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Catenary degrees of elements in numerical monoids Christopher ONeill - - PowerPoint PPT Presentation

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Catenary degrees of elements in numerical monoids Christopher ONeill Texas A&M University coneill@math.tamu.edu Joint with Vadim Ponomarenko, Reuben Tate*, and Gautam Webb* January 11, 2015 Christopher ONeill (Texas A&M

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Catenary degrees of elements in numerical monoids

Christopher O’Neill

Texas A&M University coneill@math.tamu.edu Joint with Vadim Ponomarenko, Reuben Tate*, and Gautam Webb*

January 11, 2015

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 1 / 14

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Factorial domains

Definition

An integral domain R is factorial if for each non-unit r ∈ R,

1 there is a factorization r = u1 · · · uk as a product of irreducibles, and 2 this factorization is unique (up to reordering and unit multiple). Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 2 / 14

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Factorial domains

Definition

An integral domain R is factorial if for each non-unit r ∈ R,

1 there is a factorization r = u1 · · · uk as a product of irreducibles, and 2 this factorization is unique (up to reordering and unit multiple).

Example

Z is factorial: each z = p1 · · · pk for primes p1 · · · pk.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 2 / 14

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Atomic domains

Definition

An integral domain R is atomic if for each non-unit r ∈ R,

1 there is a factorization r = u1 · · · uk as a product of irreducibles, and 2 this factorization is unique (up to reordering and unit multiple).

Example

Z is factorial: each z = p1 · · · pk for primes p1 · · · pk.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 2 / 14

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SLIDE 5

Atomic domains

Definition

An integral domain R is atomic if for each non-unit r ∈ R,

1 there is a factorization r = u1 · · · uk as a product of irreducibles, and 2 this factorization is unique (up to reordering and unit multiple).

Example

Z is factorial: each z = p1 · · · pk for primes p1 · · · pk.

Example

If R = Z[√−5], then 6 ∈ R has two distinct factorizations: 6 = 2 · 3 = (1 + √ −5)(1 − √ −5)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 2 / 14

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Atomic domains

Definition

An integral domain R is atomic if for each non-unit r ∈ R,

1 there is a factorization r = u1 · · · uk as a product of irreducibles, and 2 this factorization is unique (up to reordering and unit multiple).

Example

Z is factorial: each z = p1 · · · pk for primes p1 · · · pk.

Example

If R = Z[√−5], then 6 ∈ R has two distinct factorizations: 6 = 2 · 3 = (1 + √ −5)(1 − √ −5) To prove: define a valuation a + b√−5 → a2 + 5b2.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 2 / 14

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Atomic domains

Definition

An integral domain R is atomic if for each non-unit r ∈ R,

1 there is a factorization r = u1 · · · uk as a product of irreducibles, and 2 this factorization is unique (up to reordering and unit multiple).

Example

Z is factorial: each z = p1 · · · pk for primes p1 · · · pk.

Example

If R = Z[√−5], then 6 ∈ R has two distinct factorizations: 6 = 2 · 3 = (1 + √ −5)(1 − √ −5) To prove: define a valuation a + b√−5 → a2 + 5b2. The point: it’s involved.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 2 / 14

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Atomic domains

Definition

An integral domain R is atomic if for each non-unit r ∈ R,

1 there is a factorization r = u1 · · · uk as a product of irreducibles, and 2 this factorization is unique (up to reordering and unit multiple).

Example

Z is factorial: each z = p1 · · · pk for primes p1 · · · pk.

Example

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 2 / 14

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Atomic domains

Definition

An integral domain R is atomic if for each non-unit r ∈ R,

1 there is a factorization r = u1 · · · uk as a product of irreducibles, and 2 this factorization is unique (up to reordering and unit multiple).

Example

Z is factorial: each z = p1 · · · pk for primes p1 · · · pk.

Example

Let R = C[x2, x3].

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 2 / 14

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Atomic domains

Definition

An integral domain R is atomic if for each non-unit r ∈ R,

1 there is a factorization r = u1 · · · uk as a product of irreducibles, and 2 this factorization is unique (up to reordering and unit multiple).

