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Computing the delta set and ω-primality in numerical monoids

Christopher O’Neill

Texas A&M University coneill@math.tamu.edu Joint with Thomas Barron and Roberto Pelayo

June 13, 2015

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 1 / 13

The delta set

Definition (Numerical monoid)

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

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 2 / 13

The delta set

Definition (Numerical monoid)

A numerical monoid S is an additive submonoid of N with |N \ S| < ∞. Fix n ∈ S = n1, . . . , nk.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 2 / 13

The delta set

Definition (Numerical monoid)

A numerical monoid S is an additive submonoid of N with |N \ S| < ∞. Fix n ∈ S = n1, . . . , nk. n = a1n1 + · · · + aknk a = (a1, . . . , ak) ∈ Nk

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 2 / 13

The delta set

Definition (Numerical monoid)

A numerical monoid S is an additive submonoid of N with |N \ S| < ∞. Fix n ∈ S = n1, . . . , nk. n = a1n1 + · · · + aknk a = (a1, . . . , ak) ∈ Nk Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 2 / 13

The delta set

Definition (Numerical monoid)

A numerical monoid S is an additive submonoid of N with |N \ S| < ∞. Fix n ∈ S = n1, . . . , nk. n = a1n1 + · · · + aknk a = (a1, . . . , ak) ∈ Nk Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk} L(n) = {|a| = a1 + · · · + ak : a ∈ Z(n)}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 2 / 13

The delta set

Definition (Numerical monoid)

A numerical monoid S is an additive submonoid of N with |N \ S| < ∞. Fix n ∈ S = n1, . . . , nk. n = a1n1 + · · · + aknk a = (a1, . . . , ak) ∈ Nk Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk} L(n) = {|a| = a1 + · · · + ak : a ∈ Z(n)}

Definition (The delta set)

For L(n) = {ℓ1 < . . . < ℓr}, define ∆(n) = {ℓi − ℓi−1}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 2 / 13

The delta set

Definition (Numerical monoid)

A numerical monoid S is an additive submonoid of N with |N \ S| < ∞. Fix n ∈ S = n1, . . . , nk. n = a1n1 + · · · + aknk a = (a1, . . . , ak) ∈ Nk Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk} L(n) = {|a| = a1 + · · · + ak : a ∈ Z(n)}

Definition (The delta set)

For L(n) = {ℓ1 < . . . < ℓr}, define ∆(n) = {ℓi − ℓi−1}

Goal

Compute ∆(S) =

n∈S ∆(n).

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 2 / 13

Computing the delta set of a numerical monoid

Theorem (Chapman–Hoyer–Kaplan, 2000)

S = n1, . . . , nk. For n ≥ 2kn2n2

k, ∆(n) = ∆(n + n1nk).

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 3 / 13

Computing the delta set of a numerical monoid

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)).

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 3 / 13

Computing the delta set of a numerical monoid

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1nk)).

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 3 / 13

Computing the delta set of a numerical monoid

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk),

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 3 / 13

Computing the delta set of a numerical monoid

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 3 / 13

Computing the delta set of a numerical monoid

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk} Z(n) L(n) = {a1 + · · · + ak : a ∈ Z(n)}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 3 / 13

Computing the delta set of a numerical monoid

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk} Z(n) L(n) = {a1 + · · · + ak : a ∈ Z(n)} L(n) = {ℓ1 < . . . < ℓr} ∆(n) = {ℓi − ℓi−1}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 3 / 13

Computing the delta set of a numerical monoid

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk} Z(n) L(n) = {a1 + · · · + ak : a ∈ Z(n)} L(n) = {ℓ1 < . . . < ℓr} ∆(n) = {ℓi − ℓi−1} Compute ∆(S) =

n ∆(n).

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 3 / 13

Computing the delta set of a numerical monoid

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: → Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk} ← Z(n) L(n) = {a1 + · · · + ak : a ∈ Z(n)} L(n) = {ℓ1 < . . . < ℓr} ∆(n) = {ℓi − ℓi−1} Compute ∆(S) =

n ∆(n).

