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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 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},...,{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);
[ [ [ 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× 3+1× 7=2× 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,...,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);;
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⋅ x-msg⋅ 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,...,m_n} is the set of minimal generators of S, then L ∈ C if n-∑_i∈ L m_i∈ S ([SW86]).

This function is a generalization of the graph associated to n.

gap> s:=NumericalSemigroup(3,5,7);;
[ [  ], [ 3 ], [ 3, 7 ], [ 5 ], [ 7 ] ]


#### 4.2 Uniquely Presented Numerical Semigroups

A numerical semigroup S is uniquely presented if for any two minimal presentations σ and τ and any (a,b)∈ σ, either (a,b)∈ τ or (b,a)∈ τ, 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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