Perfect sets and f -ideals Introduction Perfect sets and f -ideals - - PowerPoint PPT Presentation

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Perfect sets and f-ideals

Author Jin Guo

(This is a joint work with professor Tongsuo Wu)

Department of Mathematics, Shanghai Jiaotong University November 2, 2013

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Outline

1 Introduction 2 Perfect sets and f-ideals of degree d 3 (n, 2)th perfect number 4 Structure of V (n, 2) 5 Further works 6 References

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

A bridge between algebra and combinatorics

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

A bridge between algebra and combinatorics

From simplicial complex to ideals

Simplicial complex ∆ − →

• 1. Stanley-Reisner ideal I∆; 2. Facet ideal I(∆).

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

A bridge between algebra and combinatorics

From simplicial complex to ideals

Simplicial complex ∆ − →

• 1. Stanley-Reisner ideal I∆; 2. Facet ideal I(∆).

From ideal to simplicial complexes

Square-free monomial ideal I − →

• 1. Stanley-Reisner complex δN (I); 2. Facet complex δF(I).

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Two simplicial complexes

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Two simplicial complexes

Simplicial complex

A simplicial complex ∆ on [n] = {1, 2, . . . , n} is a collection

• f subsets of [n], satisfying:

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Two simplicial complexes

Simplicial complex

A simplicial complex ∆ on [n] = {1, 2, . . . , n} is a collection

• f subsets of [n], satisfying:

{i} ∈ ∆ for all i ∈ [n];

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Two simplicial complexes

Simplicial complex

A simplicial complex ∆ on [n] = {1, 2, . . . , n} is a collection

• f subsets of [n], satisfying:

{i} ∈ ∆ for all i ∈ [n]; If F ∈ ∆, and G ⊆ F, then G ∈ ∆ (including ∅).

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Two simplicial complexes

Simplicial complex

Face: element of ∆;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Two simplicial complexes

Simplicial complex

Face: element of ∆; Facet: maximal face of ∆;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Two simplicial complexes

Simplicial complex

Face: element of ∆; Facet: maximal face of ∆; ∆ = F1, . . . , Fk, where F1, . . . , Fk are the facets of ∆;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Two simplicial complexes

Simplicial complex

Face: element of ∆; Facet: maximal face of ∆; ∆ = F1, . . . , Fk, where F1, . . . , Fk are the facets of ∆; dimension of a face F: |F|−1, where |F| is the number

• f vertices of F;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Two simplicial complexes

Simplicial complex

Face: element of ∆; Facet: maximal face of ∆; ∆ = F1, . . . , Fk, where F1, . . . , Fk are the facets of ∆; dimension of a face F: |F|−1, where |F| is the number

• f vertices of F;

f-vector of ∆: f(∆) = (f0, f1, . . . , fd), where fi is the number of faces of dimension i of ∆.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Two simplicial complexes

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Two simplicial complexes

Let S = K[x1, . . . , xn]. Denote by sm(S) and sm(I) the set

• f square-free monomials in S and I respectively. Denote by

sm(S)d the set of square-free monomials of degree d in S.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Two simplicial complexes

Let S = K[x1, . . . , xn]. Denote by sm(S) and sm(I) the set

• f square-free monomials in S and I respectively. Denote by

sm(S)d the set of square-free monomials of degree d in S.

A bijection between sm(S) and 2[n]

σ : xi1xi2 · · · xik → {i1, i2, . . . , ik}.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Two simplicial complexes

Let S = K[x1, . . . , xn]. Denote by sm(S) and sm(I) the set

• f square-free monomials in S and I respectively. Denote by

sm(S)d the set of square-free monomials of degree d in S.

A bijection between sm(S) and 2[n]

σ : xi1xi2 · · · xik → {i1, i2, . . . , ik}.

Facet complex

δF(I) = σ(G(I)) = {σ(g) | g ∈ G(I)}.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Two simplicial complexes

Let S = K[x1, . . . , xn]. Denote by sm(S) and sm(I) the set

• f square-free monomials in S and I respectively. Denote by

sm(S)d the set of square-free monomials of degree d in S.

A bijection between sm(S) and 2[n]

σ : xi1xi2 · · · xik → {i1, i2, . . . , ik}.

Facet complex

δF(I) = σ(G(I)) = {σ(g) | g ∈ G(I)}.

Stanley-Reisner complex

δN (I) = {σ(g) | g ∈ sm(S) \ sm(I)}. In other words, the Stanley-Reisner ideal of δN (I) is I.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Two simplicial complexes

Let S = K[x1, . . . , xn]. Denote by sm(S) and sm(I) the set

• f square-free monomials in S and I respectively. Denote by

sm(S)d the set of square-free monomials of degree d in S.

A bijection between sm(S) and 2[n]

σ : xi1xi2 · · · xik → {i1, i2, . . . , ik}.

Facet complex

δF(I) = σ(G(I)) = {σ(g) | g ∈ G(I)}.

Stanley-Reisner complex

δN (I) = {σ(g) | g ∈ sm(S) \ sm(I)}. In other words, the Stanley-Reisner ideal of δN (I) is I.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

f-ideal

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

f-ideal

f-ideal

A square-free monomial ideal I is called an f-ideal, if both δF(I) and δN (I) have the same f-vector.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

f-ideal

f-ideal

A square-free monomial ideal I is called an f-ideal, if both δF(I) and δN (I) have the same f-vector.

Background of f-ideal

Defined by G. Q. ABBASI, S. AHMAD, I. ANWAR and

• W. A. BAIG in [1];

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

f-ideal

f-ideal

A square-free monomial ideal I is called an f-ideal, if both δF(I) and δN (I) have the same f-vector.