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 3 / 13

Computing the delta set of a numerical monoid

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: → Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk} ← Z(n) L(n) = {a1 + · · · + ak : a ∈ Z(n)} L(n) = {ℓ1 < . . . < ℓr} ∆(n) = {ℓi − ℓi−1} Compute ∆(S) =

n ∆(n).

|Z(n)| ≈ nk−1

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 3 / 13

Computing the delta set of a numerical monoid

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1nk)). For n ∈ S with NS ≤ n ≤ NS + n1, compute: → Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk} ← Z(n) L(n) = {a1 + · · · + ak : a ∈ Z(n)} L(n) = {ℓ1 < . . . < ℓr} ∆(n) = {ℓi − ℓi−1} Compute ∆(S) =

n ∆(n).

|Z(n)| ≈ nk−1

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 3 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k,

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei S = 6, 9, 20:

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei S = 6, 9, 20:

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei S = 6, 9, 20:

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei S = 6, 9, 20:

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei S = 6, 9, 20:

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

S = 6, 9, 20:

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0)

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0) 18 2e1

6

3e1 {3e1, 2e2} {2, 3}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0) 18 2e1

6

3e1 {3e1, 2e2} {2, 3} e2

9

2e2

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0) 18 2e1

6

3e1 {3e1, 2e2} {2, 3} e2

9

2e2 20

20

e3 {e3} {1}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0) 18 2e1

6

3e1 {3e1, 2e2} {2, 3} e2

9

2e2 20

20

e3 {e3} {1} . . . . . . . . .

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0) 18 2e1

6

3e1 {3e1, 2e2} {2, 3} e2

9

2e2 20

20

e3 {e3} {1} . . . . . . . . .

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0) 18 2e1

6

3e1 {3e1, 2e2} {2, 3} e2

9

2e2 20

20

e3 {e3} {1} . . . . . . . . .

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0) 18 2e1

6

3e1 {3e1, 2e2} {2, 3} e2

9

2e2 20

20

e3 {e3} {1} . . . . . . . . .

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 9 12 15 18 20 . . .

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 12 15 18 20 . . .

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 15 18 20 . . .

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 {2} 1

6

2 15 18 20 . . .

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 {2} 1

6

2 15 {2} 1

6

2 18 20 . . .

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 {2} 1

6

2 15 {2} 1

6

2 1

9

2 18 20 . . .

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 {2} 1

6

2 15 {2} 1

6

2 1

9

2 18 {2, 3} 2

6

3 20 . . .

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 {2} 1

6

2 15 {2} 1

6

2 1

9

2 18 {2, 3} 2

6

3 1

9

2 20 . . .

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 {2} 1

6

2 15 {2} 1

6

2 1

9

2 18 {2, 3} 2

6

3 1

9

2 20 {1}

20

1 . . .

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

A solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 {2} 1

6

2 15 {2} 1

6

2 1

9

2 18 {2, 3} 2

6

3 1

9

2 20 {1}

20

1 . . . . . . . . .

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 4 / 13

Computing the delta set dynamically

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)).

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 5 / 13

Computing the delta set dynamically

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk} Z(n) L(n) L(n) ∆(n) Compute ∆(S) =

n ∆(n).

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 5 / 13

Computing the delta set dynamically

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk} L(n − ∗) L(n) L(n) ∆(n) Compute ∆(S) =

n ∆(n).

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 5 / 13

Computing the delta set dynamically

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk} L(n − ∗) L(n) L(n) ∆(n) Compute ∆(S) =

n ∆(n).

This is significantly faster!

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 5 / 13

Computing the delta set dynamically

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk} L(n − ∗) L(n) L(n) ∆(n) Compute ∆(S) =

n ∆(n).

This is significantly faster! |Z(n)| ≈ nk−1

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 5 / 13

Computing the delta set dynamically

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: Z(n) = {a ∈ Nk : n = a1n1 + · · · + aknk} L(n − ∗) L(n) L(n) ∆(n) Compute ∆(S) =

n ∆(n).

