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

Minimal presentations of shifted 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

Sep 26, 2016

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 1 / 23

Numerical monoids

Definition

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 2 / 23

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) Shifted numerical monoids Sep 26, 2016 2 / 23

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) Shifted numerical monoids Sep 26, 2016 2 / 23

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) Shifted numerical monoids Sep 26, 2016 2 / 23

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) Shifted numerical monoids Sep 26, 2016 2 / 23

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) Shifted numerical monoids Sep 26, 2016 2 / 23

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) Shifted numerical monoids Sep 26, 2016 2 / 23

Numerical monoids

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 3 / 23

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) Shifted numerical monoids Sep 26, 2016 3 / 23

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) Shifted numerical monoids Sep 26, 2016 3 / 23

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 3 / 23

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) Shifted numerical monoids Sep 26, 2016 3 / 23

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) Shifted numerical monoids Sep 26, 2016 3 / 23

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) Shifted numerical monoids Sep 26, 2016 3 / 23

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) Shifted numerical monoids Sep 26, 2016 3 / 23

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) Shifted numerical monoids Sep 26, 2016 3 / 23

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) Shifted numerical monoids Sep 26, 2016 3 / 23

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) Shifted numerical monoids Sep 26, 2016 3 / 23

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) Shifted numerical monoids Sep 26, 2016 3 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 4 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 4 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 4 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 4 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 4 / 23

To shift a numerical monoid. . .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 5 / 23

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) Shifted numerical monoids Sep 26, 2016 5 / 23

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) Shifted numerical monoids Sep 26, 2016 5 / 23

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) Shifted numerical monoids Sep 26, 2016 5 / 23

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) Shifted numerical monoids Sep 26, 2016 5 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 6 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 6 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 6 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 6 / 23

To shift a numerical monoid. . .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 7 / 23

To shift a numerical monoid. . .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 7 / 23

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) Shifted numerical monoids Sep 26, 2016 7 / 23

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) Shifted numerical monoids Sep 26, 2016 7 / 23

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) Shifted numerical monoids Sep 26, 2016 7 / 23

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) Shifted numerical monoids Sep 26, 2016 7 / 23

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) Shifted numerical monoids Sep 26, 2016 7 / 23

Kernel congruences and minimal presentations

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

• a = (a1, . . . , ak) ∈ Nk

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 8 / 23

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) Shifted numerical monoids Sep 26, 2016 8 / 23

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) Shifted numerical monoids Sep 26, 2016 8 / 23

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) Shifted numerical monoids Sep 26, 2016 8 / 23

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) Shifted numerical monoids Sep 26, 2016 8 / 23

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) Shifted numerical monoids Sep 26, 2016 9 / 23

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) Shifted numerical monoids Sep 26, 2016 9 / 23

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) Shifted numerical monoids Sep 26, 2016 9 / 23

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) Shifted numerical monoids Sep 26, 2016 9 / 23

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) Shifted numerical monoids Sep 26, 2016 9 / 23

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) Shifted numerical monoids Sep 26, 2016 9 / 23

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) Shifted numerical monoids Sep 26, 2016 9 / 23

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) Shifted numerical monoids Sep 26, 2016 9 / 23

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) Shifted numerical monoids Sep 26, 2016 10 / 23

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) Shifted numerical monoids Sep 26, 2016 10 / 23

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) Shifted numerical monoids Sep 26, 2016 10 / 23

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) Shifted numerical monoids Sep 26, 2016 10 / 23

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) Shifted numerical monoids Sep 26, 2016 10 / 23

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) Shifted numerical monoids Sep 26, 2016 10 / 23

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) Shifted numerical monoids Sep 26, 2016 10 / 23

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) Shifted numerical monoids Sep 26, 2016 10 / 23

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) Shifted numerical monoids Sep 26, 2016 10 / 23

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) Shifted numerical monoids Sep 26, 2016 11 / 23

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) Shifted numerical monoids Sep 26, 2016 11 / 23

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) Shifted numerical monoids Sep 26, 2016 11 / 23

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) Shifted numerical monoids Sep 26, 2016 11 / 23

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) Shifted numerical monoids Sep 26, 2016 11 / 23

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) Shifted numerical monoids Sep 26, 2016 11 / 23

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) Shifted numerical monoids Sep 26, 2016 11 / 23

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) Shifted numerical monoids Sep 26, 2016 11 / 23

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) Shifted numerical monoids Sep 26, 2016 11 / 23

Intuition: “sufficiently shifted” monoids

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

Intuition: “sufficiently shifted” monoids

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

Intuition: “sufficiently shifted” monoids

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

Intuition: “sufficiently shifted” monoids

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

Intuition: “sufficiently shifted” monoids

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

Intuition: “sufficiently shifted” monoids

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

Intuition: “sufficiently shifted” monoids

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn n + (n + r1) + · · · + (n + rk) 2 types of minimal relations a ∼ b:

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn n + (n + r1) + · · · + (n + rk) 2 types of minimal relations a ∼ b: Relations among r1, . . . , rk (cheap)

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn n + (n + r1) + · · · + (n + rk) 2 types of minimal relations a ∼ b: Relations among r1, . . . , rk (cheap) |a| = |b|

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn n + (n + r1) + · · · + (n + rk) 2 types of minimal relations a ∼ b: Relations among r1, . . . , rk (cheap) |a| = |b| Relations that change # copies of n (costly)

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

Intuition: “sufficiently shifted” monoids

Mn = n, n + r1, . . . , n + rk, πn : Nk+1 − → Mn n + (n + r1) + · · · + (n + rk) 2 types of minimal relations a ∼ b: Relations among r1, . . . , rk (cheap) |a| = |b| Relations that change # copies of n (costly) mostly ak ← − − − − − − → mostly b0

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

The shifting map

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 13 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 13 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 13 / 23

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′|
• .

