Endomorphism monoids of ω-categorical structures

Transcription

1 Endomorphism monoids of ω-categorical structures Michael Kompatscher Institute of Computer Languages Technische Universität Wien TACL - 24/06/2015

2 ω-categorical structures A structure is called ω-categorical iff its theory has exactly one countable model.

3 ω-categorical structures A structure is called ω-categorical iff its theory has exactly one countable model. Theorem (Ryll-Nardzewski 59) A countable structure A is ω-categorical iff Aut(A) is oligomorphic: Every action Aut(A) A n has only finitely many orbits.

4 ω-categorical structures A structure is called ω-categorical iff its theory has exactly one countable model. Theorem (Ryll-Nardzewski 59) A countable structure A is ω-categorical iff Aut(A) is oligomorphic: Every action Aut(A) A n has only finitely many orbits. Definable relations = unions of orbits

5 ω-categorical structures A structure is called ω-categorical iff its theory has exactly one countable model. Theorem (Ryll-Nardzewski 59) A countable structure A is ω-categorical iff Aut(A) is oligomorphic: Every action Aut(A) A n has only finitely many orbits. Definable relations = unions of orbits Countable, ω-cat. structures A and B are interdefinable iff. Aut(A) = Aut(B)

6 Interpretability A surjective partial function I : A n B is called an interpretation iff every preimage of a relation in B is definable in A.

7 Interpretability A surjective partial function I : A n B is called an interpretation iff every preimage of a relation in B is definable in A. Theorem (Ahlbrandt and Ziegler 86) Two countable ω-categorical structures A, B are bi-interpretable iff Aut(A) = T Aut(B) with the topology of pointwise convergence.

8 Interpretability A surjective partial function I : A n B is called an interpretation iff every preimage of a relation in B is definable in A. Theorem (Ahlbrandt and Ziegler 86) Two countable ω-categorical structures A, B are bi-interpretable iff Aut(A) = T Aut(B) with the topology of pointwise convergence. What about Aut(A) as abstract group?

9 Interpretability A surjective partial function I : A n B is called an interpretation iff every preimage of a relation in B is definable in A. Theorem (Ahlbrandt and Ziegler 86) Two countable ω-categorical structures A, B are bi-interpretable iff Aut(A) = T Aut(B) with the topology of pointwise convergence. What about Aut(A) as abstract group? Can we reconstruct the topology of Aut(A)?

10 Versions of interpretability More refined notion of interpretability with:

11 Versions of interpretability More refined notion of interpretability with: The endomorphisms monoid End(A): All the homomorphisms h : A A

12 Versions of interpretability More refined notion of interpretability with: The endomorphisms monoid End(A): All the homomorphisms h : A A The polymorphism clone Pol(A): All the homomorphism h : A n A for 1 n < ω

13 Versions of interpretability More refined notion of interpretability with: The endomorphisms monoid End(A): All the homomorphisms h : A A The polymorphism clone Pol(A): All the homomorphism h : A n A for 1 n < ω acting on A topologically abstract Aut(A) first-order first-order interdefinability bi-interpretability End(A) positive existential positive existential interdefinability bi-interpretability* Pol(A) primitive positive primitive positive interdefinability bi-interpretability

14 Versions of interpretability More refined notion of interpretability with: The endomorphisms monoid End(A): All the homomorphisms h : A A The polymorphism clone Pol(A): All the homomorphism h : A n A for 1 n < ω acting on A topologically abstract Aut(A) first-order first-order? interdefinability bi-interpretability End(A) positive existential positive existential? interdefinability bi-interpretability* Pol(A) primitive positive primitive positive? interdefinability bi-interpretability

15 Reconstruction Questions Can we reconstruct the topology of a closed oligomorphic permutation group transformation monoid function clone from its abstract algebraic structure?

16 Reconstruction Questions Can we reconstruct the topology of a closed oligomorphic permutation group transformation monoid function clone from its abstract algebraic structure? No! (Evans + Hewitt 90; Bodirsky + Evans + Pinsker + MK 15)

17 Profinite groups without reconstruction Is there any closed subgroup of S ω without reconstruction?

18 Profinite groups without reconstruction Is there any closed subgroup of S ω without reconstruction? ZF+DC is consistent with the statement that every isomorphism between closed subgroups of S ω is a homeomorphism.

19 Profinite groups without reconstruction Is there any closed subgroup of S ω without reconstruction? ZF+DC is consistent with the statement that every isomorphism between closed subgroups of S ω is a homeomorphism. So from now on work in ZFC.

20 Profinite groups without reconstruction Is there any closed subgroup of S ω without reconstruction? ZF+DC is consistent with the statement that every isomorphism between closed subgroups of S ω is a homeomorphism. So from now on work in ZFC. Profinite groups are closed permutation groups where every orbits contains finitely many elements. Example (Witt 54) There are two separable profinite groups G, G that are isomorphic, but not topologically isomorphic.