This is significantly faster! |Z(n)| ≈ nk−1 |L(n)| ≈ n

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 5 / 13

Runtime comparison

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 6 / 13

Runtime comparison

S NS ∆(S) Existing Dynamic 7, 15, 17, 18, 20 1935 {1, 2, 3} 1m 28s 146ms 11, 53, 73, 87 14381 {2, 4, 6, 8, 10, 22} 0m 49s 2.5s 31, 73, 77, 87, 91 31364 {2, 4, 6} 400m 12s 4.2s 100, 121, 142, 163, 284 24850 {21} ——— 0m 3.6s 1001, 1211, 1421, 1631, 2841 2063141 {10, 20, 30} ——— 1m 56s

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 6 / 13

Runtime comparison

S NS ∆(S) Existing Dynamic 7, 15, 17, 18, 20 1935 {1, 2, 3} 1m 28s 146ms 11, 53, 73, 87 14381 {2, 4, 6, 8, 10, 22} 0m 49s 2.5s 31, 73, 77, 87, 91 31364 {2, 4, 6} 400m 12s 4.2s 100, 121, 142, 163, 284 24850 {21} ——— 0m 3.6s 1001, 1211, 1421, 1631, 2841 2063141 {10, 20, 30} ——— 1m 56s

GAP Numerical Semigroups Package, available at http://www.gap-system.org/Packages/numericalsgps.html.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 6 / 13

ω-primality

As usual, n ∈ S = n1, . . . , nk.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 7 / 13

ω-primality

As usual, n ∈ S = n1, . . . , nk.

Definition (ω-primality)

ωS(n) is the minimal m such that whenever (r

i=1 xi) − n ∈ S for r > m,

there exists T ⊂ {1, . . . , r} with |T| ≤ m and (

i∈T xi) − n ∈ S.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 7 / 13

ω-primality

As usual, n ∈ S = n1, . . . , nk.

Definition (ω-primality)

ωS(n) is the minimal m such that whenever (r

i=1 xi) − n ∈ S for r > m,

there exists T ⊂ {1, . . . , r} with |T| ≤ m and (

i∈T xi) − n ∈ S.

Definition

A bullet for n ∈ S is a tuple b = (b1, . . . , bk) ∈ Nk such that (i) b1n1 + · · · + bknk − n ∈ S, and (ii) b1n1 + · · · + (bi − 1)ni + · · · + bknk − n / ∈ S for each bi > 0. The set of bullets of n is denoted bul(n).

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 7 / 13

ω-primality

As usual, n ∈ S = n1, . . . , nk.

Definition (ω-primality)

ωS(n) is the minimal m such that whenever (r

i=1 xi) − n ∈ S for r > m,

there exists T ⊂ {1, . . . , r} with |T| ≤ m and (

i∈T xi) − n ∈ S.

Definition

A bullet for n ∈ S is a tuple b = (b1, . . . , bk) ∈ Nk such that (i) b1n1 + · · · + bknk − n ∈ S, and (ii) b1n1 + · · · + (bi − 1)ni + · · · + bknk − n / ∈ S for each bi > 0. The set of bullets of n is denoted bul(n).

Proposition

ωS(n) = max{|b| : b ∈ bul(n)}.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 7 / 13

Using bullets to compute ω-primality

Algorithm: Compute bul(n), then compute ω(n) = max{|b| : b ∈ bul(n)}.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 8 / 13

Using bullets to compute ω-primality

Algorithm: Compute bul(n), then compute ω(n) = max{|b| : b ∈ bul(n)}.

Example

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

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 8 / 13

Using bullets to compute ω-primality

Algorithm: Compute bul(n), then compute ω(n) = max{|b| : b ∈ bul(n)}.

Example

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

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 8 / 13

Using bullets to compute ω-primality

Algorithm: Compute bul(n), then compute ω(n) = max{|b| : b ∈ bul(n)}.

