Pseudo-symmetric numerical semigroup
Definition
A numerical semigroup \(S\) is said to be pseudo-symmetric if it is irreducible and its Frobenius number is even.
It can be proven that \(S\) is pseudo-symmetric if, and only if,
\[ g(S) = \frac{F(S) + 2}{2}, \] where \(g(S)\) denotes the genus of \(S\).
Examples
\(\circ\) Let \(S = \langle 4, 10, 17 \rangle\). This numerical semigroup is irreducible and if we compute the first elements of \(S\), we obtain that \(S = \{0, 4, 8, 10, 12, 14, 16, 17, 18, 20, 21, 22, 24, \rightarrow \}\), so \(F(S) = 23\) and \(S\) is not a pseudo-symmetric numerical semigroup. The irreducible numerical semigroups with odd Frobenius number are called symmetric numerical semigroups.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 4, 5, 7 \rangle\), in GAP:
gap> S := NumericalSemigroup(4, 5, 7);
<Numerical semigroup with 3 generators>The functions IsPseudoSymmetric and IsPseudoSymmetricNumericalSemigroup return true or false depending on whether the numerical semigroup is pseudo-symmetric or not.
gap> IsPseudoSymmetric(S);
true
gap> IsPseudoSymmetric(S) = IsPseudoSymmetricNumericalSemigroup(S);
trueWe can also prove it by definition. First, we have to known if \(S\) is irreducible, the function IsIrreducible returns true or false depending on whether the numerical semigroup is irreducible or not.
gap> IsIrreducible(S);
trueNow, the function FrobeniusNumber computes the Frobenius number of \(S\).
gap> FrobeniusNumber(S);
6Since \(F(S)\) is even, it is concluded that \(S\) is pseudo-symmetric.
References
https://gap-packages.github.io/ numericalsgps.