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Shifting numerical monoids Christopher ONeill University of - - PowerPoint PPT Presentation

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Shifting numerical monoids Christopher ONeill University of California Davis coneill@math.ucdavis.edu Joint with Rebecca Conaway*, Felix Gotti, Jesse Horton*, Roberto Pelayo, Mesa Williams*, and Brian Wissman * = undergraduate student

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Shifting numerical monoids

Christopher O’Neill

University of California Davis coneill@math.ucdavis.edu Joint with Rebecca Conaway*, Felix Gotti, Jesse Horton*, Roberto Pelayo, Mesa Williams*, and Brian Wissman * = undergraduate student

March 21, 2017

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 1 / 25

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Numerical monoids

Definition

A numerical monoid S is an additive submonoid of N with |N \ S| < ∞.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 2 / 25

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Numerical monoids

Definition

A numerical monoid S is an additive submonoid of N with |N \ S| < ∞.

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 2 / 25

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Numerical monoids

Definition

A numerical monoid S is an additive submonoid of N with |N \ S| < ∞.

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid”

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 2 / 25

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Numerical monoids

Definition

A numerical monoid S is an additive submonoid of N with |N \ S| < ∞.

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid” Factorizations: 60 =

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 2 / 25

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SLIDE 6

Numerical monoids

Definition

A numerical monoid S is an additive submonoid of N with |N \ S| < ∞.

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid” Factorizations: 60 = 7(6) + 2(9)

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 2 / 25

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SLIDE 7

Numerical monoids

Definition

A numerical monoid S is an additive submonoid of N with |N \ S| < ∞.

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid” Factorizations: 60 = 7(6) + 2(9) = 3(20)

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 2 / 25

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SLIDE 8

Numerical monoids

Definition

A numerical monoid S is an additive submonoid of N with |N \ S| < ∞.

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid” Factorizations: 60 = 7(6) + 2(9) = 3(20)

  • (7, 2, 0)

(0, 0, 3)

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 2 / 25

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Numerical monoids

Fix a numerical monoid S = r1, . . . , rk.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 3 / 25

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Numerical monoids

Fix a numerical monoid S = r1, . . . , rk. π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk denotes the factorization homomorphism of S.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 3 / 25

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Numerical monoids

Fix a numerical monoid S = r1, . . . , rk. π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk denotes the factorization homomorphism of S. In particular, π−1(n) =

  • a ∈ Nk : n = a1r1 + · · · + akrk
  • is the set of factorizations of n ∈ S.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 3 / 25

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Numerical monoids

Fix a numerical monoid S = r1, . . . , rk. π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk denotes the factorization homomorphism of S. In particular, π−1(n) =

  • a ∈ Nk : n = a1r1 + · · · + akrk
  • is the set of factorizations of n ∈ S.

|a| = a1 + · · · + ak (length of a)

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 3 / 25

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Numerical monoids

Fix a numerical monoid S = r1, . . . , rk. π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk denotes the factorization homomorphism of S. In particular, π−1(n) =

  • a ∈ Nk : n = a1r1 + · · · + akrk
  • is the set of factorizations of n ∈ S.

|a| = a1 + · · · + ak (length of a)

Example

S = 6, 9, 20:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 3 / 25

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Numerical monoids

Fix a numerical monoid S = r1, . . . , rk. π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk denotes the factorization homomorphism of S. In particular, π−1(n) =

  • a ∈ Nk : n = a1r1 + · · · + akrk
  • is the set of factorizations of n ∈ S.

|a| = a1 + · · · + ak (length of a)

Example

S = 6, 9, 20: π−1(60) = {(10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)}

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 3 / 25

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Numerical monoids

Fix a numerical monoid S = r1, . . . , rk. π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk denotes the factorization homomorphism of S. In particular, π−1(n) =

  • a ∈ Nk : n = a1r1 + · · · + akrk
  • is the set of factorizations of n ∈ S.

|a| = a1 + · · · + ak (length of a)

Example

S = 6, 9, 20: π−1(60) = {(10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)} Possible factorization lengths for n = 60: 3, 7, 8, 9, 10.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 3 / 25

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Numerical monoids

Fix a numerical monoid S = r1, . . . , rk. π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk denotes the factorization homomorphism of S. In particular, π−1(n) =

  • a ∈ Nk : n = a1r1 + · · · + akrk
  • is the set of factorizations of n ∈ S.

|a| = a1 + · · · + ak (length of a)

Example

S = 6, 9, 20: π−1(60) = {(10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)} Possible factorization lengths for n = 60: 3, 7, 8, 9, 10. π−1(1000001) =

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 3 / 25

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Numerical monoids

Fix a numerical monoid S = r1, . . . , rk. π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk denotes the factorization homomorphism of S. In particular, π−1(n) =

  • a ∈ Nk : n = a1r1 + · · · + akrk
  • is the set of factorizations of n ∈ S.

|a| = a1 + · · · + ak (length of a)

Example

S = 6, 9, 20: π−1(60) = {(10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)} Possible factorization lengths for n = 60: 3, 7, 8, 9, 10. π−1(1000001) = {

  • shortest

, . . . ,

  • longest

}

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 3 / 25

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SLIDE 18

Numerical monoids

Fix a numerical monoid S = r1, . . . , rk. π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk denotes the factorization homomorphism of S. In particular, π−1(n) =

  • a ∈ Nk : n = a1r1 + · · · + akrk
  • is the set of factorizations of n ∈ S.

|a| = a1 + · · · + ak (length of a)