Example

S = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid” bul(60) = {(4, 4, 0), (7, 2, 0), (10, 0, 0), (1, 6, 0), (0, 8, 0), (0, 0, 3)}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 8 / 13

Using bullets to compute ω-primality

Algorithm: Compute bul(n), then compute ω(n) = max{|b| : b ∈ bul(n)}.

Example

S = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid” bul(60) = {(4, 4, 0), (7, 2, 0), (10, 0, 0), (1, 6, 0), (0, 8, 0), (0, 0, 3)} 8 · 9 − 60 = 12 ∈ S

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 8 / 13

Using bullets to compute ω-primality

Algorithm: Compute bul(n), then compute ω(n) = max{|b| : b ∈ bul(n)}.

Example

S = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid” bul(60) = {(4, 4, 0), (7, 2, 0), (10, 0, 0), (1, 6, 0), (0, 8, 0), (0, 0, 3)} 8 · 9 − 60 = 12 ∈ S 7 · 9 − 60 = 3 / ∈ S

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 8 / 13

Using bullets to compute ω-primality

Algorithm: Compute bul(n), then compute ω(n) = max{|b| : b ∈ bul(n)}.

Example

S = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid” bul(60) = {(4, 4, 0), (7, 2, 0), (10, 0, 0), (1, 6, 0), (0, 8, 0), (0, 0, 3)} 8 · 9 − 60 = 12 ∈ S 7 · 9 − 60 = 3 / ∈ S ⇒ (0, 8, 0) ∈ bul(60)

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 8 / 13

Using bullets to compute ω-primality

Algorithm: Compute bul(n), then compute ω(n) = max{|b| : b ∈ bul(n)}.

Example

S = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid” bul(60) = {(4, 4, 0), (7, 2, 0), (10, 0, 0), (1, 6, 0), (0, 8, 0), (0, 0, 3)} 8 · 9 − 60 = 12 ∈ S 7 · 9 − 60 = 3 / ∈ S ⇒ (0, 8, 0) ∈ bul(60) 1 · 6 + 6 · 9 − 60 = 0 ∈ S

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 8 / 13

Using bullets to compute ω-primality

Algorithm: Compute bul(n), then compute ω(n) = max{|b| : b ∈ bul(n)}.

Example

S = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid” bul(60) = {(4, 4, 0), (7, 2, 0), (10, 0, 0), (1, 6, 0), (0, 8, 0), (0, 0, 3)} 8 · 9 − 60 = 12 ∈ S 7 · 9 − 60 = 3 / ∈ S ⇒ (0, 8, 0) ∈ bul(60) 1 · 6 + 6 · 9 − 60 = 0 ∈ S 6 · 9 − 60 = −6 / ∈ S

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 8 / 13

Using bullets to compute ω-primality

Algorithm: Compute bul(n), then compute ω(n) = max{|b| : b ∈ bul(n)}.

Example

S = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid” bul(60) = {(4, 4, 0), (7, 2, 0), (10, 0, 0), (1, 6, 0), (0, 8, 0), (0, 0, 3)} 8 · 9 − 60 = 12 ∈ S 7 · 9 − 60 = 3 / ∈ S ⇒ (0, 8, 0) ∈ bul(60) 1 · 6 + 6 · 9 − 60 = 0 ∈ S 6 · 9 − 60 = −6 / ∈ S 1 · 6 + 5 · 9 − 60 = −9 / ∈ S

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 8 / 13

Using bullets to compute ω-primality

Algorithm: Compute bul(n), then compute ω(n) = max{|b| : b ∈ bul(n)}.

Example

S = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid” bul(60) = {(4, 4, 0), (7, 2, 0), (10, 0, 0), (1, 6, 0), (0, 8, 0), (0, 0, 3)} 8 · 9 − 60 = 12 ∈ S 7 · 9 − 60 = 3 / ∈ S ⇒ (0, 8, 0) ∈ bul(60) 1 · 6 + 6 · 9 − 60 = 0 ∈ S 6 · 9 − 60 = −6 / ∈ S 1 · 6 + 5 · 9 − 60 = −9 / ∈ S ⇒ (1, 6, 0) ∈ bul(60)

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 8 / 13

Using bullets to compute ω-primality

Algorithm: Compute bul(n), then compute ω(n) = max{|b| : b ∈ bul(n)}.

