GAP (NumericalSgps) - Chapter 4: Presentations of Numerical Semigroups

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4 Presentations of Numerical Semigroups
4.1 Presentations of Numerical Semigroups
4.1-1 MinimalPresentationOfNumericalSemigroup
4.1-2 GraphAssociatedToElementInNumericalSemigroup
4.1-3 BettiElementsOfNumericalSemigroup
4.1-4 PrimitiveElementsOfNumericalSemigroup
4.1-5 ShadedSetOfElementInNumericalSemigroup
4.2 Uniquely Presented Numerical Semigroups
4.2-1 IsUniquelyPresented
4.2-2 IsGeneric

### 4 Presentations of Numerical Semigroups

In this chapter we explain how to compute a minimal presentation of a numerical semigroup. There are three functions involved in this process.

#### 4.1 Presentations of Numerical Semigroups

##### 4.1-1 MinimalPresentationOfNumericalSemigroup

`‣ 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.

##### 4.1-2 GraphAssociatedToElementInNumericalSemigroup

`‣ 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 ] ] ]

##### 4.1-3 BettiElementsOfNumericalSemigroup

`‣ 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 ]

##### 4.1-4 PrimitiveElementsOfNumericalSemigroup

`‣ 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 ]

##### 4.1-5 ShadedSetOfElementInNumericalSemigroup

`‣ 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 ] ]

#### 4.2 Uniquely Presented Numerical Semigroups

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).

##### 4.2-1 IsUniquelyPresented

`‣ 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

##### 4.2-2 IsGeneric

`‣ 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]).

This filter implies `IsUniquelyPresentedNumericalSemigroup` (4.2-1).

    gap> s:=NumericalSemigroup(3,5,7);;
    gap> IsGenericNumericalSemigroup(s);
    true

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