Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

Geometry of the pure n-ary ab initioHrushovski construction

Omer MermelsteinBen-Gurion University of the Negev

Set Theory, Model Theory and Appplications, Eilat 201826.04.2018

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

DefinitionA combinatorial pregeometry is a (assume countable) set X with adimension function d : P(X )→ N ∪ {∞} such that for allY ,Z ∈ P(X ), x ∈ X :

1. d(Y ) ≤ |Y |2. d(Y ) ≤ d(Yx) ≤ d(Y ) + 1

3. d(Y ∪ Z ) ≤ d(Y ) + d(Z )− d(Y ∩ Z ) (submodular)

*4. d(Y ) = sup {d(Y0) | Y0 ∈ Fin(Y )} (finitary)

Examples: cardinality, linear dimension, transcendence degree.

DefinitionA set Y is closed if for any x

d(Yx) = d(Y ) =⇒ x ∈ Y .

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

DefinitionA finite set Y is independent if d(Y ) = |Y |.An infinite set Y is independent if all its finite subsets areindependent.

A pregeometry is uniquely determined by its set ofdependent/independent finite tuples.A pregeometry can be identified with a first-order structure in thelanguage {Dk | k ∈ ω} by interpreting Dk as the set of dependentk-tuples.

DefinitionA pregeometry on a set X is n-pure if Dn = ∅. Equivalently, everysubset of X of size n is independent.

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

DefinitionA finite set Y is independent if d(Y ) = |Y |.An infinite set Y is independent if all its finite subsets areindependent.

A pregeometry is uniquely determined by its set ofdependent/independent finite tuples.A pregeometry can be identified with a first-order structure in thelanguage {Dk | k ∈ ω} by interpreting Dk as the set of dependentk-tuples.

DefinitionA pregeometry on a set X is n-pure if Dn = ∅. Equivalently, everysubset of X of size n is independent.

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

DefinitionA finite set Y is independent if d(Y ) = |Y |.An infinite set Y is independent if all its finite subsets areindependent.

A pregeometry is uniquely determined by its set ofdependent/independent finite tuples.A pregeometry can be identified with a first-order structure in thelanguage {Dk | k ∈ ω} by interpreting Dk as the set of dependentk-tuples.

DefinitionA pregeometry on a set X is n-pure if Dn = ∅. Equivalently, everysubset of X of size n is independent.

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

Fraısse’s TheoremLet C be a countable (up to isomorphisms) class of finitelygenerated structures, closed under isomorphisms. Let 6 be adistinguished notion of embedding, preserved under isomorphism.Assume

HP A 6 B, B ∈ C =⇒ A ∈ C.

JEP A,B ∈ C =⇒ ∃D ∈ C s.t. A,B 6 D.

AP ∀A,B1,B2 ∈ C

B1

A D

B2

66

6 6

Then there exists a unique (up to isomorphism) countable genericstructure M with age6(M) = C such that ∀A,B ∈ C

B

A M

66

6

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

DefinitionFor any finite hypergraph A = (V ,E ) define

δ(A) = |V | − |E |

and for any induced subgraph B ⊆ A define

dA(B) = min {δ(B ′) | B ⊆ B ′ ⊆fin A}B 6 A ⇐⇒ δ(B) = dA(B)

The function dA is the dimension function of a pregeometry on Awhenever ∅ 6 A.

DefinitionSay that B strongly embeds into A if there exists an embeddingf : B → A such that f [B] 6 A.

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

Hrushovski’s constructionLet C = {A | ∅ 6 A}. The class C is a Fraısse amalgamation classwith respect to 6-embeddings. We denote its generic structure Mand its associated pregeometry G.

Variations

I Restrict C to n-uniform hypergraphs — Mn

Fact: n 6= k =⇒ Gn 6∼= Gk (Evans and Ferreira)

I Consider directed hypergraphs — M6∼Fact: G 6∼ ∼= G

I Restrict to hypergraphs whose geometries are n-pure — Mn

For a fixed n, we will construct the pregeometry Mn+1n – the

n-pure (n + 1)-ary construction – as a Fraısse-Hrushovski limit.

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

Hrushovski’s constructionLet C = {A | ∅ 6 A}. The class C is a Fraısse amalgamation classwith respect to 6-embeddings. We denote its generic structure Mand its associated pregeometry G.

