will boney university of illinois at chicago january...
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Type omission Averageable Classes Torsion Modules
Some compactness in some nonelementary classes
Will BoneyUniversity of Illinois at Chicago
January 9, 2015Beyond First Order Model Theory Miniconference
University of Texas-San Antonio
Type omission Averageable Classes Torsion Modules
Goal
The plan is to develop a framework that gives rise to compactnessin some nonelementary contexts. This allows us to develop somenonforking notions, and we specialize to the example of torsionmodules over PIDs.
Type omission Averageable Classes Torsion Modules
Prototype I: Abelian torsion groups
Have a nice elementary (model) theory of abelian groups
Torsion groups (of infinite exponent) are not first order:
∀x∨n<ω
n · x = 0
However, there is an easy way to pick out the torsion elementsfrom G :
tor(G ) := {g ∈ G : ∃n < ω.n · g = 0}
Moreover, tor(G ) is an abelian group
Key fact: given g , h and their orders, I have a bound on theorder of g + h
Type omission Averageable Classes Torsion Modules
Prototype I: Abelian torsion groups
Have a nice elementary (model) theory of abelian groups
Torsion groups (of infinite exponent) are not first order:
∀x∨n<ω
n · x = 0
However, there is an easy way to pick out the torsion elementsfrom G :
tor(G ) := {g ∈ G : ∃n < ω.n · g = 0}
Moreover, tor(G ) is an abelian group
Key fact: given g , h and their orders, I have a bound on theorder of g + h
Type omission Averageable Classes Torsion Modules
Prototype I: Abelian torsion groups
Have a nice elementary (model) theory of abelian groups
Torsion groups (of infinite exponent) are not first order:
∀x∨n<ω
n · x = 0
However, there is an easy way to pick out the torsion elementsfrom G :
tor(G ) := {g ∈ G : ∃n < ω.n · g = 0}
Moreover, tor(G ) is an abelian group
Key fact: given g , h and their orders, I have a bound on theorder of g + h
Type omission Averageable Classes Torsion Modules
Prototype II: Archimedean fields
Have a nice elementary (model) theory of ordered fields ofcharacteristic 0
Archimedean fields are not first order:
∀x∨n<ω
1 + · · ·+ 1 > n > −1− · · · − 1
However, there is an easy way to pick out the standard, finiteelements of a field:
arch(F ) := {f ∈ F : st(f ) = f }
Moreover, arch(F ) is an ordered field of characteristic 0
Key fact: given standard f , g , −f , we have 1g , f + g , and fg
are standard
Type omission Averageable Classes Torsion Modules
Prototype II: Archimedean fields
Have a nice elementary (model) theory of ordered fields ofcharacteristic 0
Archimedean fields are not first order:
∀x∨n<ω
1 + · · ·+ 1 > n > −1− · · · − 1
However, there is an easy way to pick out the standard, finiteelements of a field:
arch(F ) := {f ∈ F : st(f ) = f }
Moreover, arch(F ) is an ordered field of characteristic 0
Key fact: given standard f , g , −f , we have 1g , f + g , and fg
are standard
Type omission Averageable Classes Torsion Modules
Similarities
There are two key similarities here that will guide us in abstractingthese situations:
The types were unary
Where elements omit types lets me figure out where functionsof them omit types
I’m probably going to often use phrases like “where typeomission happens.” Each type is going to have a natural index(as we’ve seen) and the “location” of type omission is thatindex.
Type omission Averageable Classes Torsion Modules
Similarities
There are two key similarities here that will guide us in abstractingthese situations:
The types were unary
Where elements omit types lets me figure out where functionsof them omit types
I’m probably going to often use phrases like “where typeomission happens.” Each type is going to have a natural index(as we’ve seen) and the “location” of type omission is thatindex.
Type omission Averageable Classes Torsion Modules
Outline
Discuss the type omitting hull and properties that lead to itbeing well-behaved
Compactness results and ultraproducts
Averageable classes
Examples
Torsion modules
Type omission Averageable Classes Torsion Modules
Framework
We will be in the following situation:
M is an L-structure
Γ is a set of unary L-typesFor ease we enumerate Γ as follows:
Γ = {pj(x) : j < α}pj(x) = {φjk(x) : k < βj}
Type omission Averageable Classes Torsion Modules
Main definition
M is an L-structure
Γ is a set of unary L-types
Definition
Γ(M) := {m ∈ M : ∀j < α, ∃kj < βj .M � ¬φjkj (m)}
Γ(M) contains all elements of M that omit all types of Γaccording to M.
Each element has a (possibly many) witnesses to its inclusion.Namely m ∈ Γ(M) iff there is some k(m) ∈ Πβj such thatM � ¬φjk(m)(j)(m).
Type omission Averageable Classes Torsion Modules
Main definition
M is an L-structure
Γ is a set of unary L-types
Definition
Γ(M) := {m ∈ M : ∀j < α, ∃kj < βj .M � ¬φjkj (m)}
Γ(M) contains all elements of M that omit all types of Γaccording to M.