Example

S = 6, 9, 20: π−1(60) = {(10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)} Possible factorization lengths for n = 60: 3, 7, 8, 9, 10. π−1(1000001) = { (2, 1, 49999)

  • shortest

, . . . ,

  • longest

}

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 3 / 25

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Numerical monoids

Fix a numerical monoid S = r1, . . . , rk. π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk denotes the factorization homomorphism of S. In particular, π−1(n) =

  • a ∈ Nk : n = a1r1 + · · · + akrk
  • is the set of factorizations of n ∈ S.

|a| = a1 + · · · + ak (length of a)

Example

S = 6, 9, 20: π−1(60) = {(10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)} Possible factorization lengths for n = 60: 3, 7, 8, 9, 10. π−1(1000001) = { (2, 1, 49999)

  • shortest

, . . . , (166662, 1, 1)

  • longest

}

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 3 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 4 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Delta set ∆(Mn): successive factorization length differences in Mn.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 4 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Delta set ∆(Mn): successive factorization length differences in Mn.

Theorem (Chapman-Kaplan-Lemburg-Niles-Zlogar, 2014)

The delta set ∆(Mn) is singleton for n ≫ 0.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 4 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Delta set ∆(Mn): successive factorization length differences in Mn.

Theorem (Chapman-Kaplan-Lemburg-Niles-Zlogar, 2014)

The delta set ∆(Mn) is singleton for n ≫ 0. Mn = n, n + 6, n + 9, n + 20:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 4 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Delta set ∆(Mn): successive factorization length differences in Mn.

Theorem (Chapman-Kaplan-Lemburg-Niles-Zlogar, 2014)

The delta set ∆(Mn) is singleton for n ≫ 0. Mn = n, n + 6, n + 9, n + 20: ∆(Mn) = {1} for all n ≥ 48

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 4 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk.

50 100 150 200 2 4 6 8 10 12 14

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 5 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Catenary degree c(Mn): measures spread of factorizations in Mn.

50 100 150 200 2 4 6 8 10 12 14

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 5 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Catenary degree c(Mn): measures spread of factorizations in Mn. Mn = n, n + 6, n + 9, n + 20:

50 100 150 200 2 4 6 8 10 12 14

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 5 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Catenary degree c(Mn): measures spread of factorizations in Mn. Mn = n, n + 6, n + 9, n + 20:

50 100 150 200 2 4 6 8 10 12 14

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 5 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Catenary degree c(Mn): measures spread of factorizations in Mn. Mn = n, n + 6, n + 9, n + 20: c(Mn) is periodic-linear (quasilinear) for n ≥ 126.

50 100 150 200 2 4 6 8 10 12 14

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 5 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk.

50 100 150 200 500 1000 1500 2000 2500 3000 Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 6 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Betti numbers βi(Mn): Betti numbers of the defining toric ideal IMn.

50 100 150 200 500 1000 1500 2000 2500 3000 Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 6 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Betti numbers βi(Mn): Betti numbers of the defining toric ideal IMn.

Theorem (Vu, 2014)

The Betti numbers of Mn are eventually rk-periodic in n.

50 100 150 200 500 1000 1500 2000 2500 3000 Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 6 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Betti numbers βi(Mn): Betti numbers of the defining toric ideal IMn.

Theorem (Vu, 2014)

The Betti numbers of Mn are eventually rk-periodic in n. Mn = n, n + 6, n + 9, n + 20: Graded degrees for β0(Mn)

50 100 150 200 500 1000 1500 2000 2500 3000 Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 6 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 7 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Observations:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 7 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Observations: Known: the Betti numbers n → βi(Mn) are eventually rk-periodic.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 7 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Observations: Known: the Betti numbers n → βi(Mn) are eventually rk-periodic. Known: the function n → ∆(Mn) is eventually singleton.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 7 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Observations: Known: the Betti numbers n → βi(Mn) are eventually rk-periodic. Known: the function n → ∆(Mn) is eventually singleton. Observed: the function n → c(Mn) is eventually rk-quasilinear.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 7 / 25

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To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Observations: Known: the Betti numbers n → βi(Mn) are eventually rk-periodic. Known: the function n → ∆(Mn) is eventually singleton. Observed: the function n → c(Mn) is eventually rk-quasilinear. Underlying cause:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 7 / 25

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SLIDE 40

To shift a numerical monoid. . .

Fix S = r1, . . . , rk ⊂ (N, +), and let Mn = n, n + r1, . . . , n + rk. Observations: Known: the Betti numbers n → βi(Mn) are eventually rk-periodic. Known: the function n → ∆(Mn) is eventually singleton. Observed: the function n → c(Mn) is eventually rk-quasilinear. Underlying cause: minimal presentations!

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 7 / 25

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Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 8 / 25

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Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

Factorization homomorphism: π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 8 / 25

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Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

Factorization homomorphism: π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

The kernel ker π is the relation ∼ on Nk with a ∼ b whenever π(a) = π(b)

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 8 / 25

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SLIDE 44

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

Factorization homomorphism: π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

The kernel ker π is the relation ∼ on Nk with a ∼ b whenever π(a) = π(b) ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 8 / 25

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SLIDE 45

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

Factorization homomorphism: π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

The kernel ker π is the relation ∼ on Nk with a ∼ b whenever π(a) = π(b) ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation. a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 8 / 25

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SLIDE 46

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

Factorization homomorphism: π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

The kernel ker π is the relation ∼ on Nk with a ∼ b whenever π(a) = π(b) ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation. a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

slide-47
SLIDE 47

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

Factorization homomorphism: π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk Monomial map:

