Bertalan Bodor, joint work with Peter Cameron and Csaba Szab o · 2017. 9. 19. · In nitely many reducts of homogeneous structures Bertalan Bodor, joint work with Peter Cameron and

Infinitely many reducts of homogeneous structures

Bertalan Bodor,joint work with Peter Cameron and Csaba Szabó

TU Dresden

Novi Sad, 17th June 2017

Bertalan Bodor (TU Dresden) Infinitely many reducts . . . Novi Sad, 17th June 2017

Basic concepts

Structure: A = 〈A,C ,F ,R〉, where

A : underlying set

C : set of constants

F : set of functions An → AR : set of relations ⊂ An

Basic concepts

A : underlying set

Basic concepts

A : underlying set

Basic concepts

A : underlying set

F : set of functions An → A

R : set of relations ⊂ An

Basic concepts

A : underlying set

Basic concepts

Reducts

Reduct of a structure A: another structure on the same domain set;contants, functions and relations are definable in A.

Two reduct are called interdefinable iff they are reducts of one another.

Basic concepts

Reducts

Reduct of a structure A: another structure on the same domain set;contants, functions and relations are definable in A.

Two reduct are called interdefinable iff they are reducts of one another.

Reducts and automorphism groups

Facts

Aut(A): closed in Sym(A).

If B is a reduct of A, then Aut(B) ⊃ Aut(A).

All structures: countable, ω-categorical

Definition

A is ω-categorical iff for all n Aut(A) has finitely many n-orbits.

Theorem (Ryll-Nardzewski, Engeler, Svenonius)

For ω-categorcial structures B 7→ Aut(B) is a bijection between

the reducts of A (up to interdefinability) and

the closed supergroups of Aut(A).

Facts

Definition

Facts

Definition

Facts

Definition

Facts

Definition

Facts

Definition

Facts

Definition

For ω-categorcial structures B 7→ Aut(B) is a bijection betweenthe reducts of A (up to interdefinability) and

Facts

Definition

For ω-categorcial structures B 7→ Aut(B) is a bijection betweenthe reducts of A (up to interdefinability) and

Aim in general

To classify all reducts of a structure A (up to interdefinability).

Algebraic formulation

By the previous theorem: it is enough to find all closed supergroups ofAut(A).

Solved for:

(Q,

Aim in general

Solved for:

(Q,

Aim in general

Solved for:

(Q,

Aim in general

Solved for:

(Q,

Aim in general

Solved for:

(Q,

Aim in general

Solved for:

(Q,

Aim in general

Solved for:

(Q,

Aim in general

Solved for:

(Q,

All structures above: homogeneous over a finite relational language.

Definition

Homogeneous: every finite partial isomorphism can be extended to anautomorphism.

Conjecture (Thomas, 1991)

Every homogeneous, finite relational structure has finitely many reducts.

Definition

Reducts of the vector space

V = Fω2 : countably infinite dimensional vector space over F2

Difference: it is not homogeneous over a finite relational language.Proof:Suppose Aut(V) = Aut(V ,R1,R2, . . . ,Rk).Claim: (V ,R1,R2, . . . ,Rk) is not homogeneous.n > max(ar(Ri )), a1, a2, . . . , an linearly independent.Then ai 7→ ai : i < n, an 7→ a1 + a2 + · · ·+ an−1 does not extend to anautomorphism.

Difference: it is not homogeneous over a finite relational language.

Proof:Suppose Aut(V) = Aut(V ,R1,R2, . . . ,Rk).Claim: (V ,R1,R2, . . . ,Rk) is not homogeneous.n > max(ar(Ri )), a1, a2, . . . , an linearly independent.Then ai 7→ ai : i < n, an 7→ a1 + a2 + · · ·+ an−1 does not extend to anautomorphism.

Difference: it is not homogeneous over a finite relational language.Proof:

Suppose Aut(V) = Aut(V ,R1,R2, . . . ,Rk).Claim: (V ,R1,R2, . . . ,Rk) is not homogeneous.n > max(ar(Ri )), a1, a2, . . . , an linearly independent.Then ai 7→ ai : i < n, an 7→ a1 + a2 + · · ·+ an−1 does not extend to anautomorphism.

Difference: it is not homogeneous over a finite relational language.Proof:Suppose Aut(V) = Aut(V ,R1,R2, . . . ,Rk).

Claim: (V ,R1,R2, . . . ,Rk) is not homogeneous.n > max(ar(Ri )), a1, a2, . . . , an linearly independent.Then ai 7→ ai : i < n, an 7→ a1 + a2 + · · ·+ an−1 does not extend to anautomorphism.

Difference: it is not homogeneous over a finite relational language.Proof:Suppose Aut(V) = Aut(V ,R1,R2, . . . ,Rk).Claim: (V ,R1,R2, . . . ,Rk) is not homogeneous.

n > max(ar(Ri )), a1, a2, . . . , an linearly independent.Then ai 7→ ai : i < n, an 7→ a1 + a2 + · · ·+ an−1 does not extend to anautomorphism.

Difference: it is not homogeneous over a finite relational language.Proof:Suppose Aut(V) = Aut(V ,R1,R2, . . . ,Rk).Claim: (V ,R1,R2, . . . ,Rk) is not homogeneous.n > max(ar(Ri )), a1, a2, . . . , an linearly independent.

Then ai 7→ ai : i < n, an 7→ a1 + a2 + · · ·+ an−1 does not extend to anautomorphism.