Variations

I Restrict C to n-uniform hypergraphs — Mn

Fact: n 6= k =⇒ Gn 6∼= Gk (Evans and Ferreira)

I Consider directed hypergraphs — M6∼Fact: G 6∼ ∼= G

I Restrict to hypergraphs whose geometries are n-pure — Mn

For a fixed n, we will construct the pregeometry Mn+1n – the

n-pure (n + 1)-ary construction – as a Fraısse-Hrushovski limit.

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

Hrushovski’s constructionLet C = {A | ∅ 6 A}. The class C is a Fraısse amalgamation classwith respect to 6-embeddings. We denote its generic structure Mand its associated pregeometry G.

Variations

I Restrict C to n-uniform hypergraphs — Mn

Fact: n 6= k =⇒ Gn 6∼= Gk (Evans and Ferreira)

I Consider directed hypergraphs — M6∼Fact: G 6∼ ∼= G

I Restrict to hypergraphs whose geometries are n-pure — Mn

For a fixed n, we will construct the pregeometry Mn+1n – the

n-pure (n + 1)-ary construction – as a Fraısse-Hrushovski limit.

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

Hrushovski’s constructionLet C = {A | ∅ 6 A}. The class C is a Fraısse amalgamation classwith respect to 6-embeddings. We denote its generic structure Mand its associated pregeometry G.

Variations

I Restrict C to n-uniform hypergraphs — Mn

Fact: n 6= k =⇒ Gn 6∼= Gk (Evans and Ferreira)

I Consider directed hypergraphs — M6∼Fact: G 6∼ ∼= G

I Restrict to hypergraphs whose geometries are n-pure — Mn

For a fixed n, we will construct the pregeometry Mn+1n – the

n-pure (n + 1)-ary construction – as a Fraısse-Hrushovski limit.

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

Our first goal is assigning n-pure pregeometries associated to(n + 1)-ary hypergraphs a predimension.

PredimensionThe standard δ does not work – a graph of a pregeometry mayhave too many edges. The solution is to calculate predimensionusing cliques, instead of individual edges.A closed set Y of dimension n has on it all possible (n + 1)-edges.Since its dimension should be n, we define

λ(Y ) = |Y | − (|Y | − n)

Generalizing, for an n-pure pregeometry A with M(A) its set ofclosed sets of dimension (n + 1) we define

λ(A) = |A| −∑

C∈M(A)

(|C | − n)

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

Free amalgamation does not work in the case of pregeometries.

Example

Let A, B, C be (n + 1)-uniform cliques (sets of dimension n) withA ⊂ B,C , |A| > n.The hypergraph on B ∪C with set of hyperedges EB ∪ EC is not ahypergraph representing a pregeometry.

The problem is that two distinct “closed sets” have too big of anintersection, and so they must be the same closed set.Treating a union of intersecting closed sets as a single closed setsolves the problem.

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

Free amalgamation does not work in the case of pregeometries.

Example

Let A, B, C be (n + 1)-uniform cliques (sets of dimension n) withA ⊂ B,C , |A| > n.The hypergraph on B ∪C with set of hyperedges EB ∪ EC is not ahypergraph representing a pregeometry.

The problem is that two distinct “closed sets” have too big of anintersection, and so they must be the same closed set.Treating a union of intersecting closed sets as a single closed setsolves the problem.

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

DefinitionDefine Cgeo

0 to be the class of finite n-pure pregeometriesassociated to (n + 1)-uniform hypergraphs.Define Cgeo = {A � Dn+1 | A ∈ Cgeo

0 }.With respect to the predimension function λ and the alteredamalgam, Cgeo is a Fraısse-Hrushovski amalgamation class.Denote its generic Mgeo .

Proposition

The pregeometry on Mgeo given by the predimension function λ, isprecisely Gn+1

n – the pregeometry given by the predimensionfunction δ on Mn+1

n .Moreover, the hyperedges of Mgeo are exactly Dn+1 of Gn+1

n .

The dependent (n + 1)-tuples are sufficient information tocalculate the dimension of any subset of Gn+1

n , so in some sensewe could say that Gn+1

n is a Fraısse-Hrushovski limit.However, that is not exactly the case...

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

DefinitionDefine Cgeo

0 to be the class of finite n-pure pregeometriesassociated to (n + 1)-uniform hypergraphs.Define Cgeo = {A � Dn+1 | A ∈ Cgeo

0 }.With respect to the predimension function λ and the alteredamalgam, Cgeo is a Fraısse-Hrushovski amalgamation class.Denote its generic Mgeo .

Proposition

The pregeometry on Mgeo given by the predimension function λ, isprecisely Gn+1

n – the pregeometry given by the predimensionfunction δ on Mn+1

n .Moreover, the hyperedges of Mgeo are exactly Dn+1 of Gn+1

n .

