Simple homogeneous structures - helsinki.fi · Simple homogeneous structures ... 3-8 August 2015, Helsinki Vera Koponen Simple homogeneous structures. Introduction Homogeneous structures

Simple homogeneous structures

Vera Koponen

Department of MathematicsUppsala University

Logic Colloquium, 3-8 August 2015, Helsinki

Vera Koponen Simple homogeneous structures

Introduction

Homogeneous structures have interesting properties from amodel theoretic point of view. They also play a role in constraintsatisfaction, Ramsey theory, permutation group theory andtopological dynamics.

The study of simple theories/structures has developed, viastability theory, from Shelah’s classification theory of completefirst-order theories.

A central tool in this context is a “well behaved” notion ofindependence.

I will present some results, and problems, about structures that areboth simple and homogeneous.


Introduction






Introduction






Introduction






Homogeneous structures: definitions

Suppose that V is a finite and relational vocabulary/signature.

A countable V -structure M is homogeneous if the followingequivalent conditions are satisfied:

1 M has elimination of quantifiers.

2 Every isomorphism between finite substructures of M can beextended to an automorphism of M.

3 M is the Fraısse limit of an amalgamation class of finitestructures.

Examples: The random graph, or Rado graph; (Q, <); generictriangle-free graph; more generally, 2ℵ0 examples constructed byforbidding substructures (Henson 1972).

Via the Engeler, Ryll-Nardzewski, Svenonious characterization ofω-categorical theories: every infinite homogeneous structure hasω-categorical complete theory.





























Classifications of some homogeneous structures

Being homogeneous is a strong condition when restricted tocertain classes of structures.

The following classes of structures, to mention a few, have beenclassified:

1 homogeneous partial orders (Schmerl 1979).

2 homogeneous (undirected) graphs (Gardiner; Golfand, Klin;Sheehan; Lachlan, Woodrow; 1974–1980).

3 homogeneous tournaments (Lachlan 1984).

4 homogeneous directed graphs (Cherlin 1998).

5 infinite homogeneous stable V -structures for any finiterelational vocabulary V (Lachlan, Cherlin, Knight... 80ies).

6 homogeneous multipartite graphs (Jenkinson, Truss, Seidel2012).


Classifications of some homogeneous structures

Being homogeneous is a strong condition when restricted tocertain classes of structures.The following classes of structures, to mention a few, have beenclassified:

1 homogeneous partial orders (Schmerl 1979).

2 homogeneous (undirected) graphs (Gardiner; Golfand, Klin;Sheehan; Lachlan, Woodrow; 1974–1980).

3 homogeneous tournaments (Lachlan 1984).

4 homogeneous directed graphs (Cherlin 1998).

5 infinite homogeneous stable V -structures for any finiterelational vocabulary V (Lachlan, Cherlin, Knight... 80ies).

6 homogeneous multipartite graphs (Jenkinson, Truss, Seidel2012).


Homogeneous structures in general and the constraints weimpose

The class of all homogeneous structures seems to be too diverse tobe captured by some relatively uniform methods and results.

So we restrict ourselves to simple homogeneous structures,because in this context we have tools to work with.

In addition, the main results presented are about binarystructures.

If the vocabulary of M has only unary and/or binary relationsymbols, then we say that M (and its vocabulary) is binary.

Reasons for this restriction:(a) restricting the arity in a context of elimination of quantifiersmay simplify things.(b) the “independence theorem” of simple structures turns out tocombine “nicely” with the binarity assumption.























Supersimple theories/structures

We call T , as well as every M |= T , supersimple if for every typep (over any set) of T , SU(p) = α for some ordinal α.

The SU-rank of T, as well as of every M |= T , is the supremum(if it exists) of the SU-ranks of all 1-types over ∅ of T .

If the SU-rank of T is finite, then (by the Lascar inequalities)SU(p) < ω for every type p of T of any arity.

From hereon ’rank’ means ’SU-rank’.


Supersimple theories/structures

We call T , as well as every M |= T , supersimple if for every typep (over any set) of T , SU(p) = α for some ordinal α.

The SU-rank of T, as well as of every M |= T , is the supremum(if it exists) of the SU-ranks of all 1-types over ∅ of T .