Each element has a (possibly many) witnesses to its inclusion.Namely m ∈ Γ(M) iff there is some k(m) ∈ Πβj such thatM � ¬φjk(m)(j)(m).
Type omission Averageable Classes Torsion Modules
The main use
Definition
Γ(M) := {m ∈ M : ∀p ∈ Γ,∃φp ∈ p.M � ¬φp(m)}
Suppose {Mi : i ∈ I} is a collection of L-structures thatalready omit Γ
They probably also model a common theory T
U is an ultrafilter on I
Then we can formΓ(ΠMi/U)
, which will omit all of the types of Γ and give enough averaging toget some compactness results...
sometimes.
Type omission Averageable Classes Torsion Modules
The main use
Definition
Γ(M) := {m ∈ M : ∀p ∈ Γ,∃φp ∈ p.M � ¬φp(m)}
Suppose {Mi : i ∈ I} is a collection of L-structures thatalready omit Γ
They probably also model a common theory T
U is an ultrafilter on I
Then we can formΓ(ΠMi/U)
, which will omit all of the types of Γ and give enough averaging toget some compactness results...sometimes.
Type omission Averageable Classes Torsion Modules
Problems with Γ(M)
This construction turns out to be very fragile
The types of Γ must be “honestly” unary (coding,classification over a predicate, etc.)
Γ(M) might fail to be a structure
If Γ(M) is a structure, it might still fail to be an elementarysubstructure
This means, depending on the types, it might not even omit allof the types of Γ
The plan is to give examples of where this can go wrong, andthen give some sufficient conditions on when things work
Type omission Averageable Classes Torsion Modules
Problems with Γ(M)
This construction turns out to be very fragile
The types of Γ must be “honestly” unary (coding,classification over a predicate, etc.)
Γ(M) might fail to be a structure
If Γ(M) is a structure, it might still fail to be an elementarysubstructure
This means, depending on the types, it might not even omit allof the types of Γ
The plan is to give examples of where this can go wrong, andthen give some sufficient conditions on when things work
Type omission Averageable Classes Torsion Modules
Problems with Γ(M)
This construction turns out to be very fragile
The types of Γ must be “honestly” unary (coding,classification over a predicate, etc.)
Γ(M) might fail to be a structure
If Γ(M) is a structure, it might still fail to be an elementarysubstructure
This means, depending on the types, it might not even omit allof the types of Γ
The plan is to give examples of where this can go wrong, andthen give some sufficient conditions on when things work
Type omission Averageable Classes Torsion Modules
Bad example I: p-adics
M = 〈ω,+, |, 2〉I = ω, U is any non principle ultrafilter
p(x) = {(2k | x) ∧ (x 6= 0) : k < ω}
[n 7→ 1]U ∈ Γ(ΠM/U)
[n 7→ 2n − 1]U ∈ Γ(ΠM/U)
[n 7→ 2n]U 6∈ Γ(ΠM/U)
Thus p-adicly valued fields don’t get mapped to substructures
Type omission Averageable Classes Torsion Modules
Bad example I: p-adics
M = 〈ω,+, |, 2〉I = ω, U is any non principle ultrafilter
p(x) = {(2k | x) ∧ (x 6= 0) : k < ω}
[n 7→ 1]U ∈ Γ(ΠM/U)
[n 7→ 2n − 1]U ∈ Γ(ΠM/U)
[n 7→ 2n]U 6∈ Γ(ΠM/U)
Thus p-adicly valued fields don’t get mapped to substructures
Type omission Averageable Classes Torsion Modules
Bad example II: some pathology with standard naturalnumbers
M = 〈N,N′; +,×, 1; +′,×′, 1′;×∗〉I = ω, U is any non principle ultrafilter
p(x) = {N2(x) ∧ (1 + · · ·+ 1 6= x) : n < ω}
Two copies of N linked by multiplication
Two failures of Los’ Theorem
ψ(x) ≡ ∃y ∈ N2(11 ×∗ y = x)True of each n ∈ N1, but is not true of [n 7→ n]U ∈ Γ(ΠM/U)
φ ≡ ∀x ∈ N1∃y ∈ N2(11 ×∗ y = x)True in M but not in Γ(ΠM/U)
Type omission Averageable Classes Torsion Modules
Bad example II: some pathology with standard naturalnumbers
M = 〈N,N′; +,×, 1; +′,×′, 1′;×∗〉I = ω, U is any non principle ultrafilter
p(x) = {N2(x) ∧ (1 + · · ·+ 1 6= x) : n < ω}
Two copies of N linked by multiplication
Two failures of Los’ Theorem
ψ(x) ≡ ∃y ∈ N2(11 ×∗ y = x)True of each n ∈ N1, but is not true of [n 7→ n]U ∈ Γ(ΠM/U)
φ ≡ ∀x ∈ N1∃y ∈ N2(11 ×∗ y = x)True in M but not in Γ(ΠM/U)
Type omission Averageable Classes Torsion Modules
Bad example III: Archimedean fields
The construction is very fragile and does not respond well to“implicit” type omission
Archimedean fields are often defined as ordered fields omittingthe type of an infinite elementThen, the type of infinitesimal elements and two elementsinfinitely close to each other are omitted by the field axioms
However, using the Γ(F ) construction, we would not get asubstructure if Γ is just the type of an infinite element
Instead, Γ has to list each type of a nonstandard elementaround a standard real
This example shows that the unary part is crucial and can’t beavoided through simple coding
Type omission Averageable Classes Torsion Modules
When Γ(M) is a structure
There’s a straightforward condition on this:
For any F ∈ L and m0, . . . ,mn−1 ∈ M that omit the types ofΓ, there is some kj < βj for each j < α such that
M � ¬φjkj (F (m0, . . . ,mn−1))
A stronger condition is often useful in applications. It imposessome uniformity on where F of a tuple omits the types basedon where the tuple omits those types
Definition
M is Γ-closed iff for all j < α and F ∈ L, there is some
g jF :(
Πβj′)n→ βj such that, for all m0, . . . ,mn−1 ∈ Γ(M),
M � ¬φjk ′ (F (m0, . . . ,mn−2))
where k ′ = g jF (k(m0), . . . , k(mn−1)).