Example

S = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid” bul(60) = {(4, 4, 0), (7, 2, 0), (10, 0, 0), (1, 6, 0), (0, 8, 0), (0, 0, 3)} 8 · 9 − 60 = 12 ∈ S 7 · 9 − 60 = 3 / ∈ S ⇒ (0, 8, 0) ∈ bul(60) 1 · 6 + 6 · 9 − 60 = 0 ∈ S 6 · 9 − 60 = −6 / ∈ S 1 · 6 + 5 · 9 − 60 = −9 / ∈ S ⇒ (1, 6, 0) ∈ bul(60)

n ∈ S ω(n) mbul 6 3 3e3 9 3 3e3 12 3 3e3 n ∈ S ω(n) mbul 15 4 4e1 18 3 3e1 20 10 10e1 n ∈ S ω(n) mbul 21 5 5e1 24 4 4e1 26 11 11e1

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 8 / 13

Using bullets to compute ω-primality

Algorithm: Compute bul(n), then compute ω(n) = max{|b| : b ∈ bul(n)}.

Example

S = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid” bul(60) = {(4, 4, 0), (7, 2, 0), (10, 0, 0), (1, 6, 0), (0, 8, 0), (0, 0, 3)} 8 · 9 − 60 = 12 ∈ S 7 · 9 − 60 = 3 / ∈ S ⇒ (0, 8, 0) ∈ bul(60) 1 · 6 + 6 · 9 − 60 = 0 ∈ S 6 · 9 − 60 = −6 / ∈ S 1 · 6 + 5 · 9 − 60 = −9 / ∈ S ⇒ (1, 6, 0) ∈ bul(60)

n ∈ S ω(n) mbul 6 3 3e3 9 3 3e3 12 3 3e3 n ∈ S ω(n) mbul 15 4 4e1 18 3 3e1 20 10 10e1 n ∈ S ω(n) mbul 21 5 5e1 24 4 4e1 26 11 11e1

Moral of this talk: bullets behave like factorizations!

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 8 / 13

Toward a dynamic algorithm. . . the inductive step

Recall: for n ∈ S = n1, . . . nk, Z(n) = {a ∈ Nk : k

i=1 aini = n}.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 9 / 13

Toward a dynamic algorithm. . . the inductive step

Recall: for n ∈ S = n1, . . . nk, Z(n) = {a ∈ Nk : k

i=1 aini = n}.

For each i ≤ k,

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 9 / 13

Toward a dynamic algorithm. . . the inductive step

Recall: for n ∈ S = n1, . . . nk, Z(n) = {a ∈ Nk : k

i=1 aini = n}.

For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 9 / 13

Toward a dynamic algorithm. . . the inductive step

Recall: for n ∈ S = n1, . . . nk, Z(n) = {a ∈ Nk : k

i=1 aini = n}.

For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei. In particular, Z(n) =

• i≤k

φi(Z(n − ni))

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 9 / 13

Toward a dynamic algorithm. . . the inductive step

Recall: for n ∈ S = n1, . . . nk, Z(n) = {a ∈ Nk : k

i=1 aini = n}.

For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei. In particular, Z(n) =

• i≤k

φi(Z(n − ni))

Definition/Proposition (Cover morphisms)

Fix n ∈ S and i ≤ k. The i-th cover morphism for n is the map ψi : bul(n − ni) − → bul(n)

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 9 / 13

Toward a dynamic algorithm. . . the inductive step

Recall: for n ∈ S = n1, . . . nk, Z(n) = {a ∈ Nk : k

i=1 aini = n}.