If the SU-rank of T is finite, then (by the Lascar inequalities)SU(p) < ω for every type p of T of any arity.

From hereon ’rank’ means ’SU-rank’.


Examples

The following structures are ω-categorical and supersimple withfinite rank.

1 vector space over a finite field.

2 ω-categorical superstable structures (where superstable =supersimple + stable).

3 smoothly approximable structures (which contains theprevious class but need not be stable).

4 random graph/structure has rank 1 and is homogeneous.

5 random k-partite graph/structure (k < ω) has rank 1 and canbe made homogeneous by expanding with a binary relation.

6 the line graph of an infinite complete graph has rank 2 and canbe made homogeneous by expanding with a ternary relation.


More examples

1 Suppose that R is a binary random structure. Let n < ω andlet M be a binary structure with universe M = Rn such thatfor all a, a′, b, b′ ∈ M tpatM(a, a′) = tpatM(b, b′) if and only iftpR(aa′) = tpR(bb′). Then M is homogeneous andsupersimple with rank n.

2 For any n < ω one can construct a homogeneous supersimplestructure with rank n by “nesting” n different equivalencerelations (with infinitely many infinite classes).

3 The previous two ways of constructing (binary) homogeneoussupersimple structures of finite rank can be combined toproduce more complex examples.


Random structures

Let V be a finite relational vocabulary with maximal arity r andlet M be an infinite countable homogeneous V -structure.

A forbidden configuration (w.r.t. M) is a V -structure whichcannot be embedded into M.

A minimal forbidden configuration (w.r.t. M) is a forbiddenconfiguration A such that no proper substructure of A is aforbidden configuration (w.r.t. M).

M is a random structure if it does not have a minimal forbiddenconfiguration of cardinality ≥ r + 1. If M is binary then we call ita binary random structure.1

Example: Rado graph.Nonexamples: bipartite graphs (odd cycles are forbidden), thegeneric tetrahedron-free 3-hypergraph.

1The definition of a random structure (for r > 2) has some deficiencies ingeneral (so I will change the definition somewhat), but the given definitionnevertheless makes sense for the results presented here.


Random structures








Random structures








Random structures








Finiteness of rank

All known examples of simple homogeneous structures aresupersimple with finite SU-rank.

Question: Does every supersimple homogeneous structure havefinite rank?

In the case of binary simple homogeneous structures the answer isyes:

Theorem. [K14] Suppose that M is a structure which is binary,simple and homogeneous. Then M is supersimple with finite rank(which is bounded by the number of 2-types over ∅).


Finiteness of rank

All known examples of simple homogeneous structures aresupersimple with finite SU-rank.

Question: Does every supersimple homogeneous structure havefinite rank?

In the case of binary simple homogeneous structures the answer isyes:

Theorem. [K14] Suppose that M is a structure which is binary,simple and homogeneous. Then M is supersimple with finite rank(which is bounded by the number of 2-types over ∅).


1-basedness and triviality of dependence/forking

Let T be simple.

T , as well as every M |= T , is 1-based if and only if for allM |= T and all A,B ⊆Meq we have A |

CB where

C = aclMeq(A) ∩ aclMeq(B). (‘aclMeq ’ is the algebraic closure inMeq)

T , as well as every M |= T , has trivial dependence if wheneverM |= T , A,B,C ⊆Meq and A |�

CB, then there is b ∈ B such that

A |�C{b}.

Remark. All known simple homogenous structures are 1-based, orappears to be so. (Checking 1-basedness is not necessarily a trivialmatter.)



Let T be simple.


CB where




A |�C{b}.




Let T be simple.


CB where




A |�C{b}.




Suppose that M is simple and homogeneous. As aconsequence of results of Macpherson, Hart–Kim–Pillay and DePiro–Kim, the following are equivalent:

1 M is 1-based.

2 M has finite rank and trivial dependence/forking.

3 M has finite rank and every type (with respect to Meq) ofrank 1 is 1-based.

4 M has finite rank and every type (with respect to Meq) ofrank 1 has trivial pregeometry (given by algebraic closure).

If we also assume that M is binary, then – by the previous theorem– we can remove “M has finite rank” from (2)–(4).