Example

Z is factorial: each z = p1 · · · pk for primes p1 · · · pk.

Example

Let R = C[x2, x3].

1 x2 and x3 are irreducible. Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 2 / 14

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Atomic domains

Definition

An integral domain R is atomic if for each non-unit r ∈ R,

1 there is a factorization r = u1 · · · uk as a product of irreducibles, and 2 this factorization is unique (up to reordering and unit multiple).

Example

Z is factorial: each z = p1 · · · pk for primes p1 · · · pk.

Example

Let R = C[x2, x3].

1 x2 and x3 are irreducible. 2 x6 = x3 · x3 = x2 · x2 · x2. Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 2 / 14

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Atomic domains

Definition

An integral domain R is atomic if for each non-unit r ∈ R,

1 there is a factorization r = u1 · · · uk as a product of irreducibles, and 2 this factorization is unique (up to reordering and unit multiple).

Example

Z is factorial: each z = p1 · · · pk for primes p1 · · · pk.

Observation

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 2 / 14

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Atomic domains

Definition

An integral domain R is atomic if for each non-unit r ∈ R,

1 there is a factorization r = u1 · · · uk as a product of irreducibles, and 2 this factorization is unique (up to reordering and unit multiple).

Example

Z is factorial: each z = p1 · · · pk for primes p1 · · · pk.

Observation

Where’s the addition?

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 2 / 14

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Atomic domains

Definition

An integral domain R is atomic if for each non-unit r ∈ R,

1 there is a factorization r = u1 · · · uk as a product of irreducibles, and 2 this factorization is unique (up to reordering and unit multiple).

Example

Z is factorial: each z = p1 · · · pk for primes p1 · · · pk.

Observation

Where’s the addition? Factorization in (cancellative comutative) monoids:

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 2 / 14

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Atomic domains

Definition

An integral domain R is atomic if for each non-unit r ∈ R,

1 there is a factorization r = u1 · · · uk as a product of irreducibles, and 2 this factorization is unique (up to reordering and unit multiple).

Example

Z is factorial: each z = p1 · · · pk for primes p1 · · · pk.

Observation

Where’s the addition? Factorization in (cancellative comutative) monoids: (R, +, ·)

  • (R \ {0}, ·)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 2 / 14

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Atomic domains

Definition

An integral domain R is atomic if for each non-unit r ∈ R,

1 there is a factorization r = u1 · · · uk as a product of irreducibles, and 2 this factorization is unique (up to reordering and unit multiple).

Example

Z is factorial: each z = p1 · · · pk for primes p1 · · · pk.

Observation

Where’s the addition? Factorization in (cancellative comutative) monoids: (R, +, ·)

  • (R \ {0}, ·)

(C[M], +, ·)

  • (M, ·)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 2 / 14

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Numerical monoids

Definition

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

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

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Numerical monoids

Definition

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

Example

Let S = 2, 3 = {2, 3, 4, 5, . . .} under addition.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

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Numerical monoids

Definition

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

Example

Let S = 2, 3 = {2, 3, 4, 5, . . .} under addition. C[S] = C[x2, x3].

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

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Numerical monoids

Definition

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

Example

Let S = 2, 3 = {2, 3, 4, 5, . . .} under addition. C[S] = C[x2, x3]. x6 = x3 · x3 = x2 · x2 · x2

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

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Numerical monoids

Definition

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

Example

Let S = 2, 3 = {2, 3, 4, 5, . . .} under addition. C[S] = C[x2, x3]. x6 = x3 · x3 = x2 · x2 · x2

  • 6 = 3 + 3 = 2 + 2 + 2.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

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Numerical monoids

Definition

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

Example

Let S = 2, 3 = {2, 3, 4, 5, . . .} under addition. C[S] = C[x2, x3]. x6 = x3 · x3 = x2 · x2 · x2

  • 6 = 3 + 3 = 2 + 2 + 2.

Factorizations are additive!

