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Perfect sets and f -ideals Jin Guo Outline Perfect sets and f -ideals Introduction Perfect sets and f -ideals of degree d Author Jin Guo ( n, 2) th perfect number (This is a joint work with professor Tongsuo Wu) Structure of V ( n, 2)

  1. f -ideal Perfect sets f -ideal and f -ideals A square-free monomial ideal I is called an f -ideal , if both Jin Guo δ F ( I ) and δ N ( I ) have the same f -vector. Outline Introduction Background of f -ideal Perfect sets and f -ideals Defined by G. Q. ABBASI, S. AHMAD, I. ANWAR and of degree d ( n, 2) th W. A. BAIG in [1]; perfect number The authors in [1] studied the properties of f -ideals of Structure of V ( n, 2) degree 2, and presented an interesting characterization Further of such ideals; works References In [2], the authors generalized the characterization for f -ideals of degree d ( d ≥ 2),

  2. f -ideal Perfect sets f -ideal and f -ideals A square-free monomial ideal I is called an f -ideal , if both Jin Guo δ F ( I ) and δ N ( I ) have the same f -vector. Outline Introduction Background of f -ideal Perfect sets and f -ideals Defined by G. Q. ABBASI, S. AHMAD, I. ANWAR and of degree d ( n, 2) th W. A. BAIG in [1]; perfect number The authors in [1] studied the properties of f -ideals of Structure of V ( n, 2) degree 2, and presented an interesting characterization Further of such ideals; works References 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.

  3. Background of f -ideal Perfect sets and f -ideals Jin Guo Outline Introduction Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  4. Background of f -ideal Perfect sets and f -ideals Jin Guo An example of f -ideal Outline Introduction Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  5. Background of f -ideal Perfect sets and f -ideals Jin Guo An example of f -ideal Outline Let S = K [ x 1 , x 2 , x 3 , x 4 , x 5 ], and let Introduction Perfect sets and f -ideals I = � x 1 x 2 x 3 , x 1 x 2 x 4 , x 1 x 2 x 5 , x 3 x 4 x 5 , x 2 x 3 x 4 � . of degree d ( n, 2) th perfect It is not hard to check that I is an f -ideal. But the standard number primary decomposition of I is Structure of V ( n, 2) Further I = works � x 2 , x 5 � ∩ � x 2 , x 3 � ∩ � x 2 , x 4 � ∩ � x 1 , x 4 � ∩ � x 1 , x 3 � ∩ � x 3 , x 4 , x 5 � , References which shows that I is not unmixed.

  6. Four questions Perfect sets and f -ideals Jin Guo Outline Introduction Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  7. Four questions Perfect sets and f -ideals Jin Guo Four questions Outline Introduction Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  8. Four questions Perfect sets and f -ideals Jin Guo Four questions Outline Introduction How to characterize f -ideals of degree d directly? Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  9. Four questions Perfect sets and f -ideals Jin Guo Four questions Outline Introduction How to characterize f -ideals of degree d directly? Perfect sets and f -ideals of degree d How many f -ideals of degree d are there in the polyno- ( n, 2) th perfect mial ring S = K [ x 1 , . . . , x n ]? number Structure of V ( n, 2) Further works References

  10. Four questions Perfect sets and f -ideals Jin Guo Four questions Outline Introduction How to characterize f -ideals of degree d directly? Perfect sets and f -ideals of degree d How many f -ideals of degree d are there in the polyno- ( n, 2) th perfect mial ring S = K [ x 1 , . . . , x n ]? number Structure of V ( n, 2) Is there any f -ideal which is not unmixed? Further works References

  11. Four questions Perfect sets and f -ideals Jin Guo Four questions Outline Introduction How to characterize f -ideals of degree d directly? Perfect sets and f -ideals of degree d How many f -ideals of degree d are there in the polyno- ( n, 2) th perfect mial ring S = K [ x 1 , . . . , x n ]? number Structure of V ( n, 2) Is there any f -ideal which is not unmixed? Further works What can one say about f -ideals in general case? References

  12. Four questions Perfect sets and f -ideals Jin Guo Four questions Outline Introduction How to characterize f -ideals of degree d directly? Perfect sets and f -ideals of degree d How many f -ideals of degree d are there in the polyno- ( n, 2) th perfect mial ring S = K [ x 1 , . . . , x n ]? number Structure of V ( n, 2) Is there any f -ideal which is not unmixed? Further works What can one say about f -ideals in general case? References