For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei. In particular, Z(n) =

• i≤k

φi(Z(n − ni))

Definition/Proposition (Cover morphisms)

Fix n ∈ S and i ≤ k. The i-th cover morphism for n is the map ψi : bul(n − ni) − → bul(n) given by b − →

• b + ei

k

j=1 bjnj − n − ni /

∈ S b k

j=1 bjnj − n − ni ∈ S

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 9 / 13

Toward a dynamic algorithm. . . the inductive step

Recall: for n ∈ S = n1, . . . nk, Z(n) = {a ∈ Nk : k

i=1 aini = n}.

For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei. In particular, Z(n) =

• i≤k

φi(Z(n − ni))

Definition/Proposition (Cover morphisms)

Fix n ∈ S and i ≤ k. The i-th cover morphism for n is the map ψi : bul(n − ni) − → bul(n) given by b − →

• b + ei

k

j=1 bjnj − n − ni /

∈ S b k

j=1 bjnj − n − ni ∈ S

Moreover, bul(n) =

i≤k ψi(bul(n − ni)).**

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoidsJune 13, 2015 9 / 13

Toward a dynamic algorithm. . . the base case

Definition (ω-primality in numerical monoids)

Fix a numerical monoid S and n ∈ S.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 10 / 13

Toward a dynamic algorithm. . . the base case

Definition (ω-primality in numerical monoids)

Fix a numerical monoid S and n ∈ S. ωS(n) is the minimal m such that whenever (r

i=1 xi) − n ∈ S for r > m,

there exists T ⊂ {1, . . . , r} with |T| ≤ m and (

i∈T xi) − n ∈ S.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 10 / 13

Toward a dynamic algorithm. . . the base case

Definition (ω-primality in numerical monoids)

Fix a numerical monoid S and n ∈ S. ωS(n) is the minimal m such that whenever (r

i=1 xi) − n ∈ S for r > m,

there exists T ⊂ {1, . . . , r} with |T| ≤ m and (

i∈T xi) − n ∈ S.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 10 / 13

Toward a dynamic algorithm. . . the base case

Definition (ω-primality in numerical monoids)

Fix a numerical monoid S and n ∈ Z = q(S). ωS(n) is the minimal m such that whenever (r

i=1 xi) − n ∈ S for r > m,

there exists T ⊂ {1, . . . , r} with |T| ≤ m and (

i∈T xi) − n ∈ S.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 10 / 13

Toward a dynamic algorithm. . . the base case

Definition (ω-primality in numerical monoids)

Fix a numerical monoid S and n ∈ Z = q(S). ωS(n) is the minimal m such that whenever (r

i=1 xi) − n ∈ S for r > m,

there exists T ⊂ {1, . . . , r} with |T| ≤ m and (

i∈T xi) − n ∈ S.

Remark

All properties of ω extend from S to Z.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 10 / 13

Toward a dynamic algorithm. . . the base case

Definition (ω-primality in numerical monoids)

Fix a numerical monoid S and n ∈ Z = q(S). ωS(n) is the minimal m such that whenever (r

i=1 xi) − n ∈ S for r > m,

there exists T ⊂ {1, . . . , r} with |T| ≤ m and (

i∈T xi) − n ∈ S.

Remark

All properties of ω extend from S to Z.

Proposition

For n ∈ Z, the following are equivalent: (i) ω(n) = 0, (ii) bul(n) = {0}, (iii) −n ∈ S.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 10 / 13

A dynamic algorithm!

