bases and generating sets in computable algebraamelniko/melnikovtalk2015.pdf · novosibirsk, may...

Bases and generating sets in computablealgebra

Alexander Melnikov

Novosibirsk, May 2015.

Alexander Melnikov Bases and generating sets in computable algebra

A class of algebraic structures often admits a natural notion of abasis.

When a computable structure has a computable basis?

Does independence have applications in computable algebra?

How a good choice of a generating set can simplify a proof?

Preliminaries.A meta-theorem and its applications (with Montalban and

Harrison-Trainor).

Tree-bases in abelian p-groups (with Downey and Ng).

Integral domains and Σ02-trees (with Greenberg).

Root-bases of torsion-free abelian groups.

Harrison-Trainor).

Preliminaries

Definition (Mal’cev, Rabin)

A countably infinite structure A = (A, f1, . . . , fk ) is computable orconstructive if there exists an isomorphic copy B = (N,g1, . . . ,gk )of A in which g1, . . . ,gk are Turing computable functions.

Such a B is called a constructivization, acomputable presentation, or a computable copy of A.

Example 1Each “recursively presented” group (Higman) with solvableword problem has a computable copy (e.g., any finitely presented simple group,finitely generated abelian group, etc.).

Example 2Every “explicit presentation” of a field (van der Waerden) canbe turned into a constructivization.

Preliminaries

Suppose A admits a natural notion of independence.(Say, A could be a vector space or a field.)

QuestionDoes every computable copy of A have a computable basis?

Example (Mal’cev 1961)

The Q-vector space V∞ =⊕

i∈ω Q of infinite dimension has acomputable copy with no computable basis.

He actually looked at the additive group of V∞ which is a divisible torsion-free abelian

group. It does not make any difference.

Preliminaries

Clearly, V∞ also has a “nice” computable copy with acomputable basis.

Corollary (Mal′cev)V∞ has two computable copies that are

not computably isomorphic.

Corollary (Goncharov)In fact, V∞ has infinitely many computable copies that are pairwisenot computably isomorphic.

Preliminaries

Clearly, V∞ also has a “nice” computable copy with acomputable basis.

Corollary (Mal′cev)V∞ has two computable copies that are

not computably isomorphic.

Corollary (Goncharov)In fact, V∞ has infinitely many computable copies that are pairwisenot computably isomorphic.

Preliminaries

What if A is not a vector space?

QuestionDoes A have a “good” computable copy with a computable basis?

This may be tricky. For example:

Theorem (Dobrica, 1982)Every computable torsion-free abelian group has a computable copywith a computable basis.

Preliminaries

Some recent results address the following related problem:

QuestionGiven a computable copy of A, how complex a basis of A can be?

We may use the Arithmetical hierarchy to get a precise answer.

Theorem (CHKLMMSW 2012, MW 2012)

Every computable copy of the free group F∞ has a Π02 basis, and this

cannot be improved to Σ02.

We may also use the Turing jump operator.

Theorem (Downey and M. 2014)Every computable copy of a completely decomposable group has a0(4)-computable full decomposition basis, and this is sharp.

We will discuss more results later.Alexander Melnikov Bases and generating sets in computable algebra

Preliminaries

A meta-theorem and itsapplications

A framework

Often we have both a “good” and a “bad” copy.

Theorem (Mal’cev 1961 for divisible, Dobrica 1982 andNurtazin 1978 for arbitrary)Every computable torsion-free abelian group A of infinite Z-rank hastwo computable copies B and G such that:

B has no computable Z-base;

G has a computable Z-base;

B ∼=∆02

Corollary (Goncharov)Such an A has infinitely many computable copies, up to computableisomorphism (A has “infinite auto-dimension”).

A framework

Often we have both a “good” and a “bad” copy.

Theorem (Mal’cev 1961 for divisible, Dobrica 1982 andNurtazin 1978 for arbitrary)Every computable torsion-free abelian group A of infinite Z-rank hastwo computable copies B and G such that:

B has no computable Z-base;

B ∼=∆02

Corollary (Goncharov)Such an A has infinitely many computable copies, up to computableisomorphism (A has “infinite auto-dimension”).

