Most(?) theories have a Borel complete reductmath.umd.edu/~sbraunf/Laskowski_Slides.pdf · 2020....

Most(?) theories have a Borel complete reduct

Chris LaskowskiUniversity of Maryland

Joint work withDouglas Ulrich

29 October, 2020

Chris Laskowski University of Maryland Joint work with Douglas Ulrich


Teasers:

(Z,+) has a Borel complete reduct.Theory of ‘independent unary predicates’ has a Borel completereduct. [L = {Un(x) : n ∈ ω}, T says every booleancombination is consistent.]

(Sanity check:) Theory of ‘infinite sets’ in language of equalitydoes not have a Borel complete reduct. ACF does not either.



Borel complexity (to be defined later) is a measure of thecomplexity of a theory T in a countable language.

Borel complete means of maximal complexity, i.e., there is a Borelembedding of countable graphs into countable models of T .

Examples: Real closed fields (RCF) and differentially closed fields(DCF0) are Borel complete, but algebraically closed fields (ACF) isnot Borel complete.



Reducts

In model theory, a reduct of an L-structure M is like a ‘forgetfulfunctor’.

Example: (Z,+) is a reduct of (Z,+,×).

Definition

An L0-structure M0 is a reduct of an L-structure M if M0 and Mhave the same universes and every L0-definable subset of M0(without parameters) is L-definable in M (without parameters).

Equivalently, an L0-structure M0 is a reduct of an L-structure M ifthere is an expansion by definitions of M to an L0 ∪ L-structure M∗and M0 is obtained by deleting symbols from M

∗ as above.



Some reducts of (Z,+)

Note that for each n ≥ 1,

Dn = {x ∈ Z : ∃y(x = y + y + · · ·+ y) [n times]}

is definable in (Z,+).

Thus, (Z,Un)n≥1 (with unary predicates Un interpreted as Dn) is areduct of (Z,+).

Moreover, (Z,Up)p prime is a reduct of (Z,Un)n≥1, and hence of(Z,+).

Note: Th(Z,Up)p prime is ‘independent unary predicates’. We willsee that it has a Borel complete reduct.



Reducts (generally) have lower complexity

If M is decidable, then every reduct M0 (to a reasonablelanguage) is decidable. e.g., (Z,+) is a reduct of(Z,+, 0, 1,

Spaces of countable models

Fix L a countable language. Let

StrL = {L-structures M with universe ω}

Topologize: Basic open sets

Uφ(n1,...,nk ) = {M ∈ StrL : M |= φ(n1, . . . , nk)}

StrL is a standard Borel space (separable, completely metrizable ofsize continuum).

For T any L-theory, Mod(T ) = {M ∈ StrL : M |= T} is a Borelsubset.

Sym(ω) acts on StrL by σ.M = the L-structure M′ formed by

permuting ω. Note: (Mod(T ),∼=) is invariant under this action.Chris Laskowski University of Maryland Joint work with Douglas Ulrich


How complicated is (Mod(T ),∼=)?

Friedman-Stanley Given two theories T ,S (possibly in differentcountable languages L, L′) we say (Mod(T ),∼=) is Borel reducibleto (Mod(S),∼=), T ≤B S , if there is a Borel f : StrL → StrL′ suchthat for all M,N ∈ Mod(T ), f (M), f (N) ∈ Mod(S) and

M ∼= N ⇐⇒ f (M) ∼= f (N)

There is a maximal ≡B -class, containing graphs, linear orders,RCF, DCF0. A theory T is Borel complete if (Mod(T ),∼=) is inthis maximal class.



Families of equivalence relations

Let L = {En : n ∈ ω}, T asserts each En is an equivalence relation.

Old: [Ulrich-Rast-L] REF(bin)=‘binary splitting, refiningequivalence relations’ is not Borel complete, but the isomorphismrelation ⊆ Mod(T )×Mod(T ) is not Borel.

Theorem (L-Ulrich)

Let h : ω → ω \ {0} be any strictly increasing function. Let Th bethe L-theory asserting that:

Each En is an equivalence relation with h(n) classes;

The En’s cross-cut i.e., for any finite F ⊆ ω,EF (x , y) :=

∧n∈F En(x , y) has Πn∈Fh(n) classes.

Then Th is Borel complete.

Pf: www.math.umd.edu/∼laskow/Crosscutting.pdfChris Laskowski University of Maryland Joint work with Douglas Ulrich


(Z,+) has a Borel complete reduct

Corollary

(Z,Up)p prime (and hence (Z,+)) has a Borel complete reduct.