Definition

The kernel ker π is the relation ∼ on Nk with a ∼ b whenever π(a) = π(b) ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation. a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

slide-48
SLIDE 48

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

Factorization homomorphism: π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk Monomial map: ϕ : k[x1, . . . , xk] − → k[y] xi − → yri

Definition

The kernel ker π is the relation ∼ on Nk with a ∼ b whenever π(a) = π(b) ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation. a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

slide-49
SLIDE 49

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

Factorization homomorphism: π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk Monomial map: ϕ : k[x1, . . . , xk] − → k[y] xi − → yri

Definition

The kernel ker π is the relation ∼ on Nk with a ∼ b whenever π(a) = π(b) xa − xb ∈ IS = ker ϕ ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation. a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

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SLIDE 50

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

Factorization homomorphism: π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk Monomial map: ϕ : k[x1, . . . , xk] − → k[y] xi − → yri

Definition

The kernel ker π is the relation ∼ on Nk with a ∼ b whenever π(a) = π(b) xa − xb ∈ IS = ker ϕ ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c xa − xa = 0 ∈ IS that is closed under translation. a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

slide-51
SLIDE 51

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

Factorization homomorphism: π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk Monomial map: ϕ : k[x1, . . . , xk] − → k[y] xi − → yri

Definition

The kernel ker π is the relation ∼ on Nk with a ∼ b whenever π(a) = π(b) xa − xb ∈ IS = ker ϕ ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c xa − xa = 0 ∈ IS xa − xb ∈ IS ⇒ xb − xa ∈ IS that is closed under translation. a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

slide-52
SLIDE 52

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

Factorization homomorphism: π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk Monomial map: ϕ : k[x1, . . . , xk] − → k[y] xi − → yri

Definition

The kernel ker π is the relation ∼ on Nk with a ∼ b whenever π(a) = π(b) xa − xb ∈ IS = ker ϕ ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c xa − xa = 0 ∈ IS xa − xb ∈ IS ⇒ xb − xa ∈ IS (xa − xb) + (xb − xc) = xa − xc that is closed under translation. a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

slide-53
SLIDE 53

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

Factorization homomorphism: π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk Monomial map: ϕ : k[x1, . . . , xk] − → k[y] xi − → yri

Definition

The kernel ker π is the relation ∼ on Nk with a ∼ b whenever π(a) = π(b) xa − xb ∈ IS = ker ϕ ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c xa − xa = 0 ∈ IS xa − xb ∈ IS ⇒ xb − xa ∈ IS (xa − xb) + (xb − xc) = xa − xc that is closed under translation. a ∼ b ⇒ a + c ∼ b + c xa − xb ∈ IS ⇒ xc(xa − xb) ∈ IS

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

slide-54
SLIDE 54

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

slide-55
SLIDE 55

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

slide-56
SLIDE 56

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

slide-57
SLIDE 57

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20: ρ = {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} ⊂ ker π

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

slide-58
SLIDE 58

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20: ρ = {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} ⊂ ker π π−1(18):

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

slide-59
SLIDE 59

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20: ρ = {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} ⊂ ker π π−1(18): π−1(60):

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

slide-60
SLIDE 60

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20: ρ = {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} ⊂ ker π π−1(18): π−1(60):

((7, 2, 0), (4, 4, 0)) = ((3, 0, 0), (0, 2, 0)) + ((4, 2, 0), (4, 2, 0))

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

slide-61
SLIDE 61

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20: ρ = {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} ⊂ ker π π−1(18): π−1(60):

((7, 2, 0), (4, 4, 0)) = ((3, 0, 0), (0, 2, 0)) + ((4, 2, 0), (4, 2, 0)) Cong(ρ) = ker π when the graph on π−1(n) is connected for all n ∈ S.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

slide-62
SLIDE 62

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20: ρ = {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} ⊂ ker π π−1(18): π−1(60):

((7, 2, 0), (4, 4, 0)) = ((3, 0, 0), (0, 2, 0)) + ((4, 2, 0), (4, 2, 0)) Cong(ρ) = ker π when the graph on π−1(n) is connected for all n ∈ S. IS = xu − xv : (u, v) ∈ ρ

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

slide-63
SLIDE 63

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20: ρ = {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} ⊂ ker π π−1(18): π−1(60):

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

slide-64
SLIDE 64

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20: ρ = {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} ⊂ ker π π−1(18): π−1(60): All minimal presentations:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

slide-65
SLIDE 65

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20: ρ = {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} ⊂ ker π π−1(18): π−1(60): All minimal presentations:

{((3, 0, 0), (0, 2, 0)), ((10, 7, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((7, 2, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((1, 6, 0), (0, 0, 3))}

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

slide-66
SLIDE 66

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20: ρ = {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} ⊂ ker π π−1(18): π−1(60): All minimal presentations:

{((3, 0, 0), (0, 2, 0)), ((10, 7, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((7, 2, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((1, 6, 0), (0, 0, 3))}

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

slide-67
SLIDE 67

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20: ρ = {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} ⊂ ker π π−1(18): π−1(60): All minimal presentations:

{((3, 0, 0), (0, 2, 0)), ((10, 7, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((7, 2, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((1, 6, 0), (0, 0, 3))}

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

slide-68
SLIDE 68

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20: ρ = {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} ⊂ ker π π−1(18): π−1(60): All minimal presentations:

{((3, 0, 0), (0, 2, 0)), ((10, 7, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((7, 2, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((1, 6, 0), (0, 0, 3))}

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

slide-69
SLIDE 69

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20: ρ = {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} ⊂ ker π π−1(18): π−1(60): All minimal presentations:

