Properties and Consequences of th-Independence

26 Pages·0.33 MB·English

by  Alf Onshuus

PROPERTIES AND CONSEQUENCES OF
3 THORN-INDEPENDENCE
0
0
2 ALF ANGEL ONSHUUS

n
a
J

3
2 Abstra
t. In the last
ouple of de
ades, independen
e relations have
be
ome one of the
entral parts of model theory. Areas su
h as
las-

] si(cid:28)
ation theory, stability theory and simpli
ity theory use the notion
O
of (cid:16)forking independen
e(cid:17) de(cid:28)ned by Shelah as a main tool for their
L development. However, as powerful and useful as forking-independen
e
. is, itla
ks generality. O-minimalstru
tures give anexampleof a whole
h
t
lass ofstru
tures for whi
h forking independen
edoes notwork. This,
a we believe, is themain
ause whyeventhoughtheresults in o-minimal
m
theoryseemverysimilartothoseinthetheoryofstronglyminimalsets
[ (stronglyminimalsetsareinsomewaysthesmalleststru
turesforwhi
h

forkingde(cid:28)nesanindependen
erelation)theproofsofanalogousresults
2
have beenverydi(cid:27)erentin bothareas.
v
Webeginbydevelopinganewnotionofindependen
e(þ-independen
e,
4
0 read(cid:16)thorn(cid:17)-independen
e)thatarises fromafamilyofrankssuggested
0 by S
anlon (þ-ranks). We prove that in a large
lass of theories (it in-
5
ludes simple theories and o-minimal theories) this notion has manyof
0 the properties neededfor an adequategeometri
stru
ture.
2 Finally, we analyze thebehavior of þ-forking in some theories where
0 anindependen
erelationhadalreadybeenstudiedbyotherauthors. We
/
h provethatþ-independen
eagreeswiththeusualindependen
enotionsin
t stable, supersimple ando-minimaltheories. Furthermore,we give some
a
eviden
e that the equivalen
e between forking and þ-forking in simple
m
theories mightbe
losely relatedtoone ofthemainopen
onje
tures in
: simpli
ity theory, thestable forking
onje
ture. Inparti
ular, we prove
v
that in any simple theory where the stable forking
onje
ture holds,
i
X þ-independen
eand forking independen
eagree.
r
a
1. Introdu
tion
1.1. Overview. In the last de
ades independen
e notions have be
ome one
of the
entral ideas in model theory. Espe
ially in the last three de
ades
independen
e notions and ranks have been the key fa
tor (cid:28)rst in Shelah's

lassi(cid:28)
ation theory and later in understanding the geometry in the models
of theories su
h as stable, o-minimal and in the last de
ade simple theo-
ries. However, even when many of the results on the topology of o-minimal
theories were similar to some of those in geometri
stability theory, they
were developed almost independently from ea
h other mainly be
ause of the
di(cid:27)eren
e between the de(cid:28)nitions of the respe
tive independen
e notions.
1
2 ALFANGELONSHUUS
In se
tions 2 through 4 we develop a new notion of independen
e (þ-
independen
e (cid:21)read (cid:16)thorn(cid:17) independen
e(cid:21)) and a rank that is asso
iated to
this independen
e notion (þ-rank). We prove that in a large
lass of theo-
ries whi
h we shall
all (cid:16)rosy(cid:17) this notion de(cid:28)nes a geometri
independen
e
relation. This
lass of theories in
ludes simple and o-minimal theories. It
alsoin
ludes theoriesfor whi
h therewas nopreviously known independen
e
relation; forexample modelsofthetheory de(cid:28)ned by Casanovas andWagner
1
in [CW02℄ are rosy.
In se
tion 5 we study the relation between the
lassi
al independen
e
notions whi
h we had in simple and o-minimal theories and þ-independen
e.
We prove that þ-independen
e agrees with the
lassi
al independen
e notion
in o-minimal, stable and all known
ases of simple theories. This provides
a uni(cid:28)ed approa
h in two areas where, until now, the methods and proofs
have been di(cid:27)erent (even though, as mentioned above, they did go in the
same general dire
tion and similar results were obtained). It also gives an
alternative(and,inourpointofviewsimpli(cid:28)ed)de(cid:28)nitionofforkinginstable
theories whi
h might give further insight in stability theory.
1.2. Notation and Conventions. We assume the reader is familiar with
the terminology and the basi
results of model theory and, more spe
i(cid:28)
ally,
stability and simpli
ity theory.
T
As it is
ommon in stability theory, given a
omplete theory we will (cid:28)x
C T
auniverse
alled a (cid:16)monster(cid:17) model of : we
hoose some saturated model
C κ
of
ardinality and assume all sets, types and models we talk about have
κ C T