A framework

It takes more work to establish:

Theorem (Essentially Goncharov, Lempp, Solomon 2003)Every computable Archimedean ordered abelian group A of infiniteZ-rank has two computable copies B and G such that:

B has no computable Z-base (this part is actually new);

B ∼=∆02

Corollary (G.L.S.)Such an A has infinite auto-dimension.

A framework

It takes more work to establish:

Theorem (Essentially Goncharov, Lempp, Solomon 2003)Every computable Archimedean ordered abelian group A of infiniteZ-rank has two computable copies B and G such that:

B has no computable Z-base (this part is actually new);

B ∼=∆02

Corollary (G.L.S.)Such an A has infinite auto-dimension.

A framework

DefinitionWe say that a class K of countable structures with the associatednotion of independence has

the Mal′cev property

if for every A ∈ K of infinite dimension there exist computablepresentations B and G of A such that:

B has no computable base;

G has a computable base;

B ∼=∆02

Note that A has infinite auto-dimension (Goncharov 1980).

A framework

DefinitionWe say that a class K of countable structures with the associatednotion of independence has

the Mal′cev property

if for every A ∈ K of infinite dimension there exist computablepresentations B and G of A such that:

B has no computable base;

G has a computable base;

B ∼=∆02

Note that A has infinite auto-dimension (Goncharov 1980).

A framework

QuestionWhich classes have the Mal′cev property?

We want a sufficiently general and convenient localcondition that implies the Mal’cev property.

A framework

QuestionWhich classes have the Mal′cev property?

We want a sufficiently general and convenient localcondition that implies the Mal’cev property.

A framework

We proved a general meta-theorem (to be stated).

We then applied the meta-theorem to get:

Theorem (Harrison-Trainor, M., and Montalban 2014)

The following classes have the Mal′cev property:

Torsion-free abelian groups (after Mal′cev, Nurtazin, Dobrica).

Archimedean ordered abelian groups (after Goncharov, Lempp, Solomon).

Real-closed fields (new).

Differentially closed fields with δ-independence (new).

Difference closed fields with transformal independence (new).

CorollaryIn all these classes, structures of infinite dimension have infinitelymany computable presentations, up to computable isomorphism.

A framework

Pregeometries

A suitable abstraction for independence is a pregeometry.

- A pregeometry is a “span” or a “closure” operator cl : 2A → 2A satisfyingseveral natural conditions, e.g. cl cl(X) = cl(X).

- See Downey and Remmel [The Handbook of Rec. Math.] for a survey on

computable pregeometries.

We will be in a more general situation:

DefinitionA pregeometry on a computable A is r.i.c.e. if the relationsx ∈ cl({y1, . . . , yn}) are (effectively uniformly) isolated bycomputable infinitary Σ1-formulae with parameters y1, . . . , yn.

- Natural pregeometries tend to be r.i.c.e. (e.g., linear span, algebraic closure etc.).- A pregeometry is computable iff there is a computable “basis”.

Thus, all results from [Downey and Remmel] hold on a “good” copy G.

Pregeometries

The local conditions

Suppose (A, cl) is a r.i.c.e. pregeometry upon a computable structure A.

Condition G: Uniformly in c ∈ A, we can effectively list theexistential formulas ϕ(c, x) which have a solution a independentover c.

Condition B: The existential types of independent elements in A arenon-principal.

(In other words: For any existential formula ϕ(c, x) holding of any element a which is independent over c, there is

an element b which satisfies ϕ(c, x) and is dependent over c.)

The local conditions

Suppose (A, cl) is a r.i.c.e. pregeometry upon a computable structure A.

Condition G: Uniformly in c ∈ A, we can effectively list theexistential formulas ϕ(c, x) which have a solution a independentover c.

Condition B: The existential types of independent elements in A arenon-principal.

(In other words: For any existential formula ϕ(c, x) holding of any element a which is independent over c, there is

an element b which satisfies ϕ(c, x) and is dependent over c.)

The metatheorem

The metatheorem:

Theorem (Harrison-Trainor, M., Montalban 2014)

Let A be a computable structure and (A, cl) a r.i.c.e. pregeometry ofinfinite rank that satisfies Condition G and Condition B. Then (A, cl)has the Mal′cev property.

The proof uses new technical ideas such as ‘safe extensions’ that wereunnecessary in the previously known examples.

