Catenary degree in numerical. monoids (An application to numerical - - PowerPoint PPT Presentation

Catenary degree in numerical. monoids (An application to numerical monoids generated by arithmetic sequences)

• D. Llena Carrasco

´ Area de Geometr´ ıa y Topolog´ ıa Universidad de Almer´ ıa

Porto-March 2008

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

This is a joint work with

◮ S. T. Chapman, (Trinity) ◮ P

. A. Garc´ ıa-S´ anchez (Granada)

To appear in

Forum Mathematicum

2 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Set of factorizations

◮ Let S = n1, . . . , np be a numerical monoid minimally generated

by n1 < n2 < . . . < np.

◮ For every s ∈ S, we define the set of factorizations of s

Z(s) = {(a1, . . . ap) ∈ Np : a1n1 + . . . + apnp = s}

3 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Set of factorizations

◮ Let S = n1, . . . , np be a numerical monoid minimally generated

by n1 < n2 < . . . < np.

◮ For every s ∈ S, we define the set of factorizations of s

Z(s) = {(a1, . . . ap) ∈ Np : a1n1 + . . . + apnp = s}

Support

Let a = (a1, . . . ap) ∈ Zp We define the support of a as: supp(a) = {i : ai 0}

3 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Set of factorizations

◮ Let S = n1, . . . , np be a numerical monoid minimally generated

by n1 < n2 < . . . < np.

◮ For every s ∈ S, we define the set of factorizations of s

Z(s) = {(a1, . . . ap) ∈ Np : a1n1 + . . . + apnp = s}

Distance

◮ For z = (z1, . . . , zp) ∈ Zp we define its norm:

|z| = max{

• zi>0

zi,

• zi<0

|zi|} The length of a factorization a ∈ Np is |a|.

◮ For a, b ∈ Np we define the distance between a and b as

d(a, b) = |a − b| Distance is the main tool in this work. The catenary degree tries to control the distance between all different factorizations of the elements of S.

3 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Catenary degree

Given s ∈ S and a, b ∈ Z(s), then an N-chain of factorizations from a to b is a sequence a = a1, . . . , at = b, ai ∈ Z(s) such that d(ai, ai+1) ≤ N for all i.

4 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Catenary degree

Given s ∈ S and a, b ∈ Z(s), then an N-chain of factorizations from a to b is a sequence a = a1, . . . , at = b, ai ∈ Z(s) such that d(ai, ai+1) ≤ N for all i. The catenary degree of s, c(s), is the minimal N ∈ N ∪ {∞} such that for any two factorizations a, b ∈ Z(s), there is an N-chain from a to b.

4 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Catenary degree

Given s ∈ S and a, b ∈ Z(s), then an N-chain of factorizations from a to b is a sequence a = a1, . . . , at = b, ai ∈ Z(s) such that d(ai, ai+1) ≤ N for all i. The catenary degree of s, c(s), is the minimal N ∈ N ∪ {∞} such that for any two factorizations a, b ∈ Z(s), there is an N-chain from a to b. The catenary degree of S is c(S) = sup{c(s): s ∈ S}.

4 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

The catenary degree of an element

Let S = 6, 9, 11 Z(42) = {(0, 1, 3), (1, 4, 0), (4, 2, 0), (7, 0, 0)}

5 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

The catenary degree of an element

Let S = 6, 9, 11 Z(42) = {(0, 1, 3), (1, 4, 0), (4, 2, 0), (7, 0, 0)} (0, 1, 3)

• (1, 4, 0)
• (4, 2, 0)

(7, 0, 0)

5 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

The catenary degree of an element

Let S = 6, 9, 11 Z(42) = {(0, 1, 3), (1, 4, 0), (4, 2, 0), (7, 0, 0)} (0, 1, 3)

4 7

• 5

(1, 4, 0)

6 3

• (4, 2, 0)

3

(7, 0, 0)

5 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

The catenary degree of an element

Let S = 6, 9, 11 Z(42) = {(0, 1, 3), (1, 4, 0), (4, 2, 0), (7, 0, 0)} (0, 1, 3)

4 5

(1, 4, 0)

6 3

• (4, 2, 0)

3

(7, 0, 0)

5 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

The catenary degree of an element

Let S = 6, 9, 11 Z(42) = {(0, 1, 3), (1, 4, 0), (4, 2, 0), (7, 0, 0)} (0, 1, 3)

4 5

(1, 4, 0)

3

• (4, 2, 0)

3

(7, 0, 0)

5 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

The catenary degree of an element

Let S = 6, 9, 11 Z(42) = {(0, 1, 3), (1, 4, 0), (4, 2, 0), (7, 0, 0)} (0, 1, 3)

4

(1, 4, 0)

3

• (4, 2, 0)

3

(7, 0, 0) So C(42) = 4

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Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

The Tool

R-classes

Let s ∈ S and a, b ∈ Z(s) be two factorizations of s. We say that both factorizations are related aRb if there exists a sequence a = a1, . . . , at = b, ai ∈ Z(s) such that supp(ai) ∩ supp(ai+1) ∅.

