Some compactness in some nonelementary classes Will Boney - - PowerPoint PPT Presentation

Type omission Averageable Classes Torsion Modules

Some compactness in some nonelementary classes

Will Boney University of Illinois at Chicago January 9, 2015 Beyond 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 compactness in some nonelementary contexts. This allows us to develop some nonforking notions, and we specialize to the example of torsion modules 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

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 elements from G: tor(G) := {g ∈ G : ∃n < ω.n · g = 0}

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 elements from 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 the

• rder of g + h

Type omission Averageable Classes Torsion Modules

Prototype II: Archimedean fields

Have a nice elementary (model) theory of ordered fields of characteristic 0 Archimedean fields are not first order: ∀x

• n<ω

1 + · · · + 1 > n > −1 − · · · − 1

Type omission Averageable Classes Torsion Modules

Prototype II: Archimedean fields

Have a nice elementary (model) theory of ordered fields of characteristic 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, finite elements 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 1

g , f + g, and fg

are standard

Type omission Averageable Classes Torsion Modules

Similarities

There are two key similarities here that will guide us in abstracting these situations: The types were unary Where elements omit types lets me figure out where functions

• f them omit types

Type omission Averageable Classes Torsion Modules

Similarities

There are two key similarities here that will guide us in abstracting these situations: The types were unary Where elements omit types lets me figure out where functions

• f them omit types

I’m probably going to often use phrases like “where type

• mission happens.” Each type is going to have a natural index

(as we’ve seen) and the “location” of type omission is that index.

Type omission Averageable Classes Torsion Modules

Outline

Discuss the type omitting hull and properties that lead to it being 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-types For ease we enumerate Γ as follows:

Γ = {pj(x) : j < α} pj(x) = {φj

k(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 ¬φj

kj(m)}

Γ(M) contains all elements of M that omit all types of Γ according to 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 ¬φj

kj(m)}

Γ(M) contains all elements of M that omit all types of Γ according to M. Each element has a (possibly many) witnesses to its inclusion. Namely m ∈ Γ(M) iff there is some k(m) ∈ Πβj such that M ¬φj

k(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 that already omit Γ

They probably also model a common theory T

U is an ultrafilter on I Then we can form Γ(ΠMi/U) , which will omit all of the types of Γ and give enough averaging to get some compactness results...

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 that already omit Γ

They probably also model a common theory T

U is an ultrafilter on I Then we can form Γ(ΠMi/U) , which will omit all of the types of Γ and give enough averaging to get some compactness results...sometimes.

Type omission Averageable Classes Torsion Modules

Problems with Γ(M)

This construction turns out to be very fragile

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 elementary substructure

This means, depending on the types, it might not even omit all

• f the types of Γ

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 elementary substructure

This means, depending on the types, it might not even omit all

• f the types of Γ

The plan is to give examples of where this can go wrong, and then 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 = 0) : k < ω}

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 = 0) : k < ω} [n → 1]U ∈ Γ(ΠM/U) [n → 2n − 1]U ∈ Γ(ΠM/U) [n → 2n]U ∈ 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 natural numbers

M = N, N′; +, ×, 1; +′, ×′, 1′; ×∗ I = ω, U is any non principle ultrafilter p(x) = {N2(x) ∧ (1 + · · · + 1 = x) : n < ω} Two copies of N linked by multiplication

Type omission Averageable Classes Torsion Modules

Bad example II: some pathology with standard natural numbers

M = N, N′; +, ×, 1; +′, ×′, 1′; ×∗ I = ω, U is any non principle ultrafilter p(x) = {N2(x) ∧ (1 + · · · + 1 = 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 → 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 omitting the type of an infinite element Then, the type of infinitesimal elements and two elements infinitely close to each other are omitted by the field axioms However, using the Γ(F) construction, we would not get a substructure if Γ is just the type of an infinite element Instead, Γ has to list each type of a nonstandard element around a standard real This example shows that the unary part is crucial and can’t be avoided 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 of Γ, there is some kj < βj for each j < α such that M ¬φj

kj (F(m0, . . . , 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 of Γ, there is some kj < βj for each j < α such that M ¬φj

kj (F(m0, . . . , mn−1))

A stronger condition is often useful in applications. It imposes some uniformity on where F of a tuple omits the types based

• n where the tuple omits those types

Definition M is Γ-closed iff for all j < α and F ∈ L, there is some gj

F :

• Πβj′n

→ βj such that, for all m0, . . . , mn−1 ∈ Γ(M), M ¬φj

k′ (F(m0, . . . , mn−2))

where k′ = gj

F (k(m0), . . . , k(mn−1)).

