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Minimal presentations of shifted numerical monoids Christopher - - PowerPoint PPT Presentation

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Minimal presentations of shifted numerical monoids Christopher ONeill Texas A&M University coneill@math.tamu.edu Joint with Rebecca Conaway*, Felix Gotti, Jesse Horton*, Roberto Pelayo, Mesa Williams*, and Brian Wissman * =

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Minimal presentations of shifted numerical monoids

Christopher O’Neill

Texas A&M University coneill@math.tamu.edu Joint with Rebecca Conaway*, Felix Gotti, Jesse Horton*, Roberto Pelayo, Mesa Williams*, and Brian Wissman * = undergraduate student

May 7, 2016

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 1 / 15

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Numerical monoids

Definition

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

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 2 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 2 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 2 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 2 / 15

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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 = 7(6) + 2(9)

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 2 / 15

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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 = 7(6) + 2(9) = 3(20)

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 2 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 2 / 15

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To shift a numerical monoid. . .

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

50 100 150 200 250 5 10 15

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 3 / 15

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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.

50 100 150 200 250 5 10 15

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 3 / 15

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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:

50 100 150 200 250 5 10 15

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 3 / 15

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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:

50 100 150 200 250 5 10 15

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 3 / 15

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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.

50 100 150 200 250 5 10 15

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 3 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 4 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 4 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 4 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 4 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 4 / 15

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To shift a numerical monoid. . .

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

50 100 150 200 250 1000 2000 3000 4000

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 5 / 15

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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.

50 100 150 200 250 1000 2000 3000 4000

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 5 / 15

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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.

50 100 150 200 250 1000 2000 3000 4000

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 5 / 15

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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)

50 100 150 200 250 1000 2000 3000 4000

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 5 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 6 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 6 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 6 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 6 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 6 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 6 / 15

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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: minimal presentations!

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 6 / 15

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Kernel congruences and minimal presentations

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

  • a = (a1, . . . , ak) ∈ Nk

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 7 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 7 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 7 / 15

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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) ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 7 / 15

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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) 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 (Texas A&M University) Shifted numerical monoids May 7, 2016 7 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 8 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 8 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 8 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 8 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 8 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 8 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 8 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 8 / 15

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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 (Texas A&M University) Shifted numerical monoids May 7, 2016 8 / 15

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The shifting map

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

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 9 / 15

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The shifting map

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

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 9 / 15

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SLIDE 46

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk 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 (Texas A&M University) Shifted numerical monoids May 7, 2016 9 / 15

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The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk 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 (Texas A&M University) Shifted numerical monoids May 7, 2016 9 / 15

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SLIDE 48

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk 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 (Texas A&M University)

Shifted numerical monoids May 7, 2016 9 / 15

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SLIDE 49

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk 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 (Texas A&M University)

Shifted numerical monoids May 7, 2016 9 / 15

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The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk 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 (Texas A&M University)

Shifted numerical monoids May 7, 2016 9 / 15

slide-51
SLIDE 51

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk 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 (Texas A&M University) Shifted numerical monoids May 7, 2016 10 / 15

slide-52
SLIDE 52

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk given by (a, a′) − →    (a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Φn is well-defined.

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 10 / 15

slide-53
SLIDE 53

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk given by (a, a′) − →    (a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Φn is well-defined. πn(a) = a0n + k

i=1 ai(n + ri) = |a|n + k i=1 airi

πn+rk(a) = = |a|n + |a|rk + k

i=1 airi

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 10 / 15

slide-54
SLIDE 54

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk given by (a, a′) − →    (a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Φn is well-defined. πn(a) = a0n + k

i=1 ai(n + ri) = |a|n + k i=1 airi

πn+rk(a) = = |a|n + |a|rk + k

i=1 airi

Φn preserves reflexive and symmetric closure.

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 10 / 15

slide-55
SLIDE 55

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk given by (a, a′) − →    (a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Φn is well-defined. πn(a) = a0n + k

i=1 ai(n + ri) = |a|n + k i=1 airi

πn+rk(a) = = |a|n + |a|rk + k

i=1 airi

Φn preserves reflexive and symmetric closure. Φn preserves translation closure.

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 10 / 15

slide-56
SLIDE 56

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk given by (a, a′) − →    (a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Φn is well-defined. πn(a) = a0n + k

i=1 ai(n + ri) = |a|n + k i=1 airi

πn+rk(a) = = |a|n + |a|rk + k

i=1 airi

Φn preserves reflexive and symmetric closure. Φn preserves translation closure. Φn((a, a′) + (b, b)) = Φn(a, a′) + (b, b)

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 10 / 15

slide-57
SLIDE 57

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk given by (a, a′) − →    (a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Φn is well-defined. πn(a) = a0n + k

i=1 ai(n + ri) = |a|n + k i=1 airi

πn+rk(a) = = |a|n + |a|rk + k

i=1 airi

Φn preserves reflexive and symmetric closure. Φn preserves translation closure. Φn((a, a′) + (b, b)) = Φn(a, a′) + (b, b) Only missing link: transitivity.