Type omission Averageable Classes Torsion Modules
When Γ(M) is a structure
There’s a straightforward condition on this:
For any F ∈ L and m0, . . . ,mn−1 ∈ M that omit the types ofΓ, there is some kj < βj for each j < α such that
M � ¬φjkj (F (m0, . . . ,mn−1))
A stronger condition is often useful in applications. It imposessome uniformity on where F of a tuple omits the types basedon where the tuple omits those types
Definition
M is Γ-closed iff for all j < α and F ∈ L, there is some
g jF :(
Πβj′)n→ βj such that, for all m0, . . . ,mn−1 ∈ Γ(M),
M � ¬φjk ′ (F (m0, . . . ,mn−2))
where k ′ = g jF (k(m0), . . . , k(mn−1)).
Type omission Averageable Classes Torsion Modules
Universal Los’ Theorem
Peaking ahead to ultraproducts, we have the following result.
Proposition
Suppose Γ(ΠMi/U) is a structure. If φ(x) is a universal formulaand [f0]U , . . . , [fn−1]U ∈ Γ(ΠMi/U), then
{i ∈ I : Mi � φ (f0(i), . . . , fn−1(i)))} =⇒Γ(ΠMi/U) � φ ([f0]U , . . . , [fn−1]U)
Proof: Γ(ΠMi/U) ⊂ ΠMi/U so universal formulas transferdownwards. The Los’ Theorem gives the result. †The normal proof by induction works as well, which is useful toshow that a formula transferring up is closed under conjunction,disjunction, and universal quantification
Type omission Averageable Classes Torsion Modules
Universal Los’ Theorem
Peaking ahead to ultraproducts, we have the following result.
Proposition
Suppose Γ(ΠMi/U) is a structure. If φ(x) is a universal formulaand [f0]U , . . . , [fn−1]U ∈ Γ(ΠMi/U), then
{i ∈ I : Mi � φ (f0(i), . . . , fn−1(i)))} =⇒Γ(ΠMi/U) � φ ([f0]U , . . . , [fn−1]U)
Proof: Γ(ΠMi/U) ⊂ ΠMi/U so universal formulas transferdownwards. The Los’ Theorem gives the result. †
The normal proof by induction works as well, which is useful toshow that a formula transferring up is closed under conjunction,disjunction, and universal quantification
Type omission Averageable Classes Torsion Modules
Universal Los’ Theorem
Peaking ahead to ultraproducts, we have the following result.
Proposition
Suppose Γ(ΠMi/U) is a structure. If φ(x) is a universal formulaand [f0]U , . . . , [fn−1]U ∈ Γ(ΠMi/U), then
{i ∈ I : Mi � φ (f0(i), . . . , fn−1(i)))} =⇒Γ(ΠMi/U) � φ ([f0]U , . . . , [fn−1]U)
Proof: Γ(ΠMi/U) ⊂ ΠMi/U so universal formulas transferdownwards. The Los’ Theorem gives the result. †The normal proof by induction works as well, which is useful toshow that a formula transferring up is closed under conjunction,disjunction, and universal quantification
Type omission Averageable Classes Torsion Modules
Corollaries of Universal Los’
Suppose that Γ(ΠMi/U) is a structure.
Corollary
If φ(x0, . . . , xn) is a quantifier-free formula and[f0]U , . . . , [fn−1]U ∈ Γ(ΠMi/U), then
{i ∈ I : Mi |= φ(f0(i), . . . , fn−1(i))} ∈ U ⇐⇒Γ(ΠMi/U) |= φ([f0]U , . . . , [fn−1]U)
Corollary (Weak Type Omission)
If each p ∈ Γ consists of existential formulas, then Γ(ΠMi/U)omits pj .