CorollaryA has infinitely many computable copies, up to computableisomorphism.

The metatheorem

The metatheorem:

The metatheorem

The metatheorem:

Applications

The metatheorem covers all the previously known classes but with nicerproofs (we separated algebra from combinatorics!).

Corollary

The following classes have the Mal′cev property.

The trivial examples (vector spaces, ACF0, etc.).

Torsion-free abelian groups (we use Rado’s Lemma).

Archimedean ordered abelian groups (we use o-minimality of R).

In fact, the third application is a slight extension of GLS 2003.

Applications

The metatheorem covers all the previously known classes but with nicerproofs (we separated algebra from combinatorics!).

Corollary

The following classes have the Mal′cev property.

The trivial examples (vector spaces, ACF0, etc.).

Torsion-free abelian groups (we use Rado’s Lemma).

Archimedean ordered abelian groups (we use o-minimality of R).

In fact, the third application is a slight extension of GLS 2003.

Applications

The new applications include:

Real-closed fields are existentially closed ordered fields.- Model theory of RCF goes back to Tarski.- Recent effective algebra by the Notre Dame logic group.- Our proof uses cell decomposition (Knight, Pillay, Steinhorn, 1986, 1988).

Differencially closed fields are existentially closeddifferential fields.

- Differential fields are fields with a differential operator δ:

δ(xy) = δ(x)y + xδ(y).

- The natural notion of independence is δ-independence.- Much work in algebra (e.g. Kaplanski), model theory (e.g. Blum, Marker).- Recent results in effective algebra (Marker and Miller).

Difference closed fields are existentially closed differencefields.

- Difference fields are fields with a distinguished automorphism.- Model theory by Chatzidakis and Hrushovski (1999), Shelah and others.- The natural notion of independence is transformal independence.

Applications

QuestionDo existentially closed valued fields have the Mal′cev property?

Question (Goncharov-Lempp-Solomon 2003)

Do non-Archimedean ordered abelian groups have the Mal′cevproperty?

The case of infinitely many Archimedean classes has been anopen problem for over 10 years.

QuestionDo existentially closed valued fields have the Mal′cev property?

Question (Goncharov-Lempp-Solomon 2003)

Do non-Archimedean ordered abelian groups have the Mal′cevproperty?

The case of infinitely many Archimedean classes has been anopen problem for over 10 years.

Tree-bases in abelian p-groups

Not every notion of a basis corresponds to a pregeometry.

In the 1950’s and the 1960’s people wondered whethercountable abelian p-groups admit a good notion of a “basis”.

The most well-known attempt is probably Kulikov’s p-basis.However, this approach has certain limitations.

In effective algebra, we often use the notion of a p-basic tree.(L. Rogers (1977) based on Crawley and Hales (1969)).

p-Basic trees

ExampleThe labeled tree

c1 c2 c3 d

corresponds to representation (among additive abelian groups)

〈a,b, c1, c2, c3,d |pa = pb = 0,pc1 = pc2 = pc3 = a,pd = b〉

of the groupZp ⊕ Zp ⊕ Zp2 ⊕ Zp2 .

p-Basic trees

Let A be an abelian p-group.

Definition (L. Rogers)A p-basic tree of A is the collection of elements B of A with theproperties:

1 Every a ∈ A can be uniquely expressed as

a =∑b∈B

where each kb ∈ Zp;2 For every b ∈ B, either pb ∈ B or pb = 0.

Computable p-groups

Theorem (L. Rogers)Every countable abelian p-group admits a p-basic tree.

Question (Ash, Knight and Oates, ∼1990)Does every computable reduced abelian p-group have a copy with acomputable p-basic tree?

- This is directly related to the the classification problem for such groups.

- The answer is known to be YES is the Ulm type is finite (to beexplained).

Computable p-groups

Abelian p-groups

For an abelian p-group A, define its “derivative”

A′ = {a : pk |a for all k }.

Theorem (Prufer)

A1 = A/A′ splits into a direct sum of cyclic groups.

Iterate this process:A0 = A,

A(k+1) = (A(k))′,

A(ω) =⋂

The Ulm factors Aβ = A(β)/A(β+1) split into cyclic summands.