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

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Numerical monoids

Definition

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

Example

Let S = 2, 3 = {2, 3, 4, 5, . . .} under addition. C[S] = C[x2, x3]. x6 = x3 · x3 = x2 · x2 · x2

  • 6 = 3 + 3 = 2 + 2 + 2.

Factorizations are additive!

Example

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

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

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Numerical monoids

Definition

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

Example

Let S = 2, 3 = {2, 3, 4, 5, . . .} under addition. C[S] = C[x2, x3]. x6 = x3 · x3 = x2 · x2 · x2

  • 6 = 3 + 3 = 2 + 2 + 2.

Factorizations are additive!

Example

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

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

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Numerical monoids

Definition

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

Example

Let S = 2, 3 = {2, 3, 4, 5, . . .} under addition. C[S] = C[x2, x3]. x6 = x3 · x3 = x2 · x2 · x2

  • 6 = 3 + 3 = 2 + 2 + 2.

Factorizations are additive!

Example

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

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

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Numerical monoids

Definition

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

Example

Let S = 2, 3 = {2, 3, 4, 5, . . .} under addition. C[S] = C[x2, x3]. x6 = x3 · x3 = x2 · x2 · x2

  • 6 = 3 + 3 = 2 + 2 + 2.

Factorizations are additive!

Example

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

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

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Numerical monoids

Definition

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

Example

Let S = 2, 3 = {2, 3, 4, 5, . . .} under addition. C[S] = C[x2, x3]. x6 = x3 · x3 = x2 · x2 · x2

  • 6 = 3 + 3 = 2 + 2 + 2.

Factorizations are additive!

Example

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

  • (7, 2, 0)

(0, 0, 3)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

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Factorization invariants: towards the catenary degree

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 4 / 14

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Factorization invariants: towards the catenary degree

Definition

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

  • (a1, . . . , ak) ∈ Nk : n = a1n1 + · · · + aknk
  • denotes the set of factorizations of n.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 4 / 14

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Factorization invariants: towards the catenary degree

Definition

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

  • (a1, . . . , ak) ∈ Nk : n = a1n1 + · · · + aknk
  • denotes the set of factorizations of n. Equivalently, if

φ : Nk − → S

  • ei

− → ni then ZS(n) = φ−1(n).

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 4 / 14

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Factorization invariants: towards the catenary degree

Definition

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

  • (a1, . . . , ak) ∈ Nk : n = a1n1 + · · · + aknk
  • denotes the set of factorizations of n. Equivalently, if

φ : Nk − → S

  • ei

− → ni then ZS(n) = φ−1(n). For f , f ′ ∈ ZS(n), |f | = f1 + · · · + fk

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 4 / 14

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Factorization invariants: towards the catenary degree

Definition

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

  • (a1, . . . , ak) ∈ Nk : n = a1n1 + · · · + aknk
  • denotes the set of factorizations of n. Equivalently, if

φ : Nk − → S

  • ei

− → ni then ZS(n) = φ−1(n). For f , f ′ ∈ ZS(n), |f | = f1 + · · · + fk (length of f )

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 4 / 14

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Factorization invariants: towards the catenary degree

Definition

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

  • (a1, . . . , ak) ∈ Nk : n = a1n1 + · · · + aknk
  • denotes the set of factorizations of n. Equivalently, if

φ : Nk − → S

  • ei

− → ni then ZS(n) = φ−1(n). For f , f ′ ∈ ZS(n), |f | = f1 + · · · + fk (length of f ) gcd(f , f ′) = (min(f1, f ′

1), . . . , min(fk, f ′ k))

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 4 / 14

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Factorization invariants: towards the catenary degree

Definition

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

  • (a1, . . . , ak) ∈ Nk : n = a1n1 + · · · + aknk
  • denotes the set of factorizations of n. Equivalently, if

φ : Nk − → S

  • ei

− → ni then ZS(n) = φ−1(n). For f , f ′ ∈ ZS(n), |f | = f1 + · · · + fk (length of f ) gcd(f , f ′) = (min(f1, f ′

1), . . . , min(fk, f ′ k))

d(f , f ′) = max {|f − gcd(f , f ′)|, |f ′ − gcd(f , f ′)|}

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 4 / 14

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Factorization invariants: towards the catenary degree

Example

S = 4, 6, 7 ⊂ N,

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

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Factorization invariants: towards the catenary degree

Example

S = 4, 6, 7 ⊂ N, f = (3, 1, 1), f ′ = (1, 0, 3) ∈ ZS(25).