Background of f-ideal

Defined by G. Q. ABBASI, S. AHMAD, I. ANWAR and

• W. A. BAIG in [1];

The authors in [1] studied the properties of f-ideals of degree 2, and presented an interesting characterization

• f such ideals;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

f-ideal

f-ideal

A square-free monomial ideal I is called an f-ideal, if both δF(I) and δN (I) have the same f-vector.

Background of f-ideal

Defined by G. Q. ABBASI, S. AHMAD, I. ANWAR and

• W. A. BAIG in [1];

The authors in [1] studied the properties of f-ideals of degree 2, and presented an interesting characterization

• f such ideals;

In [2], the authors generalized the characterization for f-ideals of degree d (d ≥ 2),

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

f-ideal

f-ideal

A square-free monomial ideal I is called an f-ideal, if both δF(I) and δN (I) have the same f-vector.

Background of f-ideal

Defined by G. Q. ABBASI, S. AHMAD, I. ANWAR and

• W. A. BAIG in [1];

The authors in [1] studied the properties of f-ideals of degree 2, and presented an interesting characterization

• f such ideals;

In [2], the authors generalized the characterization for f-ideals of degree d (d ≥ 2), though their main result seems to be a little bit inaccurate. See the following example.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Background of f-ideal

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Background of f-ideal

An example of f-ideal

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Background of f-ideal

An example of f-ideal

Let S = K[x1, x2, x3, x4, x5], and let I = x1x2x3, x1x2x4, x1x2x5, x3x4x5, x2x3x4. It is not hard to check that I is an f-ideal. But the standard primary decomposition of I is I = x2, x5 ∩ x2, x3 ∩ x2, x4 ∩ x1, x4 ∩ x1, x3 ∩ x3, x4, x5, which shows that I is not unmixed.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Four questions

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Four questions

Four questions

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Four questions

Four questions

How to characterize f-ideals of degree d directly?

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Four questions

Four questions

How to characterize f-ideals of degree d directly? How many f-ideals of degree d are there in the polyno- mial ring S = K[x1, . . . , xn]?

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Four questions

Four questions

How to characterize f-ideals of degree d directly? How many f-ideals of degree d are there in the polyno- mial ring S = K[x1, . . . , xn]? Is there any f-ideal which is not unmixed?

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Four questions

Four questions

How to characterize f-ideals of degree d directly? How many f-ideals of degree d are there in the polyno- mial ring S = K[x1, . . . , xn]? Is there any f-ideal which is not unmixed? What can one say about f-ideals in general case?

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Four questions

Four questions

How to characterize f-ideals of degree d directly? How many f-ideals of degree d are there in the polyno- mial ring S = K[x1, . . . , xn]? Is there any f-ideal which is not unmixed? What can one say about f-ideals in general case?

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Our work

Our work

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Our work

Our work

Answer question (1);

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Our work

Our work

Answer question (1); Answer question (2) completely in the case d = 2;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Our work

Our work

Answer question (1); Answer question (2) completely in the case d = 2; Give a positive answer to question (3) in general case, and a negative answer in the case d = 2;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Our work

Our work

Answer question (1); Answer question (2) completely in the case d = 2; Give a positive answer to question (3) in general case, and a negative answer in the case d = 2; Give a preliminary answer to question (4).

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Our work

Our work

Answer question (1); Answer question (2) completely in the case d = 2; Give a positive answer to question (3) in general case, and a negative answer in the case d = 2; Give a preliminary answer to question (4).

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some definitions

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some definitions

Upper generated and lower cover set

For a subset A ⊆ sm(S), the upper generated set ⊔(A) of A is defined by ⊔(A) = {gxi | g ∈ A, xi ∤ g, 1 ≤ i ≤ n}. Dually, the lower cover set ⊓(A) of A is defined by ⊓(A) = {h | 1 = h, h = g/xi for some g ∈ A and some xi with xi | g}. Similarly, we define ⊔2(A) = ⊔(⊔(A)), and ⊔∞(A) = ∪∞

i=1 ⊔i (A), ⊓∞(A) = ∪∞ i=1 ⊓i (A).

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some definitions

Perfect set

A ⊆ sm(S)d is called

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some definitions

Perfect set

A ⊆ sm(S)d is called upper perfect: If ⊔(A) = sm(S)d+1 holds;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some definitions

Perfect set

A ⊆ sm(S)d is called upper perfect: If ⊔(A) = sm(S)d+1 holds; lower perfect: If ⊓(A) = sm(S)d−1 holds;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some definitions

Perfect set

A ⊆ sm(S)d is called upper perfect: If ⊔(A) = sm(S)d+1 holds; lower perfect: If ⊓(A) = sm(S)d−1 holds; perfect : If A is both upper perfect and lower perfect.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some definitions

Perfect set

A ⊆ sm(S)d is called upper perfect: If ⊔(A) = sm(S)d+1 holds; lower perfect: If ⊓(A) = sm(S)d−1 holds; perfect : If A is both upper perfect and lower perfect.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some definitions

Homogeneous of degree d

A monomial ideal I is called of degree d (or alternatively, homogeneous of degree d), if all monomials in G(I) have the same degree d.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some definitions

Homogeneous of degree d

A monomial ideal I is called of degree d (or alternatively, homogeneous of degree d), if all monomials in G(I) have the same degree d. If I is an f-ideal of S = K[x1, . . . , xn], and homogeneous of degree d, we also call I an (n, d)th f-ideal. Correspondingly, we can define an (n, d)th perfect sets.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some definitions

