Proportionally modular numerical semigroup
Definition
Let \(S\) be a numerical semigroup. It is said that \(S\) is proportionally modular if there exist \(a, b, c\) positive integers such that \(S = S(a,b,c)\), where \(S(a,b,c)\) denotes the set of solutions of the proportionally modular Diophantine inequality \(ax ~ (mod ~ b) \le cx\). If \(S = S(a,b,1)\) for some \(a, b\) positive integers, then it is said that \(S\) is a modular numerical semigroup.
It can be proven that for any \(a,b,c\) positive integers, \(S(a,b,c)\) is a numerical semigroup.
Examples
\(\circ\) Let \(S = \langle 3, 7, 8 \rangle = \{0, 3, 6, \rightarrow\}\). Let us prove that \(S = S(12, 32, 3)\). We divide in five cases.
If \(x = 0\), clearly \(x \in S(12,32,3)\).
If \(x \in \{1, 2\}\), then \(12x ~ (mod ~ 32) = 12x > 3x\) and \(x \not \in S(12,32,3)\).
If \(x \in \{4,5\}\), then \(x \not \in S(12,32,3)\).
If \(x \in \{3, 6, 7, 8, 9, 10\}\), checking the inequality on each number, we have that \(x \in S(12,32,3)\) for all \(x \in \{3, 6, 7, 8, 9, 10\}\).
If \(x \ge 11\), \(12x ~ (mod ~ 32) \le 31 < 3x\) and \(x \in S(12,32,3)\).
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(a, b, c\) positive integers. Te function ProportionallyModularNumericalSemigroup returns the numerical semigroup generated by the proportionally modular Diophantine inequality \(ax ~ (mod ~ b) \le cx\).
gap> a := 14;
14
gap> b := 57;
57
gap> c := 6;
6
gap> S := ProportionallyModularNumericalSemigroup(a,b,c);
<Proportionally modular numerical semigroup satisfying 14x mod 57 <= 6x >The function SmallElements returns a list with the left elements and the conductor of the numerical semigroup.
gap> SmallElements(S);
[ 0, 5, 6, 7, 9 ]\(\diamond\) Given a numerical semigroup \(S\), the function IsProportionallyModularNumericalSemigroup returns true or false depending on whether \(S\) is a proportionally modular Diophantine numerical semigroup or not.
gap> S := NumericalSemigroup(5, 6, 7, 9);
<Numerical semigroup with 4 generators>
gap> IsProportionallyModularNumericalSemigroup(S);
trueReferences
https://gap-packages.github.io/ numericalsgps.