ardinality less than and live inside . In parti
ular, by models of we
C
mean an elementary submodel of . Any automorphism will be understood
C C
to be a -automorphism. Following [Hod93℄, an equivalen
e formula of is
φ(x,y) T
a formula in the language of that de(cid:28)nes an equivalen
e relation in
C Ceq
. Unless otherwise spe
i(cid:28)ed, we will work inside in the sense of [She90℄.
a,b,c,d
By
onvention lower
ase letters will in general represent tuples (of
imaginaries) and upper
ase letters will represent sets. Greek letters su
h as
δ,σ,ψ,φ
will be used for formulas.
2. þ-Forking
2.1. De(cid:28)nitions. We will start by de(cid:28)ning the notions that we will work
with throughout this paper.
δ(x,a) A tp(a/A)
De(cid:28)nition 2.1. A formula strongly divides over if is
{δ(x,a′)}a′|=tp(a/A) k k ∈ N
non-algebrai
and is -in
onsistent for some .
δ(x,a) A c
We will say that þ-divides over if we
an (cid:28)nd some tuple su
h
δ(x,a) Ac
that strongly divides over .
A
Finally, a formula þ-forks over if it implies a ((cid:28)nite) disjun
tion of
A
formulae whi
h þ-divide over .
1
In this paper, Casanovas and Wagner
onstru
t a theory whi
h is not simple and
provides the (cid:28)rst example of a theory without the stri
t order property whi
h does not
eliminatehyperimaginaries
PROPERTIES AND CONSEQUENCES OF THORN-INDEPENDENCE 3
δ(x,a) C
Remark 2.1.1. Suppose a formula strongly divides over some set ;
p(x,C) := tp(a/C) k ∈ N
let . By de(cid:28)nition there is some su
h that
{δ(x,a′)}a′|=tp(a/C) k
is -in
onsistent.
x ,x ,...,x
1 2 k
Another way of saying this is that for all
n
p(x ,C)∪p(x ,C)∪···∪p(x ,C) |= ¬ φ(x,a ) .
1 2 k i
!
i=1
^
θ(y,c) ∈ tp(a/C)
By
ompa
tness, there is some formula su
h that
{δ(x,a′)}a′|=θ(y,c) k
is -in
onsistent.
p(x) A p(x)
We will say the type þ-divides over if there is a formula in
A a
whi
hþ-dividesover ;similarlyforþ-forking. Wesaythat isþ-independent
b A a | b tp(a/Ab) A
of over , denoted ⌣þA , if does not þ-fork over .
Note that even though þ-dividing and dividing have similar de(cid:28)nitions,
k
the fa
t that we ask for -in
osnsiten
y for a set of formulas for whi
h the
parameters vary in a de(cid:28)nable
lass as opposed to an indis
ernible sequen
e
is a signi(cid:28)
ant modi(cid:28)
ation. In parti
ular, as we will prove later, many
theories with the stri
t order property will behave ni
ely under this new
de(cid:28)nition.
2.1.1. First Results.
A,B,C C A ⊆ B ⊆ C
Lemma 2.1.2. Let be subsets (of ) su
h that and let
a,b
be tuples. Then
p B A
(1) Extension: Given a type over whi
h does not þ-fork over , we
p q C A

