Gaps
A gap of a numerical semigroup S is a nonnegative integer not belonging to S. The fundamental gaps of S are those gaps that are maximal with respect to the partial order induced by division in {\mathbb N}.

The special gaps of a numerical semigroup S, are those fundamental gaps such that if they are added to the given numerical semigroup, then the resulting set is again a numerical semigroup.

NS
is a numerical semigroup. It returns the set of gaps of NS. gap> GapsOfNumericalSemigroup(NumericalSemigroup(3,5,7)); [ 1, 2, 4 ]
S
is a numerical semigroup. It returns the set of fundamental gaps of S. gap> S := NumericalSemigroup("modular", 5,53); <Modular numerical semigroup satisfying 5x mod 53 <= x > gap> FundamentalGapsOfNumericalSemigroup(S); [ 16, 17, 18, 19, 27, 28, 29, 30, 31, 40, 41, 42 ] gap> GapsOfNumericalSemigroup(S); [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 27, 28, 29, 30, 31, 40, 41, 42 ]
S
is a numerical semigroup. It returns the special gaps of S.

gap> S := NumericalSemigroup("modular", 5,53); <Modular numerical semigroup satisfying 5x mod 53 <= x > gap> SpecialGapsOfNumericalSemigroup(S); [ 40, 41, 42 ]