Now, we claim the following.
28
Claim 2. There are the rest of points c0 , d0 , a1 , d1 , a2 , b2 so that K = hai bi ci di |i < 3i forms
a coheir sequence over M .
Proof of Claim 2. From above (‡), we have c0 , a1 , a2 , b2 such that K0_ K1_ K2 (= K012 ) forms
M -coheir sequence, where K0 = ha0 , b0 , c0 i, K1 = ha1 , b1 , c1 i, K2 = ha2 , b2 , c2 i. Write Kij for
Ki Kj . Now from (?), tp(d2 /M a0 b0 b1 c1 c2 ) is a coheir. Hence by taking a coheir extension,
we can assume that tp(d2 /M K012 ) is a coheir over M . Thus, clearly both tp(K1 /M K0 ) and
tp(K2 d2 /M K01 ) are coheirs. Now we have d01 independent from others such that
K01 d01 ≡M K02 d2 (≡M K12 d2 ).
Hence tp(K1 d01 /M K0 ) is a coheir. Now first, take a coheir extension of tp(K2 /M K01 ) over
M K01 d01 . Then move its realization to K2 over M K01 . Then we have d1 , the M K01 automorphic image of d01 , so that both tp(K1 d1 /M K0 ) and tp(K2 /M K01 d1 ) are coheirs
(??). Then again by the similar argument (taking a coheir extension of tp(d2 /M K012 ) over
M K012 d1 and possibly moving d1 over M K012 ), we can additionally assume that tp(d2 /M K012 d1 )
is a coheir, and thus from (??), so is tp(K2 d2 /M K01 d1 ). Then we have d0 such that
K01 d0 d1 ≡M K12 d1 d2 (? ? ?).
Again by the similar argument (taking coheir extensions of tp(K2 /M K01 d1 ) over M K01 d0 d1 ,
and tp(d2 /M K012 d1 ) over M K012 d0 d1 , consecutively) with (? ? ?), we can assume that K =
hai bi ci di |i < 3i is an M -coheir sequence. ¤
Due to the similar arguments in the proof of 5.11, we can further assume that hM ; Ki forms
a [3]3 -Morley sequence extending generalized sequences hM ; a0 b1 c2 i, hM ; a0 b1 d2 i, hM ; a0 c1 d2 i,
hM ; b0 c1 c2 i. Now write pure sequences K j = Kj dj (j < 3). Recall from (†) that a0 b1 c2 d3
is M -Morley, and from Claim 1(1), a0 b1 c2 , a0 b1 d2 , a0 c1 d2 , b0 c1 d2 all are equivalent over M .
Hence it can be seen that (or strictly by [9, 5.3]), there are a3 , b3 , c3 so that K 0 = K _ K 4
where K 4 = ha3 b3 c3 d3 i forms [4]3 -M -Morley. Moreover, by Claim 1 and Lemma 5.12, we can
additionally assume (by moving e if necessary) that the sequence K is eM -Morley. Hence if
K 0 infinitely extends over M , then due to K(3)-simplicity, there is e0 ≡M K e such that K 0 is
M e0 -Morley. Then e0 realizes the four types over a0 b1 c2 d3 , i.e. Theorem 5.3 follows. But, M
need not be a coheir base of K, and K 0 need not be an M -coheir sequence. However, due to
Claim 2, we can still have K i (i > 3) such that tp(K i+1 /M K 0 K 4 ...K i ) is a coheir extending
tp(K i /M K 0 ...K i−1 ) for i ≥ 4, and tp(K 4 /M K 0 ) is a coheir extending tp(K 2 /M K 0 K 1 ). Then
for each i < j < k ≤ 3, K i K j K k K 4 K 5 ... is M -Morley. (This is what Kolesnikov called the
L4 -property for K 0 in [9].) Then by exactly the same argument in [9, 6.3], it can still be seen
that Theorem 5.3 follows.
5.5. Proof of Theorem 5.4 and its variants. It is not difficult to see that by the similar
proofs we have variations of 5.3 as follows. (If necessary, in the proof of above Claim 1,
reverse the order so that find a0 last by similarly taking a coheir extension.)
Variation 1. Suppose that habcdi is a sequence infinitely extensible. Let I = hbcdi.
Then for any ap and aq such that both ap I and aq I extensible to infinite Morley sequences,
pabc , pabd , pacd , qI are generically amalgamable where pI = tp(ap /I), qI = tp(aq /I);
Variation 2. Let habcei be a sequence extensible to an infinite Morley sequence. Suppose
there are d1 and d2 such that both abcd1 (≡ abd1 e ≡ acd1 e) and bcd2 e are infinitely extensible.
Then tp(d1 /abc), tp(d1 /abe), tp(d1 /ace), tp(d2 /bce) are generically amalgamable.
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From Variation 2, we deduce the following.
Variation 3. Suppose that habcdi is a sequence infinitely extensible. Let I = hbcdi.
Then for any ap and aq such that both ap I and aq I extensible to infinite Morley sequences,
pabc , qabd , pacd , pI are generically amalgamable where pI = tp(ap /I), qI = tp(aq /I): Choose
e1 |= qabd . Now there is cp (by 4-amalgamation) such that e1 abcp ≡ e1 acp d ≡ e1 bcp d(≡ ap I).
Then from 2 above, there is
c0 |= tp(cp /e1 ab), tp(cp /e1 ad), tp(cp /e1 bd); tp(c/abd).