an extend to a type over whi
h does not þ-fork over .
δ(x,b) B
(2) Monotoni
ity: If a formula þ-forks over , then it þ-forks
A
over .
tp(a/C) A
(3) Partial right transitivity: If does not þ-fork (divide) over
tp(a/C) B tp(a/B)
then does not þ-fork (divide) over and does not
A.
þ-fork (divide) over
b | a tp(a/A) tp(a/Ab)
(4) If ⌣þA and is non-algebrai
, then is not alge-
brai
.
b | a tp(b/A) tp(b/Aa)
(5) If ⌣þA and is non-algebrai
, then is not alge-
brai
.
p(x,b) Ab A θ(x,y)
(6) If is a type over whi
h does not þ-fork over and
p(x,b) |= ∃yθ(x,y) p(x,b)∪{θ(x,y)}
is a formula su
h that . Then
A
does not strongly divide over .
a | c d d′ |=
(7) Base Extension: If ⌣þA , then for any tuple there is some
tp(d/Ac) a | c
su
h that ⌣þAd′ .
Proof. .
Γ(x) := {ψ(x)|ψ(x) ∈ L(x) ¬ψ(x) A}
1) Let and þ-divides over . As in the
p(x)∪Γ(x)
proofofextensionforforking,we(cid:28)rstneedtoprove thatthetype
is
onsistent. Suppose this was not the
ase. By
ompa
tness there would
ψ (x),...ψ (x) Γ(x) p(x) |= ∨n ¬ψ (x)
be a (cid:28)nite subset 1 n in su
h that i=1 i .
4 ALFANGELONSHUUS
p(x) A q′
By de(cid:28)nition would þ-fork over . Let be a
omplete extension of
p(x)∪Γ(x) C q q′ C
to some set
ontaining , and let be the restri
tion of to .
q A
By de(cid:28)nition does not þ-fork over .
2) In the de(cid:28)nition of þ-dividing we are allowed to add parameters to get
A
strong division, so if any formula þ-divides over an extension of then it
A
þ-divides over . The result for þ-forking follows.
3) The(cid:28)rstimpli
ationfollowsfrommonotoni
ity andtheotherfollowsfrom
the fa
t that we are
onsidering fewer formulas.
p(x,b) = tp(a/Ab) p(x,b)
4) Let . Assuming does not strongly divide and us-
tp(b/A) b ,b ,...,b ,...
1 2 n
ingnon-algebrai
ityof ,we
an(cid:28)nddistin
trealizations
tp(b/A) p(x,b ) a′
of su
hthat i i is
onsistent,realizedbysome . However, as
tp(b /A) = tp(b/A) a′b = ab p(x,b)
1 1
wSe
an assume . However, is a
omplete
b |= tp(b/Aa) tp(b/Aa)
i
type, so showing that is non algebrai
.
tp(b/Aa)
5) We will prove the
ontrapositive of the statement. Let be al-
b = b ,b ,...b tp(b/Aa)
1 2 n
gebrai
and let be all the elements that satisfy .
Then
tp(b/Aa) |= (x= b )∨(x = b )∨···∨(x = b ).
1 2 n
tp(b/A) x = b
i
If is non-algebrai
, then strongly divides (and therefore þ-
A i ≤ n tp(b/Aa) A
divides) over for all ; by de(cid:28)nition þ-forks over and
b 6| a
⌣þA .
δ(x,b) ∈ p(x,b) δ(x,b)∧θ(x,y)
6) Let us suppose there is some su
h that
A δ(x,b) ∧ ∃yθ(x,y)
strongly divides over and suppose does not strongly
A
divide over . By de(cid:28)nition (and
ompa
tness) there are some distin
t
b ,...,b ,... b |= tp(b/A) δ(x,b ) ∧ ∃yθ(x,y)
1 n su
h that i and i∈N i is
on-
a c |= θ(a,y) (a,c) |=
sistent, satis(cid:28)ed by some element . LetV . Then
δ(x,b )∧θ(x,y) i
i
for all ,
ontradi
ting our assumptions.
a′ |= tp(a/Ac) a′ | cd
7) By extension we
an (cid:28)nd some su
h that ⌣þA . Let
d′ a′ a Ac
be the image of an automorphism whi
h sends to (cid:28)xing , so that
a | cd′ a | c (cid:3)
⌣þA . By partial right transitivity, ⌣þAd′ .
Remark 2.1.3. Condition 5 above does not hold for þ-dividing in pla
e of
þ-forking, even in the theory in the language of equality that states that there
A {a,b}
are in(cid:28)nitely many elements. In fa
t, if is the unordered pair where
a≤ b tp(a/A) x∈ A
, then is algebrai
, but does not þ-divide over the empty
set.
B
Proof. Wewillprovetheremarkby
ontradi
tion. Let beanysetsu
hthat
tp(A/B) x∈ A B tp(a/Bb)
isnon-algebrai
and stronglydividesover . If was
{x ∈ A′}A′|=tp(A/B),b∈A′
non-algebrai
, then would be an in(cid:28)nite
onsistent
set of formulas whi
h would
ontradi
t strong division. This means that
b ∈ acl(Ba) a ∈ acl(Bb)
; by symmetry, . In this parti
ular theory we may
B M Meq) tp(A/B)
2
assume that is a subset of (as opposed to . But is
tp(b/B) tp(a/B)
non-algebrai
, so and must be both be non-algebrai
. By
2
This is be
ause thetheoryof equalityhas(cid:16)eliminationof imaginaries(cid:17), a propertywe
will talkabout further on
PROPERTIES AND CONSEQUENCES OF THORN-INDEPENDENCE 5
Steinitz ex
hange prin
iple (see for example [Pil96℄, de(cid:28)nition 2.1.1), there
B a acl(a) = {a}
must be some element in algebrai
over . However, , so
a∈ B a B (cid:3)
whi
h
ontradi
ts being non-algebrai
over .
This shows that, unlike the
ase of simple theories, even when we are only