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Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

The Tool

R-classes

Let s ∈ S and a, b ∈ Z(s) be two factorizations of s. We say that both factorizations are related aRb if there exists a sequence a = a1, . . . , at = b, ai ∈ Z(s) such that supp(ai) ∩ supp(ai+1) ∅.

Proposition

The distance between elements a and b in different R-clases is d(a, b) = max{|a|, |b|}. We can control the elements with more than one R-class.

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Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Ap´ ery set

◮ For every ni we define the Ap´

ery set of S respect to ni as Ap(S, ni) = {s ∈ S : s − ni S}

◮ Ap(S, ni) = {w(0), w(1), . . . , w(ni−1)} with w(j) the least element

in S congruent with j modulo ni

7 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Ap´ ery set

◮ For every ni we define the Ap´

ery set of S respect to ni as Ap(S, ni) = {s ∈ S : s − ni S}

◮ Ap(S, ni) = {w(0), w(1), . . . , w(ni−1)} with w(j) the least element

in S congruent with j modulo ni

Important Result

The elements of S with more than one R-class are of the form w + nj where w ∈ Ap(S, n1) and j = 2, . . . , p.

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Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

R-classes in S = 6, 9, 11

S = {0, 6, 9, 11, 12, 15, 17, 18, 20, 21, 22, 23, 24, 26, →}

8 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

R-classes in S = 6, 9, 11

S = {0, 6, 9, 11, 12, 15, 17, 18, 20, 21, 22, 23, 24, 26, →} The factorizations and R clases of 18, 33, 29, 36 ∈ 6, 9, 11 are: Z(18) = {(0, 2, 0)}, ∪{(3, 0, 0)} 2 R − classes Z(33) = {(0, 0, 3)}, ∪{(1, 3, 0), (4, 1, 0)} 2 R − classes Z(29) = {(0, 2, 1), (3, 0, 1)} 1 R − class Z(36) = {(6, 0, 0), (0, 4, 0), (3, 2, 0)} 1 R − class

8 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

R-classes in S = 6, 9, 11

S = {0, 6, 9, 11, 12, 15, 17, 18, 20, 21, 22, 23, 24, 26, →} The factorizations and R clases of 18, 33, 29, 36 ∈ 6, 9, 11 are: Z(18) = {(0, 2, 0)}, ∪{(3, 0, 0)} 2 R − classes Z(33) = {(0, 0, 3)}, ∪{(1, 3, 0), (4, 1, 0)} 2 R − classes Z(29) = {(0, 2, 1), (3, 0, 1)} 1 R − class Z(36) = {(6, 0, 0), (0, 4, 0), (3, 2, 0)} 1 R − class The Ap´ ery set with respect n1 is: Ap(S, 6) = {0, 31, 20, 9, 22, 11}

8 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

R-classes in S = 6, 9, 11

S = {0, 6, 9, 11, 12, 15, 17, 18, 20, 21, 22, 23, 24, 26, →} The factorizations and R clases of 18, 33, 29, 36 ∈ 6, 9, 11 are: Z(18) = {(0, 2, 0)}, ∪{(3, 0, 0)} 2 R − classes Z(33) = {(0, 0, 3)}, ∪{(1, 3, 0), (4, 1, 0)} 2 R − classes Z(29) = {(0, 2, 1), (3, 0, 1)} 1 R − class Z(36) = {(6, 0, 0), (0, 4, 0), (3, 2, 0)} 1 R − class The Ap´ ery set with respect n1 is: Ap(S, 6) = {0, 31, 20, 9, 22, 11}

Using Ap´ ery set for 6, 9, 11

9 + Ap(S, 6) = 9 40 29 18 31 20 11 + Ap(S, 6) = 11 42 31 20 33 22

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Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

66 ∈ 6, 9, 11

The factorizations of 66 ∈ 6, 9, 11 are Z(66) = {(0, 0, 6), (1, 3, 3), (2, 6, 0), (4, 1, 3), (5, 4, 0), (8, 2, 0), (11, 0, 0)} We are going to connect them using 18’s and 33’s factorizations. Remember: Z(18) = {(0, 2, 0), (3, 0, 0)} Z(33) = {(0, 0, 3), (1, 3, 0), (4, 1, 0)}

9 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

66 ∈ 6, 9, 11

The factorizations of 66 ∈ 6, 9, 11 are Z(66) = {(0, 0, 6), (1, 3, 3), (2, 6, 0), (4, 1, 3), (5, 4, 0), (8, 2, 0), (11, 0, 0)} We are going to connect them using 18’s and 33’s factorizations. Remember: Z(18) = {(0, 2, 0), (3, 0, 0)} Z(33) = {(0, 0, 3), (1, 3, 0), (4, 1, 0)} (0, 0, 6)