Abelian p-groups

Theorem (Prufer)

A(k+1) = (A(k))′,

A(ω) =⋂

Abelian p-groups

Theorem (Prufer)

A(k+1) = (A(k))′,

A(ω) =⋂

Abelian p-groups

DefinitionA countable abelian p-group is reduced if there is a (countable) βsuch that

Aβ = 0.

The least such β is called the Ulm type of A, written U(A).

Theorem (Ulm)

Let β = U(A). Then the isomorphism types of the Ulm factors Aγ ,γ < β, completely describe the isomorphism type of A.

Abelian p-groups

DefinitionA countable abelian p-group is reduced if there is a (countable) βsuch that

Aβ = 0.

The least such β is called the Ulm type of A, written U(A).

Theorem (Ulm)

Let β = U(A). Then the isomorphism types of the Ulm factors Aγ ,γ < β, completely describe the isomorphism type of A.

Suppose A is a computable reduced abelian p-group

Theorem (Ash-Knight-Oates, ∼ 1990)

If U(A) = n < ω, then A has a computable copy with a computablep-basic tree.

- The only known proof is non-uniform.

- We use a certain effective invariant of Ak , for each k ≤ n = U(A).

- These invariants are limitwise monotonic functions (l.m.f.).

Question (Ash-Knight-Oates, ∼ 1990)

What about U(A) ≥ ω?

Computable p-groups

If the l.m.f. were uniform, U(A) = ω would be easy (known).

The upper bound for

“e is an index of a ∆02n l.m.f. of An+1”

is “uniformly Π02n+3”.

Theorem (Downey, M., Ng)The upper bound above cannot be improved.

- Thus, solving the problem using any known method seems hopeless.

- The result also explains why the classification problem for such groupsis still open (l.m.f. are unavoidable).

- Our proof uses elements of the 0′′′ technique.

Computable p-groups

The upper bound for

Computable p-groups

The upper bound for

Computable p-groups

The upper bound for

Directed systems of trees, andintegral domains

Proper divisibility

Our main goal is to understand the algorithmic nature of properdivisibility in integral domains (commutative rings with 1 and no zero divisors).

DefinitionWe say that a properly divides b (written a||b) in an integral domain< if b = xa and x does not have an inverse.

- Proper divisibility plays the central role in the study of domains that are‘close‘ to UFD’s

- Virtually nothing was known about proper divisibility in computableintegral domains.

Proper divisibility

Our main goal is to understand the algorithmic nature of properdivisibility in integral domains (commutative rings with 1 and no zero divisors).

DefinitionWe say that a properly divides b (written a||b) in an integral domain< if b = xa and x does not have an inverse.

- Proper divisibility plays the central role in the study of domains that are‘close‘ to UFD’s

- Virtually nothing was known about proper divisibility in computableintegral domains.

Atomic domains

Let < be an integral domain.

Theorem (Well-known)If proper divisibility in < is well-founded, then every element of < isa product of irreducible elements. (An element a is irreducible if b||a implies b is invertible.)

An integral domain satisfying the conclusion of the Theorem iscalled atomic.

Atomic domains

Let < be an integral domain.

Theorem (Well-known)If proper divisibility in < is well-founded, then every element of < isa product of irreducible elements. (An element a is irreducible if b||a implies b is invertible.)

An integral domain satisfying the conclusion of the Theorem iscalled atomic.

Chains of proper divisions

ObservationIf < computable and is not atomic, then 0′′ can compute an finitechain of proper divisions in <.

QuestionIs 0′′ sharp?

Theorem (Greenberg, M.)YES.

There exists a computable integral domain that is not atomic but 0′ cannot compute

an infinite chain of proper divisions.

How is this related to bases/generating sets?

Proof idea

We use generating sets that naturally correspond to Σ02-trees:

Example

Suppose we start with Q[r ].1 Factorize r = ab, where a,b are fresh.2 Factorize a = xy , where x , y are fresh.

a b ⇒ a = xy b

r = ab r

Figure: Factorization of a into xy .Alexander Melnikov Bases and generating sets in computable algebra

Proof idea

ExampleMake y invertible:

a b ⇒ a ∼ x b

Figure: Inverting y . Here “a ∼ x” indicates that a and x are associates (via y ).

