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

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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+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 [R96] (see also [RG99]).

    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 1times 3+1times 7=2times 5, and so on.

##### 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_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 S, as explained in [R96].

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

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