DON’T PANIC!

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 13 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 14 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 14 / 23

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 (UC Davis)

Shifted numerical monoids Sep 26, 2016 14 / 23

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 (UC Davis)

Shifted numerical monoids Sep 26, 2016 14 / 23

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 (UC Davis)

Shifted numerical monoids Sep 26, 2016 14 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 15 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 15 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 15 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 15 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 15 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 15 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 15 / 23

Monotone chains

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 16 / 23

Monotone chains

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′|
• .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 16 / 23

Monotone chains

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′|
• .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 16 / 23

Monotone chains

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′|
• .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 16 / 23

Monotone chains

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′|
• .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 16 / 23

Monotone chains

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′|
• .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 16 / 23

Monotone chains

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′|
• .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 16 / 23

Monotone chains

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′|
• .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 16 / 23

Monotone chains

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′|
• .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 16 / 23

Monotone chains

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′|
• .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 16 / 23

Monotone chains

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′|
• .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 16 / 23

Monotone chains

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′|
• .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 17 / 23

Monotone chains

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′|
• .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 17 / 23

Monotone chains

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′|
• .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 17 / 23

Monotone chains

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′|
• .

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 17 / 23

Monotone chains

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′|
• .

Fix (a, b), (b, c) ∈ ker πn with |a| < |c| < |b|. Need: monotone chains are sufficient for transitive closure.

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 17 / 23

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.

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 18 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 18 / 23

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

Key Lemma

If n > r2

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

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 18 / 23

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

Key Lemma

If n > r2

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

The above lemma ensures:

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 18 / 23

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

Key Lemma

If n > r2

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

The above lemma ensures: Each (a, a′) ∈ ρ lies in the image of Φn.

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 18 / 23

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

Key Lemma

If n > r2

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

The above lemma ensures: Each (a, a′) ∈ ρ lies in the image of Φn. Any two factorizations are connected by a monotone ρ-chain.

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 18 / 23

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

Key Lemma

If n > r2

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

The above lemma ensures: Each (a, a′) ∈ ρ lies in the image of Φn. Any two factorizations are connected by a monotone ρ-chain.

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 18 / 23

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

Key Lemma

If n > r2

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

The above lemma ensures: Each (a, a′) ∈ ρ lies in the image of Φn. Any two factorizations are connected by a monotone ρ-chain.

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 18 / 23

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

Key Lemma

If n > r2

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

The above lemma ensures: Each (a, a′) ∈ ρ lies in the image of Φn. Any two factorizations are connected by a monotone ρ-chain.

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 18 / 23

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

Key Lemma

If n > r2

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

The above lemma ensures: Each (a, a′) ∈ ρ lies in the image of Φn. Any two factorizations are connected by a monotone ρ-chain.

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 18 / 23

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

Key Lemma

If n > r2

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

The above lemma ensures: Each (a, a′) ∈ ρ lies in the image of Φn. Any two factorizations are connected by a monotone ρ-chain.

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 18 / 23

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

Key Lemma

If n > r2

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

The above lemma ensures: Each (a, a′) ∈ ρ lies in the image of Φn. Any two factorizations are connected by a monotone ρ-chain.

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 18 / 23

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

Key Lemma

If n > r2

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

The above lemma ensures: Each (a, a′) ∈ ρ lies in the image of Φn. Any two factorizations are connected by a monotone ρ-chain.

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 18 / 23

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

Key Lemma

If n > r2

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

The above lemma ensures: Each (a, a′) ∈ ρ lies in the image of Φn. Any two factorizations are connected by a monotone ρ-chain.

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 18 / 23

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

Key Lemma

If n > r2

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

The above lemma ensures: Each (a, a′) ∈ ρ lies in the image of Φn. Any two factorizations are connected by a monotone ρ-chain.

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 18 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 19 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 19 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 19 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 19 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 19 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 20 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 20 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 20 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 20 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 20 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 20 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 20 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 20 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 20 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 20 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 20 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 21 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 21 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 21 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 21 / 23

Future work

50 100 150 200 250 1000 2000 3000 4000

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 22 / 23

Future work

Improve the bound n > r2

k .

50 100 150 200 250 1000 2000 3000 4000

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 22 / 23

Future work

Improve the bound n > r2

k .

Consider other quantities under shifting.

50 100 150 200 250 1000 2000 3000 4000

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 22 / 23

Future work

Improve the bound n > r2

k .

Consider other quantities under shifting. Frobenius number: F(S) = max(N \ S).

50 100 150 200 250 1000 2000 3000 4000

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 22 / 23

Future work

Improve the bound n > r2

k .

Consider other quantities under shifting. Frobenius number: F(S) = max(N \ S). Sneak peek for F(n, n + 6, n + 9, n + 20):

50 100 150 200 250 1000 2000 3000 4000

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 22 / 23

Future work

Improve the bound n > r2

k .

Consider other quantities under shifting. Frobenius number: F(S) = max(N \ S). Sneak peek for F(n, n + 6, n + 9, n + 20):

50 100 150 200 250 1000 2000 3000 4000

Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 22 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 23 / 23

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 (UC Davis) Shifted numerical monoids Sep 26, 2016 23 / 23