Type omission Averageable Classes Torsion Modules
A little more: ∃∀ sentences
Proposition
Suppose
Γ is finite
Γ(ΠMi/U) is a structure
T∃(∀∧¬Γ) is a complete theory of the indicated quantifiercomplexity that is modeled by each Mi
Then Γ(ΠMi/U) � T∃∀.
Proof: Given a ∃xψ(x), we can form
∃x
ψ(x) ∧∧
j<α;`<n
¬φjk j`
(x)
which is of the indicated quantifier complexity.
Type omission Averageable Classes Torsion Modules
A little more: ∃∀ sentences
Proposition
Suppose
Γ is finite
Γ(ΠMi/U) is a structure
T∃(∀∧¬Γ) is a complete theory of the indicated quantifiercomplexity that is modeled by each Mi
Then Γ(ΠMi/U) � T∃∀.
Proof: Given a ∃xψ(x), we can form
∃x
ψ(x) ∧∧
j<α;`<n
¬φjk j`
(x)
which is of the indicated quantifier complexity.
Type omission Averageable Classes Torsion Modules
When Γ(M) is an elementary substructure
There are two cases we look at:
A uniform condition similar to Γ-closed, which we call Γ-nice.
Some form of quantifier elimination plus extra work on thetheory.
Type omission Averageable Classes Torsion Modules
Γ-nice
Definition
M is Γ-nice iff for all j < α and formulas ∃xψ(x ; y), there is some
g j∃xψ(x ;y) :
(Πβj
′)n→ βj such that, for all m0, . . . ,mn−1 ∈ Γ(M),
If M � ∃xψ(x ;m), then there is n ∈ Γ(M) such that
M � ψ(n;m); and
M � ¬φjk ′ (F (m0, . . . ,mn−1)) where
k ′ = g jF (k(m0), . . . , k(mn−1)).
Type omission Averageable Classes Torsion Modules
Γ-nice
Definition
M is Γ-nice if existential formulas with parameters from Γ(M) truein M have witnesses in Γ(M) and their type omission can becalculated from the type omission of the parameters.
M is Γ-nice iff it has a Skolemization that is Γ-closed
If M is Γ-nice, then Γ(M) ≺ M
Theorem
If the input is Γ-nice, then Los’ Theorem holds.
Type omission Averageable Classes Torsion Modules
Γ-nice
Definition
M is Γ-nice if existential formulas with parameters from Γ(M) truein M have witnesses in Γ(M) and their type omission can becalculated from the type omission of the parameters.
M is Γ-nice iff it has a Skolemization that is Γ-closed
If M is Γ-nice, then Γ(M) ≺ M
Theorem
If the input is Γ-nice, then Los’ Theorem holds.
Type omission Averageable Classes Torsion Modules
Quantifier elimination
A (so far) more useful criteria comes from quantifierelimination
The basic outline is this: suppose we have some theory T so
if M � T , then Γ(M) � T (so is implicitly a structure);T has quantifier elimination of ∆-formulas; andΓ(M) is a ∆-elementary substructure of M
then Γ(M) ≺ M.
The surprising thing is that this actually occurs!
Type omission Averageable Classes Torsion Modules
Quantifier elimination
A (so far) more useful criteria comes from quantifierelimination
The basic outline is this: suppose we have some theory T so
if M � T , then Γ(M) � T (so is implicitly a structure);T has quantifier elimination of ∆-formulas; andΓ(M) is a ∆-elementary substructure of M
then Γ(M) ≺ M.
The surprising thing is that this actually occurs!
Type omission Averageable Classes Torsion Modules
Good example I: DLOGZ
Set T := Th(Q, <,+,−, 0, 1, cn)n∈Z andp(x) = {x ≤ cn or cm ≤ x : n < m ∈ Z}.A model of DLOGZ is a model of T that omits p, i.e. adense, linearly ordered group where {cn : n ∈ Z} is a discrete,cofinal sequence.
Skolem showed that T has quantifier elimination.
Moreover, if M � T , then it is not hard to show thatΓ(M) � T (can compute the bounds from the bounds of theinputs)
Then Γ(M) ≺ M
Also, (EC (T , p),≺) has amalgamation, joint embedding, andsyntactic types are Galois types
Type omission Averageable Classes Torsion Modules
Good example I: DLOGZ
Set T := Th(Q, <,+,−, 0, 1, cn)n∈Z andp(x) = {x ≤ cn or cm ≤ x : n < m ∈ Z}.A model of DLOGZ is a model of T that omits p, i.e. adense, linearly ordered group where {cn : n ∈ Z} is a discrete,cofinal sequence.
Skolem showed that T has quantifier elimination.
Moreover, if M � T , then it is not hard to show thatΓ(M) � T (can compute the bounds from the bounds of theinputs)
Then Γ(M) ≺ M
Also, (EC (T , p),≺) has amalgamation, joint embedding, andsyntactic types are Galois types
Type omission Averageable Classes Torsion Modules
Good example I: DLOGZ
Set T := Th(Q, <,+,−, 0, 1, cn)n∈Z andp(x) = {x ≤ cn or cm ≤ x : n < m ∈ Z}.A model of DLOGZ is a model of T that omits p, i.e. adense, linearly ordered group where {cn : n ∈ Z} is a discrete,cofinal sequence.