  13. Our work Perfect sets and f -ideals Jin Guo Our work Outline Introduction Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  14. Our work Perfect sets and f -ideals Jin Guo Our work Outline Introduction Answer question (1); Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  15. Our work Perfect sets and f -ideals Jin Guo Our work Outline Introduction Answer question (1); Perfect sets and f -ideals of degree d Answer question (2) completely in the case d = 2; ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  16. Our work Perfect sets and f -ideals Jin Guo Our work Outline Introduction Answer question (1); Perfect sets and f -ideals of degree d Answer question (2) completely in the case d = 2; ( n, 2) th perfect number Give a positive answer to question (3) in general case, Structure of V ( n, 2) and a negative answer in the case d = 2; Further works References

  17. Our work Perfect sets and f -ideals Jin Guo Our work Outline Introduction Answer question (1); Perfect sets and f -ideals of degree d Answer question (2) completely in the case d = 2; ( n, 2) th perfect number Give a positive answer to question (3) in general case, Structure of V ( n, 2) and a negative answer in the case d = 2; Further works Give a preliminary answer to question (4). References

  18. Our work Perfect sets and f -ideals Jin Guo Our work Outline Introduction Answer question (1); Perfect sets and f -ideals of degree d Answer question (2) completely in the case d = 2; ( n, 2) th perfect number Give a positive answer to question (3) in general case, Structure of V ( n, 2) and a negative answer in the case d = 2; Further works Give a preliminary answer to question (4). References

  19. Some definitions Perfect sets and f -ideals Jin Guo Outline Introduction Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  20. Some definitions Perfect sets and f -ideals Jin Guo Upper generated and lower cover set Outline For a subset A ⊆ sm ( S ), the upper generated set ⊔ ( A ) of A Introduction is defined by Perfect sets and f -ideals of degree d ⊔ ( A ) = { gx i | g ∈ A, x i ∤ g, 1 ≤ i ≤ n } . ( n, 2) th perfect number Dually, the lower cover set ⊓ ( A ) of A is defined by Structure of ⊓ ( A ) = { h | 1 � = h, h = g/x i for some g ∈ A and some x i V ( n, 2) Further with x i | g } . works References Similarly, we define ⊔ 2 ( A ) = ⊔ ( ⊔ ( A )), and i =1 ⊔ i ( A ), ⊓ ∞ ( A ) = ∪ ∞ i =1 ⊓ i ( A ). ⊔ ∞ ( A ) = ∪ ∞

  21. Some definitions Perfect sets and f -ideals Jin Guo Outline Perfect set Introduction Perfect sets A ⊆ sm ( S ) d is called and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  22. Some definitions Perfect sets and f -ideals Jin Guo Outline Perfect set Introduction Perfect sets A ⊆ sm ( S ) d is called and f -ideals of degree d upper perfect : If ⊔ ( A ) = sm ( S ) d +1 holds; ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  23. Some definitions Perfect sets and f -ideals Jin Guo Outline Perfect set Introduction Perfect sets A ⊆ sm ( S ) d is called and f -ideals of degree d upper perfect : If ⊔ ( A ) = sm ( S ) d +1 holds; ( n, 2) th perfect number lower perfect : If ⊓ ( A ) = sm ( S ) d − 1 holds; Structure of V ( n, 2) Further works References

  24. Some definitions Perfect sets and f -ideals Jin Guo Outline Perfect set Introduction Perfect sets A ⊆ sm ( S ) d is called and f -ideals of degree d upper perfect : If ⊔ ( A ) = sm ( S ) d +1 holds; ( n, 2) th perfect number lower perfect : If ⊓ ( A ) = sm ( S ) d − 1 holds; Structure of V ( n, 2) perfect : If A is both upper perfect and lower perfect. Further works References

  25. Some definitions Perfect sets and f -ideals Jin Guo Outline Perfect set Introduction Perfect sets A ⊆ sm ( S ) d is called and f -ideals of degree d upper perfect : If ⊔ ( A ) = sm ( S ) d +1 holds; ( n, 2) th perfect number lower perfect : If ⊓ ( A ) = sm ( S ) d − 1 holds; Structure of V ( n, 2) perfect : If A is both upper perfect and lower perfect. Further works References

