[1X9 Catenary and Tame degrees of numerical semigroups[0X [1X9.1 Factorizations in Numerical Semigroups[0X 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]). [1X9.1-1 FactorizationsElementWRTNumericalSemigroup[0m [2X> FactorizationsElementWRTNumericalSemigroup( [0X[3Xn, S[0X[2X ) _______________[0Xfunction [3XS[0m is a numerical semigroup and [3Xn[0m a nonnegative integer. The output is the set of factorizations of [3Xn[0m in terms of the minimal generating set of [3XS[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> s:=NumericalSemigroup(101,113,196,272,278,286);[0X [4X<Numerical semigroup with 6 generators>[0X [4Xgap> FactorizationsElementWRTNumericalSemigroup(1100,s);[0X [4X[ [ 0, 0, 0, 2, 2, 0 ], [ 0, 2, 3, 0, 0, 1 ], [ 0, 8, 1, 0, 0, 0 ],[0X [4X [ 5, 1, 1, 0, 0, 1 ] ][0X [4X------------------------------------------------------------------[0X [1X9.1-2 LengthsOfFactorizationsElementWRTNumericalSemigroup[0m [2X> LengthsOfFactorizationsElementWRTNumericalSemigroup( [0X[3Xn, S[0X[2X ) ______[0Xfunction [3XS[0m is a numerical semigroup and [3Xn[0m a nonnegative integer. The output is the set of lengths of the factorizations of [3Xn[0m in terms of the minimal generating set of [3XS[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> s:=NumericalSemigroup(101,113,196,272,278,286);[0X [4X<Numerical semigroup with 6 generators>[0X [4Xgap> LengthsOfFactorizationsElementWRTNumericalSemigroup(1100,s);[0X [4X[ 4, 6, 8, 9 ][0X [4X------------------------------------------------------------------[0X [1X9.1-3 ElasticityOfFactorizationsElementWRTNumericalSemigroup[0m [2X> ElasticityOfFactorizationsElementWRTNumericalSemigroup( [0X[3Xn, S[0X[2X ) ___[0Xfunction [3XS[0m is a numerical semigroup and [3Xn[0m a positive integer. The output is the maximum length divided by the minimum length of the factorizations of [3Xn[0m in terms of the minimal generating set of [3XS[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> s:=NumericalSemigroup(101,113,196,272,278,286);[0X [4X<Numerical semigroup with 6 generators>[0X [4Xgap> ElasticityOfFactorizationsElementWRTNumericalSemigroup(1100,s);[0X [4X9/4[0X [4X------------------------------------------------------------------[0X [1X9.1-4 ElasticityOfNumericalSemigroup[0m [2X> ElasticityOfNumericalSemigroup( [0X[3XS[0X[2X ) ______________________________[0Xfunction [3XS[0m is a numerical semigroup. The output is the elasticity of [3XS[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> s:=NumericalSemigroup(101,113,196,272,278,286);[0X [4X<Numerical semigroup with 6 generators>[0X [4Xgap> ElasticityOfNumericalSemigroup(s);[0X [4X286/101[0X [4X------------------------------------------------------------------[0X [1X9.1-5 DeltaSetOfFactorizationsElementWRTNumericalSemigroup[0m [2X> DeltaSetOfFactorizationsElementWRTNumericalSemigroup( [0X[3Xn, S[0X[2X ) _____[0Xfunction [3XS[0m is a numerical semigroup and [3Xn[0m a nonnegative integer. The output is the Delta set of the factorizations of [3Xn[0m in terms of the minimal generating set of [3XS[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> s:=NumericalSemigroup(101,113,196,272,278,286);[0X [4X<Numerical semigroup with 6 generators>[0X [4Xgap> DeltaSetOfFactorizationsElementWRTNumericalSemigroup(1100,s);[0X [4X[ 1, 2 ][0X [4X------------------------------------------------------------------[0X [1X9.1-6 MaximumDegreeOfElementWRTNumericalSemigroup[0m [2X> MaximumDegreeOfElementWRTNumericalSemigroup( [0X[3Xn, S[0X[2X ) ______________[0Xfunction [3XS[0m is a numerical semigroup and [3Xn[0m a nonnegative integer. The output is the maximum length of the factorizations of [3Xn[0m in terms of the minimal generating set of [3XS[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> s:=NumericalSemigroup(101,113,196,272,278,286);[0X [4X<Numerical semigroup with 6 generators>[0X [4Xgap> MaximumDegreeOfElementWRTNumericalSemigroup(1100,s);[0X [4X9[0X [4X------------------------------------------------------------------[0X [1X9.1-7 CatenaryDegreeOfNumericalSemigroup[0m [2X> CatenaryDegreeOfNumericalSemigroup( [0X[3XS[0X[2X ) __________________________[0Xfunction [3XS[0m is a numerical semigroup. The output is the catenary degree of [3XS[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> s:=NumericalSemigroup(101,113,196,272,278,286);[0X [4X<Numerical semigroup with 6 generators>[0X [4Xgap> CatenaryDegreeOfNumericalSemigroup(s);[0X [4X8[0X [4X------------------------------------------------------------------[0X [1X9.1-8 CatenaryDegreeOfElementNS[0m [2X> CatenaryDegreeOfElementNS( [0X[3Xn, S[0X[2X ) ________________________________[0Xfunction [3Xn[0m is a nonnegative integer and [3XS[0m is a numerical semigroup. The output is the catenary degree of [3Xn[0m relative to [3XS[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> CatenaryDegreeOfElementNS(157,NumericalSemigroup(13,18));[0X [4X0[0X [4Xgap> CatenaryDegreeOfElementNS(1157,NumericalSemigroup(13,18));[0X [4X18[0X [4X------------------------------------------------------------------[0X [1X9.1-9 TameDegreeOfNumericalSemigroup[0m [2X> TameDegreeOfNumericalSemigroup( [0X[3XS[0X[2X ) ______________________________[0Xfunction [3XS[0m is a numerical semigroup. The output is the tame degree of [3XS[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> s:=NumericalSemigroup(101,113,196,272,278,286);[0X [4X<Numerical semigroup with 6 generators>[0X [4Xgap> TameDegreeOfNumericalSemigroup(s);[0X [4X14[0X [4X------------------------------------------------------------------[0X