Skolem showed that T has quantifier elimination.
Moreover, if M � T , then it is not hard to show thatΓ(M) � T (can compute the bounds from the bounds of theinputs)
Then Γ(M) ≺ M
Also, (EC (T , p),≺) has amalgamation, joint embedding, andsyntactic types are Galois types
Type omission Averageable Classes Torsion Modules
Good example II: Normed spaces (and more)
Consider the two sorted structure of an abelian group B andthe ordered field structure of R, with maps between them ofscalar multiplication and norm and a constant for eachelement of RLet T be the intended theory
Let Γ = {pr (x), qr : r ∈ R ∪ {∞}}, where
p∞(x) = {R(x) ∧ (x < −n ∨ n < x) : n < ω};pr (x) = {R(x) ∧ (x 6= cr ) ∧ (cr− 1
n< x < cr+ 1
n) : n < ω} for
r ∈ R;q∞(x) = {B(x) ∧ (‖x‖ < −n ∨ n < ‖x‖) : n < ω}; andqr (x) = {B(x) ∧ (‖x‖ 6= cr ) ∧ (cr− 1
n< x < cr+ 1
n) : n < ω}.
Then universal formulas transfer (although existentials requiremore work) and we get something like the Banach spaceultraproduct
Type omission Averageable Classes Torsion Modules
Intermezzo: Γ(ΠMi/U) vs. ΠΓMi/U
Compare two definitions
Definition
Γ(ΠMi/U) = {[f ]U ∈ ΠMi/U : for every p ∈ Γ, there is φp ∈ p
so ΠMi/U � ¬φp([f ]U)}
Definition
ΠΓMi/U = {[f ]U ∈ ΠMi/U : there is Xf ∈ U such that for every
p ∈ Γ, there is φp ∈ p such that,
for all i ∈ I ,Mi � ¬φp(f (i))}
ΠΓMi/U ⊂ Γ(ΠMi/U) ⊂ ΠMi/U
Type omission Averageable Classes Torsion Modules
Averageable Classes
We now turn to averageable classes
Informally, an averageable class K = EC (T , Γ) is one whereM 7→ Γ(M) ∈ K is well-behaved according to ≺
Enough for this discussion if{Mi ∈ K : i ∈ I} 7→ Γ(ΠMi/U) ∈ K satisfies enough of Los’Theorem
The models of K are models of a first order theory T thatomit Γ and ≺ is elementary according to some good notion ofsubstructure
The key fact is that, while we don’t literally haveultraproducts (nonelementary class), we almost do and almostis enough for compactness results
Type omission Averageable Classes Torsion Modules
Good example III−: Abelian torsion groups
Fix a complete theory T of torsion groups
of infinite exponent; andthat has a torsion model
Let K be the class of torsion models of T
Let ≺ be pure subgroup
We will see that tor(ΠMi/U) satisfies Los’ Theorem, althoughwe don’t have tor(M) ≡ M in all cases
Type omission Averageable Classes Torsion Modules
Creating new models
We want to create new models using this approach
Every M ∈ K has Γ(M) = M and models of T haveΓ(M) ≺ M.
Question
When do we have M � Γ(ΠM/U)?
≺ follows from coherence/Tarski-Vaught test, so the realquestion is proper extension
Type omission Averageable Classes Torsion Modules
Creating new models
Question
When do we have M � Γ(ΠM/U)?
Answer: At least when M has an infinite subset that omits allof Γ at the same place
If infinite X ⊂ M has: for all p ∈ Γ, there is φp ∈ p so for allx ∈ X
M � ¬φp(x)
then any function picking out distinct elements of X will benew.
Type omission Averageable Classes Torsion Modules
Creating new models
Question
When do we have M � Γ(ΠM/U)?
Answer: At least when M has an infinite subset that omits allof Γ at the same place
If infinite X ⊂ M has: for all p ∈ Γ, there is φp ∈ p so for allx ∈ X
M � ¬φp(x)
then any function picking out distinct elements of X will benew.
Type omission Averageable Classes Torsion Modules
No maximal models
This gives a nice criteria for having no maximal models
Set κ = |Πj<αβj |, i. e. the number of witnesses
Proposition
K>κ has no maximal models
Corollary
If Γ is a single countable type, then K≥ℵ1 has no maximal models.
Type omission Averageable Classes Torsion Modules
Dividing line
There is actually a stronger dividing line in many cases
Theorem
Let Γ be a finite set of countable existential types and let M be astructure omitting Γ that is Γ-closed. Then, either
(a) every L structure omitting Γ and satisfying the same∃∀-theory as M is isomorphic to M; or
(b) there are extensions of M of all sizes, each satisfying the same∃∀-theory.
In our example, (a) is finite n-torsion for every n.
