Half-line
Definition
Let \(S\) be a numerical semigroup and let \(C(S)\) be the conductor of \(S\). It is said that \(S\) is half-line, or ordinary, if \(S= \{0, C(S), \rightarrow\}\). Equivalently, \(S\) is half-line, or ordinary, if \(s \ge C(S)\) for all \(s \in S \setminus \{0\}\).
Examples
\(\circ\) Let \(S\) be a numerical semigroup. Let us prove that its dominant is \(d(S) = 0\) if, and only if, \(S\) is half-line.
\(\Rightarrow\) If \(d(S) = 0\), by definition,
\[ d(S) = \max \{s \in S ~ | ~ s < C(S) \} = 0. \]
Therefore, \(S = \{0, C(S), \rightarrow \}\).
\(\Leftarrow\) If \(S = \{0, C(S), \rightarrow\}\), the only element \(s \in S\) such that \(s < C(S)\) is \(s = 0\). Thus, \(d(S) = 0\).
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 5, 7, 9 \rangle\), in GAP:
gap> S := NumericalSemigroup(5, 7, 9);
<Numerical semigroup with 3 generators>Given a numerical semigroup \(S\), the function IsOrdinary returns true or false depending on whether a numerical semigroup is ordinary or not.
gap> IsOrdinary(S);
false\(\diamond\) Let \(S = \langle 6, 7, 8, 9, 10, 11 \rangle\), in GAP:
gap> S := NumericalSemigroup(6, 7, 8, 9, 10, 11);
<Numerical semigroup with 6 generators>In this case,
gap> IsOrdinary(S);
trueIt is concluded that \(S = \{0, 6, \rightarrow\}\).
References
https://gap-packages.github.io/ numericalsgps.