Examples

Let S = K[x1, x2, x3, x4]. Consider the following three subsets of sm(S)2: A = {x1x2, x1x3, x1x4}, B = {x1x2, x1x3, x2x3}, C = {x1x2, x3x4}.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some definitions

Examples

Let S = K[x1, x2, x3, x4]. Consider the following three subsets of sm(S)2: A = {x1x2, x1x3, x1x4}, B = {x1x2, x1x3, x2x3}, C = {x1x2, x3x4}. A is lower perfect but not upper perfect, since x2x3x4 ∈ ⊔(A);

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some definitions

Examples

Let S = K[x1, x2, x3, x4]. Consider the following three subsets of sm(S)2: A = {x1x2, x1x3, x1x4}, B = {x1x2, x1x3, x2x3}, C = {x1x2, x3x4}. A is lower perfect but not upper perfect, since x2x3x4 ∈ ⊔(A); B is upper perfect but not lower perfect, since x4 ∈ ⊓(B);

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some definitions

Examples

Let S = K[x1, x2, x3, x4]. Consider the following three subsets of sm(S)2: A = {x1x2, x1x3, x1x4}, B = {x1x2, x1x3, x2x3}, C = {x1x2, x3x4}. A is lower perfect but not upper perfect, since x2x3x4 ∈ ⊔(A); B is upper perfect but not lower perfect, since x4 ∈ ⊓(B); C is perfect.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some definitions

Examples

Let S = K[x1, x2, x3, x4]. Consider the following three subsets of sm(S)2: A = {x1x2, x1x3, x1x4}, B = {x1x2, x1x3, x2x3}, C = {x1x2, x3x4}. A is lower perfect but not upper perfect, since x2x3x4 ∈ ⊔(A); B is upper perfect but not lower perfect, since x4 ∈ ⊓(B); C is perfect.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Main result of this part

Characterization of (n, d)th f-ideals

Let S = K[x1, . . . , xn], and let I be a square-free monomial ideal of S of degree d with the minimal generating set G(I). Then I is an f-ideal if and only if the followings hold:

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Main result of this part

Characterization of (n, d)th f-ideals

Let S = K[x1, . . . , xn], and let I be a square-free monomial ideal of S of degree d with the minimal generating set G(I). Then I is an f-ideal if and only if the followings hold: |G(I)| = 1

2Cd n;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Main result of this part

Characterization of (n, d)th f-ideals

Let S = K[x1, . . . , xn], and let I be a square-free monomial ideal of S of degree d with the minimal generating set G(I). Then I is an f-ideal if and only if the followings hold: |G(I)| = 1

2Cd n;

G(I) is an (n, d)th perfect set.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Main result of this part

Characterization of (n, d)th f-ideals

Let S = K[x1, . . . , xn], and let I be a square-free monomial ideal of S of degree d with the minimal generating set G(I). Then I is an f-ideal if and only if the followings hold: |G(I)| = 1

2Cd n;

G(I) is an (n, d)th perfect set.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

idea

How to construct an (n, d)th f-ideal

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

idea

How to construct an (n, d)th f-ideal

Find an (n, d)th perfect set A, such that |A| ≤ 1

2Cd n;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

idea

How to construct an (n, d)th f-ideal

Find an (n, d)th perfect set A, such that |A| ≤ 1

2Cd n;

Choose D ⊆ sm(S)d\A randomly, such that |D| = 1

2Cd n−

|A|;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

idea

How to construct an (n, d)th f-ideal

Find an (n, d)th perfect set A, such that |A| ≤ 1

2Cd n;

Choose D ⊆ sm(S)d\A randomly, such that |D| = 1

2Cd n−

|A|; Let I be the ideal generated by A ∪ D;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

idea

How to construct an (n, d)th f-ideal

Find an (n, d)th perfect set A, such that |A| ≤ 1

2Cd n;

Choose D ⊆ sm(S)d\A randomly, such that |D| = 1

2Cd n−

|A|; Let I be the ideal generated by A ∪ D; I is an (n, d)th f-ideal.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

idea

How to construct an (n, d)th f-ideal

Find an (n, d)th perfect set A, such that |A| ≤ 1

2Cd n;

Choose D ⊆ sm(S)d\A randomly, such that |D| = 1

2Cd n−

|A|; Let I be the ideal generated by A ∪ D; I is an (n, d)th f-ideal.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

How to find an (n, d)th perfect set

For a general d ≥ 2, it is not easy to find an (n, d)th perfect set, but it is not hard when d = 2.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

How to find an (n, d)th perfect set

For a general d ≥ 2, it is not easy to find an (n, d)th perfect set, but it is not hard when d = 2.

How to find an (n, 2)th perfect set

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

How to find an (n, d)th perfect set

For a general d ≥ 2, it is not easy to find an (n, d)th perfect set, but it is not hard when d = 2.

How to find an (n, 2)th perfect set

Divide [n] into two part B and C (Actually, C = B);

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

How to find an (n, d)th perfect set

For a general d ≥ 2, it is not easy to find an (n, d)th perfect set, but it is not hard when d = 2.

How to find an (n, 2)th perfect set

Divide [n] into two part B and C (Actually, C = B); Let A = {xixj | i, j ∈ B, or i, j ∈ C};

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

How to find an (n, d)th perfect set

For a general d ≥ 2, it is not easy to find an (n, d)th perfect set, but it is not hard when d = 2.

How to find an (n, 2)th perfect set

Divide [n] into two part B and C (Actually, C = B); Let A = {xixj | i, j ∈ B, or i, j ∈ C}; A is an (n, 2)th perfect set.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

How to find an (n, d)th perfect set

For a general d ≥ 2, it is not easy to find an (n, d)th perfect set, but it is not hard when d = 2.

