Proportion of a proportionally modular Diophantine inequality
Definition
Let \(a, b\) and \(c\) be positive integers and let \(ax ~ (mod ~ b) \le cx\) be a proportionally modular Diophantine inequality. It is defined the proportion of the proportionally modular Diophantine inequality as the integer \(c\). The integer \(a\) is defined as the factor and \(b\) is defined as the modulus of the proportionally modular Diophantine inequality.
If \(c = 1\), it is said that the Diophantine inequality \(ax ~ (mod ~ b) \le x\) is a modular Diophantine inequality.
Examples
\(\circ\) Let \(14x ~ (mod ~ 52) \le 7x\). Then, the factor of the proportionally modular Diophantine inequality is \(a = 14\), the modulus is \(b = 52\) and the proportion is \(c = 7\). It can be proven that the set of solutions is \(S(14, 52, 7) = \{0, 4, \rightarrow\}\).
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 ]Therefore, the set of solutions of \(14x ~ (mod ~ 57) \le 6x\) is \(S = \{0, 5, 6, 7, 9, \rightarrow\}\).
\(\diamond\) Let \(a, b\) positive integers. The function ModularNumericalSemigroup returns the numerical semigroup generated by the modular Diophantine inequality \(ax ~ (mod ~ b) \le x\).
gap> a := 6;
6
gap> b := 19;
19
gap> S := ModularNumericalSemigroup(a,b);
<Modular numerical semigroup satisfying 6x mod 19 <= x >If we use the function SmallElements,
gap> SmallElements(S);
[ 0, 7, 10, 11, 13 ]Then, the set of solutions of \(6x ~ (mod ~ 19) \le x\) is \(S = \{0, 7, 10, 11, 13, \rightarrow \}\).
References
https://gap-packages.github.io/ numericalsgps.