33

(1, 3, 3)

33

(2, 6, 0)

9 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

66 ∈ 6, 9, 11

The factorizations of 66 ∈ 6, 9, 11 are Z(66) = {(0, 0, 6), (1, 3, 3), (2, 6, 0), (4, 1, 3), (5, 4, 0), (8, 2, 0), (11, 0, 0)} We are going to connect them using 18’s and 33’s factorizations. Remember: Z(18) = {(0, 2, 0), (3, 0, 0)} Z(33) = {(0, 0, 3), (1, 3, 0), (4, 1, 0)} (0, 0, 6)

33

(1, 3, 3)

33 18

(2, 6, 0)

18

(4, 1, 3)

33

(5, 4, 0)

9 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

66 ∈ 6, 9, 11

The factorizations of 66 ∈ 6, 9, 11 are Z(66) = {(0, 0, 6), (1, 3, 3), (2, 6, 0), (4, 1, 3), (5, 4, 0), (8, 2, 0), (11, 0, 0)} We are going to connect them using 18’s and 33’s factorizations. Remember: Z(18) = {(0, 2, 0), (3, 0, 0)} Z(33) = {(0, 0, 3), (1, 3, 0), (4, 1, 0)} (0, 0, 6)

33

(1, 3, 3)

33 18

(2, 6, 0)

18

(4, 1, 3)

33

(5, 4, 0)

18

(8, 2, 0)

18

(11, 0, 0)

9 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

66 ∈ 6, 9, 11

The factorizations of 66 ∈ 6, 9, 11 are Z(66) = {(0, 0, 6), (1, 3, 3), (2, 6, 0), (4, 1, 3), (5, 4, 0), (8, 2, 0), (11, 0, 0)} We are going to connect them using 18’s and 33’s factorizations. Remember: Z(18) = {(0, 2, 0), (3, 0, 0)} Z(33) = {(0, 0, 3), (1, 3, 0), (4, 1, 0)} (0, 0, 6)

33

(1, 3, 3)

33 18

(2, 6, 0)

18

(4, 1, 3)

33

(5, 4, 0)

18

(8, 2, 0)

18

(11, 0, 0) We can see that distances between 66’s factorizations are the same as 18’s and 33’s factorizations.

9 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Catenary degree (Chapman, Garc´ ıa-S´ anchez, Ll., Ponomarenko, Rosales 2006)

◮ For every s ∈ S, let Rs 1, . . . , Rs ks be the different R-classes

10 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Catenary degree (Chapman, Garc´ ıa-S´ anchez, Ll., Ponomarenko, Rosales 2006)

◮ For every s ∈ S, let Rs 1, . . . , Rs ks be the different R-classes ◮ From each R-class, choose an element as i such that |as i | is

minimum in its R-class

10 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Catenary degree (Chapman, Garc´ ıa-S´ anchez, Ll., Ponomarenko, Rosales 2006)

◮ For every s ∈ S, let Rs 1, . . . , Rs ks be the different R-classes ◮ From each R-class, choose an element as i such that |as i | is

minimum in its R-class

◮ Define r(s) = max{|as 1|, . . . , |as ks|}

10 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Catenary degree (Chapman, Garc´ ıa-S´ anchez, Ll., Ponomarenko, Rosales 2006)

◮ For every s ∈ S, let Rs 1, . . . , Rs ks be the different R-classes ◮ From each R-class, choose an element as i such that |as i | is

minimum in its R-class

◮ Define r(s) = max{|as 1|, . . . , |as ks|} ◮ We have proved that

c(S) = max{r(s) | s ∈ S, ks > 1}

10 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Catenary degree (Chapman, Garc´ ıa-S´ anchez, Ll., Ponomarenko, Rosales 2006)

◮ For every s ∈ S, let Rs 1, . . . , Rs ks be the different R-classes ◮ From each R-class, choose an element as i such that |as i | is

minimum in its R-class

◮ Define r(s) = max{|as 1|, . . . , |as ks|} ◮ We have proved that

c(S) = max{r(s) | s ∈ S, ks > 1}

10 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Catenary degree (Chapman, Garc´ ıa-S´ anchez, Ll., Ponomarenko, Rosales 2006)

◮ For every s ∈ S, let Rs 1, . . . , Rs ks be the different R-classes ◮ From each R-class, choose an element as i such that |as i | is

minimum in its R-class

◮ Define r(s) = max{|as 1|, . . . , |as ks|} ◮ We have proved that

c(S) = max{r(s) | s ∈ S, ks > 1}

10 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Proposition 1

b = min{k ∈ N − {0}: kn1 ∈< n2, . . . , np >} ≤ c(S)

11 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Proposition 1

b = min{k ∈ N − {0}: kn1 ∈< n2, . . . , np >} ≤ c(S) Z(bn1) has two (o more) factorizations: (b, 0, . . . , 0) and (0, a2, . . . ap) such that

p

• i=2

aini = bn1

11 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Proposition 1

b = min{k ∈ N − {0}: kn1 ∈< n2, . . . , np >} ≤ c(S) Z(bn1) has two (o more) factorizations: (b, 0, . . . , 0) and (0, a2, . . . ap) such that

p

• i=2

aini = bn1 We can ensure that do not exist factorizations of bn1 joining those given above, because of the minimality of b.

