Acute numerical semigroup
Definition
Let \(S\) be a numerical semigroup, \(G(S)\) the set of gaps of \(S\) and \(C(S)\) the conductor of \(S\). Let us assume that \(S\) has positive dominant \(d(S)\). For \(s \in S\), it is defined \(g(S)\) as the cardinality of \(\{x \in G(S) ~ | ~ x < s\}\). Set
\[ c' = \min \{s \in S ~ | ~ g(s) = g(d) \}, \hspace{0.2cm} d' = \max \{s \in S ~ | ~ s < c' \}. \]
It is said that \(S\) is acute if \(C(S) - d(S) \le c' - d'\).
Examples
\(\circ\) Let \(S = \langle 4, 5, 10 \rangle = \{0, 4, 5, 8, 9, 10, 12, \rightarrow\}\). The conductor is \(C(S) = 12\) and the dominant is \(d(S) = 10\). On the other hand,
\[ c' = \min \{s \in S ~ | ~ g(s) = g(10) \} = 8, \hspace{0.2cm} d' = \max \{s \in S ~ | ~ s < 8 \} = 5. \] Finally, \(2 = d(S) - C(S) \le c' - d' = 3\) and \(S\) is acute.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 11, 12, 25, 28, 41 \rangle\), in GAP:
gap> S := NumericalSemigroup(11, 12, 25, 28, 41);
<Numerical semigroup with 5 generators>Given a numerical semigroup \(S\), the functions IsAcute and IsAcuteNumericalSemigroup return true or false depending on whether a numerical semigroup is acute or not.
gap> IsAcute(S);
true
gap> IsAcuteNumericalSemigroup(S);
trueReferences
https://gap-packages.github.io/ numericalsgps.