onsidering the language of equality there is a di(cid:27)eren
e between þ-dividing
and þ-forking.
The following results go in the same dire
tion as the ones above (showing
properties that þ-forking has in a general theory), but the proofs are a little
more elaborated.
a A ⊂ B tp(a/B)
Theorem 2.1.4. Given a tuple and sets , then þ-forks
A tp(a/acl(B)) A a | B
over if and only if þ-forks over . In other words, ⌣þA
a | acl(B)
if and only if ⌣þA .
tp(a/B) A tp(a/acl(B))
Proof. Monotoni
ityimpliesthatwhenever þ-forksover ,
A ψ(x,c) ∈ tp(a/acl(B))
þ-forks over . For the other dire
tion, suppose is
ψ(x,c) A φ (x,d )
i i
su
h that þ-forks over . Let be su
h that
ψ(x,c) ⇒ φ (x,d )∨φ(x,d )∨···∨φ(x,d )
i 1 2 m
i φ (x,d ) A c ,...,c
i i 1 n
and for any , þ-divides over . Let be the
onjugates
c B 0 ≤ j ≤ n σ B
j
of over and for let be some -automorphism su
h that
σ (c) = c tp(a/B) |= ∨n ψ(x,c ) j
j j. This means that j=1 j and for any su
h
ψ (x,c ) ⇒ ∨m φ (x,σ (d )).
j j i=1 i j i Combining this two, we get
n m
tp(a/B)|= φ (x,σ (d ))
i j i
j=1i=1
_ _
tp(a/B) A (cid:3)
and by de(cid:28)nition þ-forks over .
We
an also prove without any assumptions on the theory a version of
partial left transitivity:
a | c b | c ab | c
Lemma2.1.5. PartialLeftTransitivity: If ⌣þA and ⌣þAa then ⌣þA .
Proof. We begin with a
laim.
tp(ab/Ac) A
Claim 2.1.6. It is enough to show that does not þ-divide over .
a | c b | c ab 6| c φ(x,y,c) ∈ tp(ab/Ac)
Proof. Suppose ⌣þA , ⌣þAa and ⌣þA . Choose
ψ (x,y,c ) ψ A c¯:= (c :
whi
himplies i<n i i ,whereea
h i þ-dividesover . Put 1
i< n) a′ |= tp(a/Ac) a′ | cc¯
. ByextWensionwe
an(cid:28)ndsome su
hthat ⌣þA . Let
b b Ac a a′
2
be the image of under an automorphism (cid:28)xing whi
h sends to ,
tp(ab/Ac) = tp(a′b /Ac) b | c
so that 2 and 2 ⌣þAa′ . By extension again we
an
b′ |= tp(b /Aa′c) b′ | cc¯ a′b′
also (cid:28)nd some 2 su
h that ⌣þAa′ . So we have tuple
φ(x,y,c) ∈ tp(a′b′/Ac)
satisfying the hypothesis of the theorem. However,
φ(x,y,c) ⇒ ψ (x,y,c ) ψ (x,y,c ) ∈ tp(a′b′/Acc¯) i < n
i i i i
and , so for some ,
tp(a′b′/Acc¯) A (cid:3)
and Wþ-divides over .
6 ALFANGELONSHUUS
tp(ab/Ac) |= φ(x,y,c)
Let us suppose then that whi
h þ-divides over
A Ad
strongly dividing, say, over . By base extension, we
an (cid:28)nd some
d′ |= tp(d/Ac) a | c a | c
su
h that ⌣þAd′ , so we will assume ⌣þAd . By de(cid:28)nition
{φ(x,y,z)} k
z|=tp(c/Ad)
of strong dividing we know that is -in
onsistent and
tp(c/Ad) (4) tp(c/Ada)
is non algebrai
; by 2.1.2 so is . This means that
{φ(a,y,z)}
z|=tp(c/Ada)
k tp(c/Ada) tp(b/Aac)
isstill -in
onsistentand isnon-algebrai
. Byde(cid:28)nition
Aad Aa
strongly divides over whi
h means it þ-divides over , a
ontradi
tion.
(cid:3)
As in simple theories, this new notion of bifur
ation does have some rela-
tionwithindependentsequen
eswhi
hwillbestatedinthefollowinglemma.
However, unlike in simple theories this relation does not seem to be funda-
mentalforthedevelopment ofthetheory, perhapsbe
ausethemainfun
tion
of indis
ernible sequen
es in simple theories is to provide some uniformity
when witnessing division; su
h uniformity is provided by the de(cid:28)nability of
the parameters in the the de(cid:28)nition of þ-dividing.
A p(x,b) = tp(a/Ab).
Lemma 2.1.7. Let a,b be elements and a set. Let
Then the following
onditions are equivalent:
tp(a/Ab) A
(1) does not þ-divide over
B ⊇ A b B,
(2) For any su
h that is not algebrai
over there is some
a′ |= tp(a/Ab) Ba′ I
tuple and some in(cid:28)nite -indis
ernible sequen
e
b