Problems:

Is there a simple homogeneous structure which is not 1-based?

Is there a simple homogeneous structure with a nontrivial typeof rank 1?

(In the binary case the questions are equivalent.)


Canonically embedded structures

Suppose that A ⊆ M and that C ⊆ Meq is A-definable.

The canonically embedded structure in Meq over A withuniverse C is the structure C which, for every 0 < n < ω andA-definable relation R ⊆ Cn, has a relation symbol which isinterpreted as R (and C has no other symbols).

Note that for all 0 < n < ω and all R ⊆ Cn,R is ∅-definable in C ⇐⇒ R is A-definable in Meq.












Canonically embedded structures with rank 1

Let M and N be two structures which need not necessarily havethe same vocabulary.

N is a reduct of M if M = N and for all 0 < n < ω and allR ⊆ Mn, if R is ∅-definable in N then R is ∅-definable in M.

Theorem 3. (Ahlman and K, [AK]) Suppose that M is a binary,homogeneous, simple structure with trivial dependence. LetA ⊆ M be finite and suppose that C ⊆ Meq is A-definable andonly finitely many 1-types over ∅ are realized in C . Assume thattp(c/A) has rank 1 for every c ∈ C . Let C be the canonicallyembedded structure of Meq over A with universe C . Then C is areduct of a binary random structure.


Canonically embedded structures with rank 1

Let M and N be two structures which need not necessarily havethe same vocabulary.

N is a reduct of M if M = N and for all 0 < n < ω and allR ⊆ Mn, if R is ∅-definable in N then R is ∅-definable in M.

Theorem 3. (Ahlman and K, [AK]) Suppose that M is a binary,homogeneous, simple structure with trivial dependence. LetA ⊆ M be finite and suppose that C ⊆ Meq is A-definable andonly finitely many 1-types over ∅ are realized in C . Assume thattp(c/A) has rank 1 for every c ∈ C . Let C be the canonicallyembedded structure of Meq over A with universe C . Then C is areduct of a binary random structure.


Primitive homogeneous 1-based structures

M is primitive if there is no nontrivial equivalence relation on Mwhich is ∅-definable.

Theorem. [K15] Suppose that M is a binary, primitive,homogeneous, simple and 1-based structure. Then M has rank 1.

Theorem. (A. Aranda Lopez, [AL, Proposition 3.3.3]) Supposethat M is a binary, primitive, homogeneous and simple with rank1. Then M is a random structure.

Corollary. Suppose that M is a binary, primitive, homogeneous,simple and 1-based structure. Then M is a random structure.





















I do not know if the first theorem of the previous page holds if thebinarity condition is removed. But the binarity condition cannot beremoved from the second theorem on the same page, nor from thecorollary, as witnessed by the generic tetrahedron-free3-hypergraph.

Proposition.The generic tetrahedron-free 3-hypergraph isprimitive, homogeneous, simple, 1-based and has rank 1, but is nota random structure.

All mentioned properties of the tetrahedron-free 3-hypergraph,except the 1-basedness, are known earlier. To prove that it is1-based, a characterization of the ∅-definable equivalence relationson n-tuples (0 < n < ω) in the context of ω-categorical structureswith degenerate algebraic closure is carried out in [K15].












0-1 laws and probabilistic limits of finite substructures

Let V be a vocabulary. For every 0 < n < ω, let Kn be a set ofV -structures with universe [n] = {1, . . . , n}.

Let µn be the uniform probability measure on Kn.

The almost sure theory of K = (Kn, µn) is

T(Kn,µn) = {ϕ : limn→∞

µn(ϕ) = 1}.

We say that (Kn, µn) has a 0-1 law if for every ϕ, limn→∞ µn(ϕ)exists and is either 0 or 1. This is equivalent to T(Kn,µn) beingcomplete.

We say that a structure M is a probabilistic limit of its finitesubstructures if the following holds:

if Kn ={A : A = [n] and A ↪→M

}then (Kn, µn) has a 0-1

law and T(Kn,µn) = Th(M).







µn(ϕ) = 1}.












µn(ϕ) = 1}.