  26. Some definitions Perfect sets and f -ideals Jin Guo Outline Homogeneous of degree d Introduction Perfect sets A monomial ideal I is called of degree d (or alternatively, and f -ideals of degree d homogeneous of degree d ), if all monomials in G ( I ) have the ( n, 2) th same degree d . perfect number Structure of V ( n, 2) Further works References

  27. Some definitions Perfect sets and f -ideals Jin Guo Outline Homogeneous of degree d Introduction Perfect sets A monomial ideal I is called of degree d (or alternatively, and f -ideals of degree d homogeneous of degree d ), if all monomials in G ( I ) have the ( n, 2) th same degree d . perfect number Structure of If I is an f -ideal of S = K [ x 1 , . . . , x n ], and homogeneous of V ( n, 2) degree d , we also call I an ( n, d ) th f -ideal. Correspondingly, Further we can define an ( n, d ) th perfect sets. works References

  28. Some definitions Perfect sets and f -ideals Jin Guo Examples Outline Let S = K [ x 1 , x 2 , x 3 , x 4 ]. Consider the following Introduction three subsets of sm ( S ) 2 : A = { x 1 x 2 , x 1 x 3 , x 1 x 4 } , B = Perfect sets { x 1 x 2 , x 1 x 3 , x 2 x 3 } , C = { x 1 x 2 , x 3 x 4 } . and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  29. Some definitions Perfect sets and f -ideals Jin Guo Examples Outline Let S = K [ x 1 , x 2 , x 3 , x 4 ]. Consider the following Introduction three subsets of sm ( S ) 2 : A = { x 1 x 2 , x 1 x 3 , x 1 x 4 } , B = Perfect sets { x 1 x 2 , x 1 x 3 , x 2 x 3 } , C = { x 1 x 2 , x 3 x 4 } . and f -ideals of degree d ( n, 2) th A is lower perfect but not upper perfect, since x 2 x 3 x 4 �∈ perfect ⊔ ( A ); number Structure of V ( n, 2) Further works References

  30. Some definitions Perfect sets and f -ideals Jin Guo Examples Outline Let S = K [ x 1 , x 2 , x 3 , x 4 ]. Consider the following Introduction three subsets of sm ( S ) 2 : A = { x 1 x 2 , x 1 x 3 , x 1 x 4 } , B = Perfect sets { x 1 x 2 , x 1 x 3 , x 2 x 3 } , C = { x 1 x 2 , x 3 x 4 } . and f -ideals of degree d ( n, 2) th A is lower perfect but not upper perfect, since x 2 x 3 x 4 �∈ perfect ⊔ ( A ); number Structure of V ( n, 2) B is upper perfect but not lower perfect, since x 4 �∈ ⊓ ( B ); Further works References

  31. Some definitions Perfect sets and f -ideals Jin Guo Examples Outline Let S = K [ x 1 , x 2 , x 3 , x 4 ]. Consider the following Introduction three subsets of sm ( S ) 2 : A = { x 1 x 2 , x 1 x 3 , x 1 x 4 } , B = Perfect sets { x 1 x 2 , x 1 x 3 , x 2 x 3 } , C = { x 1 x 2 , x 3 x 4 } . and f -ideals of degree d ( n, 2) th A is lower perfect but not upper perfect, since x 2 x 3 x 4 �∈ perfect ⊔ ( A ); number Structure of V ( n, 2) B is upper perfect but not lower perfect, since x 4 �∈ ⊓ ( B ); Further works References C is perfect.

  32. Some definitions Perfect sets and f -ideals Jin Guo Examples Outline Let S = K [ x 1 , x 2 , x 3 , x 4 ]. Consider the following Introduction three subsets of sm ( S ) 2 : A = { x 1 x 2 , x 1 x 3 , x 1 x 4 } , B = Perfect sets { x 1 x 2 , x 1 x 3 , x 2 x 3 } , C = { x 1 x 2 , x 3 x 4 } . and f -ideals of degree d ( n, 2) th A is lower perfect but not upper perfect, since x 2 x 3 x 4 �∈ perfect ⊔ ( A ); number Structure of V ( n, 2) B is upper perfect but not lower perfect, since x 4 �∈ ⊓ ( B ); Further works References C is perfect.