{((3, 0, 0), (0, 2, 0)), ((10, 7, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((7, 2, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((1, 6, 0), (0, 0, 3))}

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

slide-70
SLIDE 70

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20: ρ = {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} ⊂ ker π π−1(18): π−1(60): All minimal presentations:

{((3, 0, 0), (0, 2, 0)), ((10, 7, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((7, 2, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((1, 6, 0), (0, 0, 3))}

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

slide-71
SLIDE 71

Kernel congruences and minimal presentations

Let S = r1, . . . , rk. n = a1r1 + · · · + akrk

  • a = (a1, . . . , ak) ∈ Nk

π : Nk − → r1, . . . , rk a − → a1r1 + · · · + akrk

Definition

A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π. S = 6, 9, 20: ρ = {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} ⊂ ker π π−1(18): π−1(60): All minimal presentations:

{((3, 0, 0), (0, 2, 0)), ((10, 7, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((7, 2, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((4, 4, 0), (0, 0, 3))} {((3, 0, 0), (0, 2, 0)), ((1, 6, 0), (0, 0, 3))}

β0(IS) = {18, 60}

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

slide-72
SLIDE 72

Kernel congruences and minimal presentations

S = r1, . . . , rk, π : Nk − → S

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

slide-73
SLIDE 73

Kernel congruences and minimal presentations

S = r1, . . . , rk, π : Nk − → S A larger example: S = 13, 44, 106, 120.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

slide-74
SLIDE 74

Kernel congruences and minimal presentations

S = r1, . . . , rk, π : Nk − → S A larger example: S = 13, 44, 106, 120. Minimal presentation ρ has |ρ| = 5 relations.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

slide-75
SLIDE 75

Kernel congruences and minimal presentations

S = r1, . . . , rk, π : Nk − → S A larger example: S = 13, 44, 106, 120. Minimal presentation ρ has |ρ| = 5 relations.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

slide-76
SLIDE 76

Kernel congruences and minimal presentations

S = r1, . . . , rk, π : Nk − → S A larger example: S = 13, 44, 106, 120. Minimal presentation ρ has |ρ| = 5 relations.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

slide-77
SLIDE 77

Kernel congruences and minimal presentations

S = r1, . . . , rk, π : Nk − → S A larger example: S = 13, 44, 106, 120. Minimal presentation ρ has |ρ| = 5 relations.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

slide-78
SLIDE 78

Kernel congruences and minimal presentations

S = r1, . . . , rk, π : Nk − → S A larger example: S = 13, 44, 106, 120. Minimal presentation ρ has |ρ| = 5 relations.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

slide-79
SLIDE 79

Kernel congruences and minimal presentations

S = r1, . . . , rk, π : Nk − → S A larger example: S = 13, 44, 106, 120. Minimal presentation ρ has |ρ| = 5 relations.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

slide-80
SLIDE 80

Kernel congruences and minimal presentations

S = r1, . . . , rk, π : Nk − → S A larger example: S = 13, 44, 106, 120. Minimal presentation ρ has |ρ| = 5 relations.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

slide-81
SLIDE 81

Kernel congruences and minimal presentations

S = r1, . . . , rk, π : Nk − → S A larger example: S = 13, 44, 106, 120. Minimal presentation ρ has |ρ| = 5 relations.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

slide-82
SLIDE 82

Kernel congruences and minimal presentations

S = r1, . . . , rk, π : Nk − → S A larger example: S = 13, 44, 106, 120. Minimal presentation ρ has |ρ| = 5 relations.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

slide-83
SLIDE 83

Kernel congruences and minimal presentations

S = r1, . . . , rk, π : Nk − → S A larger example: S = 13, 44, 106, 120. Minimal presentation ρ has |ρ| = 5 relations.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

slide-84
SLIDE 84

Kernel congruences and minimal presentations

S = r1, . . . , rk, π : Nk − → S A larger example: S = 13, 44, 106, 120. Minimal presentation ρ has |ρ| = 5 relations.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

slide-85
SLIDE 85

Kernel congruences and minimal presentations

S = r1, . . . , rk, π : Nk − → S A larger example: S = 13, 44, 106, 120. Minimal presentation ρ has |ρ| = 5 relations.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

slide-86
SLIDE 86

Kernel congruences and minimal presentations

S = r1, . . . , rk, π : Nk − → S A larger example: S = 13, 44, 106, 120. Minimal presentation ρ has |ρ| = 5 relations.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

slide-87
SLIDE 87

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

slide-88
SLIDE 88

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

slide-89
SLIDE 89

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

slide-90
SLIDE 90

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

slide-91
SLIDE 91

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

slide-92
SLIDE 92

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn a0n + a1(n+r1) + · · · + ak(n+rk) = b0n + b1(n+r1) + · · · + bk(n+rk)

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

slide-93
SLIDE 93

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn a0n + a1(n+r1) + · · · + ak(n+rk) = b0n + b1(n+r1) + · · · + bk(n+rk)

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

slide-94
SLIDE 94

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn a0n + a1(n+r1) + · · · + ak(n+rk) = b0n + b1(n+r1) + · · · + bk(n+rk) |a|n + a1r1 + · · · + akrk = |b|n + b1r1 + · · · + bkrk

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

slide-95
SLIDE 95

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn a0n + a1(n+r1) + · · · + ak(n+rk) = b0n + b1(n+r1) + · · · + bk(n+rk) |a|n + a1r1 + · · · + akrk = |b|n + b1r1 + · · · + bkrk 2 types of minimal relations a ∼ b:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

slide-96
SLIDE 96

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn a0n + a1(n+r1) + · · · + ak(n+rk) = b0n + b1(n+r1) + · · · + bk(n+rk) |a|n + a1r1 + · · · + akrk = |b|n + b1r1 + · · · + bkrk 2 types of minimal relations a ∼ b: Relations among r1, . . . , rk (cheap): |a| = |b|

