GAP (NumericalSgps) - Chapter 8: Numerical semigroups with maximal embedding dimension

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### 8. Numerical semigroups with maximal embedding dimension

#### 8.1 Numerical semigroups with maximal embedding dimension

If S is a numerical semigroup and m is its multiplicity (the least positive integer belonging to it), then the embedding dimension e of S (the cardinality of the minimal system of generators of S) is less than or equal to m. We say that S has maximal embedding dimension (MED for short) when e=m. The intersection of two numerical semigroups with the same multiplicity and maximal embedding dimension is again of maximal embedding dimension. Thus we define the MED closure of a non-empty subset of positive integers M=m < m_1 < cdots < m_n <cdots with gcd(M)=1 as the intersection of all MED numerical semigroups with multiplicity m.

Given a MED numerical semigroup S, we say that M=m_1 < cdots< m_k is a MED system of generators if the MED closure of M is S. Moreover, M is a minimal MED generating system for S provided that every proper subset of M is not a MED system of generators of S. Minimal MED generating systems are unique, and in general are smaller that the classical minimal generating systems (see [JB03]).

##### 8.1-1 IsMEDNumericalSemigroup

`> IsMEDNumericalSemigroup`( S ) | ( function )

S is a numerical semigroup.

Returns true if S is a MED numerical semigroup and false otherwise.

    gap> IsMEDNumericalSemigroup(NumericalSemigroup(3,5,7));
    true
    gap> IsMEDNumericalSemigroup(NumericalSemigroup(3,5));
    false

##### 8.1-2 MEDNumericalSemigroupClosure

`> MEDNumericalSemigroupClosure`( S ) | ( function )

S is a numerical semigroup.

Returns the MED closure of S.

    gap> MEDNumericalSemigroupClosure(NumericalSemigroup(3,5));
    <Numerical semigroup>
    gap> MinimalGeneratingSystemOfNumericalSemigroup(last);
    [ 3, 5, 7 ]

##### 8.1-3 MinimalMEDGeneratingSystemOfMEDNumericalSemigroup

`> MinimalMEDGeneratingSystemOfMEDNumericalSemigroup`( S ) | ( function )

S is a MED numerical semigroup.

Returns the minimal MED generating system of S.

    gap> MinimalMEDGeneratingSystemOfMEDNumericalSemigroup(
    > NumericalSemigroup(3,5,7));
    [ 3, 5 ]

#### 8.2 Numerical semigroups with the Arf property and Arf closures

Numerical semigroups with the Arf property are a special kind of numerical semigroups with maximal embedding dimension. A numerical semigroup S is Arf if for every x,y,z in S with x>= y>= z, one has that x+y-zin S.

The intersection of two Arf numerical semigroups is again Arf, and thus we can consider the Arf closure of a set of nonnegative integers with greatest common divisor equal to one. Analogously as with MED numerical semigroups, we define Arf systems of generators and minimal Arf generating system for an Arf numerical semigroup. These are also unique(see [JB04]).

##### 8.2-1 IsArfNumericalSemigroup

`> IsArfNumericalSemigroup`( S ) | ( function )

S is a numerical semigroup.

Returns true if S is an Arf numerical semigroup and false otherwise.

    gap>  IsArfNumericalSemigroup(NumericalSemigroup(3,5,7));
    true
    gap>  IsArfNumericalSemigroup(NumericalSemigroup(3,7,11));
    false
    gap> IsMEDNumericalSemigroup(NumericalSemigroup(3,7,11));
    true

##### 8.2-2 ArfNumericalSemigroupClosure

`> ArfNumericalSemigroupClosure`( S ) | ( function )

S is a numerical semigroup.

Returns the Arf closure of S.

    gap> ArfNumericalSemigroupClosure(NumericalSemigroup(3,7,11));
    <Numerical semigroup>
    gap> MinimalGeneratingSystemOfNumericalSemigroup(last);
    [ 3, 7, 8 ]

##### 8.2-3 MinimalArfGeneratingSystemOfArfNumericalSemigroup

`> MinimalArfGeneratingSystemOfArfNumericalSemigroup`( S ) | ( function )

S is an Arf numerical semigroup.

Returns the minimal MED generating system of S.

    gap> MinimalArfGeneratingSystemOfArfNumericalSemigroup(
    > NumericalSemigroup(3,7,8));
    [ 3, 7 ]

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