  33. Main result of this part Perfect sets and f -ideals Jin Guo Outline Characterization of ( n, d ) th f -ideals Introduction Perfect sets Let S = K [ x 1 , . . . , x n ], and let I be a square-free monomial and f -ideals of degree d ideal of S of degree d with the minimal generating set G ( I ). ( n, 2) th Then I is an f -ideal if and only if the followings hold: perfect number Structure of V ( n, 2) Further works References

  34. Main result of this part Perfect sets and f -ideals Jin Guo Outline Characterization of ( n, d ) th f -ideals Introduction Perfect sets Let S = K [ x 1 , . . . , x n ], and let I be a square-free monomial and f -ideals of degree d ideal of S of degree d with the minimal generating set G ( I ). ( n, 2) th Then I is an f -ideal if and only if the followings hold: perfect number | G ( I ) | = 1 2 C d n ; Structure of V ( n, 2) Further works References

  35. Main result of this part Perfect sets and f -ideals Jin Guo Outline Characterization of ( n, d ) th f -ideals Introduction Perfect sets Let S = K [ x 1 , . . . , x n ], and let I be a square-free monomial and f -ideals of degree d ideal of S of degree d with the minimal generating set G ( I ). ( n, 2) th Then I is an f -ideal if and only if the followings hold: perfect number | G ( I ) | = 1 2 C d n ; Structure of V ( n, 2) G ( I ) is an ( n, d ) th perfect set. Further works References

  36. Main result of this part Perfect sets and f -ideals Jin Guo Outline Characterization of ( n, d ) th f -ideals Introduction Perfect sets Let S = K [ x 1 , . . . , x n ], and let I be a square-free monomial and f -ideals of degree d ideal of S of degree d with the minimal generating set G ( I ). ( n, 2) th Then I is an f -ideal if and only if the followings hold: perfect number | G ( I ) | = 1 2 C d n ; Structure of V ( n, 2) G ( I ) is an ( n, d ) th perfect set. Further works References

  37. idea Perfect sets and f -ideals Jin Guo How to construct an ( n, d ) th f -ideal Outline Introduction Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  38. idea Perfect sets and f -ideals Jin Guo How to construct an ( n, d ) th f -ideal Outline Introduction Find an ( n, d ) th perfect set A , such that | A | ≤ 1 2 C d n ; Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  39. idea Perfect sets and f -ideals Jin Guo How to construct an ( n, d ) th f -ideal Outline Introduction Find an ( n, d ) th perfect set A , such that | A | ≤ 1 2 C d n ; Perfect sets and f -ideals of degree d Choose D ⊆ sm ( S ) d \ A randomly, such that | D | = 1 2 C d n − ( n, 2) th | A | ; perfect number Structure of V ( n, 2) Further works References

  40. idea Perfect sets and f -ideals Jin Guo How to construct an ( n, d ) th f -ideal Outline Introduction Find an ( n, d ) th perfect set A , such that | A | ≤ 1 2 C d n ; Perfect sets and f -ideals of degree d Choose D ⊆ sm ( S ) d \ A randomly, such that | D | = 1 2 C d n − ( n, 2) th | A | ; perfect number Structure of Let I be the ideal generated by A ∪ D ; V ( n, 2) Further works References

  41. idea Perfect sets and f -ideals Jin Guo How to construct an ( n, d ) th f -ideal Outline Introduction Find an ( n, d ) th perfect set A , such that | A | ≤ 1 2 C d n ; Perfect sets and f -ideals of degree d Choose D ⊆ sm ( S ) d \ A randomly, such that | D | = 1 2 C d n − ( n, 2) th | A | ; perfect number Structure of Let I be the ideal generated by A ∪ D ; V ( n, 2) Further I is an ( n, d ) th f -ideal. works References