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

slide-97
SLIDE 97

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn a0n + a1(n+r1) + · · · + ak(n+rk) = b0n + b1(n+r1) + · · · + bk(n+rk) |a|n + a1r1 + · · · + akrk = |b|n + b1r1 + · · · + bkrk 2 types of minimal relations a ∼ b: Relations among r1, . . . , rk (cheap): |a| = |b| Relations that change # copies of n (costly): |a| < |b|

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

slide-98
SLIDE 98

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn a0n + a1(n+r1) + · · · + ak(n+rk) = b0n + b1(n+r1) + · · · + bk(n+rk) |a|n + a1r1 + · · · + akrk = |b|n + b1r1 + · · · + bkrk 2 types of minimal relations a ∼ b: Relations among r1, . . . , rk (cheap): |a| = |b| Relations that change # copies of n (costly): |a| < |b| mostly ak ← − − − − − − → mostly b0

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

slide-99
SLIDE 99

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn a0n + a1(n+r1) + · · · + ak(n+rk) = b0n + b1(n+r1) + · · · + bk(n+rk) |a|n + a1r1 + · · · + akrk = |b|n + b1r1 + · · · + bkrk 2 types of minimal relations a ∼ b: Relations among r1, . . . , rk (cheap): |a| = |b| Relations that change # copies of n (costly): |a| < |b| mostly ak ← − − − − − − → mostly b0 In Mn = n, n + 6, n + 9, n + 20 with n = 450:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

slide-100
SLIDE 100

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn a0n + a1(n+r1) + · · · + ak(n+rk) = b0n + b1(n+r1) + · · · + bk(n+rk) |a|n + a1r1 + · · · + akrk = |b|n + b1r1 + · · · + bkrk 2 types of minimal relations a ∼ b: Relations among r1, . . . , rk (cheap): |a| = |b| Relations that change # copies of n (costly): |a| < |b| mostly ak ← − − − − − − → mostly b0 In Mn = n, n + 6, n + 9, n + 20 with n = 450: 3(n + 6) = n + 2(n + 9) is cheap 4(n + 9) + 21(n + 20) = 25n + (n + 6) is costly

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

slide-101
SLIDE 101

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 14 / 25

slide-102
SLIDE 102

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 14 / 25

slide-103
SLIDE 103

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 14 / 25

slide-104
SLIDE 104

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

DON’T PANIC!

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 14 / 25

slide-105
SLIDE 105

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 15 / 25

slide-106
SLIDE 106

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Sneak peek for Mn = n, n + 6, n + 9, n + 20 and n ≫ 0:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 15 / 25

slide-107
SLIDE 107

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Sneak peek for Mn = n, n + 6, n + 9, n + 20 and n ≫ 0: M450:

(( 0, 0, 8, 0), (3, 2, 0, 3)), (( 0, 1, 6, 0), (4, 0, 0, 3)), (( 0, 3, 0, 0), (1, 0, 2, 0)), ((20, 5, 0, 0), (0, 0, 0, 24)), ((25, 1, 0, 0), (0, 0, 4, 21)), ((26, 0, 0, 0), (0, 2, 2, 21))

  • Christopher O’Neill (UC Davis)

Shifting numerical monoids March 21, 2017 15 / 25

slide-108
SLIDE 108

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Sneak peek for Mn = n, n + 6, n + 9, n + 20 and n ≫ 0: M450:

(( 0, 0, 8, 0), (3, 2, 0, 3)), (( 0, 1, 6, 0), (4, 0, 0, 3)), (( 0, 3, 0, 0), (1, 0, 2, 0)), ((20, 5, 0, 0), (0, 0, 0, 24)), ((25, 1, 0, 0), (0, 0, 4, 21)), ((26, 0, 0, 0), (0, 2, 2, 21))

  • M470:

(( 0, 0, 8, 0), (3, 2, 0, 3)), (( 0, 1, 6, 0), (4, 0, 0, 3)), (( 0, 3, 0, 0), (1, 0, 2, 0)), ((21, 5, 0, 0), (0, 0, 0, 25)), ((26, 1, 0, 0), (0, 0, 4, 22)), ((27, 0, 0, 0), (0, 2, 2, 22))

  • Christopher O’Neill (UC Davis)

Shifting numerical monoids March 21, 2017 15 / 25

slide-109
SLIDE 109

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Sneak peek for Mn = n, n + 6, n + 9, n + 20 and n ≫ 0: M450:

(( 0, 0, 8, 0), (3, 2, 0, 3)), (( 0, 1, 6, 0), (4, 0, 0, 3)), (( 0, 3, 0, 0), (1, 0, 2, 0)), ((20, 5, 0, 0), (0, 0, 0, 24)), ((25, 1, 0, 0), (0, 0, 4, 21)), ((26, 0, 0, 0), (0, 2, 2, 21))

  • M470:

(( 0, 0, 8, 0), (3, 2, 0, 3)), (( 0, 1, 6, 0), (4, 0, 0, 3)), (( 0, 3, 0, 0), (1, 0, 2, 0)), ((21, 5, 0, 0), (0, 0, 0, 25)), ((26, 1, 0, 0), (0, 0, 4, 22)), ((27, 0, 0, 0), (0, 2, 2, 22))

  • M490:

