effective properties of abelian groupsamelniko/melnikovnsk2011.pdf · goncharov, knight, downey,...

Effective properties of abelian groups

Alexander Melnikov

The University of AucklandNovosibirsk State University

Mal’cev Meeting 2011, Novosibirsk.

Alexander Melnikov Effective properties of abelian groups

Definitions

Definition 1A structure (A, f1, . . . , fn,P1, . . . ,Pm) is computable if thedomain A is a computable set, the operations (fi)i=1,...,n and thepredicates (Pj)j=1,...,m are computable. An X -computablestructure is defined similarly.

Thus, a group is computable if the set of its elements and theoperation are computable.

The study of computable groups was initiated by Rabin. Mal’cevstarted the systematic study of computable abelian groups.

Computable abelian groups have been studied by Khisamiev,Goncharov, Knight, Downey, Dobrica, Nurtazin, Solomon,Smith, and many-many others.

Definitions

Questions

Given an abelian group (module, ordered abelian group) A, wemay ask the following questions:

Does A have a computable presentation?

What is the collection of X such that A has an X -computablepresentation? This collection is often called the degree spectrumof A.

Is A computably categorical? (Recall that a structure iscomputably categorical if every two computable presentations ofthe structure are computably isomorphic.)

What is the collection of X such that A is X -computablycategorical? (The notion of X -computable categoricity is definedsimilarly.)

The answer heavily depends on the class A belongs to.

Questions

Definitions

Recall that a Turing degree a is the n-th jump degree of astructure A if a is the least degree in

{b(n) : A is computable in b}.

The n-th jump degree is proper if the structure has no (n-1)jump degree.

A structure is ∆0n - categorical if there is a ∆0

n isomorphismbetween any two computable presentations of the structure.

Definitions

Recall that a Turing degree a is the n-th jump degree of astructure A if a is the least degree in

{b(n) : A is computable in b}.

The n-th jump degree is proper if the structure has no (n-1)jump degree.

A structure is ∆0n - categorical if there is a ∆0

n isomorphismbetween any two computable presentations of the structure.

Historical remarks

The study of computable abelian groups was initiated byMal’cev in 1960’s.

Khisamiev in 1970’s studied computable abelian p-groupsof small Ulm length and basic properties of computablypresentable torsion-free abelian groups.Metakides and Nerode in 1977 studied computable vectorspaces.Smith studied reverse mathematics of abelian groups.

Historical remarks

The study of computable abelian groups was initiated byMal’cev in 1960’s.Khisamiev in 1970’s studied computable abelian p-groupsof small Ulm length and basic properties of computablypresentable torsion-free abelian groups.

Metakides and Nerode in 1977 studied computable vectorspaces.Smith studied reverse mathematics of abelian groups.

Historical remarks

The study of computable abelian groups was initiated byMal’cev in 1960’s.Khisamiev in 1970’s studied computable abelian p-groupsof small Ulm length and basic properties of computablypresentable torsion-free abelian groups.Metakides and Nerode in 1977 studied computable vectorspaces.

Smith studied reverse mathematics of abelian groups.

Historical remarks

The study of computable abelian groups was initiated byMal’cev in 1960’s.Khisamiev in 1970’s studied computable abelian p-groupsof small Ulm length and basic properties of computablypresentable torsion-free abelian groups.Metakides and Nerode in 1977 studied computable vectorspaces.Smith studied reverse mathematics of abelian groups.

Historical remarks

Nurtazin characterized computably categorical torsion-freeabelian groups.

Goncharov and independently Smith in 1980’scharacterized computably categorical abelian p-groups.Downey and Kurtz in 1980 initiated the systematic study ofordered abelian groups.Notre Dame school of logic (Knight and herstudents/postdocs) in 1990’s - 2000’s obtained a number ofimportant results on computable abelian groups, mostlyp-groups.Downey, Coles and Slaman in 2000 showed that everytorsion-free abelian group has the least jump degree.

Historical remarks

Nurtazin characterized computably categorical torsion-freeabelian groups.Goncharov and independently Smith in 1980’scharacterized computably categorical abelian p-groups.