  42. idea Perfect sets and f -ideals Jin Guo How to construct an ( n, d ) th f -ideal Outline Introduction Find an ( n, d ) th perfect set A , such that | A | ≤ 1 2 C d n ; Perfect sets and f -ideals of degree d Choose D ⊆ sm ( S ) d \ A randomly, such that | D | = 1 2 C d n − ( n, 2) th | A | ; perfect number Structure of Let I be the ideal generated by A ∪ D ; V ( n, 2) Further I is an ( n, d ) th f -ideal. works References

  43. How to find an ( n, d ) th perfect set Perfect sets and f -ideals For a general d ≥ 2, it is not easy to find an ( n, d ) th perfect Jin Guo set, but it is not hard when d = 2. Outline Introduction Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  44. How to find an ( n, d ) th perfect set Perfect sets and f -ideals For a general d ≥ 2, it is not easy to find an ( n, d ) th perfect Jin Guo set, but it is not hard when d = 2. Outline How to find an ( n, 2) th perfect set Introduction Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  45. How to find an ( n, d ) th perfect set Perfect sets and f -ideals For a general d ≥ 2, it is not easy to find an ( n, d ) th perfect Jin Guo set, but it is not hard when d = 2. Outline How to find an ( n, 2) th perfect set Introduction Perfect sets and f -ideals Divide [ n ] into two part B and C (Actually, C = B ); of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  46. How to find an ( n, d ) th perfect set Perfect sets and f -ideals For a general d ≥ 2, it is not easy to find an ( n, d ) th perfect Jin Guo set, but it is not hard when d = 2. Outline How to find an ( n, 2) th perfect set Introduction Perfect sets and f -ideals Divide [ n ] into two part B and C (Actually, C = B ); of degree d ( n, 2) th perfect Let A = { x i x j | i, j ∈ B, or i, j ∈ C } ; number Structure of V ( n, 2) Further works References

  47. How to find an ( n, d ) th perfect set Perfect sets and f -ideals For a general d ≥ 2, it is not easy to find an ( n, d ) th perfect Jin Guo set, but it is not hard when d = 2. Outline How to find an ( n, 2) th perfect set Introduction Perfect sets and f -ideals Divide [ n ] into two part B and C (Actually, C = B ); of degree d ( n, 2) th perfect Let A = { x i x j | i, j ∈ B, or i, j ∈ C } ; number Structure of A is an ( n, 2) th perfect set. V ( n, 2) Further works References

  48. How to find an ( n, d ) th perfect set Perfect sets and f -ideals For a general d ≥ 2, it is not easy to find an ( n, d ) th perfect Jin Guo set, but it is not hard when d = 2. Outline How to find an ( n, 2) th perfect set Introduction Perfect sets and f -ideals Divide [ n ] into two part B and C (Actually, C = B ); of degree d ( n, 2) th perfect Let A = { x i x j | i, j ∈ B, or i, j ∈ C } ; number Structure of A is an ( n, 2) th perfect set. V ( n, 2) Further works Actually, this is almost the unique method to construct References a perfect set with a small cardinality.

  49. How to find an ( n, d ) th perfect set Perfect sets and f -ideals For a general d ≥ 2, it is not easy to find an ( n, d ) th perfect Jin Guo set, but it is not hard when d = 2. Outline How to find an ( n, 2) th perfect set Introduction Perfect sets and f -ideals Divide [ n ] into two part B and C (Actually, C = B ); of degree d ( n, 2) th perfect Let A = { x i x j | i, j ∈ B, or i, j ∈ C } ; number Structure of A is an ( n, 2) th perfect set. V ( n, 2) Further works Actually, this is almost the unique method to construct References a perfect set with a small cardinality.

  50. ( n, d ) th perfect number Perfect sets and f -ideals Jin Guo Definition The least number among cardinalities of ( n, d ) th perfect sets, Outline Introduction denoted by N ( n,d ) . Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  51. ( n, d ) th perfect number Perfect sets and f -ideals Jin Guo Definition The least number among cardinalities of ( n, d ) th perfect sets, Outline Introduction denoted by N ( n,d ) . Perfect sets and f -ideals of degree d ( n, 2) th perfect number ( n, 2) th perfect Let k be a positive integer, and let n ≥ 4. Then the perfect number number N ( n, 2) is given by the following rules: Structure of V ( n, 2) Further works References

