Generating Numerical Semigroups We recall some definitions from Chapter .

A numerical semigroup is a subset of the set {\mathbb N} of nonnegative integers that is closed under addition, contains 0 and whose complement in {\mathbb N} is finite.

We refer to the elements in a numerical semigroup that are less than or equal to the conductor as small elements of the semigroup.

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}.

Given a numerical semigroup S and a nonzero element s in it, one can consider for every integer i ranging from 0 to s-1, the smallest element in S congruent with i modulo s, say w(i) (this element exists since the complement of S in {\mathbb N} is finite). Clearly w(0)=0. The set {\rm Ap}(S,s)=\{ w(0),w(1),\ldots, w(s-1)\} is called the Apéry set of S with respect to s.

Let a,b,c,d be positive integers such that a/b < c/d, and let I=[a/b,c/d]. Then the set {\rm S}(I)={\mathbb N}\cap \bigcup_{n\geq 0} n I is a numerical semigroup. This class of numerical semigroups coincides with that of sets of solutions to equations of the form A x \bmod\ B \leq C x with A,B,C positive integers. A numerical semigroup in this class is said to be proportionally modular. If C = 1, then it is said to be modular.

There are different ways to specify a numerical semigroup S, namely, by its generators; by its gaps, its fundamental or special gaps by its Apéry set, just to name some. In this section we describe functions that may be used to specify, in one of these ways, a numerical semigroup in &GAP;. List is a list of nonnegative integers with greatest common divisor equal to one. These integers may be given as a list or by a sequence of individual elements. The output is the numerical semigroup spanned by List.

String does not need to be present. When it is present, it must be "generators".
gap> s1 := NumericalSemigroup(3,5,7); <Numerical semigroup with 3 generators> gap> s2 := NumericalSemigroup([3,5,7]); <Numerical semigroup with 3 generators> gap> s3 := NumericalSemigroupByGenerators(3,5,7); <Numerical semigroup with 3 generators> gap> s4 := NumericalSemigroupByGenerators([3,5,7]); <Numerical semigroup with 3 generators> gap> s5 := NumericalSemigroup("generators",3,5,7); <Numerical semigroup with 3 generators> gap> s6 := NumericalSemigroup("generators",[3,5,7]); <Numerical semigroup with 3 generators> gap> s1=s2;s2=s3;s3=s4;s4=s5;s5=s6; true true true true true
A periodic subadditive function with period m is given through the list of images of the integers from 1 to m, . The image of m has to be 0. The output is the numerical semigroup determined by this subadditive function.

In the second form, String must be "subadditive".
gap> s := NumericalSemigroupBySubAdditiveFunction([5,4,2,0]); <Numerical semigroup> gap> t := NumericalSemigroup("subadditive",[5,4,2,0]);; gap> s=t; true
List is an Apéry list. The output is the numerical semigroup whose Apéry set with respect to the length of given list is List.

In the second form, String must be "apery".
gap> s:=NumericalSemigroup(3,11);; gap> ap := AperyListOfNumericalSemigroupWRTElement(s,20); [ 0, 21, 22, 3, 24, 25, 6, 27, 28, 9, 30, 11, 12, 33, 14, 15, 36, 17, 18, 39 ] gap> t:=NumericalSemigroupByAperyList(ap);; gap> r := NumericalSemigroup("apery",ap);; gap> s=t;t=r; true true
List is the set of small elements of a numerical semigroup, that is, the set of all elements not greater than the conductor. The output is the numerical semigroup with this set of small elements. When no such semigroup exists, an error is returned.

In the second form, String must be "elements".
gap> s:=NumericalSemigroup(3,11);; gap> se := SmallElements(s); [ 0, 3, 6, 9, 11, 12, 14, 15, 17, 18, 20 ] gap> t := NumericalSemigroupBySmallElements(se);; gap> r := NumericalSemigroup("elements",se);; gap> s=t;t=r; true true gap> e := [ 0, 3, 6, 9, 11, 14, 15, 17, 18, 20 ]; [ 0, 3, 6, 9, 11, 14, 15, 17, 18, 20 ] gap> NumericalSemigroupBySmallElements(e); Error, The argument does not represent a numerical semigroup called from <function "NumericalSemigroupBySmallElements">( <arguments> ) called from read-eval loop at line 35 of *stdin* you can 'quit;' to quit to outer loop, or you can 'return;' to continue brk>
List is the set of gaps of a numerical semigroup. The output is the numerical semigroup with this set of gaps. When no semigroup exists with the given set as set of gaps, an error is returned.

