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 an element in S is the least positive integer c such
  that  for  any  two  of  its factorizations a and b, there exists a chain of
  factorizations  starting  in  a  and  ending  in  b and so that the distance
  between  two consecutive links is at most c. The catenary degree of S is the
  supremum of the catenary degrees of the elements in S.
  
  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 [PR06] for numerical semigroup (see [CL07]). 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
  [CM06] 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.
  
  ---------------------------  Example  ----------------------------
    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.
  
  ---------------------------  Example  ----------------------------
    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.
  
  ---------------------------  Example  ----------------------------
    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.
  
  ---------------------------  Example  ----------------------------
    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.
  
  ---------------------------  Example  ----------------------------
    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.
  
  ---------------------------  Example  ----------------------------
    gap> s:=NumericalSemigroup(101,113,196,272,278,286);
    <Numerical semigroup with 6 generators>
    gap> MaximumDegreeOfElementWRTNumericalSemigroup(1100,s);
    9
  ------------------------------------------------------------------
  
  9.1-7 CatenaryDegreeOfNumericalSemigroup
  
  > CatenaryDegreeOfNumericalSemigroup( S ) __________________________function
  
  S is a numerical semigroup. The output is the catenary degree of S.
  
  ---------------------------  Example  ----------------------------
    gap> s:=NumericalSemigroup(101,113,196,272,278,286);
    <Numerical semigroup with 6 generators>
    gap> CatenaryDegreeOfNumericalSemigroup(s);
    8
  ------------------------------------------------------------------
  
  9.1-8 CatenaryDegreeOfElementNS
  
  > CatenaryDegreeOfElementNS( n, S ) ________________________________function
  
  n is a nonnegative integer and S is a numerical semigroup. The output is the
  catenary degree of n relative to S.
  
  ---------------------------  Example  ----------------------------
    gap> CatenaryDegreeOfElementNS(157,NumericalSemigroup(13,18));
    0
    gap> CatenaryDegreeOfElementNS(1157,NumericalSemigroup(13,18));
    18
  ------------------------------------------------------------------
  
  9.1-9 TameDegreeOfNumericalSemigroup
  
  > TameDegreeOfNumericalSemigroup( S ) ______________________________function
  
  S is a numerical semigroup. The output is the tame degree of S.
  
  ---------------------------  Example  ----------------------------
    gap> s:=NumericalSemigroup(101,113,196,272,278,286);
    <Numerical semigroup with 6 generators>
    gap> TameDegreeOfNumericalSemigroup(s);
    14
  ------------------------------------------------------------------