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 ).

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
S is a numerical semigroup.

Returns the MED closure of S. gap> MEDNumericalSemigroupClosure(NumericalSemigroup(3,5)); <Numerical semigroup> gap> MinimalGeneratingSystemOfNumericalSemigroup(last); [ 3, 5, 7 ]
S is a MED numerical semigroup.

Returns the minimal MED generating system of S. gap> MinimalMEDGeneratingSystemOfMEDNumericalSemigroup( > NumericalSemigroup(3,5,7)); [ 3, 5 ]