ontaining .
(⇐) tp(a/Ab)
Proof. We will pro
eed by
ontradi
tion. Assume þ-divides
A B b ∈/ acl(B) δ |=
over . By de(cid:28)nition we have a with and a , su
h that
δ(a,b) {δ(x,b′)}b′|=tp(b/B) k I B
and is -in
onsistent. Let beany -indis
ernible
I {b′ |b′ |= tp(b/B)}
sequen
e. Sin
e the underlying set of is a subset of , we
{δ(x,b′)}b′∈I k a′ |= tp(a/Ab) a′ |= δ(x,b)
have that is -in
onsistent. Let , .
k 2 δ(a′,b′) b′ ∈ I
By -in
onsisten
y we have for all but (cid:28)nite whi
h implies
I Ba′
that is not indis
ernible.
(⇒) A,b,a tp(a/Ab) A b
Let be su
h that does not þ-divide over . If is
A
algebrai
over ,
ondition 2 in the lemma holds immediately so there is
b A
nothing to prove. We will assume then that is not algebrai
over .
B A b ∈/ acl(B) q(y) = tp(b/B).
Let be any set
ontaining with and let
δ(x,b) ∈ tp(a/Ab) {δ(x,b′)}b′|=tp(b/B)
We know that for any the set is not
k k
-in
onsistent for any . By
ompa
tness
q(y )∪{p(x,y ): i∈ ω}
i i
i∈ω
[
a′,I ∆
is
onsistent, and realized by , say. For any (cid:28)nite set of formulas
I ⊆ I I
∆ ∆
we
an (cid:28)nd by Ramsey's theorem some in(cid:28)nite su
h that is
∆ Ba′ Ba′
-indis
ernible over . By
ompa
tness we
an (cid:28)nd a -indis
ernible
J a′b′ |= p(x,y) ∪ q(y) b′ ∈ J σ B
sequen
e su
h that for any . If is a -
σ(b′) = b σ(a′),σ(J) (cid:3)
automorphism with then will do.
PROPERTIES AND CONSEQUENCES OF THORN-INDEPENDENCE 7
2.2. Existen
e. Given some independen
e relation, we say that su
h a re-
a A a
lation satis(cid:28)es existen
e if for any tuple and any set , is independent
A A
with over . This is a very useful property for simple theories. Until this
point we have studied the behavior of þ-forking in the most general
ontext.
In this se
tion we will only
onsider theories for whi
h þ-forking satis(cid:28)es
a A tp(a/A) A
existen
e: given a tuple and a set , does not þ-fork over .
As we mentioned before, unlike in simple theories there is a di(cid:27)eren
e
between þ-forking and þ-dividing. As we shall see in most of the proofs,
we
an usually work around this problem sin
e extension provides a way to
redu
e most of the proofs down to þ-dividing. It would be ni
e however to
have some idea of the relation between the parameters that we need for þ-
forking and those that are used in the
orresponding þ-dividing formulas, so
we
antell how mu
h dowe have to extend aþ-forking type before a
hieving
þ-division. The next lemma gives a partial answer to this question.
δ(x,a) A
Lemma 2.2.1. nIf ψ(x,a )is
onsistent aδn(xd,aþ-)forks over ψ, a(sx,wait)nessed by
a disjun
tion i=1 i implied by , su
h that i i strongly
Ac a Aac¯ c¯ = (c : i < n)
i i i
divides over W, then is algebrai
over where .
i a Aac
i i
Even more, for at least one , is algebrai
over . (We are assuming
ψ δ(x,a) ; ψ(x,a )
that there are no (cid:16)extra(cid:17) i's: i.e. that j∈I ij for any
I ( {1,2,...,n}
.) W
b b |= δ(x,a)
Proof. Let be any element su
h that . By existen
e and de(cid:28)ni-
tp(b/Aa) tp(b/Aaa¯c¯) b | a¯c¯
tion of þ-forking we
an extend to i so that ⌣þAa i .
b | a
In parti
ular, by partial transitivity ⌣þAaci i
b |= ψ (x,a ) i a′ |= tp(a /Abac )
We know that i i for some ; for any i i i we
C |= ψ(b,a′) a′ |= tp(a /Ac )
have i and i i i . By the de(cid:28)nition of strong dividing
a′ b
there
annot be in(cid:28)nitely many su
h i's ( would witness the
onsisten
y).
tp(a /Abc ) tp(a /Abac )
i i i i
Thus must be algebrai
and therefore so is . But we
b | a (4) tp(a /Aac )
know that ⌣þAaci i, so by lemma 2.1.2 this means that i i is
algebrai
.
tp(b/Aa)
To (cid:28)nish the proof, we just have to be
areful when extending to
Aaa δ(x,a,a ) δ(x,a)∧¬ψ (x,a ) δ(x,a,a )
i i i i i
;let be so impliesthedisjun
tion
ψ (x,a ) k a
j6=i j j . Sin
e -in
onsisten
y is preserved, either there is some j
Aa δ(x,a,a ) Aa
i i i
Walgebrai
over or þ-forks over in whi
h
ase we
an repeat
a Aa c a
j i j j
the pro
ess and get some algebrai
over . Either way we get
Aa c j a Ac a
i j i i j
algebrai
over for some . However, is algebrai
over so is
Ac c (cid:3)
i j
algebrai
over .
p(x,b) Ab
Theorem 2.2.2. Let be a type over whi
h is non-þ-forking over
A θ(x,y) p(x,b) |= ∃yθ(x,y) p(x,b)∪
and let be a formula su
h that . Then
{θ(x,y)} A
does not þ-fork over .
A p(x,b) ∪ {θ(x,y)} |=
Prnoofψ. S(uxp,yp,oase) that itψd(oxe,sy,þa-f)ork over so Athat
i=1 i i where i i þ-divides over by strongly dividing over
Ac
i
W .
8 ALFANGELONSHUUS
p(x,b) A
By hypothesis does not þ-fork over , so using extension we
an
a |= p(x,b) a | ba¯c¯

