Numerical semigroups in Sage Christopher ONeill University of - - PowerPoint PPT Presentation

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Numerical semigroups in Sage Christopher ONeill University of - - PowerPoint PPT Presentation

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Numerical semigroups in Sage Christopher ONeill University of California Davis coneill@math.ucdavis.edu Oct 26, 2016 Christopher ONeill (UC Davis) Numerical semigroups Oct 26, 2016 1 / 9 Numerical monoids Definition A numerical

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Numerical semigroups in Sage

Christopher O’Neill

University of California Davis coneill@math.ucdavis.edu

Oct 26, 2016

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 1 / 9

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Numerical monoids

Definition

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

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

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Numerical monoids

Definition

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

Example

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

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

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Numerical monoids

Definition

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

Example

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

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

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Numerical monoids

Definition

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

Example

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

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

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Numerical monoids

Definition

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

Example

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

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

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Numerical monoids

Definition

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

Example

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

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

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SLIDE 8

Numerical monoids

Definition

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

Example

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

  • (7, 2, 0)

(0, 0, 3)

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

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SLIDE 9

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) Unique minimal generating set:

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

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SLIDE 10

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) Unique minimal generating set: 15, 17, 22, 32, 40, 42, 56, 58

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

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Numerical monoids

Definition

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

Example

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

  • (7, 2, 0)

(0, 0, 3) Unique minimal generating set: 15, 17, 22, 32, 40, 42, 56, 58 15, 17, 22, 40, 42, 58

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

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Numerical monoids

Definition

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

Example

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

  • (7, 2, 0)

(0, 0, 3) Unique minimal generating set: 15, 17, 22, 32, 40, 42, 56, 58 15, 17, 22, 40, 42, 58 32 = 15 + 17 56 = 2 · 17 + 22

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

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Factorization invariants

Fix a numerical monoid S = n1, . . . , nk. For n ∈ S,

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 3 / 9

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Factorization invariants

Fix a numerical monoid S = n1, . . . , nk. For n ∈ S, Max and min factorization length: M(n), m(n) ∈ Z≥0

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 3 / 9

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Factorization invariants

Fix a numerical monoid S = n1, . . . , nk. For n ∈ S, Max and min factorization length: M(n), m(n) ∈ Z≥0 Delta set: ∆(n) ⊂ Z≥1 ∆(S) =

m∈S ∆(m)

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 3 / 9

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Factorization invariants

Fix a numerical monoid S = n1, . . . , nk. For n ∈ S, Max and min factorization length: M(n), m(n) ∈ Z≥0 Delta set: ∆(n) ⊂ Z≥1 ∆(S) =

m∈S ∆(m)

Catenary degree: c(n) ∈ Z≥1 c(S) = max{c(m) : m ∈ S}

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 3 / 9

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Factorization invariants

Fix a numerical monoid S = n1, . . . , nk. For n ∈ S, Max and min factorization length: M(n), m(n) ∈ Z≥0 Delta set: ∆(n) ⊂ Z≥1 ∆(S) =

m∈S ∆(m)

Catenary degree: c(n) ∈ Z≥1 c(S) = max{c(m) : m ∈ S} ω-primality: ω(n) ∈ Z≥1

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 3 / 9

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Factorization invariants

Invariant behavior for large numerical monoid elements:

10 20 30 40 50 60 2 4 6 8 10 20 30 40 50 60 1 2 3 4 5 6 7

M : 6, 9, 20 → N m : 6, 9, 20 → N

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 4 / 9

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Factorization invariants

Invariant behavior for large numerical monoid elements:

50 100 150 1 2 3 4

∆ : 6, 9, 20 → 2N

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 5 / 9

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Factorization invariants

Invariant behavior for large numerical monoid elements:

50 100 150 200 250 5 10 15

n − → c(n, n + 6, n + 9, n + 20)

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 6 / 9

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Software!

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 7 / 9

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Software!

Input: Numerical monoid S = n1, . . . , nk Monoid element n ∈ S

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 7 / 9

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Software!

Input: Numerical monoid S = n1, . . . , nk Monoid element n ∈ S Output: Invariant value Delta set, catenary degree, list of factorizations, . . .

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 7 / 9

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Software!