  52. ( n, d ) th perfect number Perfect sets and f -ideals Jin Guo Definition The least number among cardinalities of ( n, d ) th perfect sets, Outline Introduction denoted by N ( n,d ) . Perfect sets and f -ideals of degree d ( n, 2) th perfect number ( n, 2) th perfect Let k be a positive integer, and let n ≥ 4. Then the perfect number number N ( n, 2) is given by the following rules: Structure of V ( n, 2) Further k = k 2 − k ; If n = 2 k , then N ( n, 2) = C 2 k + C 2 works References

  53. ( n, d ) th perfect number Perfect sets and f -ideals Jin Guo Definition The least number among cardinalities of ( n, d ) th perfect sets, Outline Introduction denoted by N ( n,d ) . Perfect sets and f -ideals of degree d ( n, 2) th perfect number ( n, 2) th perfect Let k be a positive integer, and let n ≥ 4. Then the perfect number number N ( n, 2) is given by the following rules: Structure of V ( n, 2) Further k = k 2 − k ; If n = 2 k , then N ( n, 2) = C 2 k + C 2 works References If n = 2 k + 1, then N ( n, 2) = C 2 k + C 2 k +1 = k 2 .

  54. ( n, d ) th perfect number Perfect sets and f -ideals Jin Guo Definition The least number among cardinalities of ( n, d ) th perfect sets, Outline Introduction denoted by N ( n,d ) . Perfect sets and f -ideals of degree d ( n, 2) th perfect number ( n, 2) th perfect Let k be a positive integer, and let n ≥ 4. Then the perfect number number N ( n, 2) is given by the following rules: Structure of V ( n, 2) Further k = k 2 − k ; If n = 2 k , then N ( n, 2) = C 2 k + C 2 works References If n = 2 k + 1, then N ( n, 2) = C 2 k + C 2 k +1 = k 2 .

  55. Existence of ( n, 2) th f -ideal Perfect sets and f -ideals Existence Jin Guo V ( n, 2) � = ∅ if and only if 2 | C 2 n , i.e., if and only if n = 4 k or Outline n = 4 k + 1 for some positive integer k . Introduction Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  56. Existence of ( n, 2) th f -ideal Perfect sets and f -ideals Existence Jin Guo V ( n, 2) � = ∅ if and only if 2 | C 2 n , i.e., if and only if n = 4 k or Outline n = 4 k + 1 for some positive integer k . Introduction Perfect sets Note that: When n = 4 k and f -ideals of degree d N ( n, 2) = 4 k 2 − 2 k and C 2 n / 2 = 4 k 2 − k ; ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  57. Existence of ( n, 2) th f -ideal Perfect sets and f -ideals Existence Jin Guo V ( n, 2) � = ∅ if and only if 2 | C 2 n , i.e., if and only if n = 4 k or Outline n = 4 k + 1 for some positive integer k . Introduction Perfect sets Note that: When n = 4 k and f -ideals of degree d N ( n, 2) = 4 k 2 − 2 k and C 2 n / 2 = 4 k 2 − k ; ( n, 2) th perfect number N ( n, 2) ≤ C 2 n / 2. Structure of V ( n, 2) Further works References

  58. Existence of ( n, 2) th f -ideal Perfect sets and f -ideals Existence Jin Guo V ( n, 2) � = ∅ if and only if 2 | C 2 n , i.e., if and only if n = 4 k or Outline n = 4 k + 1 for some positive integer k . Introduction Perfect sets Note that: When n = 4 k and f -ideals of degree d N ( n, 2) = 4 k 2 − 2 k and C 2 n / 2 = 4 k 2 − k ; ( n, 2) th perfect number N ( n, 2) ≤ C 2 n / 2. Structure of V ( n, 2) When n = 4 k + 1 Further works N ( n, 2) = 4 k 2 and C 2 n / 2 = 4 k 2 + k ; References

  59. Existence of ( n, 2) th f -ideal Perfect sets and f -ideals Existence Jin Guo V ( n, 2) � = ∅ if and only if 2 | C 2 n , i.e., if and only if n = 4 k or Outline n = 4 k + 1 for some positive integer k . Introduction Perfect sets Note that: When n = 4 k and f -ideals of degree d N ( n, 2) = 4 k 2 − 2 k and C 2 n / 2 = 4 k 2 − k ; ( n, 2) th perfect number N ( n, 2) ≤ C 2 n / 2. Structure of V ( n, 2) When n = 4 k + 1 Further works N ( n, 2) = 4 k 2 and C 2 n / 2 = 4 k 2 + k ; References N ( n, 2) ≤ C 2 n / 2.

