Microinvariants
Definition
Let \(S\) be a numerical semigroup with multiplicity \(m\), and let \(B(S)\) be the blowup. It is defined the set of microinvariants of \(S\), denoted by \(micr(S)\), as
\[ micr(S) = \left \{ \frac{w - w'}{m} ~ | ~ w \in Ap(S, m), ~ w' \in Ap(B(S), m)\right \}, \]
where \(Ap(S, m)\) denotes the Apéry set of \(S\) in \(m\) and both are ordered by module, that is,
\[ Ap(S, m) = \{w(0) = 0, \ldots, w(m-1)\}, ~~ Ap(B(S), m) = \{w'(0) = 0, \ldots, w'(m-1)\}, \]
where \(w(i)\) (resp. \(w'(i)\)) is the least element in \(S\) (resp. \(B(S)\)) such that \(w(i) \equiv i ~ (mod ~ m)\) (resp. \(w'(i) \equiv i ~ (mod ~ m)\)).
It can be proven that \(micr(S) \subseteq \mathbb{N}\) for any numerical semigroup.
Examples
\(\circ\) Let \(S = \langle 5, 7, 11 \rangle\) and \(M = S \setminus \{0\}\) its maximal ideal. The blowup of \(S\) is defined as \(B(S) = r(M)*M - m(M)r(M)\), where \(r(M)\) is the reduction number of \(M\), \(m(M)\) is the multiplicity ideal of \(M\). With few computations, it is obtained that \(r(M) = 2\) and \(B(S) = \{0, 2, 4, \rightarrow\}\). Then, taking into account that \(m(S) = 5\),
\[ Ap(S, 5) := \{w \in S ~ | ~ w - 5 \not \in S \} = \{0, 11, 7, 18, 14\}, ~~ \text{and} ~~ Ap(B(S), 5) = \{w \in B(S) ~ | ~ w - S \not \in B(S)\} = \{0, 6, 2, 8, 4\}. \]
Therefore, the set of microinvariants of \(S\) is
\[ micr(S) = \{0, 1, 1, 2, 2\}. \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 12, 21, 30, 39, 56, 64 \rangle\), in GAP:
gap> S := NumericalSemigroup(12, 21, 30, 39, 56, 64);
<Numerical semigroup with 6 generators>Given a numerical semigroup \(S\), the functions MicroInvariants and MicroInvariantsOfNumericalSemigroup return the set of microinvariants of \(S\).
gap> MicroInvariants(S);
[ 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2 ]With functions Multiplicity, AperyList and BlowUp can be calculated the multiplicity, the Apéry set and the blowup of a given numerical semigroup, respectively.
gap> m := Multiplicity(S);
12
gap> Ap := AperyList(S,m);
[ 0, 85, 86, 39, 64, 77, 30, 103, 56, 21, 94, 95 ]
gap> ApB := AperyList(BlowUp(S), m);
[ 0, 61, 62, 27, 52, 53, 18, 79, 44, 9, 70, 71 ]
gap> MicroInvariants(S) = (Ap - ApB)/m;
trueReferences
https://gap-packages.github.io/ numericalsgps.