Some random structures, such as the Rado graph, areprobabilistic limits of their finite substructures.

Other (even binary) random structures are not.

Example. Let V = {P,Q} where P is unary and Q is binary andlet M be a countable V -structure such that

both P(M) and ¬P(M) are infinite,

if P(x) and P(y) then ¬Q(x , y), and

M�¬P(M) is the Rado graph.

Then M is a binary random structure but not a probabilistic limitof its finite substructures, because with probability approaching 0there will be no element satisfying P.



Some random structures, such as the Rado graph, areprobabilistic limits of their finite substructures.

Other (even binary) random structures are not.

Example. Let V = {P,Q} where P is unary and Q is binary andlet M be a countable V -structure such that

both P(M) and ¬P(M) are infinite,

if P(x) and P(y) then ¬Q(x , y), and

M�¬P(M) is the Rado graph.

Then M is a binary random structure but not a probabilistic limitof its finite substructures, because with probability approaching 0there will be no element satisfying P.



Theorem. [Ahlman, article in preparation] Suppose that M isbinary, ω-categorical and simple with SU-rank 1 and degeneratealgebraic closure. Then there is a binary random structure M′

such that

(a) M is a reduct of M′ and

(b) M′ is a probabilistic limit of its finite substructures.

Remark. (i) Suppose that M satisfies all assumptions of theabove theorem and, in addition, is primitive. Then M is a randomstructure and a probabilistic limit of its finite substructures.

(ii) It also follows from [Ahlman] that if M satisfies the hypothesesof the above theorem, then M is homogenizable (one gets ahomogeneous structure from M by expanding it, if necessary, withfinitely many unary and/or binary relations symbols).




such that








such that







If we remove the binarity assumption from the previous remark,then it fails, in two ways. For example:

Let H be the generic tetrahedron-free 3-hypergraph. Then

H is not a random structure, and

H is not a probabilistic limit of its finite substructures,because even if Kn = {A : A = [n],A ↪→ H} has a 0-1 law2

we have TK 6= Th(M) (which follows from Theorem 3.4 in[K12]).

2The answer is unknown.Vera Koponen Simple homogeneous structures


Problem. Characterize the primitive (nonbinary) homogeneous (orω-categorical with degenerate algebraic closure) simple structuresM with rank 1 that are probabilistic limits of their finitesubstructures.

By Theorem 3.4 and Remark 3.5 in [K12] we can exclude thosehomogeneous M which are defined by forbidding someindecomposable A1, . . . ,Ak in the weak substructure/embeddingsense.

Problem. Suppose that M is a primitive (nonbinary)homogeneous (or ω-categorical with degenerate algebraic closure)simple structure which is not a probabilistic limit of its finitesubstructures. Under what conditions does {A : A = [n],A ↪→M}nevertheless have a 0-1 law? What properties does the almost suretheory have (if it is complete)?













For example: Let H be the generic tetrahedron-free3-hypergraph and let Kn = {A : A = [n],A ↪→ H}. We know thatH is not a probabilistic limit of its finite substructures. But watcan we say about the asymptotic behaviour of Kn as n→∞?


References

[AK] O. Ahlman, V. Koponen, On sets with rank one in simplehomogeneous structures, Fundamenta Mathematicae, Vol. 228 (2015)223–250.

[AL] A. Aranda Lopez, Omega-categorical Simple Theories, Ph.D. thesis,University of Leeds, 2014.

[K12] V. Koponen, Asymptotic probabilities of extension properties andrandom l-colourable structures, Annals of Pure and Applied Logic, Vol.163 (2012) 391-438.

The following articles can be found via the linkhttp://www2.math.uu.se/~vera/research/index.html

and on arXiv:

[K14] V. Koponen, Binary simple homogeneous structures aresupersimple with finite rank. to appear in Proceedings of the AmericanMathematical Society.

[K15] V. Koponen, Binary primitive homogeneous one-based structures.

Thanks for listening!


http://www2.math.uu.se/~vera/research/index.html

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Simple homogeneous structures - helsinki.fi · Simple homogeneous structures ... 3-8 August 2015, Helsinki Vera Koponen Simple homogeneous structures. Introduction Homogeneous structures