How to find an (n, 2)th perfect set

Divide [n] into two part B and C (Actually, C = B); Let A = {xixj | i, j ∈ B, or i, j ∈ C}; A is an (n, 2)th perfect set. Actually, this is almost the unique method to construct a perfect set with a small cardinality.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

How to find an (n, d)th perfect set

For a general d ≥ 2, it is not easy to find an (n, d)th perfect set, but it is not hard when d = 2.

How to find an (n, 2)th perfect set

Divide [n] into two part B and C (Actually, C = B); Let A = {xixj | i, j ∈ B, or i, j ∈ C}; A is an (n, 2)th perfect set. Actually, this is almost the unique method to construct a perfect set with a small cardinality.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

(n, d)th perfect number

Definition

The least number among cardinalities of (n, d)th perfect sets, denoted by N(n,d).

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

(n, d)th perfect number

Definition

The least number among cardinalities of (n, d)th perfect sets, denoted by N(n,d).

(n, 2)th perfect number

Let k be a positive integer, and let n ≥ 4. Then the perfect number N(n,2) is given by the following rules:

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

(n, d)th perfect number

Definition

The least number among cardinalities of (n, d)th perfect sets, denoted by N(n,d).

(n, 2)th perfect number

Let k be a positive integer, and let n ≥ 4. Then the perfect number N(n,2) is given by the following rules: If n = 2k, then N(n,2) = C2

k + C2 k = k2 − k;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

(n, d)th perfect number

Definition

The least number among cardinalities of (n, d)th perfect sets, denoted by N(n,d).

(n, 2)th perfect number

Let k be a positive integer, and let n ≥ 4. Then the perfect number N(n,2) is given by the following rules: If n = 2k, then N(n,2) = C2

k + C2 k = k2 − k;

If n = 2k + 1, then N(n,2) = C2

k + C2 k+1 = k2.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

(n, d)th perfect number

Definition

The least number among cardinalities of (n, d)th perfect sets, denoted by N(n,d).

(n, 2)th perfect number

Let k be a positive integer, and let n ≥ 4. Then the perfect number N(n,2) is given by the following rules: If n = 2k, then N(n,2) = C2

k + C2 k = k2 − k;

If n = 2k + 1, then N(n,2) = C2

k + C2 k+1 = k2.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Existence of (n, 2)th f-ideal

Existence

V (n, 2) = ∅ if and only if 2 | C2

n, i.e., if and only if n = 4k or

n = 4k + 1 for some positive integer k.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Existence of (n, 2)th f-ideal

Existence

V (n, 2) = ∅ if and only if 2 | C2

n, i.e., if and only if n = 4k or

n = 4k + 1 for some positive integer k. Note that: When n = 4k N(n,2) = 4k2 − 2k and C2

n/2 = 4k2 − k;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Existence of (n, 2)th f-ideal

Existence

V (n, 2) = ∅ if and only if 2 | C2

n, i.e., if and only if n = 4k or

n = 4k + 1 for some positive integer k. Note that: When n = 4k N(n,2) = 4k2 − 2k and C2

n/2 = 4k2 − k;

N(n,2) ≤ C2

n/2.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Existence of (n, 2)th f-ideal

Existence

V (n, 2) = ∅ if and only if 2 | C2

n, i.e., if and only if n = 4k or

n = 4k + 1 for some positive integer k. Note that: When n = 4k N(n,2) = 4k2 − 2k and C2

n/2 = 4k2 − k;

N(n,2) ≤ C2

n/2.

When n = 4k + 1 N(n,2) = 4k2 and C2

n/2 = 4k2 + k;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Existence of (n, 2)th f-ideal

Existence

V (n, 2) = ∅ if and only if 2 | C2

n, i.e., if and only if n = 4k or

n = 4k + 1 for some positive integer k. Note that: When n = 4k N(n,2) = 4k2 − 2k and C2

n/2 = 4k2 − k;

N(n,2) ≤ C2

n/2.

When n = 4k + 1 N(n,2) = 4k2 and C2

n/2 = 4k2 + k;

N(n,2) ≤ C2

n/2.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Existence of (n, 2)th f-ideal

Existence

V (n, 2) = ∅ if and only if 2 | C2

n, i.e., if and only if n = 4k or

n = 4k + 1 for some positive integer k. Note that: When n = 4k N(n,2) = 4k2 − 2k and C2

n/2 = 4k2 − k;

N(n,2) ≤ C2

n/2.

When n = 4k + 1 N(n,2) = 4k2 and C2

n/2 = 4k2 + k;

N(n,2) ≤ C2

n/2.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some notations

Two Part Complete Structure

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some notations

Two Part Complete Structure

For a subset B of [n], denote WB = {xixj | i, j ∈ B or i, j ∈ B};

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some notations

Two Part Complete Structure

For a subset B of [n], denote WB = {xixj | i, j ∈ B or i, j ∈ B}; Clearly WB = WB holds, and WB is an (n, 2)th perfect set;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some notations

Two Part Complete Structure

For a subset B of [n], denote WB = {xixj | i, j ∈ B or i, j ∈ B}; Clearly WB = WB holds, and WB is an (n, 2)th perfect set; A subset A of sm(S)2 is called satisfying Two Part Com- plete Structure, abbreviated as TPCS, if there exists a B ⊆ [n], such that WB ⊆ A;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some notations