  60. Existence of ( n, 2) th f -ideal Perfect sets and f -ideals Existence Jin Guo V ( n, 2) � = ∅ if and only if 2 | C 2 n , i.e., if and only if n = 4 k or Outline n = 4 k + 1 for some positive integer k . Introduction Perfect sets Note that: When n = 4 k and f -ideals of degree d N ( n, 2) = 4 k 2 − 2 k and C 2 n / 2 = 4 k 2 − k ; ( n, 2) th perfect number N ( n, 2) ≤ C 2 n / 2. Structure of V ( n, 2) When n = 4 k + 1 Further works N ( n, 2) = 4 k 2 and C 2 n / 2 = 4 k 2 + k ; References N ( n, 2) ≤ C 2 n / 2.

  61. Some notations Perfect sets Two Part Complete Structure and f -ideals Jin Guo Outline Introduction Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  62. Some notations Perfect sets Two Part Complete Structure and f -ideals Jin Guo For a subset B of [ n ], denote W B = { x i x j | i, j ∈ Outline B or i, j ∈ B } ; Introduction Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  63. Some notations Perfect sets Two Part Complete Structure and f -ideals Jin Guo For a subset B of [ n ], denote W B = { x i x j | i, j ∈ Outline B or i, j ∈ B } ; Introduction Clearly W B = W B holds, and W B is an ( n, 2) th perfect Perfect sets and f -ideals of degree d set; ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  64. Some notations Perfect sets Two Part Complete Structure and f -ideals Jin Guo For a subset B of [ n ], denote W B = { x i x j | i, j ∈ Outline B or i, j ∈ B } ; Introduction Clearly W B = W B holds, and W B is an ( n, 2) th perfect Perfect sets and f -ideals of degree d set; ( n, 2) th perfect A subset A of sm ( S ) 2 is called satisfying Two Part Com- number plete Structure , abbreviated as TPCS, if there exists a Structure of V ( n, 2) B ⊆ [ n ], such that W B ⊆ A ; Further works References

  65. Some notations Perfect sets Two Part Complete Structure and f -ideals Jin Guo For a subset B of [ n ], denote W B = { x i x j | i, j ∈ Outline B or i, j ∈ B } ; Introduction Clearly W B = W B holds, and W B is an ( n, 2) th perfect Perfect sets and f -ideals of degree d set; ( n, 2) th perfect A subset A of sm ( S ) 2 is called satisfying Two Part Com- number plete Structure , abbreviated as TPCS, if there exists a Structure of V ( n, 2) B ⊆ [ n ], such that W B ⊆ A ; Further works If further | B | = l , then A is called satisfying l th TPCS; References

  66. Some notations Perfect sets Two Part Complete Structure and f -ideals Jin Guo For a subset B of [ n ], denote W B = { x i x j | i, j ∈ Outline B or i, j ∈ B } ; Introduction Clearly W B = W B holds, and W B is an ( n, 2) th perfect Perfect sets and f -ideals of degree d set; ( n, 2) th perfect A subset A of sm ( S ) 2 is called satisfying Two Part Com- number plete Structure , abbreviated as TPCS, if there exists a Structure of V ( n, 2) B ⊆ [ n ], such that W B ⊆ A ; Further works If further | B | = l , then A is called satisfying l th TPCS; References An f -ideal I is called of l type , if G ( I ) satisfies l th TPCS;

  67. Some notations Perfect sets Two Part Complete Structure and f -ideals Jin Guo For a subset B of [ n ], denote W B = { x i x j | i, j ∈ Outline B or i, j ∈ B } ; Introduction Clearly W B = W B holds, and W B is an ( n, 2) th perfect Perfect sets and f -ideals of degree d set; ( n, 2) th perfect A subset A of sm ( S ) 2 is called satisfying Two Part Com- number plete Structure , abbreviated as TPCS, if there exists a Structure of V ( n, 2) B ⊆ [ n ], such that W B ⊆ A ; Further works If further | B | = l , then A is called satisfying l th TPCS; References An f -ideal I is called of l type , if G ( I ) satisfies l th TPCS; Denote by W l the set of f -ideals of l type in S .