(( 0, 0, 8, 0), (3, 2, 0, 3)), (( 0, 1, 6, 0), (4, 0, 0, 3)), (( 0, 3, 0, 0), (1, 0, 2, 0)), ((22, 5, 0, 0), (0, 0, 0, 26)), ((27, 1, 0, 0), (0, 0, 4, 23)), ((28, 0, 0, 0), (0, 2, 2, 23))

  • Christopher O’Neill (UC Davis)

Shifting numerical monoids March 21, 2017 15 / 25

slide-110
SLIDE 110

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 16 / 25

slide-111
SLIDE 111

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Congruence properties preserved by Φn: Reflexive and symmetric closure

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 16 / 25

slide-112
SLIDE 112

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Congruence properties preserved by Φn: Reflexive and symmetric closure Translation closure

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 16 / 25

slide-113
SLIDE 113

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Congruence properties preserved by Φn: Reflexive and symmetric closure Translation closure Φn((a, a′) + (b, b)) = Φn(a, a′) + (b, b)

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 16 / 25

slide-114
SLIDE 114

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Congruence properties preserved by Φn: Reflexive and symmetric closure Translation closure Φn((a, a′) + (b, b)) = Φn(a, a′) + (b, b) Only missing link: transitivity

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 16 / 25

slide-115
SLIDE 115

Transitivity before/after shifting

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 17 / 25

slide-116
SLIDE 116

Transitivity before/after shifting

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

For Mn = n, n + 6, n + 9, n + 20:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 17 / 25

slide-117
SLIDE 117

Transitivity before/after shifting

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

For Mn = n, n + 6, n + 9, n + 20:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 17 / 25

slide-118
SLIDE 118

Transitivity before/after shifting

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

For Mn = n, n + 6, n + 9, n + 20:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 17 / 25

slide-119
SLIDE 119

Transitivity before/after shifting

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

For Mn = n, n + 6, n + 9, n + 20:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 17 / 25

slide-120
SLIDE 120

Transitivity before/after shifting

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

For Mn = n, n + 6, n + 9, n + 20:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 17 / 25

slide-121
SLIDE 121

Transitivity before/after shifting

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

For Mn = n, n + 6, n + 9, n + 20:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 17 / 25

slide-122
SLIDE 122

Transitivity before/after shifting

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

For Mn = n, n + 6, n + 9, n + 20:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 17 / 25

slide-123
SLIDE 123

Transitivity before/after shifting

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

For Mn = n, n + 6, n + 9, n + 20:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 17 / 25

slide-124
SLIDE 124

Transitivity before/after shifting

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

For Mn = n, n + 6, n + 9, n + 20:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 17 / 25

slide-125
SLIDE 125

Transitivity before/after shifting

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

For Mn = n, n + 6, n + 9, n + 20: translate after shifting to build a chain!

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 17 / 25

slide-126
SLIDE 126

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 18 / 25

slide-127
SLIDE 127

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 18 / 25

slide-128
SLIDE 128

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 18 / 25

slide-129
SLIDE 129

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 18 / 25

slide-130
SLIDE 130

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 18 / 25

slide-131
SLIDE 131

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 18 / 25

slide-132
SLIDE 132

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 18 / 25

slide-133
SLIDE 133

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 18 / 25

slide-134
SLIDE 134

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|: translate Φn(a, b) and Φn(b, c) ⇒ obtain Φn(a, c).

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 18 / 25

slide-135
SLIDE 135

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|: translate Φn(a, b) and Φn(b, c) ⇒ obtain Φn(a, c).

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 18 / 25

slide-136
SLIDE 136

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|: translate Φn(a, b) and Φn(b, c) ⇒ obtain Φn(a, c).

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 18 / 25

slide-137
SLIDE 137

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|: translate Φn(a, b) and Φn(b, c) ⇒ obtain Φn(a, c).

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 18 / 25

slide-138
SLIDE 138

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|: translate Φn(a, b) and Φn(b, c) ⇒ obtain Φn(a, c).

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 18 / 25

slide-139
SLIDE 139

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|: translate Φn(a, b) and Φn(b, c) ⇒ obtain Φn(a, c). If |a| < |c| < |b|:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 19 / 25

slide-140
SLIDE 140

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|: translate Φn(a, b) and Φn(b, c) ⇒ obtain Φn(a, c). If |a| < |c| < |b|:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 19 / 25

slide-141
SLIDE 141

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|: translate Φn(a, b) and Φn(b, c) ⇒ obtain Φn(a, c). If |a| < |c| < |b|:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 19 / 25

slide-142
SLIDE 142

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|: translate Φn(a, b) and Φn(b, c) ⇒ obtain Φn(a, c). If |a| < |c| < |b|:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 19 / 25

slide-143
SLIDE 143

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|: translate Φn(a, b) and Φn(b, c) ⇒ obtain Φn(a, c). If |a| < |c| < |b|: chaos!

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 19 / 25

slide-144
SLIDE 144

Monotone chains

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk is given by (a, a′) − →

    

(a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Fix (a, b), (b, c), (a, c) ∈ ker πn. If |a| < |b| < |c|: translate Φn(a, b) and Φn(b, c) ⇒ obtain Φn(a, c). If |a| < |c| < |b|: chaos! Need: monotone chains for Φn to preserve transitive closure.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 19 / 25

slide-145
SLIDE 145

The main result

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 20 / 25

slide-146
SLIDE 146

The main result

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 20 / 25

slide-147
SLIDE 147

The main result

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk

Key Lemma

If n > r 2

k and (a, a′) ∈ ρ with |a| > |a′| (costly), then a0 > 0 and a′ k > 0.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 20 / 25

slide-148
SLIDE 148

The main result

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk

Key Lemma

If n > r 2

k and (a, a′) ∈ ρ with |a| > |a′| (costly), then a0 > 0 and a′ k > 0.