Example

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

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} −43 1 {e1, e2, e3} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} −43 1 {e1, e2, e3} −42 {0} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} −43 1 {e1, e2, e3} −42 {0} . . . . . . . . . −38 {0} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} −43 1 {e1, e2, e3} −42 {0} . . . . . . . . . −38 {0} −37 2 {2e1, e2, e3} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} −43 1 {e1, e2, e3} −42 {0} . . . . . . . . . −38 {0} −37 2 {2e1, e2, e3} −36 {0} −35 {0} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} −43 1 {e1, e2, e3} −42 {0} . . . . . . . . . −38 {0} −37 2 {2e1, e2, e3} −36 {0} −35 {0} −34 2 {e1, 2e2, e3} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} −43 1 {e1, e2, e3} −42 {0} . . . . . . . . . −38 {0} −37 2 {2e1, e2, e3} −36 {0} −35 {0} −34 2 {e1, 2e2, e3} −33 {0} −32 {0} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} −43 1 {e1, e2, e3} −42 {0} . . . . . . . . . −38 {0} −37 2 {2e1, e2, e3} −36 {0} −35 {0} −34 2 {e1, 2e2, e3} −33 {0} −32 {0} −31 3 {3e1, e2, e3} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} −43 1 {e1, e2, e3} −42 {0} . . . . . . . . . −38 {0} −37 2 {2e1, e2, e3} −36 {0} −35 {0} −34 2 {e1, 2e2, e3} −33 {0} −32 {0} −31 3 {3e1, e2, e3} . . . . . . . . . n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} −43 1 {e1, e2, e3} −42 {0} . . . . . . . . . −38 {0} −37 2 {2e1, e2, e3} −36 {0} −35 {0} −34 2 {e1, 2e2, e3} −33 {0} −32 {0} −31 3 {3e1, e2, e3} . . . . . . . . . n ∈ Z ω(n) bul(n) 1 5 {5e1, (2, 1, 0), . . .}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} −43 1 {e1, e2, e3} −42 {0} . . . . . . . . . −38 {0} −37 2 {2e1, e2, e3} −36 {0} −35 {0} −34 2 {e1, 2e2, e3} −33 {0} −32 {0} −31 3 {3e1, e2, e3} . . . . . . . . . n ∈ Z ω(n) bul(n) 1 5 {5e1, (2, 1, 0), . . .} 2 7 {7e1, 6e2, . . .}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} −43 1 {e1, e2, e3} −42 {0} . . . . . . . . . −38 {0} −37 2 {2e1, e2, e3} −36 {0} −35 {0} −34 2 {e1, 2e2, e3} −33 {0} −32 {0} −31 3 {3e1, e2, e3} . . . . . . . . . n ∈ Z ω(n) bul(n) 1 5 {5e1, (2, 1, 0), . . .} 2 7 {7e1, 6e2, . . .} 3 3 {3e3, 2e2, . . .} 4 4 {4e1, 4e2, . . .} 5 9 {9e1, (6, 1, 0), . . .} 6 3 {3e3, 2e2, . . .} 7 6 {6e1, (3, 1, 0), . . .} 8 8 {8e1, (5, 2, 0), . . . , } 9 3 {3e1, 3e3, . . .} . . . . . . . . .