  68. Some notations Perfect sets Two Part Complete Structure and f -ideals Jin Guo For a subset B of [ n ], denote W B = { x i x j | i, j ∈ Outline B or i, j ∈ B } ; Introduction Clearly W B = W B holds, and W B is an ( n, 2) th perfect Perfect sets and f -ideals of degree d set; ( n, 2) th perfect A subset A of sm ( S ) 2 is called satisfying Two Part Com- number plete Structure , abbreviated as TPCS, if there exists a Structure of V ( n, 2) B ⊆ [ n ], such that W B ⊆ A ; Further works If further | B | = l , then A is called satisfying l th TPCS; References An f -ideal I is called of l type , if G ( I ) satisfies l th TPCS; Denote by W l the set of f -ideals of l type in S .

  69. l type Perfect sets Question: and f -ideals Jin Guo Outline Introduction Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  70. l type Perfect sets Question: and f -ideals Jin Guo Is there any f -ideal who is of no l type? Outline Example Introduction Perfect sets and f -ideals of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

  71. l type Perfect sets Question: and f -ideals Jin Guo Is there any f -ideal who is of no l type? Outline Example Introduction Perfect sets and f -ideals Let S = K [ x 1 , x 2 , x 3 , x 4 , x 5 ]. Consider the ideal of degree d ( n, 2) th perfect I = � x 1 x 2 , x 2 x 3 , x 3 x 4 , x 4 x 5 , x 1 x 5 � number Structure of V ( n, 2) Further works References

  72. l type Perfect sets Question: and f -ideals Jin Guo Is there any f -ideal who is of no l type? Outline Example Introduction Perfect sets and f -ideals Let S = K [ x 1 , x 2 , x 3 , x 4 , x 5 ]. Consider the ideal of degree d ( n, 2) th perfect I = � x 1 x 2 , x 2 x 3 , x 3 x 4 , x 4 x 5 , x 1 x 5 � number Structure of V ( n, 2) Further It is direct to check that I is an f -ideal. works References

  73. l type Perfect sets Question: and f -ideals Jin Guo Is there any f -ideal who is of no l type? Outline Example Introduction Perfect sets and f -ideals Let S = K [ x 1 , x 2 , x 3 , x 4 , x 5 ]. Consider the ideal of degree d ( n, 2) th perfect I = � x 1 x 2 , x 2 x 3 , x 3 x 4 , x 4 x 5 , x 1 x 5 � number Structure of V ( n, 2) Further It is direct to check that I is an f -ideal. works References but I is not of l type for any l .

  74. l type Perfect sets Question: and f -ideals Jin Guo Is there any f -ideal who is of no l type? Outline Example Introduction Perfect sets and f -ideals Let S = K [ x 1 , x 2 , x 3 , x 4 , x 5 ]. Consider the ideal of degree d ( n, 2) th perfect I = � x 1 x 2 , x 2 x 3 , x 3 x 4 , x 4 x 5 , x 1 x 5 � number Structure of V ( n, 2) Further It is direct to check that I is an f -ideal. works References but I is not of l type for any l . Such kind of f -ideal of K [ x 1 , x 2 , x 3 , x 4 , x 5 ] is called C 5 (5-cycle).

  75. l type Perfect sets Question: and f -ideals Jin Guo Is there any f -ideal who is of no l type? Outline Example Introduction Perfect sets and f -ideals Let S = K [ x 1 , x 2 , x 3 , x 4 , x 5 ]. Consider the ideal of degree d ( n, 2) th perfect I = � x 1 x 2 , x 2 x 3 , x 3 x 4 , x 4 x 5 , x 1 x 5 � number Structure of V ( n, 2) Further It is direct to check that I is an f -ideal. works References but I is not of l type for any l . Such kind of f -ideal of K [ x 1 , x 2 , x 3 , x 4 , x 5 ] is called C 5 (5-cycle).

  76. l type Perfect sets and f -ideals Jin Guo Outline Introduction Perfect sets and f -ideals Another question: of degree d ( n, 2) th perfect number Structure of V ( n, 2) Further works References

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