Two Part Complete Structure

For a subset B of [n], denote WB = {xixj | i, j ∈ B or i, j ∈ B}; Clearly WB = WB holds, and WB is an (n, 2)th perfect set; A subset A of sm(S)2 is called satisfying Two Part Com- plete Structure, abbreviated as TPCS, if there exists a B ⊆ [n], such that WB ⊆ A; If further |B| = l, then A is called satisfying lth TPCS;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some notations

Two Part Complete Structure

For a subset B of [n], denote WB = {xixj | i, j ∈ B or i, j ∈ B}; Clearly WB = WB holds, and WB is an (n, 2)th perfect set; A subset A of sm(S)2 is called satisfying Two Part Com- plete Structure, abbreviated as TPCS, if there exists a B ⊆ [n], such that WB ⊆ A; If further |B| = l, then A is called satisfying lth TPCS; An f-ideal I is called of l type, if G(I) satisfies lth TPCS;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some notations

Two Part Complete Structure

For a subset B of [n], denote WB = {xixj | i, j ∈ B or i, j ∈ B}; Clearly WB = WB holds, and WB is an (n, 2)th perfect set; A subset A of sm(S)2 is called satisfying Two Part Com- plete Structure, abbreviated as TPCS, if there exists a B ⊆ [n], such that WB ⊆ A; If further |B| = l, then A is called satisfying lth TPCS; An f-ideal I is called of l type, if G(I) satisfies lth TPCS; Denote by Wl the set of f-ideals of l type in S.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Some notations

Two Part Complete Structure

For a subset B of [n], denote WB = {xixj | i, j ∈ B or i, j ∈ B}; Clearly WB = WB holds, and WB is an (n, 2)th perfect set; A subset A of sm(S)2 is called satisfying Two Part Com- plete Structure, abbreviated as TPCS, if there exists a B ⊆ [n], such that WB ⊆ A; If further |B| = l, then A is called satisfying lth TPCS; An f-ideal I is called of l type, if G(I) satisfies lth TPCS; Denote by Wl the set of f-ideals of l type in S.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

l type

Question:

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

l type

Question:

Is there any f-ideal who is of no l type?

Example

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

l type

Question:

Is there any f-ideal who is of no l type?

Example

Let S = K[x1, x2, x3, x4, x5]. Consider the ideal I = x1x2, x2x3, x3x4, x4x5, x1x5

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

l type

Question:

Is there any f-ideal who is of no l type?

Example

Let S = K[x1, x2, x3, x4, x5]. Consider the ideal I = x1x2, x2x3, x3x4, x4x5, x1x5 It is direct to check that I is an f-ideal.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

l type

Question:

Is there any f-ideal who is of no l type?

Example

Let S = K[x1, x2, x3, x4, x5]. Consider the ideal I = x1x2, x2x3, x3x4, x4x5, x1x5 It is direct to check that I is an f-ideal. but I is not of l type for any l.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

l type

Question:

Is there any f-ideal who is of no l type?

Example

Let S = K[x1, x2, x3, x4, x5]. Consider the ideal I = x1x2, x2x3, x3x4, x4x5, x1x5 It is direct to check that I is an f-ideal. but I is not of l type for any l. Such kind of f-ideal of K[x1, x2, x3, x4, x5] is called C5 (5-cycle).

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

l type

Question:

Is there any f-ideal who is of no l type?

Example

Let S = K[x1, x2, x3, x4, x5]. Consider the ideal I = x1x2, x2x3, x3x4, x4x5, x1x5 It is direct to check that I is an f-ideal. but I is not of l type for any l. Such kind of f-ideal of K[x1, x2, x3, x4, x5] is called C5 (5-cycle).

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

l type

Another question:

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

l type

Another question:

Is there any other f-ideal who is of no l type?

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Bijection τ

τ: From 2sm(S)2 to the set of Graphs with n vertices.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Bijection τ

τ: From 2sm(S)2 to the set of Graphs with n vertices.

A ⊆ sm(S)2, where S = K[x1, . . . , xn];

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Bijection τ

τ: From 2sm(S)2 to the set of Graphs with n vertices.

A ⊆ sm(S)2, where S = K[x1, . . . , xn]; T = τ(A) is a graph whose vertices are v1, . . . , vn;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Bijection τ

τ: From 2sm(S)2 to the set of Graphs with n vertices.

A ⊆ sm(S)2, where S = K[x1, . . . , xn]; T = τ(A) is a graph whose vertices are v1, . . . , vn; vivj ∈ E(T) holds if and only if xixj ∈ A, where E(T) is the edge set of T.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Bijection τ

τ: From 2sm(S)2 to the set of Graphs with n vertices.

A ⊆ sm(S)2, where S = K[x1, . . . , xn]; T = τ(A) is a graph whose vertices are v1, . . . , vn; vivj ∈ E(T) holds if and only if xixj ∈ A, where E(T) is the edge set of T.

Example

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Bijection τ

τ: From 2sm(S)2 to the set of Graphs with n vertices.

A ⊆ sm(S)2, where S = K[x1, . . . , xn]; T = τ(A) is a graph whose vertices are v1, . . . , vn; vivj ∈ E(T) holds if and only if xixj ∈ A, where E(T) is the edge set of T.

Example

Let S = K[x1, x2, x3, x4, x5], and let I = x1x2, x2x3, x3x4, x4x5, x1x5. Note that τ(G(I)) = v1 − v2 − v3 − v4 − v5 − v1 is a 5-cycle.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Bijection τ

τ: From 2sm(S)2 to the set of Graphs with n vertices.

A ⊆ sm(S)2, where S = K[x1, . . . , xn]; T = τ(A) is a graph whose vertices are v1, . . . , vn; vivj ∈ E(T) holds if and only if xixj ∈ A, where E(T) is the edge set of T.