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

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Factorization invariants: towards the catenary degree

Example

S = 4, 6, 7 ⊂ N, f = (3, 1, 1), f ′ = (1, 0, 3) ∈ ZS(25).

6 7 7 7 7 4 4 4 4 (3,1,1) (1,0,3)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

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Factorization invariants: towards the catenary degree

Example

S = 4, 6, 7 ⊂ N, f = (3, 1, 1), f ′ = (1, 0, 3) ∈ ZS(25). g = gcd(f , f ′)

6 7 7 7 7 4 4 4 4 (3,1,1) (1,0,3)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

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Factorization invariants: towards the catenary degree

Example

S = 4, 6, 7 ⊂ N, f = (3, 1, 1), f ′ = (1, 0, 3) ∈ ZS(25). g = gcd(f , f ′)

6 7 7 7 7 4 4 4 4 (3,1,1) (1,0,3)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

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Factorization invariants: towards the catenary degree

Example

S = 4, 6, 7 ⊂ N, f = (3, 1, 1), f ′ = (1, 0, 3) ∈ ZS(25). g = gcd(f , f ′) = (1, 0, 1).

6 7 7 7 7 4 4 4 4 (3,1,1) (1,0,3)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

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Factorization invariants: towards the catenary degree

Example

S = 4, 6, 7 ⊂ N, f = (3, 1, 1), f ′ = (1, 0, 3) ∈ ZS(25). g = gcd(f , f ′) = (1, 0, 1). d(f , f ′)

6 7 7 7 7 4 4 4 4 (3,1,1) (1,0,3)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

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Factorization invariants: towards the catenary degree

Example

S = 4, 6, 7 ⊂ N, f = (3, 1, 1), f ′ = (1, 0, 3) ∈ ZS(25). g = gcd(f , f ′) = (1, 0, 1). d(f , f ′) = max {|f − g|, |f ′ − g|}

6 7 7 7 7 4 4 4 4 (3,1,1) (1,0,3)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

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Factorization invariants: towards the catenary degree

Example

S = 4, 6, 7 ⊂ N, f = (3, 1, 1), f ′ = (1, 0, 3) ∈ ZS(25). g = gcd(f , f ′) = (1, 0, 1). d(f , f ′) = max {|f − g|, |f ′ − g|}

6 7 7 7 7 4 4 4 4 (3,1,1) (1,0,3)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

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Factorization invariants: towards the catenary degree

Example

S = 4, 6, 7 ⊂ N, f = (3, 1, 1), f ′ = (1, 0, 3) ∈ ZS(25). g = gcd(f , f ′) = (1, 0, 1). d(f , f ′) = max {|f − g|, |f ′ − g|} = 3.

6 7 7 7 7 4 4 4 4 (3,1,1) (1,0,3)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

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The catenary degree

Definition

Fix a numerical monoid S = n1, . . . , nk. For n ∈ S, define the catenary degree c(n) as follows:

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 6 / 14

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The catenary degree

Definition

Fix a numerical monoid S = n1, . . . , nk. For n ∈ S, define the catenary degree c(n) as follows:

1 Construct a complete graph G with vertex set ZS(n) where each edge

(f , f ′) has label d(f , f ′) (catenary graph).

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 6 / 14

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SLIDE 47

The catenary degree

Definition

Fix a numerical monoid S = n1, . . . , nk. For n ∈ S, define the catenary degree c(n) as follows:

1 Construct a complete graph G with vertex set ZS(n) where each edge

(f , f ′) has label d(f , f ′) (catenary graph).