11 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Proposition 1

b = min{k ∈ N − {0}: kn1 ∈< n2, . . . , np >} ≤ c(S) Z(bn1) has two (o more) factorizations: (b, 0, . . . , 0) and (0, a2, . . . ap) such that

p

• i=2

aini = bn1 We can ensure that do not exist factorizations of bn1 joining those given above, because of the minimality of b. The above is equivalent to say that bn1 − n1 − ni S We see next examples in which this bound is reached.

11 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Generated by arithmetic sequences

¿From now on, we consider S = a, a + d, . . . , a + cd with 1 ≤ c < a and gcd(a, d) = 1.

12 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Generated by arithmetic sequences

¿From now on, we consider S = a, a + d, . . . , a + cd with 1 ≤ c < a and gcd(a, d) = 1.

Main theorem

c(S) = a c

• + d

Where ⌈q⌉ = min{n ∈ N: n ≥ q}

12 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Sketch of the proof First, we control membership to S and the factorizations of the elements in the Ap´ ery set of the multiplicity

The elements

Every element in S can be written as ka + rd. Note that ad can be appear as d times a or a times d. So we can assume that r < a. This will be the key for the following.

13 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Sketch of the proof First, we control membership to S and the factorizations of the elements in the Ap´ ery set of the multiplicity

The elements

Every element in S can be written as ka + rd. Note that ad can be appear as d times a or a times d. So we can assume that r < a. This will be the key for the following. If r < a, n = ka + rd ∈ S ⇔ 0 ≤ r ≤ kc In the minimal case n is multiple of a. In the maximal case n is multiple of a + cd.

13 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Apery Set in arithmetic sequences

Let w ∈ Ap(S, a) and let z ∈ Z(w). Then |z| ≤ a c

• 14 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Apery Set in arithmetic sequences

Let w ∈ Ap(S, a) and let z ∈ Z(w). Then |z| ≤ a c

• Factorizations of elements with more than one R-class

Let n = w + (a + jd) with w ∈ Ap(S, a) and j ∈ {1, . . . c}. If z ∈ Z(n) then |z| ≤

• a

c

• + d + 1

14 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Apery Set in arithmetic sequences

Let w ∈ Ap(S, a) and let z ∈ Z(w). Then |z| ≤ a c

• Factorizations of elements with more than one R-class

Let n = w + (a + jd) with w ∈ Ap(S, a) and j ∈ {1, . . . c}. If z ∈ Z(n) then |z| ≤

• a

c

• + d + 1

Hence c(S) ≤ a c

• + d + 1

14 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Besides recall that that min{k ∈ N − {0}: ka ∈< a + d, . . . , a + cd >} ≤ c(S)

15 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Besides recall that that min{k ∈ N − {0}: ka ∈< a + d, . . . , a + cd >} ≤ c(S)

Computing this bound

min{k ∈ N − {0}: ka ∈< a + d, . . . , a + cd >} =

• a

c

• + d

15 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Besides recall that that min{k ∈ N − {0}: ka ∈< a + d, . . . , a + cd >} ≤ c(S)

Computing this bound

min{k ∈ N − {0}: ka ∈< a + d, . . . , a + cd >} =

• a

c

• + d

Thus a c

• + d ≤ c(S) ≤

a c

• + d + 1

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Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

Besides recall that that min{k ∈ N − {0}: ka ∈< a + d, . . . , a + cd >} ≤ c(S)

Computing this bound

min{k ∈ N − {0}: ka ∈< a + d, . . . , a + cd >} =

• a

c

• + d

Thus a c

• + d ≤ c(S) ≤

a c

• + d + 1

The bound is tight

Now to finish the proof of main theorem. We only need to prove that c(S)

• a

c

• + d + 1

15 / 16

Catenary degree Credits Introduction

Definitions Catenary degree

Computing the catenary degree

• f the monoid

example example Characterizations of the catenary degree

Numerical Semigroups Generated By Arithmetic Sequences

A new result. Generalized arithmetic sequences

Omidali has been to proof that if S = a, ha + d, . . . , ha + cd then: c(S) = a c

• h + d

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