In this chapter we explain how to compute a minimal presentation of a numerical semigroup. There are three functions involved in this process.
‣ MinimalPresentationOfNumericalSemigroup( S ) | ( function ) |
‣ MinimalPresentation( S ) | ( operation ) |
S 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 \( \{ \{x_1,y_1\},\ldots,\{x_k,y_k\}\} \) is the output and \( \{m_1,\ldots,m_n\} \) is the minimal system of generators of the numerical semigroup, then \( \{x_i,y_i\}=\{\{a_{i_1},\ldots,a_{i_n}\},\{b_{i_1},\ldots,b_{i_n}\}\}\) and \( a_{i_1}m_1+\cdots+a_{i_n}m_n= b_{i_1}m_1+ \cdots +b_{i_n}m_n.\)
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 [RGS99b]).
gap> s:=NumericalSemigroup(3,5,7); <Numerical semigroup with 3 generators> gap> MinimalPresentationOfNumericalSemigroup(s); [ [ [ 0, 2, 0 ], [ 1, 0, 1 ] ], [ [ 3, 1, 0 ], [ 0, 0, 2 ] ], [ [ 4, 0, 0 ], [ 0, 1, 1 ] ] ]
The first element in the list means that \( 1\times 3+1\times 7=2\times 5 \), and the others have similar meanings.
‣ GraphAssociatedToElementInNumericalSemigroup( n, S ) | ( function ) |
S is a numerical semigroup and n is an element in S.
The output is a pair. If \( \{m_1,\ldots,m_n\} \) is the set of minimal generators of S, then the first component is the set of vertices of the graph associated to n in S, that is, the set \(\{ m_i \ |\ n-m_i\in S\} \), 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 S, as explained in [Ros96a].
gap> s:=NumericalSemigroup(3,5,7);; gap> GraphAssociatedToElementInNumericalSemigroup(10,s); [ [ 3, 5, 7 ], [ [ 3, 7 ] ] ]
‣ BettiElementsOfNumericalSemigroup( S ) | ( function ) |
‣ BettiElements( S ) | ( operation ) |
S is a numerical semigroup.
The output is the set of elements in S whose associated graph is nonconnected [GSO10].
gap> s:=NumericalSemigroup(3,5,7);; gap> BettiElementsOfNumericalSemigroup(s); [ 10, 12, 14 ]
‣ PrimitiveElementsOfNumericalSemigroup( S ) | ( function ) |
S is a numerical semigroup.
The output is the set of elements \(s\) in S such that there exists a minimal solution to \(msg\cdot x-msg\cdot y = 0\), such that \(x,y\) are factorizations of \(s\), and \(msg\) is the minimal generating system of S. Betti elements are primitive, but not the way around in general.
gap> s:=NumericalSemigroup(3,5,7);; gap> PrimitiveElementsOfNumericalSemigroup(s); [ 3, 5, 7, 10, 12, 14, 15, 21, 28, 35 ]
‣ ShadedSetOfElementInNumericalSemigroup( n, S ) | ( function ) |
S is a numerical semigroup and n is an element in S.
The output is a simplicial complex \(C\). If \( \{m_1,\ldots,m_n\} \) is the set of minimal generators of S, then \(L \in C\) if \(n-\sum_{i\in L} m_i\in S\) ([SW86]).
This function is a generalization of the graph associated to n.
gap> s:=NumericalSemigroup(3,5,7);; gap> ShadedSetOfElementInNumericalSemigroup(10,s); [ [ ], [ 3 ], [ 3, 7 ], [ 5 ], [ 7 ] ]
A numerical semigroup \(S\) is uniquely presented if for any two minimal presentations \(\sigma\) and \(\tau\) and any \((a,b)\in \sigma\), either \((a,b)\in \tau\) or \((b,a)\in \tau\), that is, there is essentially a unique minimal presentation (up to arrangement of the components of the pairs in it).
‣ IsUniquelyPresented( S ) | ( property ) |
‣ IsUniquelyPresentedNumericalSemigroup( S ) | ( property ) |
S is a numerical semigroup.
The output is true if S has uniquely presented. The implementation is based on (see [GSO10]).
gap> s:=NumericalSemigroup(3,5,7);; gap> IsUniquelyPresentedNumericalSemigroup(s); true
‣ IsGeneric( S ) | ( property ) |
‣ IsGenericNumericalSemigroup( S ) | ( property ) |
S is a numerical semigroup.
The output is true if S has a generic presentation, that is, in every minimal relation all generators occur. These semigroups are uniquely presented (see [BGSG11]).
gap> s:=NumericalSemigroup(3,5,7);; gap> IsGenericNumericalSemigroup(s); true
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