2 Locate the largest edge weight e in G. Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 6 / 14

slide-48
SLIDE 48

The catenary degree

Definition

Fix a numerical monoid S = n1, . . . , nk. For n ∈ S, define the catenary degree c(n) as follows:

1 Construct a complete graph G with vertex set ZS(n) where each edge

(f , f ′) has label d(f , f ′) (catenary graph).

2 Locate the largest edge weight e in G. 3 Remove all edges from G with weight e. Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 6 / 14

slide-49
SLIDE 49

The catenary degree

Definition

Fix a numerical monoid S = n1, . . . , nk. For n ∈ S, define the catenary degree c(n) as follows:

1 Construct a complete graph G with vertex set ZS(n) where each edge

(f , f ′) has label d(f , f ′) (catenary graph).

2 Locate the largest edge weight e in G. 3 Remove all edges from G with weight e. 4 If G is disconnected, return e. Otherwise, return to step 2. Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 6 / 14

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SLIDE 50

The catenary degree

Definition

Fix a numerical monoid S = n1, . . . , nk. For n ∈ S, define the catenary degree c(n) as follows:

1 Construct a complete graph G with vertex set ZS(n) where each edge

(f , f ′) has label d(f , f ′) (catenary graph).

2 Locate the largest edge weight e in G. 3 Remove all edges from G with weight e. 4 If G is disconnected, return e. Otherwise, return to step 2.

If |ZS(n)| = 1, define c(n) = 0.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 6 / 14

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A Big Example

S = 11, 36, 39, n = 450

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

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A Big Example

S = 11, 36, 39, n = 450

27 24 24 21 21 21 19 18 18 18 17 17 16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

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SLIDE 53

A Big Example

S = 11, 36, 39, n = 450

27 24 24 21 21 21 19 18 18 18 17 17 16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

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SLIDE 54

A Big Example

S = 11, 36, 39, n = 450

24 24 21 21 21 19 18 18 18 17 17 16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

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SLIDE 55

A Big Example

S = 11, 36, 39, n = 450

24 24 21 21 21 19 18 18 18 17 17 16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

slide-56
SLIDE 56

A Big Example

S = 11, 36, 39, n = 450

21 21 21 19 18 18 18 17 17 16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

slide-57
SLIDE 57

A Big Example

S = 11, 36, 39, n = 450

21 21 21 19 18 18 18 17 17 16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

slide-58
SLIDE 58

A Big Example

S = 11, 36, 39, n = 450

19 18 18 18 17 17 16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

slide-59
SLIDE 59

A Big Example

S = 11, 36, 39, n = 450

19 18 18 18 17 17 16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

slide-60
SLIDE 60

A Big Example

S = 11, 36, 39, n = 450

18 18 18 17 17 16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

slide-61
SLIDE 61

A Big Example

S = 11, 36, 39, n = 450

18 18 18 17 17 16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

slide-62
SLIDE 62

A Big Example

S = 11, 36, 39, n = 450

17 17 16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

slide-63
SLIDE 63

A Big Example

S = 11, 36, 39, n = 450

17 17 16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

slide-64
SLIDE 64

A Big Example

S = 11, 36, 39, n = 450

16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

slide-65
SLIDE 65

A Big Example

S = 11, 36, 39, n = 450

16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

slide-66
SLIDE 66

A Big Example

S = 11, 36, 39, n = 450

12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

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SLIDE 67

A Big Example

S = 11, 36, 39, n = 450, c(n) = 16

12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

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SLIDE 68

A Big Example, Method 2

S = 11, 36, 39, n = 450

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 8 / 14

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SLIDE 69

A Big Example, Method 2

S = 11, 36, 39, n = 450

(0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 8 / 14

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SLIDE 70

A Big Example, Method 2

S = 11, 36, 39, n = 450

4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 8 / 14

slide-71
SLIDE 71

A Big Example, Method 2

S = 11, 36, 39, n = 450

8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 8 / 14

slide-72
SLIDE 72

A Big Example, Method 2

S = 11, 36, 39, n = 450

12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 8 / 14

slide-73
SLIDE 73

A Big Example, Method 2

S = 11, 36, 39, n = 450

16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 8 / 14

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SLIDE 74

A Big Example, Method 2

S = 11, 36, 39, n = 450, c(n) = 16

16 12 12 8 8 8 8 4 4 4 4 4 4 (0,6,6) (27,1,3) (24,3,2) (21,5,1) (18,7,0) (9,0,9) (6,2,8) (3,4,7)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 8 / 14

