The definitions
NS is a numerical semigroup. Returns the multiplicity of NS, which is the smallest positive integer belonging to NS. gap> S := NumericalSemigroup("modular", 7,53); <Modular numerical semigroup satisfying 7x mod 53 <= x > gap> MultiplicityOfNumericalSemigroup(S); 8
S is a numerical semigroup. GeneratorsOfNumericalSemigroup returns a set of generators of S, which may not be minimal.

GeneratorsOfNumericalSemigroupNC
returns the set of generators recorded in S!.generators, which may not be minimal.

MinimalGeneratingSystemOfNumericalSemigroup
returns the minimal set of generators of S.

gap> S := NumericalSemigroup("modular", 5,53); <Modular numerical semigroup satisfying 5x mod 53 <= x > gap> GeneratorsOfNumericalSemigroup(S); [ 11, 12, 13, 32, 53 ] gap> S := NumericalSemigroup(3, 5, 53); <Numerical semigroup with 3 generators> gap> GeneratorsOfNumericalSemigroup(S); [ 3, 5, 53 ] gap> MinimalGeneratingSystemOfNumericalSemigroup(S); [ 3, 5 ]
NS
is a numerical semigroup. It returns the list of small elements of NS. Of course, the time consumed to return a result may depend on the way the semigroup is given. gap> SmallElementsOfNumericalSemigroup(NumericalSemigroup(3,5,7)); [ 0, 3, 5 ]
S is a numerical semigroup and m is a positive element of S. Computes the Apéry list of S wrt m. It contains for every i\in \{0,\ldots,m-1\}, in the i+1th position, the smallest element in the semigroup congruent with i modulo m.

gap> S := NumericalSemigroup("modular", 5,53); <Modular numerical semigroup satisfying 5x mod 53 <= x > gap> AperyListOfNumericalSemigroupWRTElement(S,12); [ 0, 13, 26, 39, 52, 53, 54, 43, 32, 33, 22, 11 ]
ap is the Apéry list of a numerical semigroup. This function draws the graph (ap, E) where the edge u -> v is in E iff v - u is in ap. To use this function, Graphviz (http://www.graphviz.org) should be installed and also Evince (http://www.gnome.org/projects/evince/) or ggv (http://directory.fsf.org/ggv.html).
ap is the Apéry list of a numerical semigroup. This function returns the adjacency list of the graph (ap, E) where the edge u -> v is in E iff v - u is in ap. The 0 is ignored.

gap> s:=NumericalSemigroup(3,7); <Numerical semigroup with 2 generators> gap> AperyListOfNumericalSemigroupWRTElement(s,10); [ 0, 21, 12, 3, 14, 15, 6, 7, 18, 9 ] gap> AperyListOfNumericalSemigroupAsGraph(last); [ ,, [ 3, 6, 9, 12, 15, 18, 21 ],,, [ 6, 9, 12, 15, 18, 21 ], [ 7, 14, 21 ],, [ 9, 12, 15, 18, 21 ],,, [ 12, 15, 18, 21 ],, [ 14, 21 ], [ 15, 18, 21 ],,, [ 18, 21 ],,, [ 21 ] ]