The dependent (n + 1)-tuples are sufficient information tocalculate the dimension of any subset of Gn+1

n , so in some sensewe could say that Gn+1

n is a Fraısse-Hrushovski limit.However, that is not exactly the case...

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

DefinitionDefine Cgeo

0 to be the class of finite n-pure pregeometriesassociated to (n + 1)-uniform hypergraphs.Define Cgeo = {A � Dn+1 | A ∈ Cgeo

0 }.With respect to the predimension function λ and the alteredamalgam, Cgeo is a Fraısse-Hrushovski amalgamation class.Denote its generic Mgeo .

Proposition

The pregeometry on Mgeo given by the predimension function λ, isprecisely Gn+1

n – the pregeometry given by the predimensionfunction δ on Mn+1

n .Moreover, the hyperedges of Mgeo are exactly Dn+1 of Gn+1

n .

The dependent (n + 1)-tuples are sufficient information tocalculate the dimension of any subset of Gn+1

n , so in some sensewe could say that Gn+1

n is a Fraısse-Hrushovski limit.However, that is not exactly the case...

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

DefinitionDefine Cgeo

0 to be the class of finite n-pure pregeometriesassociated to (n + 1)-uniform hypergraphs.Define Cgeo = {A � Dn+1 | A ∈ Cgeo

0 }.With respect to the predimension function λ and the alteredamalgam, Cgeo is a Fraısse-Hrushovski amalgamation class.Denote its generic Mgeo .

Proposition

The pregeometry on Mgeo given by the predimension function λ, isprecisely Gn+1

n – the pregeometry given by the predimensionfunction δ on Mn+1

n .Moreover, the hyperedges of Mgeo are exactly Dn+1 of Gn+1

n .

The dependent (n + 1)-tuples are sufficient information tocalculate the dimension of any subset of Gn+1

n , so in some sensewe could say that Gn+1

n is a Fraısse-Hrushovski limit.However, that is not exactly the case...

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

The structures differ in language: Mgeo is an n + 1-uniformhypergraph, whereas Gn+1

n is an expansion that has edges of allarities.Combinatorially, the two hypergraphs hold the exact sameinformation, but this information is not first-order.

Example

Mgeo is ω-stable and saturated.Gn+1

n is not saturated.Consider the type of a dependent (n + 2)-tuple whose every(n + 1)-subtuple is independent.

In particular, the relations {Di}i>n+1 are not definable in Mgeo .They are strictly V -definable.

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

The structures differ in language: Mgeo is an n + 1-uniformhypergraph, whereas Gn+1

n is an expansion that has edges of allarities.Combinatorially, the two hypergraphs hold the exact sameinformation, but this information is not first-order.

Example

Mgeo is ω-stable and saturated.Gn+1

n is not saturated.Consider the type of a dependent (n + 2)-tuple whose every(n + 1)-subtuple is independent.

In particular, the relations {Di}i>n+1 are not definable in Mgeo .They are strictly V -definable.

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

The structures differ in language: Mgeo is an n + 1-uniformhypergraph, whereas Gn+1

n is an expansion that has edges of allarities.Combinatorially, the two hypergraphs hold the exact sameinformation, but this information is not first-order.

Example

Mgeo is ω-stable and saturated.Gn+1

n is not saturated.Consider the type of a dependent (n + 2)-tuple whose every(n + 1)-subtuple is independent.

In particular, the relations {Di}i>n+1 are not definable in Mgeo .They are strictly V -definable.

Geometry of the puren-ary Hrushovski

construction

Omer Mermelstein

Preliminaries

Combinatorialpregeometries

Fraısse amalgamation

Ab initio construction

Geometric construction

Predimension

Amalgamation

The generic structure

Fine print

Questions

Questions - first order

1. Is Th(Gn+1n ) ω-stable?

2. Is Th(Gn+1n ) = Th(Gm+1

m ) for m, n ≥ 3?

3. Does Gn+1n have any proper (geometrically) non-trivial

reducts?

When the purity and arity differ, say Gn+2n , naive generalizations of

λ fail submodularity. The magic in the case of (n, n + 1) is thatthe class of n-pure (n + 1)-ary pregeometries is closed undersubstructures.The construction of Gn+2

n can be carried out from a class ofpregeometries (Evans-Ferreira) that is not closed undersubstructures.

Question - predimension construction

Can the construction of Gn+2n be carried out as a

Fraısse-Hrushovski predimension construction?