Type omission Averageable Classes Torsion Modules

Universal Lo´ s’ Theorem

Peaking ahead to ultraproducts, we have the following result. Proposition Suppose Γ(ΠMi/U) is a structure. If φ(x) is a universal formula and [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)

Type omission Averageable Classes Torsion Modules

Universal Lo´ s’ Theorem

Peaking ahead to ultraproducts, we have the following result. Proposition Suppose Γ(ΠMi/U) is a structure. If φ(x) is a universal formula and [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 transfer

• downwards. The

Lo´ s’ Theorem gives the result. †

Type omission Averageable Classes Torsion Modules

Universal Lo´ s’ Theorem

Peaking ahead to ultraproducts, we have the following result. Proposition Suppose Γ(ΠMi/U) is a structure. If φ(x) is a universal formula and [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 transfer

• downwards. The

Lo´ s’ Theorem gives the result. † The normal proof by induction works as well, which is useful to show that a formula transferring up is closed under conjunction, disjunction, and universal quantification

Type omission Averageable Classes Torsion Modules

Corollaries of Universal Lo´ s’

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)

• mits pj.

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A little more: ∃∀ sentences

Proposition Suppose Γ is finite Γ(ΠMi/U) is a structure T∃(∀∧¬Γ) is a complete theory of the indicated quantifier complexity that is modeled by each Mi Then Γ(ΠMi/U) T∃∀.

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 quantifier complexity that is modeled by each Mi Then Γ(ΠMi/U) T∃∀. Proof: Given a ∃xψ(x), we can form ∃x  ψ(x) ∧

• j<α;ℓ<n

¬φj

kj

ℓ

(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 the theory.

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 gj

∃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 ¬φj

k′ (F(m0, . . . , mn−1)) where

k′ = gj

F (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) true in M have witnesses in Γ(M) and their type omission can be calculated from the type omission of the parameters.

Type omission Averageable Classes Torsion Modules

Γ-nice

Definition M is Γ-nice if existential formulas with parameters from Γ(M) true in M have witnesses in Γ(M) and their type omission can be calculated 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 Lo´ s’ Theorem holds.

Type omission Averageable Classes Torsion Modules

Quantifier elimination

A (so far) more useful criteria comes from quantifier elimination 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; and Γ(M) is a ∆-elementary substructure of M

then Γ(M) ≺ M.

Type omission Averageable Classes Torsion Modules

Quantifier elimination

A (so far) more useful criteria comes from quantifier elimination 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; and Γ(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 and p(x) = {x ≤ cn or cm ≤ x : n < m ∈ Z}. A model of DLOGZ is a model of T that omits p, i.e. a dense, linearly ordered group where {cn : n ∈ Z} is a discrete, cofinal sequence.

Type omission Averageable Classes Torsion Modules

Good example I: DLOGZ

Set T := Th(Q, <, +, −, 0, 1, cn)n∈Z and p(x) = {x ≤ cn or cm ≤ x : n < m ∈ Z}. A model of DLOGZ is a model of T that omits p, i.e. a dense, 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 that Γ(M) T (can compute the bounds from the bounds of the inputs)

Type omission Averageable Classes Torsion Modules

Good example I: DLOGZ

Set T := Th(Q, <, +, −, 0, 1, cn)n∈Z and p(x) = {x ≤ cn or cm ≤ x : n < m ∈ Z}. A model of DLOGZ is a model of T that omits p, i.e. a dense, 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 that Γ(M) T (can compute the bounds from the bounds of the inputs) Then Γ(M) ≺ M Also, (EC(T, p), ≺) has amalgamation, joint embedding, and syntactic 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 and the ordered field structure of R, with maps between them of scalar multiplication and norm and a constant for each element of R Let 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 = cr) ∧ (cr− 1

n < x < cr+ 1 n ) : n < ω} for

r ∈ R; q∞(x) = {B(x) ∧ (x < −n ∨ n < x) : n < ω}; and qr(x) = {B(x) ∧ (x = cr) ∧ (cr− 1

n < x < cr+ 1 n ) : n < ω}.

Then universal formulas transfer (although existentials require more work) and we get something like the Banach space ultraproduct

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 where M → Γ(M) ∈ K is well-behaved according to ≺

Enough for this discussion if {Mi ∈ K : i ∈ I} → Γ(ΠMi/U) ∈ K satisfies enough of Lo´ s’ Theorem

The models of K are models of a first order theory T that

• mit Γ and ≺ is elementary according to some good notion of

substructure The key fact is that, while we don’t literally have ultraproducts (nonelementary class), we almost do and almost is enough for compactness results

Type omission Averageable Classes Torsion Modules

Good example III−: Abelian torsion groups

Fix a complete theory T of torsion groups

• f infinite exponent; and

that 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 Lo´ s’ Theorem, although we 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 have Γ(M) ≺ M. Question When do we have M Γ(ΠM/U)? ≺ follows from coherence/Tarski-Vaught test, so the real question 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 all

• f Γ at the same place

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 all

• f Γ at the same place

If infinite X ⊂ M has: for all p ∈ Γ, there is φp ∈ p so for all x ∈ X M ¬φp(x) then any function picking out distinct elements of X will be new.