hoose some su
h that ⌣þA i i whi
h implies by partial tran-
a | a
sitivity that ⌣þAci i. On the other hand, using existen
e and extension we
c|= θ(a,y) c | ba¯c¯

an
hoosesome su
h that ⌣þAa i i. Usingpartial transitivity
c | a
again we get ⌣þAaci i.
(a,c) |= p(x,b)∪θ{(x,y)} (a,c) |= ψ (x,y,a )
j j
We know that so for some
j a Aacc
j j
. By de(cid:28)nition, is algebrai
over . However, using lemma 2.1.2
c | a a Aac
with ⌣þAacj j we get that j is algebrai
over j. Using 2.1.2 one more
a Ac
j j
time will give us algebrai
over ,
ontradi
ting the de(cid:28)nition of strong
dividing.
(cid:3)
| 0
⌣
De(cid:28)nition 2.2. A notion of independen
e has the strong extension
a | 0 b c c′ |= tp(c/Aa)
property if whenever ⌣A then for any there is there is
ac′ | 0 b c b′ |= tp(b/Aa) ac | 0 b′
with ⌣A , or equivalently, forany there is with ⌣A .
Corollary 2.2.3. þ-independen
e has the strong extension property.
p(x,b) p ↾ A
Proof. It is enough to prove that if is a non forking extension of
q(x,y) A p ↾ A p(x,b)∪q(x,y)
and is a
onsistent type over
ontaining then
A
does not þ-fork over . This follows from theorem 2.2.2, as
onsisten
y of
(p ↾ A)∪q p∪q (cid:3)
implies
onsisten
y of .
A
De(cid:28)nition 2.3. A sequen
e of elements is a þ-Morley sequen
e over if it
A
is indis
ernible and þ-independent over .
p(x) B ⊃ A
Claim 2.2.4. Let be a
omplete type over whi
h does not þ-fork
A A
over . Then there is a þ-Morley sequen
e over with all of its elements
p(x)
realizing .
The
laim is true for any independent notion that has extension. The
proof is exa
tly the same as the one given in the simple theoreti