Example

Let S = K[x1, x2, x3, x4, x5], and let I = x1x2, x2x3, x3x4, x4x5, x1x5. Note that τ(G(I)) = v1 − v2 − v3 − v4 − v5 − v1 is a 5-cycle.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

translation from combinatorics to graph theory

Perfect set

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

translation from combinatorics to graph theory

Perfect set

Let A ⊆ sm(S)2. Then the followings hold: A is upper perfect if and only if ω(τ(A)) ≤ 2 holds, where τ(A) is the complement graph of τ(A).

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

translation from combinatorics to graph theory

Perfect set

Let A ⊆ sm(S)2. Then the followings hold: A is upper perfect if and only if ω(τ(A)) ≤ 2 holds, where τ(A) is the complement graph of τ(A). A is lower perfect if and only if for each i ∈ [n], d(vi) < n − 1 holds in the graph τ(A).

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

translation from combinatorics to graph theory

Perfect set

Let A ⊆ sm(S)2. Then the followings hold: A is upper perfect if and only if ω(τ(A)) ≤ 2 holds, where τ(A) is the complement graph of τ(A). A is lower perfect if and only if for each i ∈ [n], d(vi) < n − 1 holds in the graph τ(A).

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

translation from combinatorics to graph theory

Perfect set

Let A ⊆ sm(S)2. Then the followings hold: A is upper perfect if and only if ω(τ(A)) ≤ 2 holds, where τ(A) is the complement graph of τ(A). A is lower perfect if and only if for each i ∈ [n], d(vi) < n − 1 holds in the graph τ(A).

l type

If I is an (n, 2)th f-ideal, then I is of l type for some 1 ≤ l ≤ ⌊n/2⌋ if and only if τ(G(I)) is a bipartite graph.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

translation from combinatorics to graph theory

Perfect set

Let A ⊆ sm(S)2. Then the followings hold: A is upper perfect if and only if ω(τ(A)) ≤ 2 holds, where τ(A) is the complement graph of τ(A). A is lower perfect if and only if for each i ∈ [n], d(vi) < n − 1 holds in the graph τ(A).

l type

If I is an (n, 2)th f-ideal, then I is of l type for some 1 ≤ l ≤ ⌊n/2⌋ if and only if τ(G(I)) is a bipartite graph.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Four conditions

Four conditions

I is an (n, 2)th f-ideal which is not of l type for any l, if and

• nly if τ(G(I)) satisfies the following four conditions (abbre-

viated as FC in what follows):

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Four conditions

Four conditions

I is an (n, 2)th f-ideal which is not of l type for any l, if and

• nly if τ(G(I)) satisfies the following four conditions (abbre-

viated as FC in what follows): For each i ∈ [n], d(vi) < n − 1 holds in τ(G(I)).

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Four conditions

Four conditions

I is an (n, 2)th f-ideal which is not of l type for any l, if and

• nly if τ(G(I)) satisfies the following four conditions (abbre-

viated as FC in what follows): For each i ∈ [n], d(vi) < n − 1 holds in τ(G(I)). ω(τ(G(I))) = 2.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Four conditions

Four conditions

I is an (n, 2)th f-ideal which is not of l type for any l, if and

• nly if τ(G(I)) satisfies the following four conditions (abbre-

viated as FC in what follows): For each i ∈ [n], d(vi) < n − 1 holds in τ(G(I)). ω(τ(G(I))) = 2. |E(τ(G(I)))| = C2

n

2 .

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Four conditions

Four conditions

I is an (n, 2)th f-ideal which is not of l type for any l, if and

• nly if τ(G(I)) satisfies the following four conditions (abbre-

viated as FC in what follows): For each i ∈ [n], d(vi) < n − 1 holds in τ(G(I)). ω(τ(G(I))) = 2. |E(τ(G(I)))| = C2

n

2 .

τ(G(I)) is not a bipartite graph.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Four conditions

Four conditions

I is an (n, 2)th f-ideal which is not of l type for any l, if and

• nly if τ(G(I)) satisfies the following four conditions (abbre-

viated as FC in what follows): For each i ∈ [n], d(vi) < n − 1 holds in τ(G(I)). ω(τ(G(I))) = 2. |E(τ(G(I)))| = C2

n

2 .

τ(G(I)) is not a bipartite graph.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Main result

Structure of V(n, 2)

If n = 5, then V (n, 2) = ⌊n/2⌋

l=1

Wl, which is a mutually disjoint union of the Wl’s.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Main result

Structure of V(n, 2)

If n = 5, then V (n, 2) = ⌊n/2⌋

l=1

Wl, which is a mutually disjoint union of the Wl’s. Proof: Note that V (n, 2) = ∪⌊n/2⌋

l=1 Wl holds true, if and only

if each f-ideal is of l type for some l; and the latter holds if and only if, there is no graph satisfying the FC. We will show that a graph will not satisfy condition (3) if it satisfies conditions (2) and (4), except for the case n = 5.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Main result

Assume that T is a graph satisfying conditions (2) and (4). Since T is not a bipartite graph, there exists at least an odd cycle in T. Assume that D is a minimal odd cycle of T, with |V (D)| = 2i + 1. Note that ω(T) = 2, so i ≥ 2. Denote by |E(D)| the edge number of the subgraph induced on D, and denote by |E(B, C)| the number of edges, each of which has end vertices in B and C respectively. It is clear that |E(T)| = |E(D)| + |E(T \ D)| + |E(D, T \ D)|.

