Saturated numerical semigroups Saturated numerical semigroups with the Arf property are a special kind of numerical semigroups with maximal embedding dimension. A numerical semigroup S is saturated if the following condition holds: s, s_1 , \ldots , s_r in S are such that s_i \leq s for all i in \{1, \ldots , r\} and z_1 , \ldots , z_r in \mathbb Z are such that z_1 s_1 + \cdots + z_r s_r\geq 0, then s + z_1 s_1 + \cdots + z_r s_r in S.

The intersection of two saturated numerical semigroups is again saturated, and thus we can consider the saturated closure of a set of nonnegative integers with greatest common divisor equal to one (see ). S is a numerical semigroup.

Returns true if S is a saturated numerical semigroup and false otherwise.
gap> IsSaturatedNumericalSemigroup(NumericalSemigroup(4,6,9,11)); true gap> IsSaturatedNumericalSemigroup(NumericalSemigroup(8, 9, 12, 13, 15, 19 )); false
S is a numerical semigroup.

Returns the saturated closure of S.
gap> SaturatedNumericalSemigroupClosure(NumericalSemigroup(8, 9, 12, 13, 15)); <Numerical semigroup> gap> MinimalGeneratingSystemOfNumericalSemigroup(last); [ 8 .. 15 ]
f is an integer greater than or equal to -1. The output is the set of all Saturated numerical semigroups with Frobenius number f.
gap> SaturatedNumericalSemigroupsWithFrobeniusNumber(10); [ <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup> ] gap> List(last,MinimalGeneratingSystemOfNumericalSemigroup); [ [ 3, 11, 13 ], [ 4, 11, 13, 14 ], [ 6, 9, 11, 13, 14, 16 ], [ 6, 11, 13, 14, 15, 16 ], [ 7, 11, 12, 13, 15, 16, 17 ], [ 8, 11, 12, 13, 14, 15, 17, 18 ], [ 9, 11, 12, 13, 14, 15, 16, 17, 19 ], [ 11 .. 21 ] ]