<Section> <Heading>Presentations of Numerical Semigroups</Heading> <ManSection> <Func Arg="L" Name="FortenTruncatedNCForNumericalSemigroups"></Func> <Description> <A>L</A> 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 <Cite Key="MR1283022" />). Returns <C>fail</C> if no solution exists. <Example><![CDATA[ gap> FortenTruncatedNCForNumericalSemigroups([ -57, 3 ]); [ 1, 19 ] gap> FortenTruncatedNCForNumericalSemigroups([ -57, 33 ]); fail gap> FortenTruncatedNCForNumericalSemigroups([ -57, 19 ]); [ 1, 3 ] ]]></Example> </Description> </ManSection> <ManSection> <Func Arg="S" Name="MinimalPresentationOfNumericalSemigroup"></Func> <Description> <A>S</A> 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 <M> \{ \{x_1,y_1\},\ldots,\{x_k,y_k\}\} </M> is the output and <M> \{m_1,\ldots,m_n\} </M> is the minimal system of generators of the numerical semigroup, then <M> \{x_i,y_i\}=\{\{a_{i_1},\ldots,a_{i_n}\},\{b_{i_1},\ldots,b_{i_n}\}\}</M> and <M> a_{i_1}m_1+\cdots+a_{i_n}m_n= b_{i_1}m_1+ \cdots +b_{i_n}m_n.</M> <P/> Any other relation among the minimal generators of the semigroup can be deduced from the ones given in the output. <P/> The algorithm implemented is described in <Cite Key="Ros96"></Cite> (see also <Cite Key="RGS99"></Cite>). <Example><![CDATA[ gap> s:=NumericalSemigroup(3,5,7); <Numerical semigroup with 3 generators> gap> MinimalPresentationOfNumericalSemigroup(s); [ [ [ 1, 0, 1 ], [ 0, 2, 0 ] ], [ [ 4, 0, 0 ], [ 0, 1, 1 ] ], [ [ 3, 1, 0 ], [ 0, 0, 2 ] ] ] ]]> </Example> The first element in the list means that <M> 1\times 3+1\times 7=2\times 5 </M>, and so on. </Description> </ManSection> <ManSection> <Func Arg="n, S" Name="GraphAssociatedToElementInNumericalSemigroup"></Func> <Description> <A>S</A> is a numerical semigroup and <A>n</A> is an element in <A>S</A>. <P/> The output is a pair. If <M> \{m_1,\ldots,m_n\} </M> is the set of minimal generators of <A>S</A>, then the first component is the set of vertices of the graph associated to <A>n</A> in <A>S</A>, that is, the set <M>\{ m_i \ |\ n-m_i\in S\} </M>, and the second component is the set of edges of this graph, that is, <M> \{ \{m_i,m_j\} \ |\ n-(m_i+m_j)\in S\}.</M> <P/> This function is used to compute a minimal presentation of the numerical semigroup <A>S</A>, as explained in <Cite Key="Ros96"></Cite>. <Example><![CDATA[ gap> s:=NumericalSemigroup(3,5,7); <Numerical semigroup with 3 generators> gap> GraphAssociatedToElementInNumericalSemigroup(10,s); [ [ 3, 5, 7 ], [ [ 3, 7 ] ] ] ]]> </Example> </Description> </ManSection> </Section>