Type omission Averageable Classes Torsion Modules
Tameness and coheir
In averageable classes, we can redo many elementaryarguments or nonelementary arguments with more completeultraproducts
For instance,
Theorem
Galois types are determined by finite restrictions
Theorem
If K has amalgamation, doesn’t have the weak Galois orderproperty, and every model is ℵ0-saturated, then Galois coheir is astable independence relation
There is a similar theorem for syntactic coheir
Type omission Averageable Classes Torsion Modules
Tameness and coheir
In averageable classes, we can redo many elementaryarguments or nonelementary arguments with more completeultraproducts
For instance,
Theorem
Galois types are determined by finite restrictions
Theorem
If K has amalgamation, doesn’t have the weak Galois orderproperty, and every model is ℵ0-saturated, then Galois coheir is astable independence relation
There is a similar theorem for syntactic coheir
Type omission Averageable Classes Torsion Modules
Good example III: Torsion modules over PIDs
Fix a (usually commutative) ring R
LR = 〈+,−, 0, r ·〉r∈R and TR is the theory of (left) R-modules
tor(x) = {r · x 6= 0 : regular r ∈ R}tor(ΠMi/U) is the torsion submodule of the true ultraproduct
This is a module if the ring is commutative (or at least∀x∀y∃z(xy = zx))
Type omission Averageable Classes Torsion Modules
Torsion modules over PIDs
We’re now focusing on these results applied to torsion modules.
First, we use p. p. elimination of quantifiers to showtor(ΠMi/U) ≺ ΠMi/U
Second, we explore some examples
Third, we explore independence in this nonelementary class
Type omission Averageable Classes Torsion Modules
P. p. elimination of quantifiers
Definition
φ(x) is p.p. iff it is ∃y(Ay + Bx = 0), for A and B appropriatelysized matrices over R.
Fact
If R is a PID, then φ(x) is p.p. iff∧j<m
(∃yj .p
njj · yj = τj(x)
)∧∧j<m′
σj(x) = 0
Essentially diagonal matrices
Fact (Baur)
In a complete theory of modules, every formula is equivalent to aboolean combination of p. p. formula.
Type omission Averageable Classes Torsion Modules
P. p. elimination of quantifiers
Definition
φ(x) is p.p. iff it is ∃y(Ay + Bx = 0), for A and B appropriatelysized matrices over R.
Fact
If R is a PID, then φ(x) is p.p. iff∧j<m
(∃yj .p
njj · yj = τj(x)
)∧∧j<m′
σj(x) = 0
Essentially diagonal matrices
Fact (Baur)
In a complete theory of modules, every formula is equivalent to aboolean combination of p. p. formula.
Type omission Averageable Classes Torsion Modules
Better p. p. elimination of quantifiers
Given p. p. φ(x), ψ(x) and M,
Inv(M, φ, ψ) := |φ(M)/φ(M) ∩ ψ(M)|
An invariants condition is
Inv(M, φ, ψ) ≥ k or Inv(M, φ, ψ) < k
Expressing these is first order.
Fact
Given φ(x), there is a boolean combination of invariants conditionsσ and a boolean combination of p. p. formulas ψ(x) such thatφ(x) and σ ∧ ψ(x) are equivalent modulo the (incomplete) theoryof R-modules.
Type omission Averageable Classes Torsion Modules
Better p. p. elimination of quantifiers
Given p. p. φ(x), ψ(x) and M,
Inv(M, φ, ψ) := |φ(M)/φ(M) ∩ ψ(M)|
An invariants condition is
Inv(M, φ, ψ) ≥ k or Inv(M, φ, ψ) < k
Expressing these is first order.
Fact
Given φ(x), there is a boolean combination of invariants conditionsσ and a boolean combination of p. p. formulas ψ(x) such thatφ(x) and σ ∧ ψ(x) are equivalent modulo the (incomplete) theoryof R-modules.
Type omission Averageable Classes Torsion Modules
Los’ Theorem
We want to show Los’ Theorem holds in this context
Enough to show it for invariants conditions, p.p. formulas,and their negations (negations of invariants conditions areinvariants conditions)
Don’t have this exactly, but good enough
:
Theorem
Let Mi be elementary equivalent torsion modules over a PID R.For any [f0]U , . . . , [fn−1]U ∈ ΠtorMi/U and formula φ(x),
{i ∈ I : Mi � φ (f0(i), . . . , fn−1(i))} ⇐⇒ΠtorMi/U � φ ([f0]U , . . . , [fn−1]U)
Type omission Averageable Classes Torsion Modules
Los’ Theorem
We want to show Los’ Theorem holds in this context
Enough to show it for invariants conditions, p.p. formulas,and their negations (negations of invariants conditions areinvariants conditions)
Don’t have this exactly, but good enough:
Theorem
Let Mi be elementary equivalent torsion modules over a PID R.For any [f0]U , . . . , [fn−1]U ∈ ΠtorMi/U and formula φ(x),
{i ∈ I : Mi � φ (f0(i), . . . , fn−1(i))} ⇐⇒ΠtorMi/U � φ ([f0]U , . . . , [fn−1]U)
Type omission Averageable Classes Torsion Modules
Los’ Theorem for p. p. formulas and their negations
Theorem
Let Mi be elementary equivalent torsion modules over a PID R.For any [f0]U , . . . , [fn−1]U ∈ ΠtorMi/U and any booleancombination of p.p. formulas φ(x),
{i ∈ I : Mi � φ (f0(i), . . . , fn−1(i))} ⇐⇒ΠtorMi/U � φ ([f0]U , . . . , [fn−1]U)
¬φ(x) is universal, so this is known.