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SLIDE 75

Betti elements

Definition

For an element n ∈ S = n1, . . . , nk, let ∇n denote the subgraph of the catenary graph in which only edges (f , f ′) with gcd(f , f ′) = 0 are drawn.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 9 / 14

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SLIDE 76

Betti elements

Definition

For an element n ∈ S = n1, . . . , nk, let ∇n denote the subgraph of the catenary graph in which only edges (f , f ′) with gcd(f , f ′) = 0 are drawn. We say n is a Betti element of S if ∇n is disconnected.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 9 / 14

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SLIDE 77

Betti elements

Definition

For an element n ∈ S = n1, . . . , nk, let ∇n denote the subgraph of the catenary graph in which only edges (f , f ′) with gcd(f , f ′) = 0 are drawn. We say n is a Betti element of S if ∇n is disconnected.

Example

S = 10, 15, 17 has Betti elements 30 and 85.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 9 / 14

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SLIDE 78

Betti elements

Definition

For an element n ∈ S = n1, . . . , nk, let ∇n denote the subgraph of the catenary graph in which only edges (f , f ′) with gcd(f , f ′) = 0 are drawn. We say n is a Betti element of S if ∇n is disconnected.

Example

S = 10, 15, 17 has Betti elements 30 and 85.

∇30 : ∇85 :

(0,2,0) (3,0,0) (0,0,5) (7,1,0) (4,3,0) (1,5,0)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 9 / 14

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SLIDE 79

Maximal catenary degree in S

Theorem

max{c(n) : n ∈ S} = max{c(b) : b Betti element of S}.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

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SLIDE 80

Maximal catenary degree in S

Theorem

max{c(n) : n ∈ S} = max{c(b) : b Betti element of S}. Key concept: Cover morphisms.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

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SLIDE 81

Maximal catenary degree in S

Theorem

max{c(n) : n ∈ S} = max{c(b) : b Betti element of S}. Key concept: Cover morphisms. ZS(n) ZS(n + ni)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

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SLIDE 82

Maximal catenary degree in S

Theorem

max{c(n) : n ∈ S} = max{c(b) : b Betti element of S}. Key concept: Cover morphisms. ZS(n) ֒ − − − − − − − − − − − → ZS(n + ni)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

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SLIDE 83

Maximal catenary degree in S

Theorem

max{c(n) : n ∈ S} = max{c(b) : b Betti element of S}. Key concept: Cover morphisms. ZS(n) ֒ − − − − − − − − − − − → ZS(n + ni) f

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

slide-84
SLIDE 84

Maximal catenary degree in S

Theorem

max{c(n) : n ∈ S} = max{c(b) : b Betti element of S}. Key concept: Cover morphisms. ZS(n) ֒ − − − − − − − − − − − → ZS(n + ni) f − − − − − − − − − − − − → f + ei

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

slide-85
SLIDE 85

Maximal catenary degree in S

Theorem

max{c(n) : n ∈ S} = max{c(b) : b Betti element of S}. Key concept: Cover morphisms.

2 3 5 3

ZS(n) ֒ − − − − − − − − − − − → ZS(n + ni) f − − − − − − − − − − − − → f + ei

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

slide-86
SLIDE 86

Maximal catenary degree in S

Theorem

max{c(n) : n ∈ S} = max{c(b) : b Betti element of S}. Key concept: Cover morphisms.