In the second form, String must be "gaps".
gap> g := [ 1, 2, 4, 5, 7, 8, 10, 13, 16 ];; gap> s := NumericalSemigroupByGaps(g);; gap> t := NumericalSemigroup("gaps",g);; gap> s=t; true gap> h := [ 1, 2, 5, 7, 8, 10, 13, 16 ];; gap> NumericalSemigroupByGaps(h); Error, The argument does not represent the gaps of a numerical semigroup called from <function "NumericalSemigroupByGaps">( <arguments> ) called from read-eval loop at line 34 of *stdin* you can 'quit;' to quit to outer loop, or you can 'return;' to continue brk>
List is the set of fundamental gaps of a numerical semigroup, . The output is the numerical semigroup determined by these gaps. When the given set contains elements (which will be gaps) that are not fundamental gaps, they are silently removed.

In the second form, String must be "fundamentalgaps".
gap> fg := [ 11, 14, 17, 20, 23, 26, 29, 32, 35 ];; gap> NumericalSemigroupByFundamentalGaps(fg); <Numerical semigroup> gap> NumericalSemigroup("fundamentalgaps",fg); <Numerical semigroup> gap> last=last2; true gap> gg := [ 11, 17, 20, 22, 23, 26, 29, 32, 35 ];; #22 is not fundamental gap> NumericalSemigroup("fundamentalgaps",fg); <Numerical semigroup>
Given three nonnegative integers a, b and c, with a,c>0 and \gcd(b,c)=1, this function returns the least (with respect to set order inclusion) numerical semigroup containing c and closed under the map x\mapsto ax+b. The procedure is explained in .

In the second form, String must be "affinemap".
gap> s:=NumericalSemigroupByAffineMap(3,1,3); <Numerical semigroup with 3 generators> gap> SmallElements(s); [ 0, 3, 6, 9, 10, 12, 13, 15, 16, 18 ] gap> t:=NumericalSemigroup("affinemap",3,1,3);; gap> s=t; true
Given two positive integers a and b, this function returns a modular numerical semigroup satisfying ax \bmod\ b \le x, .

In the second form, String must be "modular".
gap> ModularNumericalSemigroup(3,7); <Modular numerical semigroup satisfying 3x mod 7 <= x > gap> NumericalSemigroup("modular",3,7); <Modular numerical semigroup satisfying 3x mod 7 <= x >
Given three positive integers a, b and c, this function returns a proportionally modular numerical semigroup satisfying ax\bmod\ b \le cx, .

In the second form, String must be "propmodular".
gap> ProportionallyModularNumericalSemigroup(3,7,12); <Proportionally modular numerical semigroup satisfying 3x mod 7 <= 12x > gap> NumericalSemigroup("propmodular",3,7,12); <Proportionally modular numerical semigroup satisfying 3x mod 7 <= 12x >
When c=1, the semigroup is seen as a modular numerical semigroup.
gap> NumericalSemigroup("propmodular",67,98,1); <Modular numerical semigroup satisfying 67x mod 98 <= x >

Numerical semigroups generated by an interval of positive integers are known to be proportionally modular, and thus they are treated as such, since membership and other problems can be solved efficiently for these semigroups.

The input is a list of rational numbers defining a closed interval. The output is the semigroup of numerators of all rational numbers in this interval, .

String does not need to be present. When it is present, it must be "interval".
gap> NumericalSemigroupByInterval(7/5,5/3); <Proportionally modular numerical semigroup satisfying 25x mod 35 <= 4x > gap> NumericalSemigroup("interval",[7/5,5/3]); <Proportionally modular numerical semigroup satisfying 25x mod 35 <= 4x > gap> SmallElements(last); [ 0, 3, 5 ]
The input is a list of rational numbers defining an open interval. The output is the semigroup of numerators of all rational numbers in this interval, .

String does not need to be present. When it is present, it must be "openinterval".
gap> NumericalSemigroupByOpenInterval(7/5,5/3); <Numerical semigroup> gap> NumericalSemigroup("openinterval",[7/5,5/3]); <Numerical semigroup> gap> SmallElements(last); [ 0, 3, 6, 8 ]