ontext
(see [Wag00℄). We will prove later that in the theories that will a
tually
interest us, þ-Morley sequen
es will provide an alternative de(cid:28)nition of þ-
forking.
3. þ-Rank and Rosy Theories
3.1. De(cid:28)nition of þ-rank. We will now de(cid:28)ne a notion of rank that will

ode þ-forking.
φ, ∆ x;y
De(cid:28)nition3.1. Givena formula aset of formulas in the variables ,
Π y;z z
a set of formulae in the variables (with possibly of in(cid:28)nite length)
k, (φ,∆,Π,k)
and a number we de(cid:28)ne þ indu
tively as follows:
(φ,∆,Π,k) ≥ 0 φ
(1) þ if is
onsistent.
λ (φ,∆,Π,k) ≥ λ (φ,∆,Π,k) ≥ α
(2) For limit ordinal, þ if and only if þ
α< λ
for all
PROPERTIES AND CONSEQUENCES OF THORN-INDEPENDENCE 9
(φ,∆,Π,k) ≥ α+1 δ ∈ ∆ π(y;z) ∈ Π
(3) þ if and only if there is a , some
c
and parameters su
h that
(φ∧δ(x,a),∆,Π,k) ≥ α a|= π(y;c)
(a) þ for in(cid:28)nitely many
{δ(x,a)} k−
(b) a|=π(y;c) is in
onsistent
p
As usual, for a type , we de(cid:28)ne
(p,∆,Π,k) = min{ (φ(x),∆,Π,k) |φ(x)∈ p}.
þ þ
Remark 3.1.1. We
an give a de(cid:28)nition dire
tly for types,
hanging all
φ p
instan
es of forsome type whi
h isthe
ase we will usually use. However,
when dealing with types, we will use an alternative version of
ondition 3(a).
p A
Let be a
omplete type over and let us assume that we are witnessing
c
the rank going up as in
ondition 3, with being a tuple
ontaining all the
π q(y)
parameters in . We
an always (cid:28)nd some non-algebrai