Proof idea

Now the main theorem splits into:

Proposition

There exists an effective procedure that transforms any binary Σ02-tree

T into an integral domain A(T ). Furthermore:1 T is finite if and only if A(T ) is atomic.2 Every infinite sequence of proper divisions in A(T ) computes a

properly decreasing sequence of multisets of T .

Thus, we can completely strip algebra off and work directly with trees.

Proposition

There exists an infinite Σ02-tree with no infinite properly decreasing

∆02-sequence of multisets.

Our proof is a non-uniform infinite injury argument, or a Σ03 argument.

Proof idea

Proposition

Proof idea

Proposition

Proof idea

Proposition

Combining the two technical proposition above, we get thedesired sharpness result:

Theorem (Greenberg, M.)

There exists a computable integral domain that is not atomic but 0′

cannot compute an infinite chain of proper divisions.

The technique of Σ02-tree bases allowed us to

separate algebra from recursion theory.Without this separation the proof would be a total mess.The technique and the result has applications to reversemathematics of integral domains (beyond this talk).

Root-bases of torsion-freeabelian groups

Why TFAGs?

1 Our understanding of torsion-free abelian groups (TFAGs)is quite limited.

2 The class contains enough counter-intuitive examples.3 Nonetheless, computable TFAGs are not effectively

universal (essentially Goncharov ∼1980).4 There are several nice structural theorems about broad

subclasses of TFAGs.5 This in-between nature of TFAGs makes the class very

attractive.

Degree spectra

DefinitionA countably infinite structure A is X -computable (constructive) if itselements can be indexed by N so that the operations becomeX -computable on the indices.

DefinitionThe degree spectrum of a countable A is

{deg(X ) : X computes a copy of A}.

Much work has been done on degree spectra of:

- Graphs, universal structures (inludes fields, two-step nilpotent groups) andalmost universal structures (DCF0,RCF ,OAGr ).

- Boolean algebras.

- Linear orders.

- Abelian p-groups.

Degree spectra

- Boolean algebras.

- Linear orders.

- Abelian p-groups.

Degree spectra

- Boolean algebras.

- Linear orders.

- Abelian p-groups.

Degree Spectra of TFAGs

QuestionWhich spectra can be realized by a TFAG?

The “standard” complicated spectra include, for every α < ω1CK :

1. A spectrum DSp(A) such that

DSp(α)(A) = {X(α) : X ∈ DSp(A)}

is a cone.- If α = 0 then A has a degree.

- If DSp(β) is not a cone for β < α, then A has a proper α’th jump degree.

2. DSp(A) = non-lowα.- The case α = 0 corresponds to the Slaman-Wehner spectrum.

These spectra correspond to effective definability results (interpreting sets andfamilies of sets, respectively).

DSp(α)(A) = {X(α) : X ∈ DSp(A)}

The two results are:

Theorem (Andersen, Kach, M., Solomon, 2012)For every computable α there exists a TFAG having a proper α’thjump degree.

Theorem (M., 2014)For every computable α of the form δ + 2n + 1, where either δ = 0or δ is a limit ordinal, there exists a TFAG with spectrum non-lowα.

In both results we heavily use special bases of TFAGs.

QuestionWhat about non-lowα for “even” α?

A discussion

Our technique relies on a special choice of a basis (a root basis).

The idea goes back to Pontrjagin, but had been used only to constructindecomposable groups (see Fuchs, vol. II).

Used in descriptive set theory (Hjorth). We exploit an extension ofHjorth’s technique.

With such a basis, a group admits a “nicer” presentation by divisibilityconditions.

We have a machinery to deal with the complicated combinatorics.

We don’t know how to build such examples without using a root-basis.

ProblemDevelop a sufficiently general meta-theory of root-bases and rootedgroups that would (at least) unite the known proofs.

A discussion

Spasibo

Thanks!

Biboiography

The talk was based on:

1 Independence in Effective Algebra. Harrison-Trainor, M., and

Montalban. Journal of Algebra (to appear).

2 Iterated Effective Embeddings of Abelian p-Groups. Downey,

M., Ng. International Journal of Algebra and Computation (2014).

3 Proper Divisibility in Computable Rings. Greenberg and M.

Preprint.

4 New Degree Spectra of Abelian Groups. M. Notre Dame Journal

of Formal Logic (to appear).

Go to my homepage (click here) for preprints.

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