Downey and Kurtz in 1980 initiated the systematic study ofordered abelian groups.Notre Dame school of logic (Knight and herstudents/postdocs) in 1990’s - 2000’s obtained a number ofimportant results on computable abelian groups, mostlyp-groups.Downey, Coles and Slaman in 2000 showed that everytorsion-free abelian group has the least jump degree.

Historical remarks

Nurtazin characterized computably categorical torsion-freeabelian groups.Goncharov and independently Smith in 1980’scharacterized computably categorical abelian p-groups.Downey and Kurtz in 1980 initiated the systematic study ofordered abelian groups.

Notre Dame school of logic (Knight and herstudents/postdocs) in 1990’s - 2000’s obtained a number ofimportant results on computable abelian groups, mostlyp-groups.Downey, Coles and Slaman in 2000 showed that everytorsion-free abelian group has the least jump degree.

Historical remarks

Nurtazin characterized computably categorical torsion-freeabelian groups.Goncharov and independently Smith in 1980’scharacterized computably categorical abelian p-groups.Downey and Kurtz in 1980 initiated the systematic study ofordered abelian groups.Notre Dame school of logic (Knight and herstudents/postdocs) in 1990’s - 2000’s obtained a number ofimportant results on computable abelian groups, mostlyp-groups.

Downey, Coles and Slaman in 2000 showed that everytorsion-free abelian group has the least jump degree.

Historical remarks

Nurtazin characterized computably categorical torsion-freeabelian groups.Goncharov and independently Smith in 1980’scharacterized computably categorical abelian p-groups.Downey and Kurtz in 1980 initiated the systematic study ofordered abelian groups.Notre Dame school of logic (Knight and herstudents/postdocs) in 1990’s - 2000’s obtained a number ofimportant results on computable abelian groups, mostlyp-groups.Downey, Coles and Slaman in 2000 showed that everytorsion-free abelian group has the least jump degree.

Historical remarks

The modern theory of computable abelian groups can beroughly divided into several sub-fields:

The study of computable torsion-free abelian groups.

The study of computable abelian p-groups.

Very few general results (e.g., every computable abelian grouphas a computable divisible hull (folklore), etc.).

Related structures: computable ordered abelian groups (Downeyand Kurtz), computable vector spaces (Metakides and Nerode),computable modules (very few results).

Other classes of computable abelian groups (e.g., classes ofmixed groups, torsion groups which are not p-groups, etc.).

Historical remarks

Abelian p-groups and monotonic approximations

Khisamiev (independently Ash, Knight, Oates (unpubl.))obtained a characterization of computably presented reducedabelian p-groups of small Ulm length using the following notion:

Definition 2A set S is limitwise monotonic if there is a total computablefunction f (x , s) such that S = range sups f (x , s).

The general case of arbitrary Ulm length (< ωCK1 ) is still open.

Question: Can we build an abelian group (not necessarily ap-group) having an X -computable copy iff X > 0?

Theorem 3 (Kalimullin, Khoussainov, Melnikov 2010)1 There is a non-computable torsion abelian group (not a

p-group) G such that G has an X-computable copy, forevery hyperimmune X.

2 There is a non-computable abelian p-group A such that Ahas a computable copy relative to every non-computableX ≤T 0′.

We introduce a new concept of limitwise monotonic sequencewhich can be viewed as a uniform version of limitwisemonotonicity.

Theorem 4 (Kalimullin, Khoussainov, Melnikov 2010)Let G be of the form ⊕

(⊕s∈Sn

Zpsn).

If G has an X-computable copy for every set X , except at mostcountably many, then G has a computable copy.

Limitwise monotonic functions have found applications in othertopics of computable model theory:

models of ω1-categorical not ω-categorical theories,

equivalence structures,linear orderings,the notion has connections to degree theory.

More about limitwise monotonic sets can be found in PhDthesis of Wallbaum, a recent paper by Downey, Kach andTuretsky, or our paper available on-line. More about theeffective use of Ulm invariants can be found in PhD thesis ofCalvert.

models of ω1-categorical not ω-categorical theories,equivalence structures,

linear orderings,the notion has connections to degree theory.

models of ω1-categorical not ω-categorical theories,equivalence structures,linear orderings,

the notion has connections to degree theory.

models of ω1-categorical not ω-categorical theories,equivalence structures,linear orderings,the notion has connections to degree theory.