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} −43 1 {e1, e2, e3} −42 {0} . . . . . . . . . −38 {0} −37 2 {2e1, e2, e3} −36 {0} −35 {0} −34 2 {e1, 2e2, e3} −33 {0} −32 {0} −31 3 {3e1, e2, e3} . . . . . . . . . n ∈ Z ω(n) bul(n) 1 5 {5e1, (2, 1, 0), . . .} 2 7 {7e1, 6e2, . . .} 3 3 {3e3, 2e2, . . .} 4 4 {4e1, 4e2, . . .} 5 9 {9e1, (6, 1, 0), . . .} 6 3 {3e3, 2e2, . . .} 7 6 {6e1, (3, 1, 0), . . .} 8 8 {8e1, (5, 2, 0), . . . , } 9 3 {3e1, 3e3, . . .} . . . . . . . . . 148 28 {28e1, . . .}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} −43 1 {e1, e2, e3} −42 {0} . . . . . . . . . −38 {0} −37 2 {2e1, e2, e3} −36 {0} −35 {0} −34 2 {e1, 2e2, e3} −33 {0} −32 {0} −31 3 {3e1, e2, e3} . . . . . . . . . n ∈ Z ω(n) bul(n) 1 5 {5e1, (2, 1, 0), . . .} 2 7 {7e1, 6e2, . . .} 3 3 {3e3, 2e2, . . .} 4 4 {4e1, 4e2, . . .} 5 9 {9e1, (6, 1, 0), . . .} 6 3 {3e3, 2e2, . . .} 7 6 {6e1, (3, 1, 0), . . .} 8 8 {8e1, (5, 2, 0), . . . , } 9 3 {3e1, 3e3, . . .} . . . . . . . . . 148 28 {28e1, . . .} 149 33 {33e1, . . .}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) ≤ −44 {0} −43 1 {e1, e2, e3} −42 {0} . . . . . . . . . −38 {0} −37 2 {2e1, e2, e3} −36 {0} −35 {0} −34 2 {e1, 2e2, e3} −33 {0} −32 {0} −31 3 {3e1, e2, e3} . . . . . . . . . n ∈ Z ω(n) bul(n) 1 5 {5e1, (2, 1, 0), . . .} 2 7 {7e1, 6e2, . . .} 3 3 {3e3, 2e2, . . .} 4 4 {4e1, 4e2, . . .} 5 9 {9e1, (6, 1, 0), . . .} 6 3 {3e3, 2e2, . . .} 7 6 {6e1, (3, 1, 0), . . .} 8 8 {8e1, (5, 2, 0), . . . , } 9 3 {3e1, 3e3, . . .} . . . . . . . . . 148 28 {28e1, . . .} 149 33 {33e1, . . .} 150 25 {25e1, . . .}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

A dynamic algorithm!

Example

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

n ∈ Z ω(n) bul(n) n ∈ Z ω(n) bul(n) 6 3 {3e3, 2e2, . . .} 9 3 {3e1, 3e3, . . .} . . . . . . . . . 148 28 {28e1, . . .} 149 33 {33e1, . . .} 150 25 {25e1, . . .}

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 11 / 13

Runtime comparison

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 12 / 13

Runtime comparison

S n ∈ S ωS(n) Existing Dynamic 6, 9, 20 1000 170 1m 1.3s 6ms 11, 13, 15 1000 97 0m 10.7s 5ms 11, 13, 15 3000 279 14m 34.7s 15ms 11, 13, 15 10000 915 ——— 42ms 15, 27, 32, 35 1000 69 3m 54.7s 9ms 100, 121, 142, 163, 284 25715 308 ——— 0m 27s 1001, 1211, 1421, 1631, 2841 357362 405 ——— 57m 27s GAP Numerical Semigroups Package, available at http://www.gap-system.org/Packages/numericalsgps.html.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 12 / 13

References

• C. O’Neill, R. Pelayo (2014)

How do you measure primality? American Mathematical Monthly, 122 (2014), no. 2, 121–137.

• J. Garc´

ıa-Garc´ ıa, M. Moreno-Fr´ ıas, A. Vigneron-Tenorio (2014) Computation of delta sets of numerical monoids. preprint.

• T. Barron, C. O’Neill, R. Pelayo (2015)

On the computation of delta sets and ω-primality in numerical monoids. preprint.

• M. Delgado, P. Garc´

ıa-S´ anchez, J. Morais GAP Numerical Semigroups Package http://www.gap-system.org/Packages/numericalsgps.html.

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 13 / 13

References

• C. O’Neill, R. Pelayo (2014)

How do you measure primality? American Mathematical Monthly, 122 (2014), no. 2, 121–137.

• J. Garc´

ıa-Garc´ ıa, M. Moreno-Fr´ ıas, A. Vigneron-Tenorio (2014) Computation of delta sets of numerical monoids. preprint.

• T. Barron, C. O’Neill, R. Pelayo (2015)

On the computation of delta sets and ω-primality in numerical monoids. preprint.

• M. Delgado, P. Garc´

ıa-S´ anchez, J. Morais GAP Numerical Semigroups Package http://www.gap-system.org/Packages/numericalsgps.html. Thanks!

Christopher O’Neill (Texas A&M University)Computing the delta set and ω-primality in numerical monoids June 13, 2015 13 / 13