Given ∧∃yj .p
njj · yj = τj(x) ∧
∧σj ′(x)
we can calculate the/an order of τj(x) based on the orders ofx and this order annihilates any witness yjIf a sequence with constant (on a U-large set) order is put in,then there’s a witnessing sequence that has constant orderThis completes the proof. Boolean combinations easilyfollows.
Type omission Averageable Classes Torsion Modules
Los’ Theorem for p. p. formulas and their negations
Theorem
Let Mi be elementary equivalent torsion modules over a PID R.For any [f0]U , . . . , [fn−1]U ∈ ΠtorMi/U and any booleancombination of p.p. formulas φ(x),
{i ∈ I : Mi � φ (f0(i), . . . , fn−1(i))} ⇐⇒ΠtorMi/U � φ ([f0]U , . . . , [fn−1]U)
¬φ(x) is universal, so this is known.Given ∧
∃yj .pnjj · yj = τj(x) ∧
∧σj ′(x)
we can calculate the/an order of τj(x) based on the orders ofx and this order annihilates any witness yj
If a sequence with constant (on a U-large set) order is put in,then there’s a witnessing sequence that has constant orderThis completes the proof. Boolean combinations easilyfollows.
Type omission Averageable Classes Torsion Modules
Los’ Theorem for p. p. formulas and their negations
Theorem
Let Mi be elementary equivalent torsion modules over a PID R.For any [f0]U , . . . , [fn−1]U ∈ ΠtorMi/U and any booleancombination of p.p. formulas φ(x),
{i ∈ I : Mi � φ (f0(i), . . . , fn−1(i))} ⇐⇒ΠtorMi/U � φ ([f0]U , . . . , [fn−1]U)
¬φ(x) is universal, so this is known.Given ∧
∃yj .pnjj · yj = τj(x) ∧
∧σj ′(x)
we can calculate the/an order of τj(x) based on the orders ofx and this order annihilates any witness yjIf a sequence with constant (on a U-large set) order is put in,then there’s a witnessing sequence that has constant order
This completes the proof. Boolean combinations easilyfollows.
Type omission Averageable Classes Torsion Modules
Los’ Theorem for p. p. formulas and their negations
Theorem
Let Mi be elementary equivalent torsion modules over a PID R.For any [f0]U , . . . , [fn−1]U ∈ ΠtorMi/U and any booleancombination of p.p. formulas φ(x),
{i ∈ I : Mi � φ (f0(i), . . . , fn−1(i))} ⇐⇒ΠtorMi/U � φ ([f0]U , . . . , [fn−1]U)
¬φ(x) is universal, so this is known.Given ∧
∃yj .pnjj · yj = τj(x) ∧
∧σj ′(x)
we can calculate the/an order of τj(x) based on the orders ofx and this order annihilates any witness yjIf a sequence with constant (on a U-large set) order is put in,then there’s a witnessing sequence that has constant orderThis completes the proof. Boolean combinations easilyfollows.
Type omission Averageable Classes Torsion Modules
Los’ Theorem for invariants conditions
Invariants conditions are first order expressible
Inv(M, φ, ψ) ≥ k ≡“∃v0, . . . , vk−1 (
∧i<k
φ(vi ) ∧∧
j<i<k
¬ψ(vj − vi ))′′
Inv(M, φ, ψ) < k ≡“∀v0, . . . , vk−1 (
∨i<k
¬φ(vi ) ∨∨
j<i<k
ψ(vj − vi ))′′
The first is ∃∀, so transfers up
The second is universal over something that transfers up, sotransfers up
Type omission Averageable Classes Torsion Modules
Los’ Theorem for invariants conditions
Invariants conditions are first order expressible
Inv(M, φ, ψ) ≥ k ≡“∃v0, . . . , vk−1 (
∧i<k
φ(vi ) ∧∧
j<i<k
¬ψ(vj − vi ))′′
Inv(M, φ, ψ) < k ≡“∀v0, . . . , vk−1 (
∨i<k
¬φ(vi ) ∨∨
j<i<k
ψ(vj − vi ))′′
The first is ∃∀, so transfers up
The second is universal over something that transfers up, sotransfers up
Type omission Averageable Classes Torsion Modules
Los’ Theorem for Torsion modules
Theorem
Let Mi be elementary equivalent torsion modules over a PID R.For any [f0]U , . . . , [fn−1]U ∈ ΠtorMi/U and formula φ(x),
{i ∈ I : Mi � φ (f0(i), . . . , fn−1(i))} ⇐⇒ΠtorMi/U � φ ([f0]U , . . . , [fn−1]U)
Unclear if PID or elementary equivalence are really necessary
∃∀-equivalence is enough, but this is equivalent to fullelementary equivalence
Type omission Averageable Classes Torsion Modules
Los’ Theorem for Torsion modules
Theorem
Let Mi be elementary equivalent torsion modules over a PID R.For any [f0]U , . . . , [fn−1]U ∈ ΠtorMi/U and formula φ(x),
{i ∈ I : Mi � φ (f0(i), . . . , fn−1(i))} ⇐⇒ΠtorMi/U � φ ([f0]U , . . . , [fn−1]U)
Unclear if PID or elementary equivalence are really necessary
∃∀-equivalence is enough, but this is equivalent to fullelementary equivalence
Type omission Averageable Classes Torsion Modules
Uninteresting examples
Recall our dividing line. Here, this means that there must be someelement that annihilates infinitely many elements. The followingtorsion groups (and any direct sum of finitely many of them) donot grow via the torsion ultraproduct.
