Tarski problems for algebras

Olga Kharlampovich and Alexei Miasnikov

Oaxaca, January 27, 2015

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Tarski-type questions

Alfred Tarski

Tarski’s type problems for a given group (or a ring) G :

First-order classification: Describe groups H such thatTh(G ) = Th(H).

Decidability: Is the theory Th(G ) decidable?

Here Th(G ) is decidable if there exists an algorithm which for anygiven sentence � decides if � 2 Th(G ) or not.

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Model theory of algebras

General intuition:

The situation in ”free-like” associative algebras is very di↵erentfrom the one in ”free-like” groups (free or torsion-free hyperbolic)groups.

In free-like groups there is more geometry and topology, moreabout equations and their solutions,

In free-like associative algebras it is more about algebra,arithmetic, about describing (interpreting) by means of logic someclassical commutative objects sitting in algebras. It is more like inthe case of nilpotent and solvable groups.

Free Lie algebras are a di↵erent story.

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Model theory of algebras

General intuition:

The situation in ”free-like” associative algebras is very di↵erentfrom the one in ”free-like” groups (free or torsion-free hyperbolic)groups.

In free-like groups there is more geometry and topology, moreabout equations and their solutions,

In free-like associative algebras it is more about algebra,arithmetic, about describing (interpreting) by means of logic someclassical commutative objects sitting in algebras. It is more like inthe case of nilpotent and solvable groups.

Free Lie algebras are a di↵erent story.

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Definability

Let F be a field, X a set of variables, and F [X ] a ring ofcommutative polynomials in variables X with coe�cients in F .

The following formula defines F in F [X ]:

�(x) = (x = 0) _ 9y(xy = 1)

The operations + and · in F are the restrictions of the ones fromF [X ], so they are also definable in F [X ] by formulas.

The field F is definable in the ring F [X ].Notice, that the formula all the formulas that define F in F [X ] donot depend on F or X . F is definable uniformly in F and X .

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Important consequences

Since F is definable in F [X ] for every first-order sentence � offields one can e↵ectively construct a sentence �⇤ of rings such that

F |= �() F [X ] |= �⇤.

Implications:

If Th(F ) is undecidable then Th(F [X ]) is undecidable;

for any fields F and K

F [X ] ⌘ K [Y ] =) F ⌘ K

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Important consequences

Since F is definable in F [X ] for every first-order sentence � offields one can e↵ectively construct a sentence �⇤ of rings such that

F |= �() F [X ] |= �⇤.

Implications:

If Th(F ) is undecidable then Th(F [X ]) is undecidable;

for any fields F and K

F [X ] ⌘ K [Y ] =) F ⌘ K

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Definability of Arithmetic in polynomial rings

Lemma

Let F be an arbitrary field. The arithmetic N = hN; +, ·, 0, 1i isinterpretable in F [t] uniformly in F .

Sketch of the proof in char 0:

N is definable in F [X ]: a 2 F belongs to N if and only if it satisfiesthe following formula.

8u 62 F9v(u | v ^(8b 2 F ((u+b) | v ! (u+b+1) | v _(b = a))))

If a 2 N we take v = u(u + 1)(u + 2) . . . (u + a).If a 2 F and a 62 N, and a satisfies the formula, then v hasinfinitely many non-associated divisors, this is impossible in a UFD.⇤

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Non-zero characteristic

If char of F is not zero, then N does not sit inside F . In this caseit is not definable in F [X ] - it is interpretable.

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Interpretability

Definition

An algebraic structure A is interpretable in a structure B if thereare formulas �,E (x , y),�

f

,�P

in the language of B such that

�A

defines a subset A⇤ ✓ Bm,

E (x , y) defines an equivalence relation ⇠ on A⇤,

For every operation f and predicate P in A there are formulas�f

and �P

in B that define operations f ⇤ and predicates P⇤

on A⇤/ ⇠, so A⇤ = hA/ ⇠; f , . . . ,P . . .i is a structure in thelanguage of A.

There is an isomorphism µ : A ! A⇤.

Interpretation is uniformly in a class of structures C if the formulas�,E (x , y),�

f

,�P

are the same for every structure B from C .

