Irreducible numerical semigroup
Definition
A numerical semigroup \(S\) is said to be irreducible if it cannot be expressed as the intersection of two numerical semigroups properly containing it. The set of irreducible numerical semigroups is divided in two big families: symmetric numerical semigroups and pseudo-symmetric numerical semigroups, depending on whether \(F(S)\) is odd (symmetric case) or even (pseudo-symmetric case).
It is well known that the intersection of two numerical semigroups is also a numerical semigroup.
Examples
\(\circ\) Let \(S = \langle 5, 7, 9 \rangle\) and the numerical semigroups \(S_1 = \langle 5, 7, 8, 9 \rangle\) and \(S_2 = \langle 5, 7, 9, 11 \rangle\). Clearly, \(S \subsetneq S_1\), \(S \subsetneq S_2\) and \(S = S_1 \cap S_2\), since \(S_1 = S \cup \{8, 13\}\) and \(S_2 = S \cup \{11\}\).
\(\circ\) Let \(S = \langle 2, 7 \rangle = \{0, 2, 4, 6, \rightarrow \}\), let’s prove that \(S\) is irreducible. Let us consider the set of oversemigroups of \(S\),
\[ \mathcal{O}(S) = \{\mathbb{N}, \{0, 2, \rightarrow\}, \{0, 2, 4, \rightarrow\}, S\} = \{\mathbb{N}, S \cup \{3, 5\}, S \cup \{5\}, S\} = \{\mathbb{N}, T_1, T_2, S\}. \]
If \(S\) is not irreducible, the only possibility is \(S = T_1 \cap T_2\), but \(T_1 \cap T_2 = S \cup \{5\} \ne S\), therefore \(S\) is irreducible.
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 4, 10, 17 \rangle\), in GAP:
gap> S := NumericalSemigroup(4, 10, 17);
<Numerical semigroup with 3 generators>We can check if \(S\) is irreducible numerical semigroup with the function IsIrreducible.
gap> IsIrreducible(S);
true\(\diamond\) The function IrreducibleNumericalSemigroupsWithFrobeniusNumber returns the set of all irreducible numerical semigroups with fixed Frobenius number \(f\).
gap> Length(IrreducibleNumericalSemigroupsWithFrobeniusNumber(16));
7We can also fix the multiplicity with the function IrreducibleNumericalSemigroupsWithFrobeniusNumberAndMultiplicity.
gap> Length(IrreducibleNumericalSemigroupsWithFrobeniusNumberAndMultiplicity(16, 3));
1\(\diamond\) Given a numerical semigroup \(S\), the function DecomposeIntoIrreducibles returns a list of numerical semigroups \([S_1, \ldots, S_k]\) such that \(S = S_1 \cap \ldots \cap S_k\).
gap> S := NumericalSemigroup(13, 17, 21, 40, 58);
<Numerical semigroup with 5 generators>
gap> D := DecomposeIntoIrreducibles(S);
[ <Numerical semigroup with 12 generators>,
<Numerical semigroup with 12 generators>,
<Numerical semigroup with 8 generators>,
<Numerical semigroup with 8 generators> ]
gap> List(D, T -> MinimalGenerators(T));
[ [ 13, 17, 21, 24, 25, 28, 29, 32, 33, 35, 36, 40 ],
[ 13, 17, 21, 25, 28, 29, 32, 33, 36, 37, 40, 44 ],
[ 13, 17, 21, 33, 35, 37, 40, 44 ],
[ 13, 17, 21, 35, 36, 40, 44, 45 ] ]References
https://gap-packages.github.io/ numericalsgps.