• holds. Note that |E(D)| = 2i + 1 holds, since D is a minimal
• cycle. Since there exists no triangles in T, it is not hard to

see that |E(D, T \ D)| ≤ (n − 2i − 1)i holds, since D is an odd cycle. We will discuss |E(T \ D)| in the following two subcases:

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Main result

If n = 2k for some positive k, then |V (T \ D)| = 2k − 2i − 1 holds. It follows from Turan theorem that |E(T \ D)| ≤ (k − i)(k − i − 1) hold, hence we get |E(T)| = |E(D)| + |E(T \ D)| + |E(D, T \ D)| ≤ (2i+1)+(2k−2i−1)i+(k−i)(k−i−1) = k2−k−i2+2i+1. Note that C2

n/2 = k2 − k/2, thus

C2

n/2 − |E(T)| ≥ k/2 + i2 − 2i − 1 = k/2 + (i − 1)2 − 2

• holds. Since i ≥ 2 and 2k > 2i + 1, C2

n/2 − |E(T)| > 0 holds.

This shows that there is no graph satisfying FC when n = 2k.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Main result

If n = 2k + 1, then |V (T \ D)| = 2k − 2i holds. Again by Turan theorem, |E(T \ D)| ≤ (k − i)2 holds, hence we have |E(T)| = |E(D)| + |E(T \ D)| + |E(D, T \ D)| ≤ (2i + 1) + (2k − 2i)i + (k − i)2 = k2 − i2 + 2i + 1. Note that C2

n/2 = k2 + k/2, thus

C2

n/2 − |E(T)| ≥ k/2 + i2 − 2i − 1 = k/2 + (i − 1)2 − 2

holds true. Then we have C2

n/2 − |E(T)| ≥ 0, since i ≥ 2 and

k ≥ i hold by assumption. Note further that the equality holds if and only if k = i = 2. Thus in this case, there is no graph satisfying FC except n = 5. This completes the proof.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Structure of V (n, 2)

Structure of V (n, 2)

Let k be a positive integer. Then the following equalities hold true: V (n, 2) = ∪

0≤i≤ √ k

W2k−i, if n = 4k;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Structure of V (n, 2)

Structure of V (n, 2)

Let k be a positive integer. Then the following equalities hold true: V (n, 2) = ∪

0≤i≤ √ k

W2k−i, if n = 4k; V (n, 2) = ∪

0≤i≤

√1+4k−1 2

W2k−i, if n = 4k + 1(k = 1);

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Structure of V (n, 2)

Structure of V (n, 2)

Let k be a positive integer. Then the following equalities hold true: V (n, 2) = ∪

0≤i≤ √ k

W2k−i, if n = 4k; V (n, 2) = ∪

0≤i≤

√1+4k−1 2

W2k−i, if n = 4k + 1(k = 1); V (n, 2) = W2 ∪ C5, if n = 5;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Structure of V (n, 2)

Structure of V (n, 2)

Let k be a positive integer. Then the following equalities hold true: V (n, 2) = ∪

0≤i≤ √ k

W2k−i, if n = 4k; V (n, 2) = ∪

0≤i≤

√1+4k−1 2

W2k−i, if n = 4k + 1(k = 1); V (n, 2) = W2 ∪ C5, if n = 5; V (n, 2) = ∅, if n = 4k + 2 or n = 4k + 3.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Structure of V (n, 2)

Structure of V (n, 2)

Let k be a positive integer. Then the following equalities hold true: V (n, 2) = ∪

0≤i≤ √ k

W2k−i, if n = 4k; V (n, 2) = ∪

0≤i≤

√1+4k−1 2

W2k−i, if n = 4k + 1(k = 1); V (n, 2) = W2 ∪ C5, if n = 5; V (n, 2) = ∅, if n = 4k + 2 or n = 4k + 3.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Cardinality of V (n, 2)

Cardinality of V (n, 2)

Let k be a positive integer. Then the following equalities hold true: |V (n, 2)| = 1

2C2k 4kCk 4k2 +

• 1≤i≤

√ k

C2k−i

4k

Ck−i2

4k2−i2, if n = 4k;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Cardinality of V (n, 2)

Cardinality of V (n, 2)

Let k be a positive integer. Then the following equalities hold true: |V (n, 2)| = 1

2C2k 4kCk 4k2 +

• 1≤i≤

√ k

C2k−i

4k

Ck−i2

4k2−i2, if n = 4k;

|V (n, 2)| =

• 0≤i≤

√1+4k−1 2

C2k−i

4k+1Ck−i−i2 4k2+2k−i−i2, if n = 4k +

1(k = 1);

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Cardinality of V (n, 2)

Cardinality of V (n, 2)

Let k be a positive integer. Then the following equalities hold true: |V (n, 2)| = 1

2C2k 4kCk 4k2 +

• 1≤i≤

√ k

C2k−i

4k

Ck−i2

4k2−i2, if n = 4k;

|V (n, 2)| =

• 0≤i≤

√1+4k−1 2

C2k−i

4k+1Ck−i−i2 4k2+2k−i−i2, if n = 4k +

1(k = 1); |V (n, 2)| = 72, if n = 5;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Cardinality of V (n, 2)

Cardinality of V (n, 2)

Let k be a positive integer. Then the following equalities hold true: |V (n, 2)| = 1

2C2k 4kCk 4k2 +

• 1≤i≤

√ k

C2k−i

4k

Ck−i2

4k2−i2, if n = 4k;

|V (n, 2)| =

• 0≤i≤

√1+4k−1 2

C2k−i

4k+1Ck−i−i2 4k2+2k−i−i2, if n = 4k +

1(k = 1); |V (n, 2)| = 72, if n = 5; |V (n, 2)| = 0, if n = 4k + 2 or n = 4k + 3.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Cardinality of V (n, 2)

