[1X4 Presentations of Numerical Semigroups[0X In this chapter we explain how to compute a minimal presentation of a numerical semigroup. There are three functions involved in this process. [1X4.1 Presentations of Numerical Semigroups[0X [1X4.1-1 FortenTruncatedNCForNumericalSemigroups[0m [2X> FortenTruncatedNCForNumericalSemigroups( [0X[3XL[0X[2X ) _____________________[0Xfunction [3XL[0m contains the list of coefficients of a single linear equation. This function gives a minimal generator of the affine semigroup of nonnegative solutions of this equation with the first coordinate equal to one (see [CD94]). Returns [10Xfail[0m if no solution exists. [4X--------------------------- Example ----------------------------[0X [4Xgap> FortenTruncatedNCForNumericalSemigroups([ -57, 3 ]);[0X [4X[ 1, 19 ][0X [4Xgap> FortenTruncatedNCForNumericalSemigroups([ -57, 33 ]);[0X [4Xfail[0X [4Xgap> FortenTruncatedNCForNumericalSemigroups([ -57, 19 ]);[0X [4X[ 1, 3 ][0X [4X------------------------------------------------------------------[0X [1X4.1-2 MinimalPresentationOfNumericalSemigroup[0m [2X> MinimalPresentationOfNumericalSemigroup( [0X[3XS[0X[2X ) _____________________[0Xfunction [3XS[0m is a numerical semigroup. The output is a list of lists with two elements. Each list of two elements represents a relation between the minimal generators of the numerical semigroup. If { {x_1,y_1},...,{x_k,y_k}} is the output and {m_1,...,m_n} is the minimal system of generators of the numerical semigroup, then {x_i,y_i}={{a_i_1,...,a_i_n},{b_i_1,...,b_i_n}} and a_i_1m_1+cdots+a_i_nm_n= b_i_1m_1+ cdots +b_i_nm_n. Any other relation among the minimal generators of the semigroup can be deduced from the ones given in the output. The algorithm implemented is described in [Ros96a] (see also [RG99]). [4X--------------------------- Example ----------------------------[0X [4Xgap> s:=NumericalSemigroup(3,5,7);[0X [4X<Numerical semigroup with 3 generators>[0X [4Xgap> MinimalPresentationOfNumericalSemigroup(s);[0X [4X[ [ [ 1, 0, 1 ], [ 0, 2, 0 ] ], [ [ 4, 0, 0 ], [ 0, 1, 1 ] ],[0X [4X [ [ 3, 1, 0 ], [ 0, 0, 2 ] ] ][0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X The first element in the list means that 1x 3+1x 7=2x 5, and so on. [1X4.1-3 GraphAssociatedToElementInNumericalSemigroup[0m [2X> GraphAssociatedToElementInNumericalSemigroup( [0X[3Xn, S[0X[2X ) _____________[0Xfunction [3XS[0m is a numerical semigroup and [3Xn[0m is an element in [3XS[0m. The output is a pair. If {m_1,...,m_n} is the set of minimal generators of [3XS[0m, then the first component is the set of vertices of the graph associated to [3Xn[0m in [3XS[0m, that is, the set { m_i | n-m_iin S}, and the second component is the set of edges of this graph, that is, { {m_i,m_j} | n-(m_i+m_j)in S}. This function is used to compute a minimal presentation of the numerical semigroup [3XS[0m, as explained in [Ros96a]. [4X--------------------------- Example ----------------------------[0X [4Xgap> s:=NumericalSemigroup(3,5,7);[0X [4X<Numerical semigroup with 3 generators>[0X [4Xgap> GraphAssociatedToElementInNumericalSemigroup(10,s);[0X [4X[ [ 3, 5, 7 ], [ [ 3, 7 ] ] ][0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X