Hence if we choose e such that eabcd ≡ e1 abc0 d, then e |= pabc , qabd , pacd , pI .
We are ready to show Theorem 5.4. Proof will follow the same pattern. Assume that
there exist sequences I = bcd, aI, ap I, aq I, all extensible to infinite Morley, and types
pI = tp(ap /I), qI = tp(aq /I). Let g = tp(abcd). There is an infinite Morley sequence J =
hai , bi , ci , di |i ∈ ωi such that a0 b1 c2 d3 , a0 b0 c1 d2 (:= I3 ), a0 b0 c2 d2 (:= I2 ), a0 b1 c2 d2 (:= I1 ) |= g.
Then by K(3)-simplicity, 5.3 and Variation 3, there are
a1 |= pa0 b1 c2 , pa0 b1 d2 , pa0 c2 d2 , pb1 c2 d2 ; a2 |= pa0 b0 c2 , qa0 b0 d2 , pa0 c2 d2 , pb0 c2 d2 ;
a3 |= pa0 b0 c1 , qa0 b0 d2 , qa0 c1 d2 , qb0 c1 d2 .
Then tp(a1 /I1 ), tp(a2 /I2 ) and tp(a3 /I3 ) are compatible, hence so by 4-amalgamation we have
a generic realization a0 of the 3 types so that,
a0 |= pa0 b1 c2 ∪ pa0 b1 d2 ∪ qa0 c1 d2 ∪ qb0 c1 d2
and moreover,
a0 b0 c1 ≡a0 a0 b0 c2 and b0 c2 d2 ≡a0 b1 c2 d2 .
Now by applying 5.12 to J3 = hai , bi , ci , di |i < 3i and applying K(3)-simplicity to J, we can
additionally assume that J is a0 -Morley. Thus a0 |= pa0 b1 c2 ∪ pa0 b1 d3 ∪ qa0 c2 d3 ∪ qb1 c2 d3 and
Theorem 5.4 follows.
5.6. Proof of Theorem 5.2. We argue that the theorems 5.3, 5.4 imply 5.2. Hence suppose
that there are sequences habcdi, hap bcdi, haq bcdi, har bcdi, has bcdi all extensible to infinite
Morley, and types pI = tp(ap /I), qI = tp(aq /I), rI = tp(ar /I), sI = tp(as /I). We let g =
tp(abcd). Now clearly, we have an infinite Morley sequence J = hai , bi , ci , di |i ∈ ωi such that
a0 b1 c2 d3 , a0 b0 c1 d2 (:= I3 ), a0 b0 c2 d2 (:= I2 ), a0 b1 c2 d2 (:= I1 ) |= g.
Then by 5.4 and 5.3 with its variants, there are
a1 |= pa0 b1 c2 , qa0 b1 d2 , qa0 c2 d2 , qb1 c2 d2 ; a2 |= ra0 b0 c2 , ra0 b0 d2 , qa0 c2 d2 , qb0 c2 d2 ;
a3 |= ra0 b0 c1 , ra0 b0 d2 , ra0 c1 d2 , sb0 c1 d2 .
Note that ai realizes types on Ii . Then tp(a1 /I1 ), tp(a2 /I2 ) and tp(a3 /I3 ) are compatible,
hence so by 4-amalgamation we have a generic realization a0 of the 3 types so that,
a0 |= pa0 b1 c2 ∪ qa0 b1 d2 ∪ ra0 c1 d2 ∪ sb0 c1 d2
and a0 b0 b1 c1 c2 d2 has the Morley condition of a0 , i.e.
a0 b0 c1 ≡a0 a0 b0 c2 and b0 c2 d2 ≡a0 b1 c2 d2 .
Now the rest is usual, applying again 5.12 and K(3)-simplicity to the sequence J.
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6. Model-n-simplicity
As pointed out, some stable theory need not be even 2-simple. But there is a variant of
n-simplicity, which is implied by stability for each n. That property, pointed out in [3], we
call model-n-complete amalgamation is implying n-complete amalgamation over models, but
is implied by both (n + 1)-complete amalgamation over models, and n-CA.
Definition 6.1. We say T has model-n-complete amalgamation if the following holds: Let
un = {1, ..., n}, and Wn = P(un+1 )− \ {un }. Let W be a subset of Wn , closed under subsets.
For each w ∈ W , complete type rw (xw ) over a model M is given where xw is possibly an
infinite set of variables. Suppose that
(1) for w ⊆ w0 , xw ⊆ xw0 and rw ⊆ rw0 .
Moreover for any aw |= rw ,
(2) {a{i} |i ∈ w} is M -independent,
(3) aw is as a set bdd(∪i∈w a{i} M ) (and the map aw → xw is a bijection).
Then there is a complete type run+1 (xun+1 ) over M such that (1),(2),(3) hold for all w ∈
W ∪ {un+1 }.
By the definition, the notion of model-n-CA is placed in between that of n-CA over models
and (n + 1)-CA over models. In the same spirit as before, we may call model-n-CA as model(n − 2)-simplicity. It is also clear that n-simplicity implies model-n-simplicity. Stability
implies model-n-CA for all n, as well [3]. Moreover what Hrushovski (and Chatzidakis)
proved such properties for all n for ACFA and PSF-structures [2, 4].
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Department of Mathematics, Yonsei University, 134 Shinchon-dong, Seodaemun-gu, Seoul
120-749, Korea
E-mail address: [email protected]
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
E-mail address: [email protected]
Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
E-mail address: [email protected]
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