Searching for template structures for the class of ...omer/BPGMT18Presentation.pdf · Intuition...

Searching for ab initioTemplates

Omer Mermelstein

Preliminaries

Intuition

Combinatorial geometries

Fraısse amalgamation

The ab initioconstruction

Generalizing thepredimension

Motivation

Clique predimension

The geometricconstruction

Searching for template structures for theclass of Hrushovski ab initio geometries

Omer MermelsteinBen-Gurion University of the Negev

BPGMTC 2018

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

Intuition

δ(A) = |A| − |E |

d(A,B) = inf {δ(B ′) | A ⊆ B ′ ⊆fin B}

A 6 B ⇐⇒ δ(A) = d(A,B)

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

Intuition

δ(A) = |A| − |E |

d(A,B) = inf {δ(B ′) | A ⊆ B ′ ⊆fin B}

A 6 B ⇐⇒ δ(A) = d(A,B)

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

Intuition

δ(A) = |A| − |E |

d(A,B) = inf {δ(B ′) | A ⊆ B ′ ⊆fin B}

A 6 B ⇐⇒ δ(A) = d(A,B)

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

Intuition

δ(A) = |A| − |E |

d(A,B) = inf {δ(B ′) | A ⊆ B ′ ⊆fin B}

A 6 B ⇐⇒ δ(A) = d(A,B)

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

Pregeometry - definition

Examples: cardinality, linear dimension, transcendence degree.

DefinitionA combinatorial pregeometry is a (assume countable) set X with adimension function d : P(X )→ N ∪ {∞} such that for allY1,Y2 ∈ P(X ):

1. d(Y1) ≤ |Y1|2. d(Y1) ≤ d(Y1 ∪ Y2) ≤ d(Y1) + d(Y2)

3. d(Y1 ∪ Y2) ≤ d(Y1) + d(Y2)− d(Y1 ∩ Y2) (submodular)

*4. d(Y1) = sup {d(Y0) | Y0 ∈ Fin(Y1)} (finitary)

More definitionsA finite set Y is independent if d(Y ) = |Y |.(X , d) is n-pure if all Y ∈ [X ]≤n are independent.A pregeometry is called a geometry if it is 2-pure.

Each pregeometry has a canonically associated geometry.

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

Pregeometry - definition

Examples: cardinality, linear dimension, transcendence degree.

DefinitionA combinatorial pregeometry is a (assume countable) set X with adimension function d : P(X )→ N ∪ {∞} such that for allY1,Y2 ∈ P(X ):

1. d(Y1) ≤ |Y1|2. d(Y1) ≤ d(Y1 ∪ Y2) ≤ d(Y1) + d(Y2)

3. d(Y1 ∪ Y2) ≤ d(Y1) + d(Y2)− d(Y1 ∩ Y2) (submodular)

*4. d(Y1) = sup {d(Y0) | Y0 ∈ Fin(Y1)} (finitary)

More definitionsA finite set Y is independent if d(Y ) = |Y |.(X , d) is n-pure if all Y ∈ [X ]≤n are independent.A pregeometry is called a geometry if it is 2-pure.

Each pregeometry has a canonically associated geometry.

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

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

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

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

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

δ(A) = |V | − |E |

and for any induced subgraph B ⊆ A define

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

The function d(·,A) defines a pregeometry whenever ∅ 6 A.

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

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

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 geometry 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

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

Variations

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

Variations

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

AxiomWe want to classify the geometries of strongly minimal structures(regular types).

(Buzzwords: Zilber’s conjecture, Hrushovski fusion)

FactEvery reduct of a strongly minimal structure is strongly minimal. 1

Questions

I What are the non-trivial reducts of M?

I Is any reduct of an ab initio construction again an ab initioconstruction?

1Disregard the fact that M is not strongly minimal

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

Questions

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

Questions

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

Questions

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

Example

Consider the reduct of M to the formula:

ϕ(x1, . . . , xn) := ∃!y1, . . . yn−1

n∧i=1

E (y1, . . . , yn−1, xi )

Where x1, . . . , xn, y1, . . . , yn−1 are distinct elements.

I The reduct 〈M, ϕ(M)〉 is non-trivial.

I The number of ϕ-edges on a set is not linearly bounded – astandard δ-function cannot capture the dimension.

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

Example

Consider the reduct of M to the formula:

ϕ(x1, . . . , xn) := ∃!y1, . . . yn−1

n∧i=1

E (y1, . . . , yn−1, xi )

Where x1, . . . , xn, y1, . . . , yn−1 are distinct elements.

I The reduct 〈M, ϕ(M)〉 is non-trivial.

I The number of ϕ-edges on a set is not linearly bounded – astandard δ-function cannot capture the dimension.

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

SolutionGeneralize the predimension function: make a ϕ-clique havepredimension n − 1.

From here on, we restrict the discussion to n-uniform hypergraphs,for some fixed big n.

The λ function

I K ⊆ A is a clique if [K ]n ⊆ E .

I Define M(A) to be the set of maximal cliques in A.

I s(A) =∑

K∈M(A)(|K | − (n − 1))

I λ(A) = |A| − s(A)

Redefining 6 using λ, the class C expands, but remains a Fraısseamalgamation class (modulo some technicalities swept under therug).

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

The λ function

I s(A) =∑

K∈M(A)(|K | − (n − 1))

I λ(A) = |A| − s(A)

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

The λ function

I s(A) =∑

K∈M(A)(|K | − (n − 1))

I λ(A) = |A| − s(A)

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

DefinitionConsider a pregeometry (X , d) as a hypergraph by defining

E = {Y ⊆fin X | d(Y ) < Y }

Now (X , d) can be seen as a first order structure in the languageL = {En}n∈N.

The λ function estimates the dimension of an n-uniformhypergraph associated to a pregeometry.

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

DefinitionDefine Cgeo = age(Gn

n � En).(Mn

n is Hrushovski’s construction for n-uniform hypergraphs whosegeometry is n-pure).

TheoremCgeo is a Fraısse amalgamation class with respect to6-embeddings with a generic structure Mgeo .Moreover, Mgeo ∼= Gn

n � En, and under this identificationGgeo = Gn

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

Questions?

Thank you!

Omer Mermelstein

Preliminaries

Intuition

Motivation

Clique predimension

Questions?

Thank you!

Searching for template structures for the class of ...omer/BPGMT18Presentation.pdf · Intuition...

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