Torsion-free groups: the easy case of finite rank.

Each subgroup of 〈Q,+〉 can be associated with a sequence ofnatural numbers and symbols∞. Say, if a subgroup isgenerated by {1/p : p prime}, then the correspondingsequence is (1,1, . . . ,1, . . .).

Theorem 5 (Baer)

Let G and H be subgroups of 〈Q,+〉. Then G and H areisomorphic if and only if the corresponding sequences are thesame up to a finite difference.

Theorem 6 (Mal’cev; Knight, Downey)

A subgroup of 〈Q,+〉 has a computable copy iff thecorresponding sequence has a computable monotonicapproximation from below iff a certain invariant set is c.e.

Theorem 5 (Baer)

Theorem 7 (Downey, Coles, Slaman)Every set has a least jump enumeration.

Corollary 8

Every subgroup of 〈Q,+〉 has a jump degree. There is atorsion-free abelian group of rank 1 having a proper jumpdegree.

(Melnikov 2007) The result above hold for torsion-free abeliangroups of any finite rank.

(Recall that a torsion-free abelian group has rank n if it can beembedded into Qn and can not be embedded into Q(n+1).)

Corollary 8

(Easy) Every torsion-free abelian group of finite rank iscomputably categorical.

Conclusion: We have complete satisfactory answers for theconsidered questions in the case of finite rank.

(Easy) Every torsion-free abelian group of finite rank iscomputably categorical.

Conclusion: We have complete satisfactory answers for theconsidered questions in the case of finite rank.

Torsion-free completely decomposable groups.

A torsion-free abelian group is completely decomposable if it isisomorphic to

⊕i∈N Ai , where Ai 5 Q.

Question. Which completely decomposable groups arecomputable?

The case when all the summands are isomorphic is easy, butthe general case is wide open.

Problem of Khisamiev.

Let p be a prime number. Consider the following subgroup of〈Q,+〉:

Q(p) = { npk : n ∈ Z and k ∈ N}.

Now let P be a set of primes, and define the following group:

GP =⊕p∈P

Problem(Khisamiev 199?). Describe all sets P for which GP iscomputable.

Problem of Khisamiev

Theorem 9 (Khisamiev 2002)The group GP is computable in the signature augmented bycertain membership predicates (one for each elementarysummand) if and only if P is Σ0

2 and is not quasi-hhimmune.

Problem of Khisamiev

Theorem 10 (Downey, Goncharov, Kach, Knight, Kudinov, M.,Turetsky 2010)

GP is computable if and only if P ∈ Σ03.

Recall the following:

Theorem 11 (M. 2007)There exists a torsion-free abelian group which has anX-computable presentation if and only if X ′ > 0′ (that is, it hasexactly non-low copies).

In contrast to the case of finite rank, there is a completelydecomposable group having no jump degree. There exist alsocompletely decomposable groups A and B having a propersecond and third jump degree, respectively (M. 2007).

Question: Is it true that every completely decomposable grouphas a third jump degree?

Categoricity of completely decomposable groups.

A group of the form⊕

i∈ω Hi , where Hi 5 (Q,+), is nevercomputably categorical.

Question: Is it true that there exists n (or a computableα) suchthat every completely decomposable group is ∆0

n-categorical(∆0

α -categorical)?

Categoricity of completely decomposable groups.

A group of the form⊕

i∈ω Hi , where Hi 5 (Q,+), is nevercomputably categorical.

Question: Is it true that there exists n (or a computableα) suchthat every completely decomposable group is ∆0

n-categorical(∆0

α -categorical)?

Categoricity of homogeneous completelydecomposable groups.

A completely decomposable group is homogeneous if all thesummands in its complete decomposition are isomorphic:⊕

i∈ω H, where H 5 (Q,+).

Theorem 12 (Downey, M., 2011)Every homogeneous completely decomposable group is∆0

3-categorical, and this is sharp.