Cardinality of V (n, 2)

Let k be a positive integer. Then the following equalities hold true: |V (n, 2)| = 1

2C2k 4kCk 4k2 +

• 1≤i≤

√ k

C2k−i

4k

Ck−i2

4k2−i2, if n = 4k;

|V (n, 2)| =

• 0≤i≤

√1+4k−1 2

C2k−i

4k+1Ck−i−i2 4k2+2k−i−i2, if n = 4k +

1(k = 1); |V (n, 2)| = 72, if n = 5; |V (n, 2)| = 0, if n = 4k + 2 or n = 4k + 3.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Other results

In general, an f-ideal may be not unmixed. But when d = 2, we have:

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Other results

In general, an f-ideal may be not unmixed. But when d = 2, we have:

f-ideals of degree 2 is unmixed

If I is an f-ideal, then I is unmixed.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Other results

In general, an f-ideal may be not unmixed. But when d = 2, we have:

f-ideals of degree 2 is unmixed

If I is an f-ideal, then I is unmixed. Corresponding to V (n, 2) = ∅, we have:

V (n, d) = ∅

For any integer d ≥ 2 and any integer n ≥ d + 2 such that 2 | Cd

n, V (n, d) = ∅.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Other results

Denote G(I) = ∪k

i=1Gdi, in which Gdi consists of the genera-

tors of degree di.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Other results

Denote G(I) = ∪k

i=1Gdi, in which Gdi consists of the genera-

tors of degree di.

f-ideals in general case

Let I be a square-free monomial ideal of S = K[x1, . . . , xn], with the minimal generating set G(I) = ∪k

i=1Gdi. Then I is

an f-ideal if and only if |Gl| = 1 2(Cl

n − | ∪di>l (⊓di−l(Gdi))| − | ∪di<l (⊔l−di(Gdi))| )

holds for each l ∈ [n].

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Other results

unmixed f-ideals

I is an (n, d)th unmixed f-ideal if and only if |G(I)| = Cd

n/2;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Other results

unmixed f-ideals

I is an (n, d)th unmixed f-ideal if and only if |G(I)| = Cd

n/2;

G(I) is perfect;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Other results

unmixed f-ideals

I is an (n, d)th unmixed f-ideal if and only if |G(I)| = Cd

n/2;

G(I) is perfect; sm(S)d \ G(I) is lower perfect.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Other results

unmixed f-ideals

I is an (n, d)th unmixed f-ideal if and only if |G(I)| = Cd

n/2;

G(I) is perfect; sm(S)d \ G(I) is lower perfect.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Other results

unmixed f-ideals

I is an (n, d)th unmixed f-ideal if and only if |G(I)| = Cd

n/2;

G(I) is perfect; sm(S)d \ G(I) is lower perfect. if and only if |G(I)| = Cd

n/2;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Other results

unmixed f-ideals

I is an (n, d)th unmixed f-ideal if and only if |G(I)| = Cd

n/2;

G(I) is perfect; sm(S)d \ G(I) is lower perfect. if and only if |G(I)| = Cd

n/2;

ldeg(Iσ(G(I))) = d;

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Other results

unmixed f-ideals

I is an (n, d)th unmixed f-ideal if and only if |G(I)| = Cd

n/2;

G(I) is perfect; sm(S)d \ G(I) is lower perfect. if and only if |G(I)| = Cd

n/2;

ldeg(Iσ(G(I))) = d; σ(u) | u ∈ sm(S)d \ G(I) is a d-flag complex.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Other results

unmixed f-ideals

I is an (n, d)th unmixed f-ideal if and only if |G(I)| = Cd

n/2;

G(I) is perfect; sm(S)d \ G(I) is lower perfect. if and only if |G(I)| = Cd

n/2;

ldeg(Iσ(G(I))) = d; σ(u) | u ∈ sm(S)d \ G(I) is a d-flag complex.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Further works

Questions

How to calculate the perfect number N(n,d)?

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Further works

Questions

How to calculate the perfect number N(n,d)? What about the structure of V (n, d)?

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Further works

Questions

How to calculate the perfect number N(n,d)? What about the structure of V (n, d)? What about nonhomogeneous f-ideal?

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Further works

Questions

How to calculate the perfect number N(n,d)? What about the structure of V (n, d)? What about nonhomogeneous f-ideal?

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

References

[1]

• G. Q. Abbasi, S. Ahmad, I. Anwar, W. A. Baig, f-Ideals of

degree 2, Algebra Collo- quium, 19(Spec1)(2012), 921 − 926. [2]

• I. Anwar, H. Mahmood, M. A. Binyamin and M. K. Zafar, On

the Characterization of f-Ideals, Comm. Algebra (accepted), see also arXiv: 1309.3765 (Sep. 15, 2013 ). [3]

• S. Faridi, The facet ideal of a simplicial complex, Manuscripta

Mathematica, 109(2002), 159 − 174. [4]

• J. Guo, T. S. Wu and Q. Liu, Perfect sets and f-Ideals,

preprint. [5]

• J. Herzog and T. Hibi, Monomial Ideals. Springer-Verlag Lon-

don Limited, 2011.

Perfect sets and f-ideals Jin Guo Outline Introduction Perfect sets and f-ideals

• f degree d

(n, 2)th perfect number Structure of V (n, 2) Further works References

Thank you!!!

Author: Guo Jin Address: Department of Mathematics Shanghai Jiaotong University Shanghai, 200240, China Email: guojinecho@163.com