2 3 5 3 2 3 5 3

ZS(n) ֒ − − − − − − − − − − − → ZS(n + ni) f − − − − − − − − − − − − → f + ei

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

slide-87
SLIDE 87

Maximal catenary degree in S

Theorem

max{c(n) : n ∈ S} = max{c(b) : b Betti element of S}.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 11 / 14

slide-88
SLIDE 88

Maximal catenary degree in S

Theorem

max{c(n) : n ∈ S} = max{c(b) : b Betti element of S}. Idea for proof: Certain edges (determined by Betti elements) connect the catenary graph of each n ∈ S.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 11 / 14

slide-89
SLIDE 89

Maximal catenary degree in S

Theorem

max{c(n) : n ∈ S} = max{c(b) : b Betti element of S}. Idea for proof: Certain edges (determined by Betti elements) connect the catenary graph of each n ∈ S.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 11 / 14

slide-90
SLIDE 90

Maximal catenary degree in S

Theorem

max{c(n) : n ∈ S} = max{c(b) : b Betti element of S}. Idea for proof: Certain edges (determined by Betti elements) connect the catenary graph of each n ∈ S.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 11 / 14

slide-91
SLIDE 91

Maximal catenary degree in S

Theorem

max{c(n) : n ∈ S} = max{c(b) : b Betti element of S}. Idea for proof: Certain edges (determined by Betti elements) connect the catenary graph of each n ∈ S.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 11 / 14

slide-92
SLIDE 92

Maximal catenary degree in S

Theorem

max{c(n) : n ∈ S} = max{c(b) : b Betti element of S}. Idea for proof: Certain edges (determined by Betti elements) connect the catenary graph of each n ∈ S.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 11 / 14

slide-93
SLIDE 93

Minimal (nonzero) catenary degree in S

Conjecture

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

slide-94
SLIDE 94

Minimal (nonzero) catenary degree in S

Conjecture Theorem (O., Ponomarenko, Tate, Webb)

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

slide-95
SLIDE 95

Minimal (nonzero) catenary degree in S

Conjecture Theorem (O., Ponomarenko, Tate, Webb)

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}. B = min{c(b) : b Betti element of S}.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

slide-96
SLIDE 96

Minimal (nonzero) catenary degree in S

Conjecture Theorem (O., Ponomarenko, Tate, Webb)

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}. B = min{c(b) : b Betti element of S}.

Lemma

If f , f ′ ∈ ZS(n)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

slide-97
SLIDE 97

Minimal (nonzero) catenary degree in S

Conjecture Theorem (O., Ponomarenko, Tate, Webb)

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}. B = min{c(b) : b Betti element of S}.

Lemma

If f , f ′ ∈ ZS(n) and d(f , f ′) < B,

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

slide-98
SLIDE 98

Minimal (nonzero) catenary degree in S

Conjecture Theorem (O., Ponomarenko, Tate, Webb)

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}. B = min{c(b) : b Betti element of S}.

Lemma

If f , f ′ ∈ ZS(n) and d(f , f ′) < B, then there exists f ′′ ∈ ZS(n)

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

slide-99
SLIDE 99

Minimal (nonzero) catenary degree in S

Conjecture Theorem (O., Ponomarenko, Tate, Webb)

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}. B = min{c(b) : b Betti element of S}.

Lemma

If f , f ′ ∈ ZS(n) and d(f , f ′) < B, then there exists f ′′ ∈ ZS(n) with max {|f |, |f ′|} < |f ′′|.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

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SLIDE 100

Minimal (nonzero) catenary degree in S

Conjecture Theorem (O., Ponomarenko, Tate, Webb)

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

slide-101
SLIDE 101

Minimal (nonzero) catenary degree in S

Conjecture Theorem (O., Ponomarenko, Tate, Webb)

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}. Proof of theorem:

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

slide-102
SLIDE 102

Minimal (nonzero) catenary degree in S

Conjecture Theorem (O., Ponomarenko, Tate, Webb)

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}. Proof of theorem: Fix n ∈ S

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

slide-103
SLIDE 103

Minimal (nonzero) catenary degree in S

Conjecture Theorem (O., Ponomarenko, Tate, Webb)

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}. Proof of theorem: Fix n ∈ S Catenary graph of n:

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

slide-104
SLIDE 104

Minimal (nonzero) catenary degree in S

Conjecture Theorem (O., Ponomarenko, Tate, Webb)