We introduce a new algebraic concept of S-independencewhich generalizes the classical notion of p-independence. Ingeneral, it is rather unusual that a notion similar top-independence is useful in the torsion-free case. Also, we usea bit of module theory.

Question: What can be said about ∆02-categorical

homogeneous completely decomposable groups?

Theorem 13 (Downey, M., 2011)A homogeneous completely decomposable group G is∆0

2-categorical if and only if it encodes a c.e. set with a semi-lowcomplement: G ∼=

⊕i∈ω Q(P), where Q(P) is generated by

{ 1pn : p ∈ P and n ∈ ω}, and P has a semi-low complement.

A set S is semi-low if HS = {e : We ∩ P} is ∆02.

The proof splits into several cases, in each case we need adifferent argument.

We introduce a new notion of computable setting time of amonotonic approximation which seems to be crucial for theproof.

That was Kalimullin who suggested us to try semi-low for somespecial case. Interestingly, it turned to be the only case whenthe group is ∆0

2-categorical.

Completely decomposable groups.

Question: Is it true that there exists n (computable α) such thatevery computable completely decomposable group is∆0

n-categorical (∆0α -categorical)? (Open.)

This is not true for torsion-free abelian groups in general, as aconsequence of a recent result of Andersen, Kach, M.,Solomon (2011). This is not true if we add a linear order evento the free abelian group (M., 2010).

(M., 2010) Many of the results for linear orders can betransferred to ordered abelian groups and ordered fields.

Question: Can we apply limitwise monotonic sequences andthe notion of computable setting time in other areas ofcomputable model theory?

In general, not much is known about computable completelydecomposable groups (degree spectra, index sets, the effectivecontent of p- and S- independence, reverse mathematics, etc.).More about completely decomposable groups can be learnedfrom papers of Khisamiev, Khisamiev and Krykpaeva, and frompapers available from my homepage.

Question: Can we apply limitwise monotonic sequences andthe notion of computable setting time in other areas ofcomputable model theory?

In general, not much is known about computable completelydecomposable groups (degree spectra, index sets, the effectivecontent of p- and S- independence, reverse mathematics, etc.).More about completely decomposable groups can be learnedfrom papers of Khisamiev, Khisamiev and Krykpaeva, and frompapers available from my homepage.

Torsion-free groups in general.

Theorem 14 (Downey, Montalban 2008)The isomorphism problem for computable torsion-free abeliangroups is as complex as possible (Σ1

1 - complete).

The proof is based on a coding of trees into torsion-free abeliangroups suggested by Hjorth.

The ideas were developed by Knight, Fokina, M., Quinn,Safransky (2011). We showed that for a certain class of treesthe coding preserves the isomorphism type. This is not true forall trees in general (M. 2011). Our result found an application(Fokina, Friedman, Harizanov, Knight, McCoy, Montalban(201?).

1 - complete).

We could also encode families of subgroups, using the ideas ofFuchs.

Theorem 15 (Andresen, Kach, M., Solomon 2011)For every computable α there is a torsion-free abelian grouphaving a proper α jump degree.

We could also encode families of subgroups, using the ideas ofFuchs.

Theorem 15 (Andresen, Kach, M., Solomon 2011)For every computable α there is a torsion-free abelian grouphaving a proper α jump degree.

Given a computable α, we can use the main construction tobuild a computable group Gα which is not ∆0

α categorical.Unfortunately, the construction is too heavy to obtain an answerto the following natural question:

Is Gα ∆0α+1-categorical?

At the moments we can not even find β such that Gα

∆0α+β-categorical (we believe β should be finite).

Question: Can we simplify the main construction of the α jumpdegree result?

Problem: Apply the suggested invariants to other questions inthe theory of computable torsion-free abelian groups, or findother invariants of computable torsion-free abelian groupswhich are easier to deal with.

Question: Can we simplify the main construction of the α jumpdegree result?

Problem: Apply the suggested invariants to other questions inthe theory of computable torsion-free abelian groups, or findother invariants of computable torsion-free abelian groupswhich are easier to deal with.

Thank you!

effective properties of abelian groupsamelniko/melnikovnsk2011.pdf · goncharov, knight, downey,...

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