<Section> <Heading> Numerical semigroups with maximal embedding dimension </Heading> If <M>S</M> is a numerical semigroup and <M>m</M> is its multiplicity (the least positive integer belonging to it), then the embedding dimension <M>e</M> of <M>S</M> (the cardinality of the minimal system of generators of <M>S</M>) is less than or equal to <M>m</M>. We say that <M>S</M> has maximal embedding dimension (MED for short) when <M>e=m</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 < m_1 < \cdots < m_n <\cdots\}</M> with <M>\gcd(M)=1</M> as the intersection of all MED numerical semigroups with multiplicity <M>m</M>. <P/> Given a MED numerical semigroup <M>S</M>, we say that <M>M=\{m_1 < \cdots< m_k\}</M> is a MED system of generators if the MED closure of <M>M</M> is <M>S</M>. Moreover, <M>M</M> is a minimal MED generating system for <M>S</M> provided that every proper subset of <M>M</M> is not a MED system of generators of <M>S</M>. Minimal MED generating systems are unique, and in general are smaller that the classical minimal generating systems (see <Cite Key="RGGB03"></Cite>). <ManSection> <Func Arg="S" Name="IsMEDNumericalSemigroup"></Func> <Description> <A>S</A> is a numerical semigroup. <P/> Returns true if <A>S</A> is a MED numerical semigroup and false otherwise. <Example><![CDATA[ gap> IsMEDNumericalSemigroup(NumericalSemigroup(3,5,7)); true gap> IsMEDNumericalSemigroup(NumericalSemigroup(3,5)); false ]]> </Example> </Description> </ManSection> <ManSection> <Func Arg="S" Name="MEDNumericalSemigroupClosure"></Func> <Description> <A>S</A> is a numerical semigroup. <P/> Returns the MED closure of <A>S</A>. <Example><![CDATA[ gap> MEDNumericalSemigroupClosure(NumericalSemigroup(3,5)); <Numerical semigroup> gap> MinimalGeneratingSystemOfNumericalSemigroup(last); [ 3, 5, 7 ] ]]> </Example> </Description> </ManSection> <ManSection> <Func Arg="S" Name="MinimalMEDGeneratingSystemOfMEDNumericalSemigroup"></Func> <Description> <A>S</A> is a MED numerical semigroup. <P/> Returns the minimal MED generating system of <A>S</A>. <Example><![CDATA[ gap> MinimalMEDGeneratingSystemOfMEDNumericalSemigroup( > NumericalSemigroup(3,5,7)); [ 3, 5 ] ]]> </Example> </Description> </ManSection> </Section>