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}. Proof of theorem: Fix n ∈ S Draw edges with weight < B Catenary graph of n:

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

slide-105
SLIDE 105

Minimal (nonzero) catenary degree in S

Conjecture Theorem (O., Ponomarenko, Tate, Webb)

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}. Proof of theorem: Fix n ∈ S Draw edges with weight < B f ∈ ZS(n) with |f | maximal Catenary graph of n:

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

slide-106
SLIDE 106

Minimal (nonzero) catenary degree in S

Conjecture Theorem (O., Ponomarenko, Tate, Webb)

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}. Proof of theorem: Fix n ∈ S Draw edges with weight < B f ∈ ZS(n) with |f | maximal f ′ ∈ ZS(n) with d(f , f ′) < B Catenary graph of n:

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

slide-107
SLIDE 107

Minimal (nonzero) catenary degree in S

Conjecture Theorem (O., Ponomarenko, Tate, Webb)

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}. Proof of theorem: Fix n ∈ S Draw edges with weight < B f ∈ ZS(n) with |f | maximal f ′ ∈ ZS(n) with d(f , f ′) < B Lemma ⇒ |f ′′| > |f | Catenary graph of n:

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

slide-108
SLIDE 108

Minimal (nonzero) catenary degree in S

Conjecture Theorem (O., Ponomarenko, Tate, Webb)

min{c(n) > 0 : n ∈ S} = min{c(b) : b Betti element of S}. Proof of theorem: Fix n ∈ S Draw edges with weight < B f ∈ ZS(n) with |f | maximal f ′ ∈ ZS(n) with d(f , f ′) < B Lemma ⇒ |f ′′| > |f | maximality of |f | ⇒ f ′′ has no edges! Catenary graph of n:

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

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SLIDE 109

Future directions: catenary sets

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 13 / 14

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SLIDE 110

Future directions: catenary sets

50 100 150 200 250 300 2 4 6 8 10 12 14

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 13 / 14

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SLIDE 111

Future directions: catenary sets

50 100 150 200 250 300 2 4 6 8 10 12 14

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 13 / 14

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SLIDE 112

Future directions: catenary sets

50 100 150 200 250 300 2 4 6 8 10 12 14

Problem

Find a (canonical) finite set on which every catenary degree is achieved.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 13 / 14

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SLIDE 113

References

Alfred Geroldinger, Franz Halter-Koch (2006) Nonunique factorization: Algebraic, Combinatorial, and Analytic Theory. Chapman & Hall/CRC, Boca Raton, FL, 2006. Scott Champan, Pedro Garc´ ıa-S´ anchez, David Llena, Vadim Ponomarenko, Jos´ e Rosales (2006) The catenary and tame degree in finitely generated cancellative commutative monoids.

  • Manus. Math, 120 (2006) 253 – 264.

Christopher O’Neill, Vadim Ponomarenko, Reuben Tate, Gautam Webb (2014) On the set of catenary degrees in numerical monoids. In preparation. Manuel Delgado, Pedro Garc´ ıa-S´ anchez, Jose Morais GAP Numerical Semigroups Package http://www.gap-system.org/Packages/numericalsgps.html.

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 14 / 14

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SLIDE 114

References

Alfred Geroldinger, Franz Halter-Koch (2006) Nonunique factorization: Algebraic, Combinatorial, and Analytic Theory. Chapman & Hall/CRC, Boca Raton, FL, 2006. Scott Champan, Pedro Garc´ ıa-S´ anchez, David Llena, Vadim Ponomarenko, Jos´ e Rosales (2006) The catenary and tame degree in finitely generated cancellative commutative monoids.

  • Manus. Math, 120 (2006) 253 – 264.

Christopher O’Neill, Vadim Ponomarenko, Reuben Tate, Gautam Webb (2014) On the set of catenary degrees in numerical monoids. In preparation. Manuel Delgado, Pedro Garc´ ıa-S´ anchez, Jose Morais GAP Numerical Semigroups Package http://www.gap-system.org/Packages/numericalsgps.html. Thanks!

Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 14 / 14