This lemma ensures:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 20 / 25

slide-149
SLIDE 149

The main result

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk

Key Lemma

If n > r 2

k and (a, a′) ∈ ρ with |a| > |a′| (costly), then a0 > 0 and a′ k > 0.

This lemma ensures: Any two factorizations are connected by a monotone chain.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 20 / 25

slide-150
SLIDE 150

The main result

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk

Key Lemma

If n > r 2

k and (a, a′) ∈ ρ with |a| > |a′| (costly), then a0 > 0 and a′ k > 0.

This lemma ensures: Any two factorizations are connected by a monotone chain. The image Φn(ρ) generates ker πn+rk.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 20 / 25

slide-151
SLIDE 151

The main result

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 21 / 25

slide-152
SLIDE 152

The main result

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk Consequences:

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 21 / 25

slide-153
SLIDE 153

The main result

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk Consequences: The Betti numbers n → βj(Mn) are eventually rk-periodic: Graded degrees for β0(Mn) are πn(a) for each (a, a′) ∈ ρ (Control over minimal syzygies ⇒ higher Betti numbers)

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 21 / 25

slide-154
SLIDE 154

The main result

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk Consequences: The Betti numbers n → βj(Mn) are eventually rk-periodic: Graded degrees for β0(Mn) are πn(a) for each (a, a′) ∈ ρ (Control over minimal syzygies ⇒ higher Betti numbers) The function n → ∆(Mn) is eventually singleton: ∆(Mn) = {d} when ||a| − |a′|| ∈ {0, d} for all (a, a′) ∈ ρ

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 21 / 25

slide-155
SLIDE 155

The main result

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk Consequences: The Betti numbers n → βj(Mn) are eventually rk-periodic: Graded degrees for β0(Mn) are πn(a) for each (a, a′) ∈ ρ (Control over minimal syzygies ⇒ higher Betti numbers) The function n → ∆(Mn) is eventually singleton: ∆(Mn) = {d} when ||a| − |a′|| ∈ {0, d} for all (a, a′) ∈ ρ The function n → c(Mn) is eventually rk-quasilinear: c(Mn) is determined by {minimal presentations of Mn}

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 21 / 25

slide-156
SLIDE 156

Application: computing minimal presentations

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 22 / 25

slide-157
SLIDE 157

Application: computing minimal presentations

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk S = 1234, 1240, 1243, 1254

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 22 / 25

slide-158
SLIDE 158

Application: computing minimal presentations

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk S = 1234, 1240, 1243, 1254 = n, n + r1, . . . , n + rk

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 22 / 25

slide-159
SLIDE 159

Application: computing minimal presentations

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk S = 1234, 1240, 1243, 1254 = n, n + r1, . . . , n + rk n = 1234

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 22 / 25

slide-160
SLIDE 160

Application: computing minimal presentations

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk S = 1234, 1240, 1243, 1254 = n, n + r1, . . . , n + rk n = 1234 r3 = 1254 − 1234 = 20

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 22 / 25

slide-161
SLIDE 161

Application: computing minimal presentations

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk S = 1234, 1240, 1243, 1254 = n, n + r1, . . . , n + rk n = 1234 r3 = 1254 − 1234 = 20 Verify n > r 2

k :

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 22 / 25

slide-162
SLIDE 162

Application: computing minimal presentations

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk S = 1234, 1240, 1243, 1254 = n, n + r1, . . . , n + rk n = 1234 r3 = 1254 − 1234 = 20 Verify n > r 2

k : 1234 > 400

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 22 / 25

slide-163
SLIDE 163

Application: computing minimal presentations

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk S = 1234, 1240, 1243, 1254 = n, n + r1, . . . , n + rk n = 1234 r3 = 1254 − 1234 = 20 Verify n > r 2

k : 1234 > 400

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 22 / 25

slide-164
SLIDE 164

Application: computing minimal presentations

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk S = 1234, 1240, 1243, 1254 = n, n + r1, . . . , n + rk n = 1234 r3 = 1254 − 1234 = 20 Verify n > r 2

k : 1234 > 400

414, 420, 423, 434 :

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 22 / 25

slide-165
SLIDE 165

Application: computing minimal presentations

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk S = 1234, 1240, 1243, 1254 = n, n + r1, . . . , n + rk n = 1234 r3 = 1254 − 1234 = 20 Verify n > r 2

k : 1234 > 400

414, 420, 423, 434 : ((0, 0, 8, 0), (3, 2, 0, 3)), ((0, 1, 6, 0), (4, 0, 0, 3)), ((0, 3, 0, 0), (1, 0, 2, 0)), ((21, 1, 0, 0), (0, 0, 0, 21)), ((25, 0, 0, 0), (0, 0, 6, 18))

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 22 / 25

slide-166
SLIDE 166

Application: computing minimal presentations

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk S = 1234, 1240, 1243, 1254 = n, n + r1, . . . , n + rk n = 1234 r3 = 1254 − 1234 = 20 Verify n > r 2

k : 1234 > 400

414, 420, 423, 434 : ((0, 0, 8, 0), (3, 2, 0, 3)), ((0, 1, 6, 0), (4, 0, 0, 3)), ((0, 3, 0, 0), (1, 0, 2, 0)), ((21, 1, 0, 0), (0, 0, 0, 21)), ((25, 0, 0, 0), (0, 0, 6, 18))

  • 1234, 1240, 1243, 1254 :

