Bases and generating sets in computable algebraamelniko/MelnikovTalk2015.pdf · Novosibirsk, May...
Transcript of 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?
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?
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?
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?
Alexander Melnikov Bases and generating sets in computable algebra
PLAN:
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.
Alexander Melnikov Bases and generating sets in computable algebra
PLAN:
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.
Alexander Melnikov Bases and generating sets in computable algebra
PLAN:
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.
Alexander Melnikov Bases and generating sets in computable algebra
PLAN:
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.
Alexander Melnikov Bases and generating sets in computable algebra
PLAN:
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.
Alexander Melnikov Bases and generating sets in computable algebra
Preliminaries
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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
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
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
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
A meta-theorem and itsapplications
Alexander Melnikov Bases and generating sets in computable algebra
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
G.
Corollary (Goncharov)Such an A has infinitely many computable copies, up to computableisomorphism (A has “infinite auto-dimension”).
Alexander Melnikov Bases and generating sets in computable algebra
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
G.
Corollary (Goncharov)Such an A has infinitely many computable copies, up to computableisomorphism (A has “infinite auto-dimension”).
Alexander Melnikov Bases and generating sets in computable algebra
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);
G has a computable Z-base;
B ∼=∆02
G.
Corollary (G.L.S.)Such an A has infinite auto-dimension.
Alexander Melnikov Bases and generating sets in computable algebra
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);
G has a computable Z-base;
B ∼=∆02
G.
Corollary (G.L.S.)Such an A has infinite auto-dimension.
Alexander Melnikov Bases and generating sets in computable algebra
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
G.
Note that A has infinite auto-dimension (Goncharov 1980).
Alexander Melnikov Bases and generating sets in computable algebra
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
G.
Note that A has infinite auto-dimension (Goncharov 1980).
Alexander Melnikov Bases and generating sets in computable algebra
A framework
QuestionWhich classes have the Mal′cev property?
We want a sufficiently general and convenient localcondition that implies the Mal’cev property.
Alexander Melnikov Bases and generating sets in computable algebra
A framework
QuestionWhich classes have the Mal′cev property?
We want a sufficiently general and convenient localcondition that implies the Mal’cev property.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.)
Alexander Melnikov Bases and generating sets in computable algebra
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.)
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
Tree-bases in abelian p-groups
Alexander Melnikov Bases and generating sets in computable algebra
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)).
Alexander Melnikov Bases and generating sets in computable algebra
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)).
Alexander Melnikov Bases and generating sets in computable algebra
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)).
Alexander Melnikov Bases and generating sets in computable algebra
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)).
Alexander Melnikov Bases and generating sets in computable algebra
p-Basic trees
ExampleThe labeled tree
0
a b
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 .
Alexander Melnikov Bases and generating sets in computable algebra
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
kbb,
where each kb ∈ Zp;2 For every b ∈ B, either pb ∈ B or pb = 0.
Alexander Melnikov Bases and generating sets in computable algebra
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).
Alexander Melnikov Bases and generating sets in computable algebra
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).
Alexander Melnikov Bases and generating sets in computable algebra
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).
Alexander Melnikov Bases and generating sets in computable algebra
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) =⋂
n
A(n)
. . .
The Ulm factors Aβ = A(β)/A(β+1) split into cyclic summands.
Alexander Melnikov Bases and generating sets in computable algebra
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) =⋂
n
A(n)
. . .
The Ulm factors Aβ = A(β)/A(β+1) split into cyclic summands.
Alexander Melnikov Bases and generating sets in computable algebra
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) =⋂
n
A(n)
. . .
The Ulm factors Aβ = A(β)/A(β+1) split into cyclic summands.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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?
Alexander Melnikov Bases and generating sets in computable algebra
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?
Alexander Melnikov Bases and generating sets in computable algebra
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?
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
Directed systems of trees, andintegral domains
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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?
Alexander Melnikov Bases and generating sets in computable algebra
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?
Alexander Melnikov Bases and generating sets in computable algebra
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?
Alexander Melnikov Bases and generating sets in computable algebra
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?
Alexander Melnikov Bases and generating sets in computable algebra
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.
x y
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:
x y
a b ⇒ a ∼ x b
r r
Figure: Inverting y . Here “a ∼ x” indicates that a and x are associates (via y ).
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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).
Alexander Melnikov Bases and generating sets in computable algebra
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).
Alexander Melnikov Bases and generating sets in computable algebra
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).
Alexander Melnikov Bases and generating sets in computable algebra
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).
Alexander Melnikov Bases and generating sets in computable algebra
Root-bases of torsion-freeabelian groups
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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).
Alexander Melnikov Bases and generating sets in computable algebra
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).
Alexander Melnikov Bases and generating sets in computable algebra
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).
Alexander Melnikov Bases and generating sets in computable algebra
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).
Alexander Melnikov Bases and generating sets in computable algebra
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).
Alexander Melnikov Bases and generating sets in computable algebra
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).
Alexander Melnikov Bases and generating sets in computable algebra
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).
Alexander Melnikov Bases and generating sets in computable algebra
Degree Spectra of TFAGs
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?
Alexander Melnikov Bases and generating sets in computable algebra
Degree Spectra of TFAGs
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?
Alexander Melnikov Bases and generating sets in computable algebra
Degree Spectra of TFAGs
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?
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra
Spasibo
Thanks!
Alexander Melnikov Bases and generating sets in computable algebra
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.
Alexander Melnikov Bases and generating sets in computable algebra