Depth and remainder
Definition
Let \(S\) be a numerical semigroup with conductor \(C(S)\) and multiplicity \(m(S)\). It is defined the depth of \(S\), denoted by \(q(S)\), as the value \(q(S) = \lceil C(S)/m(S) \rceil\). On the other hand, the remainder of \(S\), denoted by \(\rho(S)\), is defined as \(\rho(S) = q(S)m(S) - C(S)\).
Examples
\(\circ\) Let \(S = \langle 6,7,11 \rangle = \{0, 6, 7, 11, 12, 13, 14, 17, \rightarrow \}\). The conductor of \(S\) is \(C(S) = 17\) and the multiplicity is \(m(S) = 6\). Therefore, the depth of \(S\) is \(q(S) = \left \lceil \frac{17}{6} \right \rceil = 3\) and the remainder is \(\rho(S) = 1\).
Examples with GAP
Nowadays, there are no functions in NumericalSgps related to depth or remainder of a numerical semigroup. However, given a numerical semigroup, the following functions return the depth and the remainder of \(S\), respectively.
gap> DepthOfNumericalSemigroup := function(S)
> if not IsNumericalSemigroup(S) then
> Error("The argument must be a Numerical Semigroup");
> fi;
>
> return CeilingOfRational(Conductor(S)/Multiplicity(S));
> end;
function( S ) ... endgap> RemainderOfNumericalSemigroup := function(S)
> if not IsNumericalSemigroup(S) then
> Error("The argument must be a Numerical Semigroup");
> fi;
>
> return DepthOfNumericalSemigroup(S)*Multiplicity(S) - Conductor(S);
> end;
function( S ) ... end\(\diamond\) Let \(S = \langle 27, 29, 41, 66, 94 \rangle\), in GAP:
gap> S := NumericalSemigroup(27, 29, 41, 66, 94);
<Numerical semigroup with 5 generators>Given a numerical semigroup, the functions Conductor and Multiplicity return the conductor and the multiplicity of \(S\), respectively.
gap> C := Conductor(S);
213
gap> m := Multiplicity(S);
27
gap> C/m;
71/9If we use the functions defined above,
gap> DepthOfNumericalSemigroup(S);
8
gap> RemainderOfNumericalSemigroup(S);
3Then, the depth of \(S\) is \(q(S) = 8\) and the remainder is \(\rho(S) = 3\).
References
https://gap-packages.github.io/ numericalsgps.