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 a structure 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 elementary arguments or nonelementary arguments with more complete ultraproducts

Type omission Averageable Classes Torsion Modules

Tameness and coheir

In averageable classes, we can redo many elementary arguments or nonelementary arguments with more complete ultraproducts For instance, Theorem Galois types are determined by finite restrictions Theorem If K has amalgamation, doesn’t have the weak Galois order property, and every model is ℵ0-saturated, then Galois coheir is a stable 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 = 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 show tor(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 appropriately sized matrices over R. Fact If R is a PID, then φ(x) is p.p. iff

• j<m
• ∃yj.pnj

j · yj = τj(x)

• ∧
• j<m′

σj(x) = 0 Essentially diagonal matrices

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 appropriately sized matrices over R. Fact If R is a PID, then φ(x) is p.p. iff

• j<m
• ∃yj.pnj

j · yj = τj(x)

• ∧
• j<m′

σj(x) = 0 Essentially diagonal matrices Fact (Baur) In a complete theory of modules, every formula is equivalent to a boolean 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.

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 conditions σ and a boolean combination of p. p. formulas ψ(x) such that φ(x) and σ ∧ ψ(x) are equivalent modulo the (incomplete) theory

• f R-modules.

Type omission Averageable Classes Torsion Modules

• Lo´

s’ Theorem

We want to show Lo´ s’ Theorem holds in this context Enough to show it for invariants conditions, p.p. formulas, and their negations (negations of invariants conditions are invariants conditions) Don’t have this exactly, but good enough

Type omission Averageable Classes Torsion Modules

• Lo´

s’ Theorem

We want to show Lo´ s’ Theorem holds in this context Enough to show it for invariants conditions, p.p. formulas, and their negations (negations of invariants conditions are invariants 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

• Lo´

s’ 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 boolean combination 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.

Type omission Averageable Classes Torsion Modules

• Lo´

s’ 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 boolean combination 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.pnj

j · yj = τj(x) ∧

• σj′(x)

we can calculate the/an order of τj(x) based on the orders of x and this order annihilates any witness yj

Type omission Averageable Classes Torsion Modules

• Lo´

s’ 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 boolean combination 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.pnj

j · yj = τj(x) ∧

• σj′(x)

we can calculate the/an order of τj(x) based on the orders of x 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 order

Type omission Averageable Classes Torsion Modules

• Lo´

s’ 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 boolean combination 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.pnj

j · yj = τj(x) ∧

• σj′(x)

we can calculate the/an order of τj(x) based on the orders of x 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 order This completes the proof. Boolean combinations easily

Type omission Averageable Classes Torsion Modules

• Lo´

s’ 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))′′

Type omission Averageable Classes Torsion Modules

• Lo´

s’ 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, so transfers up

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• Lo´

s’ 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)

Type omission Averageable Classes Torsion Modules

• Lo´

s’ 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 full elementary equivalence

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Uninteresting examples

Recall our dividing line. Here, this means that there must be some element that annihilates infinitely many elements. The following torsion groups (and any direct sum of finitely many of them) do not grow via the torsion ultraproduct. ⊕p primeZp Z(p∞), the Prufer p-group [think all pk roots of unity] Q/Z

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Interesting example I: ⊕n<ωZ2n

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Interesting example I: ⊕n<ωZ2n

Any 2n annihilates infinitely many elements, so the tor-ultraproduct creates new models This process gives rise to countable, elementarily equivalent, torsion groups that are not isomorphic to the original group

Type omission Averageable Classes Torsion Modules

Interesting example I: ⊕n<ωZ2n

Any 2n annihilates infinitely many elements, so the tor-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

• therwise

This has order 2, but is divisible by all the powers of two. Any countable subgroup of ΠtorG/U containing [f ]U is as advertised

Type omission Averageable Classes Torsion Modules

Interesting example II: ⊕n<ωZ(p∞)

Type omission Averageable Classes Torsion Modules

Interesting example II: ⊕n<ωZ(p∞)

Any pk annihilates infinitely many elements, so the tor-ultraproduct creates new models

Type omission Averageable Classes Torsion Modules

Interesting example II: ⊕n<ωZ(p∞)

Any pk annihilates infinitely many elements, so the tor-ultraproduct creates new models Fact Every divisible group is isomorphic to a direct sum of copies of Q and Z(q∞). Thus, every torsion module elementarily equivalent to this group is isomorphic to ⊕i<κZ(p∞) Thus the nonelementary class is categorical, while the elementary class is not!

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Nonforking

Definition Let M be a torsion module. Then KM is the class of torsion modules elementarily equivalent to M and ≺ is pure submodule.

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Nonforking

Definition Let M be a torsion module. Then KM is the class of torsion modules 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 copies

• f 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 Lo´ s’ Theorem hold for modules over just commutative rings?

Type omission Averageable Classes Torsion Modules

Thanks!

Any questions?