⊕p primeZp
Z(p∞), the Prufer p-group [think all pk roots of unity]
Q/Z
Type omission Averageable Classes Torsion Modules
Interesting example I: ⊕n<ωZ2n
Any 2n annihilates infinitely many elements, so thetor -ultraproduct creates new models
This process gives rise to countable, elementarily equivalent,torsion groups that are not isomorphic to the original group:take the element [f ]U given by
f (i)(n) =
{2i−1 if i = n
0 otherwise
This has order 2, but is divisible by all the powers of two.
Any countable subgroup of ΠtorG/U containing [f ]U is asadvertised
Type omission Averageable Classes Torsion Modules
Interesting example I: ⊕n<ωZ2n
Any 2n annihilates infinitely many elements, so thetor -ultraproduct creates new models
This process gives rise to countable, elementarily equivalent,torsion groups that are not isomorphic to the original group
:take the element [f ]U given by
f (i)(n) =
{2i−1 if i = n
0 otherwise
This has order 2, but is divisible by all the powers of two.
Any countable subgroup of ΠtorG/U containing [f ]U is asadvertised
Type omission Averageable Classes Torsion Modules
Interesting example I: ⊕n<ωZ2n
Any 2n annihilates infinitely many elements, so thetor -ultraproduct creates new models
This process gives rise to countable, elementarily equivalent,torsion groups that are not isomorphic to the original group:take the element [f ]U given by
f (i)(n) =
{2i−1 if i = n
0 otherwise
This has order 2, but is divisible by all the powers of two.
Any countable subgroup of ΠtorG/U containing [f ]U is asadvertised
Type omission Averageable Classes Torsion Modules
Interesting example II: ⊕n<ωZ(p∞)
Any pk annihilates infinitely many elements, so thetor -ultraproduct creates new models
Fact
Every divisible group is isomorphic to a direct sum of copies of Qand Z(q∞).
Thus, every torsion module elementarily equivalent to thisgroup is isomorphic to
⊕i<κZ(p∞)
Thus the nonelementary class is categorical, while theelementary class is not!
Type omission Averageable Classes Torsion Modules
Interesting example II: ⊕n<ωZ(p∞)
Any pk annihilates infinitely many elements, so thetor -ultraproduct creates new models
Fact
Every divisible group is isomorphic to a direct sum of copies of Qand Z(q∞).
Thus, every torsion module elementarily equivalent to thisgroup is isomorphic to
⊕i<κZ(p∞)
Thus the nonelementary class is categorical, while theelementary class is not!
Type omission Averageable Classes Torsion Modules
Interesting example II: ⊕n<ωZ(p∞)
Any pk annihilates infinitely many elements, so thetor -ultraproduct creates new models
Fact
Every divisible group is isomorphic to a direct sum of copies of Qand Z(q∞).
Thus, every torsion module elementarily equivalent to thisgroup is isomorphic to
⊕i<κZ(p∞)
Thus the nonelementary class is categorical, while theelementary class is not!
Type omission Averageable Classes Torsion Modules
Nonforking
Definition
Let M be a torsion module. Then KM is the class of torsionmodules elementarily equivalent to M and ≺ is pure submodule.
KM has amalgamation and joint embedding
Galois types are syntactic
KM has no maximal models or it consists of isomorphic copiesof M (M countable)
KM is stable and coheir is a stable independence relation
Type omission Averageable Classes Torsion Modules
Nonforking
Definition
Let M be a torsion module. Then KM is the class of torsionmodules elementarily equivalent to M and ≺ is pure submodule.
KM has amalgamation and joint embedding
Galois types are syntactic
KM has no maximal models or it consists of isomorphic copiesof M (M countable)
KM is stable and coheir is a stable independence relation
Type omission Averageable Classes Torsion Modules
Future work
Possible extensions:
Apply construction to more contexts
Extend the construction to non-unary types
Extend construction to expanded languages
L(Q)
Does Los’ Theorem hold for modules over just commutativerings?
Type omission Averageable Classes Torsion Modules
Thanks!
Any questions?