((0, 0, 8, 0), (3, 2, 0, 3)), ((0, 1, 6, 0), (4, 0, 0, 3)), ((0, 3, 0, 0), (1, 0, 2, 0)), ((62, 1, 0, 0), (0, 0, 0, 62)), ((66, 0, 0, 0), (0, 0, 6, 59))

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 22 / 25

slide-167
SLIDE 167

Application: computing minimal presentations

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 23 / 25

slide-168
SLIDE 168

Application: computing minimal presentations

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk n Mn

  • Min. Pres. Runtime

50 50, 56, 59, 70 1 ms 200 200, 206, 209, 220 40 ms 400 400, 406, 409, 420 210 ms 1000 1000, 1006, 1009, 1020 3 sec 3000 3000, 3006, 3009, 3020 2 min 5000 5000, 5006, 5009, 5020 18 min 10000 10000, 10006, 10009, 10020 4.2 hr

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 23 / 25

slide-169
SLIDE 169

Application: computing minimal presentations

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk n Mn

  • Min. Pres. Runtime

50 50, 56, 59, 70 1 ms 200 200, 206, 209, 220 40 ms 400 400, 406, 409, 420 210 ms 1000 1000, 1006, 1009, 1020 3 sec 210 ms 3000 3000, 3006, 3009, 3020 2 min 210 ms 5000 5000, 5006, 5009, 5020 18 min 210 ms 10000 10000, 10006, 10009, 10020 4.2 hr 210 ms

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 23 / 25

slide-170
SLIDE 170

Application: computing minimal presentations

Theorem (Conaway–Gotti–Horton–O.–Pelayo–Williams–Wissman)

For any n > r 2

k , the image Φn(ker πn) generates ker πn+rk as a congruence.

{minimal presentations of Mn}

  • {minimal presentations of Mn+rk}

ρ ⊂ ker πn − → Φn(ρ) ⊂ ker πn+rk n Mn

  • Min. Pres. Runtime

50 50, 56, 59, 70 1 ms 200 200, 206, 209, 220 40 ms 400 400, 406, 409, 420 210 ms 1000 1000, 1006, 1009, 1020 3 sec 210 ms 3000 3000, 3006, 3009, 3020 2 min 210 ms 5000 5000, 5006, 5009, 5020 18 min 210 ms 10000 10000, 10006, 10009, 10020 4.2 hr 210 ms GAP Numerical Semigroups Package, available at http://www.gap-system.org/Packages/numericalsgps.html.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 23 / 25

slide-171
SLIDE 171

Future shifty work

20 40 60 80 100 120 140 500 1000 1500 2000

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 24 / 25

slide-172
SLIDE 172

Future shifty work

Frobenius number: F(S) = max(N \ S).

Example

If S = 6, 9, 20, then F(S) = 43 since N \ S = {1, 2, 3, 4, 5, 7, 8, 10, 11, 13, . . . , 31, 34, 37, 43}.

20 40 60 80 100 120 140 500 1000 1500 2000

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 24 / 25

slide-173
SLIDE 173

Future shifty work

Frobenius number: F(S) = max(N \ S).

Example

If S = 6, 9, 20, then F(S) = 43 since N \ S = {1, 2, 3, 4, 5, 7, 8, 10, 11, 13, . . . , 31, 34, 37, 43}. Sneak peek for F(n, n + 6, n + 9, n + 20):

20 40 60 80 100 120 140 500 1000 1500 2000

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 24 / 25

slide-174
SLIDE 174

Future shifty work

Frobenius number: F(S) = max(N \ S).

Example

If S = 6, 9, 20, then F(S) = 43 since N \ S = {1, 2, 3, 4, 5, 7, 8, 10, 11, 13, . . . , 31, 34, 37, 43}. Sneak peek for F(n, n + 6, n + 9, n + 20):

20 40 60 80 100 120 140 500 1000 1500 2000

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 24 / 25

slide-175
SLIDE 175

References

  • S. Chapman, N. Kaplan, T. Lemburg, A. Niles, and C. Zlogar (2014),

Shifts of generators and delta sets of numerical monoids,

  • Internat. J. Algebra Comput. 24 (2014), no. 5, 655–669.
  • T. Vu (2014),

Periodicity of Betti numbers of monomial curves, Journal of Algebra 418 (2014) 66–90.

  • R. Conaway, F. Gotti, J. Horton, C. O’Neill, R. Pelayo, M. Williams, and
  • B. Wissman (2016)

Minimal presentations of shifted numerical monoids,

  • submitted. Available at [arXiv:1701.08555].
  • M. Delgado, P. Garc´

ıa-S´ anchez, and J. Morais, GAP numerical semigroups package http://www.gap-system.org/Packages/numericalsgps.html.

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 25 / 25

slide-176
SLIDE 176

References

  • S. Chapman, N. Kaplan, T. Lemburg, A. Niles, and C. Zlogar (2014),

Shifts of generators and delta sets of numerical monoids,

  • Internat. J. Algebra Comput. 24 (2014), no. 5, 655–669.
  • T. Vu (2014),

Periodicity of Betti numbers of monomial curves, Journal of Algebra 418 (2014) 66–90.

  • R. Conaway, F. Gotti, J. Horton, C. O’Neill, R. Pelayo, M. Williams, and
  • B. Wissman (2016)

Minimal presentations of shifted numerical monoids,

  • submitted. Available at [arXiv:1701.08555].
  • M. Delgado, P. Garc´

ıa-S´ anchez, and J. Morais, GAP numerical semigroups package http://www.gap-system.org/Packages/numericalsgps.html. Thanks!

Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 25 / 25