omplete type
Ac π a′ |= q(y) (φ∧δ(x,a′),∆,Π,k) ≥
over
ontaining su
h that for any , þ
α
. Therefore, we
an
hange
ondition 3(a) by
(p∪{δ(x,a)},∆,Π,k) ≥ α a |= π tp(a/Ac)
3(a)'. þ where and is non-
algebrai
.
p(x;y)
Remark 3.1.2. Given any type (not ne
essarily
omplete), formulas
φ,π k,n
and integers , the set
{b| (p(x,b),φ,π,k) ≥ n}
þ
is type de(cid:28)nable.
This has ni
e
onsequen
es for the stru
ture of the þ-rank. For exam-
(p∪{δ(x,a)},∆,Π,k)
ple,
ompa
tness implies that þ is (cid:28)nite whenever it is
de(cid:28)ned.
Proof. Noti
e that
(p(x,b),φ,π,k) ≥ n
þ
φ(x,a)
is witnessed by a tree where ea
h of the nodes is a formula , its
n i ≤ n c i
i
height is , for ea
h there is some su
h that the level
ontains
φ(x,a) k
all a|=π(x,ci), the union any of su
h formulas (all in the same level)
p
is in
onsistent and the union of any bran
h is
onsistent with . All theses
(cid:3)
properties
an be des
ribed by formulas, and we get type de(cid:28)nability.
3.2. Properties of the þ-rank.
Theorem 3.2.1. This thorn rank has the following properties:
∆ ⊆ ∆′ p⊇ p′, Π ⊆ Π′
(1) Monotoni
ity: If , and then
p′,∆′,Π′,k ≥ (p,∆,Π,k).
þ þ
(cid:0)p ⊆ q ⊆ r, (cid:1) (r,∆,Π,k) = (p,∆,Π,k)
(2) Transitivity: If then þ þ if and
(r,∆,Π,k) = (q,∆,Π,k) (q,∆,Π,k) = (p,∆,Π,k)
onlyif þ þ and þ þ .
((θ∨ψ),∆,Π,k) = max{ (θ,∆,Π,k), (ψ,∆,Π,k)}
(3) Additivity: þ þ þ
10 ALFANGELONSHUUS
Proof. .
λ λ ≤ (p,∆,Π,k) (p′,∆′,Π′,k) ≥
1)Wewillprovebyindu
tionon thatif þ thenþ
λ. λ = 0
For , the proof is
lear, as is the indu
tion step for the
ase
λ λ = α
when is a limit ordinal. Now, suppose it is true for and let us
(p,∆,Π,k) ≥ α + 1 ψ p′
assume that þ . Let be any formula in ; by hy-
ψ ∈ p (ψ,∆,Π,k) ≥ α + 1 φ ∈ ∆
pothesis so þ . We
an therefore (cid:28)nd a
π Π (ψ∧φ(x,a),∆,Π,k) ≥ α
and a formula in su
h that þ for in(cid:28)nitely
a |= π {φ(x,a)} k .
many and a|=π is -in
onsistent By indu
tion hypothesis
(ψ∧φ(x,a),∆′,Π′,k) ≥ α ψ ∈ p′ φ ∈ ∆′ π ∈ Π′ φ
þ . But , and so a
tually
(ψ,∆′,Π′,k) ≥ α+1 ψ ∈ p′
witnesses þ for all .
(r,∆,Π,k) ≤ (q,∆,Π,k) ≤ (r,∆,Π,k) ≥
2) By monotoni
ity, þ þ þ ; transi-
tivity follows from transitivity for equality.
3) By monotoni
ity we have
((θ∨ψ),∆,Π,k) ≥ max{ (θ,∆,Π,k), (ψ,∆,Π,k)}.
þ þ þ
α
For the other dire
tion we will prove by indu
tion on that if
((θ∨ψ),∆,Π,k) ≥ α
þ
(θ,∆,Π,k) ≥ α (ψ,∆,Π,k) ≥ α.
then either þ or þ On
e again, the only
α
di(cid:30)
ult step is the indu
tion step. Let us assume it is true for and that
((θ∨ψ),∆,Π,k) ≥ α+1. δ ∈ ∆ π ∈ Π
þ We
an then (cid:28)nd a and a formula
su
h that
((θ∨ψ)∧δ(x,a ),∆,Π,k) ≥ α
i
þ
a |= π {φ(x,a )} k
for in(cid:28)nitely many i and i ai|=π is -in
onsistent. By indu
-
a (θ∧δ(x,a ),∆,Π,k) ≥ α (ψ∧δ(x,a ),∆,Π,k) ≥
i i i
tion,foranysu
h eitherþ orþ
α ψ θ θ a′s
;butthenforoneof or (letusassume )wehavein(cid:28)nitelymany i su
h
(θ∧δ(x,a ),∆,Π,k) ≥ α (θ,∆,Π,k) ≥ α+1. (cid:3)
i
that þ and by de(cid:28)nition þ
∆ Π
Corollary 3.2.2. Extension for (cid:28)xed and :
ξ A ∆ Π
For any partial type de(cid:28)ned over a set , (cid:28)nite sets of formulas and
k, ξ p A (p,∆,Π,k) =
andany we
anextend toa
ompletetype over su
hthatþ
(ξ,∆,Π,k).
þ
Proof. It is just the usual appli
ation of additivity, the de(cid:28)nition of þ-rank
(cid:3)
fortypesand Zorn'slemma asitisusedin simpletheories (see[Kim96℄).
One of the properties that we will prove here is that þ-independen
e and
D
þ-ranks are related in the same way that forking and the -ranks are. The
next theorem is one of the dire
tions of the relation we will prove.
p B ⊇ A φ,Π,k,
Theorem 3.2.3. Let be a type over su
h that for every
(p ↾ A,φ,Π,k) = (p,φ,Π,k) .
3
þ þ
p A.
Then does not þ-fork over
(p,φ,Π,k) (p,{φ},Π,k)
3
By þ we really meanþ .