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Interpretability

Definition

An algebraic structure A is interpretable in a structure B if thereare formulas �,E (x , y),�

f

,�P

in the language of B such that

�A

defines a subset A⇤ ✓ Bm,

E (x , y) defines an equivalence relation ⇠ on A⇤,

For every operation f and predicate P in A there are formulas�f

and �P

in B that define operations f ⇤ and predicates P⇤

on A⇤/ ⇠, so A⇤ = hA/ ⇠; f , . . . ,P . . .i is a structure in thelanguage of A.

There is an isomorphism µ : A ! A⇤.

Interpretation is uniformly in a class of structures C if the formulas�,E (x , y),�

f

,�P

are the same for every structure B from C .

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Important implications

If A is interpretable in B then for every formula (x) of LA

onecan e↵ectively construct a formula ⇤(y) of L

B

such thatA |= (a) () B |= ⇤(µ(a))

In particular, we have the same result as above for sentences:

If A is interpretable in B then for every first-order sentence � inthe language of A one can e↵ectively construct a sentence �⇤ inthe language of B such that

A |= �() B |= �⇤.

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Undecidability of Th(F [X ])

Since the arithmetic N is interpretable in F [X ] and Th(N) isundecidable the following result holds.

Theorem [Robinson, ...]

For any field F the elementary theory of F [X ] is undecidable.

This is typical use of interpretability.

It solves the Tarski’s problem on decidability of the theory forpolynomial rings F [X ].

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First-order classification of polynomial rings

Theorem

F [X ] ⌘ K [Y ] =) |X | = |Y | and F ⌘ K .

Sketch of the proof: If F [X ] ⌘ K [Y ], then they have the sameKrull dimension, hence |Y | = |X |.

Since F is definable in F [X ] uniformly in F and X it follows thatF ⌘ K .

But the converse is not true!

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First-order classification of polynomial rings

Theorem

F [X ] ⌘ K [Y ] =) |X | = |Y | and F ⌘ K .

Sketch of the proof: If F [X ] ⌘ K [Y ], then they have the sameKrull dimension, hence |Y | = |X |.

Since F is definable in F [X ] uniformly in F and X it follows thatF ⌘ K .

But the converse is not true!

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First-order classification of polynomial rings

Theorem

F [X ] ⌘ K [Y ] =) |X | = |Y | and F ⌘ K .

Sketch of the proof: If F [X ] ⌘ K [Y ], then they have the sameKrull dimension, hence |Y | = |X |.

Since F is definable in F [X ] uniformly in F and X it follows thatF ⌘ K .

But the converse is not true!

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Hereditary finite superstructures

For a set A let Pf (A) be the set of all finite subsets of A.

Definition of hereditary finite sets over A

HF0(A) = A,

HFn+1(A) = HF

n

(A) [ Pf (HFn

(A)),

HF (A) =S

n2! HFn

(A).

For a ring R = (A; +,⇥, 0, 1) define a new structure

HF (R) = hHF (A);PA

,+,⇥, 0, 1,2i,

where PA

is the predicate defining A in HF (A), +,⇥, 0, 1 aredefined on A as before and 2 is the membership predicate onHF (A).

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Hereditary finite superstructures

For a set A let Pf (A) be the set of all finite subsets of A.

Definition of hereditary finite sets over A

HF0(A) = A,

HFn+1(A) = HF

n

(A) [ Pf (HFn

(A)),

HF (A) =S

n2! HFn

(A).

For a ring R = (A; +,⇥, 0, 1) define a new structure

HF (R) = hHF (A);PA

,+,⇥, 0, 1,2i,

where PA

is the predicate defining A in HF (A), +,⇥, 0, 1 aredefined on A as before and 2 is the membership predicate onHF (A).

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The weak second-order theory

The main point:

The first-order theory of HF (R) has the same expressiveness as theweak second order theory of R .

For example: one can use arithmetic in HF (R), finite sequences ofof elements from R , define the lengths of the sequences byformulas, take their components, concatenations, etc. - extremelypowerful language.

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First-order classification of polynomial rings

Theorem [Bauval]

If F is infinite, then HF (F ) is interpretable in F [t] uniformlyin F .

F [X ] is interpretable in HF (F ) uniformly in |X |.

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Theorem

Rings of polynomials of finite number of variables over infinitefields F [X ] andK [Y ] are elementarily equivalent if and only if|X | = |Y | and HF (F ) ⌘ HF (K ).

