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### 9. Catenary and Tame degrees of numerical semigroups

#### 9.1 Factorizations in Numerical Semigroups

Let S be a numerical semigroup minimally generated by m_1,...,m_n. A factorization of an element sin S is an n-tuple a=(a_1,...,a_n) of nonnegative integers such that n=a_1 n_1+cdots+a_n m_n. The lenght of a is |a|=a_1+cdots+a_n. Given two factorizations a and b of n, the distance between a and b is d(a,b)=max |a-gcd(a,b)|,|b-gcd(a,b)|, where gcd((a_1,...,a_n),(b_1,...,b_n))=(min(a_1,b_1),...,min(a_n,b_n)).

If l_1>cdots > l_k are the lenghts of all the factorizations of s in S, the Delta set associated to s is Delta(s)=l_1-l_2,...,l_k-l_k-1.

The catenary degree of S is the least positive integer c such that for any two factorizations a and b of an element in S, there exists a chain of factorizations staring in a and ending in b and so that the distance between two consecutive links is at most c.

The tame degree of S is the least positive integer t for any factorization a of an element s in S, and any i such that s-m_iin S, there exists another factorization b of s so that the distance to a is at most t and b_inot = 0.

The basic properties of these constants can be found in [GH06]. The algorithm used to compute the catenary and tame degree is an adaptation of the algorithms appearing in [SR06] for numerical semigroup (see [SL]). The computation of the elascitiy of a numerical semigroup reduces to m/n with m the multiplicity of the semigroup and n its largest minimal generator (see [SM] or [GH06]).

##### 9.1-1 FactorizationsElementWRTNumericalSemigroup

`> FactorizationsElementWRTNumericalSemigroup`( n, S ) | ( function )

S is a numerical semigroup and n a nonnegative integer. The output is the set of factorizations of n in terms of the minimal generating set of S.

    gap> s:=NumericalSemigroup(101,113,196,272,278,286);
    <Numerical semigroup with 6 generators>
    gap> FactorizationsElementWRTNumericalSemigroup(1100,s);
    [ [ 0, 0, 0, 2, 2, 0 ], [ 0, 2, 3, 0, 0, 1 ], [ 0, 8, 1, 0, 0, 0 ],
      [ 5, 1, 1, 0, 0, 1 ] ]

##### 9.1-2 LengthsOfFactorizationsElementWRTNumericalSemigroup

`> LengthsOfFactorizationsElementWRTNumericalSemigroup`( n, S ) | ( function )

S is a numerical semigroup and n a nonnegative integer. The output is the set of lengths of the factorizations of n in terms of the minimal generating set of S.

    gap> s:=NumericalSemigroup(101,113,196,272,278,286);
    <Numerical semigroup with 6 generators>
    gap> LengthsOfFactorizationsElementWRTNumericalSemigroup(1100,s);
    [ 4, 6, 8, 9 ]

##### 9.1-3 ElasticityOfFactorizationsElementWRTNumericalSemigroup

`> ElasticityOfFactorizationsElementWRTNumericalSemigroup`( n, S ) | ( function )

S is a numerical semigroup and n a positive integer. The output is the maximum length divided by the minimum length of the factorizations of n in terms of the minimal generating set of S.

    gap> s:=NumericalSemigroup(101,113,196,272,278,286);
    <Numerical semigroup with 6 generators>
    gap> ElasticityOfFactorizationsElementWRTNumericalSemigroup(1100,s);
    9/4

##### 9.1-4 ElasticityOfNumericalSemigroup

`> ElasticityOfNumericalSemigroup`( S ) | ( function )

S is a numerical semigroup. The output is the elasticity of S.

    gap> s:=NumericalSemigroup(101,113,196,272,278,286);
    <Numerical semigroup with 6 generators>
    gap> ElasticityOfNumericalSemigroup(s);
    286/101

##### 9.1-5 DeltaSetOfFactorizationsElementWRTNumericalSemigroup

`> DeltaSetOfFactorizationsElementWRTNumericalSemigroup`( n, S ) | ( function )

S is a numerical semigroup and n a nonnegative integer. The output is the Delta set of the factorizations of n in terms of the minimal generating set of S.

    gap> s:=NumericalSemigroup(101,113,196,272,278,286);
    <Numerical semigroup with 6 generators>
    gap> DeltaSetOfFactorizationsElementWRTNumericalSemigroup(1100,s);
    [ 1, 2 ]

##### 9.1-6 MaximumDegreeOfElementWRTNumericalSemigroup

`> MaximumDegreeOfElementWRTNumericalSemigroup`( n, S ) | ( function )

S is a numerical semigroup and n a nonnegative integer. The output is the maximum length of the factorizations of n in terms of the minimal generating set of S.

    gap> s:=NumericalSemigroup(101,113,196,272,278,286);
    <Numerical semigroup with 6 generators>
    gap> MaximumDegreeOfElementWRTNumericalSemigroup(1100,s);
    9

##### 9.1-7 CatenaryDegreeNumericalSemigroup

`> CatenaryDegreeNumericalSemigroup`( S ) | ( function )

S is a numerical semigroup. The output is the catenary degree of S.

    gap> s:=NumericalSemigroup(101,113,196,272,278,286);
    <Numerical semigroup with 6 generators>
    gap> CatenaryDegreeNumericalSemigroup(s);
    8

##### 9.1-8 TameDegreeNumericalSemigroup

`> TameDegreeNumericalSemigroup`( S ) | ( function )

S is a numerical semigroup. The output is the tame degree of S.

    gap> s:=NumericalSemigroup(101,113,196,272,278,286);
    <Numerical semigroup with 6 generators>
    gap> TameDegreeNumericalSemigroup(s);
    14

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