Difference: it is not homogeneous over a finite relational language.Proof:Suppose Aut(V) = Aut(V ,R1,R2, . . . ,Rk).Claim: (V ,R1,R2, . . . ,Rk) is not homogeneous.n > max(ar(Ri )), a1, a2, . . . , an linearly independent.Then ai 7→ ai : i < n, an 7→ a1 + a2 + · · ·+ an−1 does not extend to anautomorphism.

Theorem (B., Kalina, Szabó) (Bossière, Bodirsky)

V = Fω2 has exactly 4 reducts.

Model theoretical formulation

1 the vector space V itself

2 the countably infinite set

3 the countably infinitedimensional affine space

4 the countably infinite setwith a constant 0

1 Aut(V), the automorphismgroup of V

2 Sym(V), the symmetricgroup

3 Aff(V) = Aut(V) n Tr, thegroup of affinetransformations on V

4 Sym(V)0, the stabilizer of 0in Sym(V)

Model theoretical formulation

1 the vector space V itself

Model theoretical formulation1 the vector space V itself

Theorem (B., Kalina, Szabó)

What if we add a constant?

Theorem (B., Cameron, Szabó)

(V, c) has infinitely many reducts.

In fact: there exists an infinite ascending chain of reducts.

The constructionAlgebraic description

We want: Aut(V, c) ≤ G0 < G1 < . . . closed groups.

Construction:

1 V = W ⊕ 〈c〉

2 Wn ≤W , codim(Wn) = n3 hn: flipping along Wn (u ↔ u + c iff u ∈Wn ⊕ 〈c〉)4 Gn = 〈Aut(V, c), hn〉

Observations:

Gn only depends on n.

Gn ⊂ Gn+1

Why are they different?

We want: Aut(V, c) ≤ G0 < G1 < . . . closed groups.Construction:

1 V = W ⊕ 〈c〉2 Wn ≤W , codim(Wn) = n

3 hn: flipping along Wn (u ↔ u + c iff u ∈Wn ⊕ 〈c〉)4 Gn = 〈Aut(V, c), hn〉

Observations:

Gn ⊂ Gn+1

1 V = W ⊕ 〈c〉

2 Wn ≤W , codim(Wn) = n3 hn: flipping along Wn (u ↔ u + c iff u ∈Wn ⊕ 〈c〉)

4 Gn = 〈Aut(V, c), hn〉

Observations:

Gn ⊂ Gn+1

1 V = W ⊕ 〈c〉2 Wn ≤W , codim(Wn) = n

3 hn: flipping along Wn (u ↔ u + c iff u ∈Wn ⊕ 〈c〉)4 Gn = 〈Aut(V, c), hn〉

Observations:

Gn ⊂ Gn+1

1 V = W ⊕ 〈c〉2 Wn ≤W , codim(Wn) = n3 hn: flipping along Wn (u ↔ u + c iff u ∈Wn ⊕ 〈c〉)

4 Gn = 〈Aut(V, c), hn〉Observations:

Gn ⊂ Gn+1

1 V = W ⊕ 〈c〉2 Wn ≤W , codim(Wn) = n3 hn: flipping along Wn (u ↔ u + c iff u ∈Wn ⊕ 〈c〉)4 Gn = 〈Aut(V, c), hn〉

Observations:

Gn ⊂ Gn+1Why are they different?

Observations:

Gn ⊂ Gn+1

Observations:

Definition

(x1, . . . , x2n) ∈ Rn iff

x1, x1 + c , . . . , x2n , x2n + c is a subspace

|{i : xi ∈W }| is even.

Remark

This is not a first-order definition.But! Aut(V, c) preserves Rn. Hence Rn is definable in (V, c).

hn preserves Rn+1. In fact: Gn = Aut(Rn+1)hn+1 does not preserve Rn+1.Consequence: Gn 6= Gn+1.

Definition

(x1, . . . , x2n) ∈ Rn iffx1, x1 + c , . . . , x2n , x2n + c is a subspace

Remark

Definition

Remark

Definition

Remark

This is not a first-order definition.

But! Aut(V, c) preserves Rn. Hence Rn is definable in (V, c).

Definition

Remark

This is not a first-order definition.But! Aut(V, c) preserves Rn.

Hence Rn is definable in (V, c).

Definition

Remark

Definition

Remark

hn preserves Rn+1.

In fact: Gn = Aut(Rn+1)hn+1 does not preserve Rn+1.Consequence: Gn 6= Gn+1.

Definition

Remark

hn preserves Rn+1.

In fact: Gn = Aut(Rn+1)

hn+1 does not preserve Rn+1.

Consequence: Gn 6= Gn+1.

Definition

Remark

hn preserves Rn+1.

In fact: Gn = Aut(Rn+1)

hn+1 does not preserve Rn+1.Consequence: Gn 6= Gn+1.

Definition

Remark

Remarks, future work

Remarks:

this result shows the limits of Thomas’ Conjecture

the construction is related to Reed–Muller codes

the construction works over Fqcountable atomless Boolean algebra has infinitely many reducts

Future work:

find all the reducts of (V, c)modify construction to disprove Thomas’ Conjecture

Remarks:

Future work:

Remarks:

the construction works over Fq

countable atomless Boolean algebra has infinitely many reducts

Future work:

Remarks:

Future work:

Remarks:

Future work:

find all the reducts of (V, c)

modify construction to disprove Thomas’ Conjecture

Remarks:

Future work:

Thank you for your attention!

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Bertalan Bodor, joint work with Peter Cameron and Csaba Szab o · 2017. 9. 19. · In nitely many reducts of homogeneous structures Bertalan Bodor, joint work with Peter Cameron and