21 Encoding profinite groups with oligomorphic groups Lift the result to oligomorphic groups:

22 Encoding profinite groups with oligomorphic groups Lift the result to oligomorphic groups: Lemma (Hrushovski) There is a oligomorphic Φ such that for every separable profinite group R there is an oligomorphic Σ R : Σ R /Φ = T R. Φ is the intersection of open subgroups of finite index in Σ R

23 Encoding profinite groups with oligomorphic groups Lift the result to oligomorphic groups: Lemma (Hrushovski) There is a oligomorphic Φ such that for every separable profinite group R there is an oligomorphic Σ R : Σ R /Φ = T R. Φ is the intersection of open subgroups of finite index in Σ R Proof idea: R n 1 Sym(n).

24 Encoding profinite groups with oligomorphic groups Lift the result to oligomorphic groups: Lemma (Hrushovski) There is a oligomorphic Φ such that for every separable profinite group R there is an oligomorphic Σ R : Σ R /Φ = T R. Φ is the intersection of open subgroups of finite index in Σ R Proof idea: R n 1 Sym(n). Look at finite sets. Partition the n-tuples into partition classes P n 1, Pn 2,... Pn n for all n 1. This gives us a Fraïssé-class.

25 Encoding profinite groups with oligomorphic groups Let A = (A, (P n i ) i,n) be the Fraïssé-limit; Φ = Aut(A)

26 Encoding profinite groups with oligomorphic groups Let A = (A, (P n i ) i,n) be the Fraïssé-limit; Φ = Aut(A) Forget about the labelling equivalence relations E n

27 Encoding profinite groups with oligomorphic groups Let A = (A, (P n i ) i,n) be the Fraïssé-limit; Φ = Aut(A) Forget about the labelling equivalence relations E n Σ = Aut(A, (E n ) n N )

28 Encoding profinite groups with oligomorphic groups Let A = (A, (P n i ) i,n) be the Fraïssé-limit; Φ = Aut(A) Forget about the labelling equivalence relations E n Σ = Aut(A, (E n ) n N ) We can think of Σ acting on the partition classes P n 1, Pn 2,... Pn n.

29 Encoding profinite groups with oligomorphic groups Let A = (A, (P n i ) i,n) be the Fraïssé-limit; Φ = Aut(A) Forget about the labelling equivalence relations E n Σ = Aut(A, (E n ) n N ) We can think of Σ acting on the partition classes P n 1, Pn 2,... Pn n. This gives us Σ/Φ = T n N Sym(n).

30 Permutation groups Idea Use the encoding lemma to show:

31 Permutation groups Idea Use the encoding lemma to show: G T G Σ G T Σ G

32 Permutation groups Idea Use the encoding lemma to show: G T G Σ G T Σ G G = G Σ G = ΣG

33 Permutation groups Idea Use the encoding lemma to show: G T G Σ G T Σ G G = G Σ G = ΣG Problem: We do not know if Σ G = ΣG for G = G.

34 Permutation groups Idea Use the encoding lemma to show: G T G Σ G T Σ G G = G Σ G = ΣG Problem: We do not know if Σ G = ΣG for G = G. The real proof deviates from the above.

35 Lifting to the monoid closure Let Σ R be the topological closure of Σ R in ω ω.

36 Lifting to the monoid closure Let Σ R be the topological closure of Σ R in ω ω. Lemma The quotient homomorphism Σ R R extends to a continuous monoid homomorphism Σ R R with kernel Φ.

37 Lifting to the monoid closure Let Σ R be the topological closure of Σ R in ω ω. Lemma The quotient homomorphism Σ R R extends to a continuous monoid homomorphism We get: Result for monoids Σ R R with kernel Φ. Σ G and Σ G are isomorphic, but not topologically isomorphic.

38 Oligomorphic clones Observation Let I : Γ be a monoid homomorphism. If I sends constants to constants, it has a natural extension to a clone homomorphism Clo(Γ) Clo( ).

39 Oligomorphic clones Observation Let I : Γ be a monoid homomorphism. If I sends constants to constants, it has a natural extension to a clone homomorphism Clo(Γ) Clo( ). Result for clones The clones Clo(Σ G ) and Clo(Σ G ) are isomorphic but not topologically isomorphic.

40 Oligomorphic clones Observation Let I : Γ be a monoid homomorphism. If I sends constants to constants, it has a natural extension to a clone homomorphism Clo(Γ) Clo( ). Result for clones The clones Clo(Σ G ) and Clo(Σ G ) are isomorphic but not topologically isomorphic. This answers a question by Bodirsky, Pinsker and Pongrácz.

41 Thank you!