Pf: Partition Primes=⊔{Fn : n ∈ ω}, where each Fn is finite, but

|Fn+1| > |Fn| for all n. For each n, let

En(x , y) :=∧p∈Fn

Up(x)↔ Up(y)

So (Z,En)n∈ω is a reduct of (Z,Up)p prime (and hence of (Z,+)).Each En is an equivalence relation with 2

|Fn| classes, and the family{En} is cross-cutting. By Theorem, (Z,En)n∈ω is Borel complete.



Extensions

Theorem

Let M be any structure (in a countable language) withuncountably many 1-types consistent with Th(M). Then M has aBorel complete reduct.

Corollary

Suppose Th(M) is not small (i.e., for some n ≥ 1, there areuncountably many n-types). Then Meq has a Borel completereduct.

Corollary

If Th(M) is not ω-stable, then the elementary diagram ElDiag(M)has a Borel complete reduct.



Further extensions

The previous results are not characterizations:

REF(bin) is small and not Borel complete, but has a Borelcomplete reduct. [Build a tree of strong types.]

There are extremely nice 2-dimensional, ω-stable structures Mthat have Borel complete reducts. [Start with the ‘standardeni-DOP example’ M0. Form an expansion M by adding afamily of commuting functions enforcing that any two boxeshave the same cardinality.]

But... if M is ω-stable of Morley rank 1 [e.g., (C,+,×)] thenevery reduct has only countably many countable models(hence is not Borel complete).



λ-Borel sets and functions

Not all Borel complete theories have the same complexity!

Fix L a countable language and λ an infinite cardinal. Let

StrλL = {L-structures M with universe λ}

Topologize: Basic open sets

Uφ(α1,...,αk ) = {M ∈ StrλL : M |= φ(α1, . . . , αk)}(α1, . . . , αk) ∈ λk

Call a set X λ-Borel if it is in the (≤ λ)-algebra generated by theopen sets, and a function f : X → Y λ-Borel if the inverse imageof every basic open set is λ-Borel.



λ-Borel reducibility

Given two theories T , S (possibly in different countable languagesL, L′) we say T is λ-Borel reducible to S if there is a λ-Borelf : StrλL → StrλL′ such that for all M,N |= T with universe λ,f (M), f (N) |= S and

M ≡λ+,ω N ⇐⇒ f (M) ≡λ+,ω f (N)

There is a maximal ≡λB -class, containing graphs, linear orders,RCF, DCF0. A theory T is λ-Borel complete if T is in thismaximal class.



Borel complete does not imply λ-Borel complete for large λ

Fact: If T is weakly minimal, then T is not λ-Borel complete forλ ≥ i2. [Every model N is back-and-forth equivalent to asubmodel of size at most 2ℵ0 .]

Corollary

Suppose h : ω → ω \ {0} is strictly increasing and Th is the theoryof cross-cutting equivalence relations where En has h(n) classes.Then Th is Borel complete, but not λ-Borel complete for λ ≥ i2.

Note: A more complicated example of a Borel complete T that isnot λ-Borel complete for large λ appears in [Ulrich-Rast-L].



Classifying cross-cutting equivalence relations

Let L = {En : n ∈ ω} and let h : ω → (ω \ {0, 1}) ∪ {∞}.Th := Theory of cross-cutting equivalence relations, where En hash(n) classes.Model theory facts:

Every Th admits QE and is stable.

If h(n)

Three test questions for Th

A. Isomorphism on Mod(Th)×Mod(Th) is not Borel?

⇑

B. Th is Borel complete?

⇑

C. Th is λ-Borel complete for all infinite λ?



All h(n)

h(n) =∞ for exactly one n.

Case: h(0) =∞, but h(n) 0. [Th is superstable ofR∞-rank 2.]

Theorem

1 Iso on Mod(Th) is not Borel iff h(n) > 2 for infinitely many n.

2 Th is Borel complete iff {h(n) : n > 0} is unbounded.3 Th is not λ-Borel complete for large λ.

Exactly the same result!



h(n) =∞ for at least two n.

Case: h(n) =∞ for at least two n’s.

Theorem

Th is λ-Borel complete for all λ.

Pf: Use two relations with infinitely many classes to code eni-DOP.



Thanks for listening!

Proof of main theorem atwww.math.umd.edu/∼laskow/Crosscutting.pdf



Most(?) theories have a Borel complete reductmath.umd.edu/~sbraunf/Laskowski_Slides.pdf · 2020....

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