will boney university of illinois at chicago january...

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


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.


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


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


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.


Outline

Discuss the type omitting hull and properties that lead to itbeing well-behaved

Compactness results and ultraproducts

Averageable classes

Examples

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}


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).


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.


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.


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


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


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)


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


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)).


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




Proposition



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




Proposition



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


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 .


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.


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.


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)).


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.


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!


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


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


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


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


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


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


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.


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.


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.


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


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))


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


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.


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.


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)


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




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),


¬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.



Theorem



¬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.



Theorem





∧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.



Theorem





∧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.


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


Los’ Theorem for Torsion modules

Theorem



Unclear if PID or elementary equivalence are really necessary

∃∀-equivalence is enough, but this is equivalent to fullelementary equivalence


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


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




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 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




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!


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


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?


Thanks!

Any questions?

will boney university of illinois at chicago january...

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