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 10 / 15

slide-58
SLIDE 58

The shifting map

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn Shifting map Φn : ker πn − → ker πn+rk given by (a, a′) − →    (a, a′) |a| = |a′| (a + ℓek, a′ + ℓe0) |a| < |a′| (a + ℓe0, a′ + ℓek) |a| > |a′| where ℓ =

  • |a| − |a′|
  • .

Φn is well-defined. πn(a) = a0n + k

i=1 ai(n + ri) = |a|n + k i=1 airi

πn+rk(a) = = |a|n + |a|rk + k

i=1 airi

Φn preserves reflexive and symmetric closure. Φn preserves translation closure. Φn((a, a′) + (b, b)) = Φn(a, a′) + (b, b) Only missing link: transitivity. Φn only preserves monotone chain connectivity.

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 10 / 15

slide-59
SLIDE 59

The main result

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

For any n > r2

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 (Texas A&M University) Shifted numerical monoids May 7, 2016 11 / 15

slide-60
SLIDE 60

The main result

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

For any n > r2

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 (Texas A&M University) Shifted numerical monoids May 7, 2016 11 / 15

slide-61
SLIDE 61

The main result

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

For any n > r2

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 → β0(Mn) are eventually rk-periodic: Graded degrees for β0(Mn) are πn(a) for each (a, a′) ∈ ρ

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 11 / 15

slide-62
SLIDE 62

The main result

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

For any n > r2

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 → β0(Mn) are eventually rk-periodic: Graded degrees for β0(Mn) are πn(a) for each (a, a′) ∈ ρ The function n → ∆(Mn) is eventually singleton: ∆(Mn) = {d} when ||a| − |a′|| ∈ {0, d} for all (a, a′) ∈ ρ

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 11 / 15

slide-63
SLIDE 63

The main result

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

For any n > r2

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 → β0(Mn) are eventually rk-periodic: Graded degrees for β0(Mn) are πn(a) for each (a, a′) ∈ ρ 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 (Texas A&M University) Shifted numerical monoids May 7, 2016 11 / 15

slide-64
SLIDE 64

Application: computing minimal presentations

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

For any n > r2

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 (Texas A&M University) Shifted numerical monoids May 7, 2016 12 / 15

slide-65
SLIDE 65

Application: computing minimal presentations

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

For any n > r2

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 (Texas A&M University) Shifted numerical monoids May 7, 2016 12 / 15

slide-66
SLIDE 66

Application: computing minimal presentations

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

For any n > r2

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 (Texas A&M University) Shifted numerical monoids May 7, 2016 12 / 15

slide-67
SLIDE 67

Application: computing minimal presentations

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

For any n > r2

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 (Texas A&M University) Shifted numerical monoids May 7, 2016 12 / 15

slide-68
SLIDE 68

Application: computing minimal presentations

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

For any n > r2

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 (Texas A&M University) Shifted numerical monoids May 7, 2016 12 / 15

slide-69
SLIDE 69

Application: computing minimal presentations

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

For any n > r2

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 > r2

k :

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 12 / 15

slide-70
SLIDE 70

Application: computing minimal presentations

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

For any n > r2

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 > r2

k : 1234 > 400

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 12 / 15

slide-71
SLIDE 71

Application: computing minimal presentations

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

For any n > r2

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 > r2

k : 1234 > 400

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 12 / 15

slide-72
SLIDE 72

Application: computing minimal presentations

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

For any n > r2

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 > r2

k : 1234 > 400

414, 420, 423, 434 :

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 12 / 15

slide-73
SLIDE 73

Application: computing minimal presentations

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

For any n > r2

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 > r2

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 (Texas A&M University) Shifted numerical monoids May 7, 2016 12 / 15

slide-74
SLIDE 74

Application: computing minimal presentations

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

For any n > r2

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 > r2

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 (Texas A&M University) Shifted numerical monoids May 7, 2016 12 / 15

slide-75
SLIDE 75

Application: computing minimal presentations

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

For any n > r2

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 (Texas A&M University) Shifted numerical monoids May 7, 2016 13 / 15

slide-76
SLIDE 76

Application: computing minimal presentations

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

For any n > r2

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 (Texas A&M University) Shifted numerical monoids May 7, 2016 13 / 15

slide-77
SLIDE 77

Application: computing minimal presentations

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

For any n > r2

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 (Texas A&M University) Shifted numerical monoids May 7, 2016 13 / 15

slide-78
SLIDE 78

Application: computing minimal presentations

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

For any n > r2

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 (Texas A&M University) Shifted numerical monoids May 7, 2016 13 / 15

slide-79
SLIDE 79

Future work

50 100 150 200 250 1000 2000 3000 4000

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 14 / 15

slide-80
SLIDE 80

Future work

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

50 100 150 200 250 1000 2000 3000 4000

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 14 / 15

slide-81
SLIDE 81

Future work

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

50 100 150 200 250 1000 2000 3000 4000

Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 14 / 15

slide-82
SLIDE 82

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. in preparation.

  • 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 (Texas A&M University) Shifted numerical monoids May 7, 2016 15 / 15

slide-83
SLIDE 83

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. in preparation.

  • 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 (Texas A&M University) Shifted numerical monoids May 7, 2016 15 / 15