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### 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 FortenTruncatedNCForNumericalSemigroups
 `‣ FortenTruncatedNCForNumericalSemigroups`( L ) ( function )

L 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 `fail` if no solution exists.

```gap> FortenTruncatedNCForNumericalSemigroups([ -57, 3 ]);
[ 1, 19 ]
gap> FortenTruncatedNCForNumericalSemigroups([ -57, 33 ]);
fail
gap> FortenTruncatedNCForNumericalSemigroups([ -57, 19 ]);
[ 1, 3 ]
```

##### 4.1-2 MinimalPresentationOfNumericalSemigroup
 `‣ MinimalPresentationOfNumericalSemigroup`( S ) ( function )

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},...,{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+⋯+a_i_nm_n= b_i_1m_1+ ⋯ +b_i_nm_n.

Any other relation among the minimal generators of the semigroup can be deduced from the ones given in the output.

```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 ] ] ]

```

The first element in the list means that 1× 3+1× 7=2× 5, and the others have similar meanings.

##### 4.1-3 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,...,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∈ S}, and the second component is the set of edges of this graph, that is, { {m_i,m_j} | n-(m_i+m_j)∈ 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);
<Numerical semigroup with 3 generators>
gap> GraphAssociatedToElementInNumericalSemigroup(10,s);
[ [ 3, 5, 7 ], [ [ 3, 7 ] ] ]

```

##### 4.1-4 BettiElementsOfNumericalSemigroup
 `‣ BettiElementsOfNumericalSemigroup`( S ) ( function )

S is a numerical semigroup.

The output is the set of elements in S whose associated graph is nonconnected [GO10].

```gap> s:=NumericalSemigroup(3,5,7);
<Numerical semigroup with 3 generators>
gap> BettiElementsOfNumericalSemigroup(s);
[ 10, 12, 14 ]

```
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