Corollary

If F and K are computable (for instance, Q, and f.g. extensions ofQ, or algebraic closure of Q), then the polynomial rings F [X ] andK [Y ] are elementarily equivalent i↵ they are isomorphic.

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Theorem

Rings of polynomials of finite number of variables over infinitefields F [X ] andK [Y ] are elementarily equivalent if and only if|X | = |Y | and HF (F ) ⌘ HF (K ).

Corollary

If F and K are computable (for instance, Q, and f.g. extensions ofQ, or algebraic closure of Q), then the polynomial rings F [X ] andK [Y ] are elementarily equivalent i↵ they are isomorphic.

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By Bauval’s theorem the model HF (F ) of hereditary finite setsover F is uniformly definable in the ring F [x ]. Therefore HF (F ) ininterpretable in F [X ] using the constant a.If A is interpretable in B then for every sentence � of the languageLA one can e↵ectively construct a sentence �⇤ of LB such that

A |= � () B |= �⇤.

If the interpretation of A in B depends on some constants, saya1, . . . , a

k

then �⇤ contains a1, . . . , ak

.So in our case

HF (F ) |= � () F [X ] |= 9a(a 62 F ) ^ �⇤(a).

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Hence

HF (F ) |= � () A |= 8a(a 6= 0 ^ ¬9x(ax = 1)) ! �⇤(a).

Hence F [X ] ⌘ K [Y ] =) HF (F ) ⌘ HF (K ), and

HF (F ) ⌘ HF (K ) () F ⌘w .s.o.l . K .

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Conversely, we can enumerate all the monomials in F [X ] andrepresent each element in F [X ] as a finite set of coe�cients in F .Multiplication in F [X ] is interpretable in the weak second orderlogic of F . Therefore the theory of F [X ] is interpretable in the theweak second order theory of F uniformly on F [X ] .

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Associative algebras

Denote by AF

(X ) a free associative unital algbera over a field Fwith basis X .

Approach to Tarski’s problems for AF

(X ):

Do the same as for commutative polynomials

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Definability of F [t]

Theorem [Bergman]

The centralizer in AF

(X ) of a non-scalar polynomial is isomorphicto F [t].

Corollary

The ring F [t] is definable in AF

(X ) uniformly in F and X .

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Decidability

Theorem

The first-order theory of AF

(X ) is undecidable for any F and X .

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First-order classification

Theorem

Free associative algebras AF1(X ) and A

F2(Y ) of finite rank overinfinite fields F1,F2 are elementarily equivalent if and only if theirranks are the same and HF (F1) ⌘ HF (F2).

Corollary

If F1 and F2 are computable (for instance, Q, and f.g. extensionsof Q, or algebraic closure of Q), then the algebras A

F1(X ) andAF2(Y ) are elementarily equivalent i↵ they are isomorphic.

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Defining bases AF (X )

Let X be finite.

Theorem

The following holds in AF

(X ):

The set of free bases is definable AF

(X ) uniformly in F ;

The rank of AF

(X ) is a first-order invariant;

The group of automorphisms Aut(AF

(X )) is interpretable inAF

(X ).

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Non-unitary free associative algebras

Let A0F

(X ) be a free associative algebra with basis X without unity( non-commutative monomials on X without constant terms).

Main Problem: there is no subfield F in A0F

(X )!

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Non-unitary free associative algebras

Theorem

The field F and its action on A0F

(X ) are interpretable in A0F

(X )uniformly in F .

Corollary

Algebra AF

(X ) is definable in A0F

(X ).

Indeed, A0F

(X ) = 1 · F � A0F

(X ).

Theorem

Free associative algebras A0F1(X ) and A0

F2(Y ) of finite rank over

infinite fields F1,F2 are elementarily equivalent if and only if theirranks are the same and HF (F1) ⌘ HF (F2).

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Non-unitary free associative algebras

Theorem

The field F and its action on A0F

(X ) are interpretable in A0F

(X )uniformly in F .

Corollary

Algebra AF

(X ) is definable in A0F

(X ).

Indeed, A0F

(X ) = 1 · F � A0F

(X ).

Theorem

Free associative algebras A0F1(X ) and A0

F2(Y ) of finite rank over

infinite fields F1,F2 are elementarily equivalent if and only if theirranks are the same and HF (F1) ⌘ HF (F2).