Input: Numerical monoid S = n1, . . . , nk Monoid element n ∈ S Output: Invariant value Delta set, catenary degree, list of factorizations, . . . GAP Numerical Semigroups Package, available at http://www.gap-system.org/Packages/numericalsgps.html.

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 7 / 9

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Software!

Input: Numerical monoid S = n1, . . . , nk Monoid element n ∈ S Output: Invariant value Delta set, catenary degree, list of factorizations, . . . GAP Numerical Semigroups Package, available at http://www.gap-system.org/Packages/numericalsgps.html. Let’s see it in action!

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 7 / 9

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GAP from within Sage

Using GAP within Sage via gap console()

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GAP from within Sage

Using GAP within Sage via gap console() Conflated message

50 100 150 200 50 100 150 200 Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 8 / 9

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GAP from within Sage

Using GAP within Sage via gap console() Conflated message Less intuitive language

50 100 150 200 50 100 150 200 Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 8 / 9

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GAP from within Sage

Using GAP within Sage via gap console() Conflated message Less intuitive language Port data to Sage for plots:

50 100 150 200 50 100 150 200 Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 8 / 9

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GAP from within Sage

Using GAP within Sage via gap console() Conflated message Less intuitive language Port data to Sage for plots: Conjecture: For all S, s ∈ N \ S, there exists a height 2 Leamer s-atom.

50 100 150 200 50 100 150 200 Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 8 / 9

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GAP from within Sage

Using GAP within Sage via gap console() Conflated message Less intuitive language Port data to Sage for plots: Conjecture: For all S, s ∈ N \ S, there exists a height 2 Leamer s-atom.

50 100 150 200 50 100 150 200

S = 14, 17

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 8 / 9

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Sage wrapper

Solution: NumericalSemigroup Sage class!

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 9 / 9

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Sage wrapper

Solution: NumericalSemigroup Sage class! NumericalSemigroup instance GAP object

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 9 / 9

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Sage wrapper

Solution: NumericalSemigroup Sage class! NumericalSemigroup instance GAP object Benefits:

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 9 / 9

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Sage wrapper

Solution: NumericalSemigroup Sage class! NumericalSemigroup instance GAP object Benefits: Returns “Sage types”

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 9 / 9

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Sage wrapper

Solution: NumericalSemigroup Sage class! NumericalSemigroup instance GAP object Benefits: Returns “Sage types” Shorter function names

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 9 / 9

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Sage wrapper

Solution: NumericalSemigroup Sage class! NumericalSemigroup instance GAP object Benefits: Returns “Sage types” Shorter function names Some computed values stored internally

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 9 / 9

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Sage wrapper

Solution: NumericalSemigroup Sage class! NumericalSemigroup instance GAP object Benefits: Returns “Sage types” Shorter function names Some computed values stored internally Calling S.CatenaryDegree(500) from Sage:

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 9 / 9

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Sage wrapper

Solution: NumericalSemigroup Sage class! NumericalSemigroup instance GAP object Benefits: Returns “Sage types” Shorter function names Some computed values stored internally Calling S.CatenaryDegree(500) from Sage: Call CatenaryDegreeOfElementInNumericalSemigroup(500,-)

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 9 / 9

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Sage wrapper

Solution: NumericalSemigroup Sage class! NumericalSemigroup instance GAP object Benefits: Returns “Sage types” Shorter function names Some computed values stored internally Calling S.CatenaryDegree(500) from Sage: Call CatenaryDegreeOfElementInNumericalSemigroup(500,-) Convert result to “Sage int”

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 9 / 9

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Sage wrapper

Solution: NumericalSemigroup Sage class! NumericalSemigroup instance GAP object Benefits: Returns “Sage types” Shorter function names Some computed values stored internally Calling S.CatenaryDegree(500) from Sage: Call CatenaryDegreeOfElementInNumericalSemigroup(500,-) Convert result to “Sage int” Archive value and return

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 9 / 9

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Sage wrapper

Solution: NumericalSemigroup Sage class! NumericalSemigroup instance GAP object Benefits: Returns “Sage types” Shorter function names Some computed values stored internally Calling S.CatenaryDegree(500) from Sage: Call CatenaryDegreeOfElementInNumericalSemigroup(500,-) Convert result to “Sage int” Archive value and return Let’s see it in action!

Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 9 / 9