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Rings elementarily equivalent to free associative algebras

Theorem

If a ring B is elementarily equivalent to a free associative algebraA0F

(X ) of rank n, then B is an associative algebra over a field F1,such that:

F1 is elementarily equivalent to F ,

B/Bn ⌘ Cn

, where Cn

is a free n-nilpotent associative algebrawith basis X over the field F1.

In particular, if B is residually nilpotent, then B is para-free.

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Lie Algebras

Let LF

(X ) be a free Lie algebra with basis X with coe�cients in F .

Theorem

The field F and its action on LF

(X ) are definable in LF

(X )uniformly in F .

Corollary

The theory of LF

(X ) over Q (any field with undecidable theory) isundecidable.

Theorem

If two free Lie algebras of finite rank over fields are elementarilyequivalent, then the ranks are the same and the fields areelementarily equivalent.

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Lie Algebras

Theorem

If a ring B is elementarily equivalent to a free lie algebra LF

(X ) ofrank n , then B is a Lie algebra over a field F1, such that:

F1 is elementarily equivalent to F ,

B/Bn ⌘ Cn

, where Cn

is a free n-nilpotent Lie algebra withbasis X over the field F1.

In particular, if B is residually nilpotent, then B is para-free.

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Ordered groups

A group G is called Left Orderable (LO) if there is a linear orderingon G which respects left multiplication in G .

Free groups, special groups, etc., are LO.

It is not known if every torsion-free hyperbolic group is LO or not.

Fact

Every LO group satisfies the Kaplansky unit conjecture, i.e., thegroup of units in the group ring K (G ) is precisely K · G .

Corollary

If G is LO then G is definable in K (G ).

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Interpretability of the field action

Theorem

Let G be a torsion free non-abelian hyperbolic group. Then thefield K and its action on K (G ) are interpretable in K (G ) uniformlyin K and G .

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Centralizers

Centralizers

Let G be a commutative transitive and torsion free group andg 2 G such that C

G

(g) = Zn. ThenCK(G)(g) = K (Zn) = K [t1, . . . , tn, t

�11 , . . . , t�1

n

].

Corollary

If G is torsion-free hyperbolic, then for any g 2 G the centralizerCK(G)(x) = K [t, t�1] is the ring of Laurent polynomials in one

variable.

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Centralizers

Centralizers

Let G be a commutative transitive and torsion free group andg 2 G such that C

G

(g) = Zn. ThenCK(G)(g) = K (Zn) = K [t1, . . . , tn, t

�11 , . . . , t�1

n

].

Corollary

If G is torsion-free hyperbolic, then for any g 2 G the centralizerCK(G)(x) = K [t, t�1] is the ring of Laurent polynomials in one

variable.

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Interpretability of arithmetic in Laurent polynomials

Theorem

The following holds for every ring of Laurant polynomialsK [x , x�1] over a field K of characteristic 0:

The arithmetic N = hN | +, ·, 0, 1i is interpretable inK [x , x�1] uniformly in K .

HF (K ) is interpretable in K [x , x�1] uniformly in K .

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Interpretability of arithmetic in group rings

Theorem

Let G be torsion free hyperbolic or total relatively hyperbolic,charK = 0, then Th(K (G )) is undecidable.

Theorem

Let G1 is LO and hyperbolic. Then K1(G1) ⌘ K2(G2) thenG1 ⌘ G2 and HF (K1) ⌘ HF (K2).

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Open problems

Problem

Describe para-free Lie (associative) algebras which areelementarily equivalent to a free Lie (associative) algebra.

Decidability of equations in free associative or Lie algebrasover a ”good” field.

Are free associative or Lie algebras equationally Noetherian?

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Open problems

Problem

Describe para-free Lie (associative) algebras which areelementarily equivalent to a free Lie (associative) algebra.

Decidability of equations in free associative or Lie algebrasover a ”good” field.

Are free associative or Lie algebras equationally Noetherian?

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Open problems

Problem

Describe para-free Lie (associative) algebras which areelementarily equivalent to a free Lie (associative) algebra.

Decidability of equations in free associative or Lie algebrasover a ”good” field.

Are free associative or Lie algebras equationally Noetherian?

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