Geometrical illustration of numerical semigroups and of some of their invariants.

*(English)*Zbl 1325.20053Let \(H\) be a numerical semigroup, that is, a submonoid of the set of nonnegative integers \(\mathbb N\) with finite complement in \(\mathbb N\). Take \(p,q\in H\). We have then \(H_{pq}:=\langle p,q\rangle\subseteq H\), and consequently \(H\) is an oversemigroup of \(H_{pq}\).

Let \(c\) be the conductor of \(H_{pq}\), that is, the least nonnegative integer fulfilling \(c+\mathbb N\subseteq H_{pq}\). Every gap \(l\) in \(H_{pq}\) (positive integer not in \(H_{pq}\)) can be expressed uniquely as \(l=c-1-(ap+bq)\), for some \((a,b)\in\mathbb N^2\), defining in this way a bijection between \(\mathbb N\setminus H_{pq}\) and \(\Delta_0=\{(a,b)\in\mathbb N\mid aX+bY\leq c-1\}\).

Since \(H_{pq}\subseteq H\), we can obtain \(H\) from \(H_{pq}\) by filling some gaps. Also every gap of \(H\) is a gap of \(H_{pq}\), and thus corresponds to a point in \(\Delta_0\). The fact that \(s\in H\) implies that \(s+p,s+q\in H\), implies that the region corresponding to elements of \(H\) in \(\Delta_0\) is a union of rectangles. Consequently, the set of gaps of \(H\) is determined by the region of \(\mathbb N^2\) of elements between a path in \(\mathbb N^2\) (with the shape of a stair) and the line \(ax+by=c-1\).

Not all paths in \(\Delta_0\) correspond to a numerical semigroup, whence a path that corresponds to a numerical semigroup containing \(p\) and \(q\) is called admissible path. Hence counting paths allows counting numerical semigroups (this is not the first time that paths are used to count numerical semigroups [see M. Bras-Amorós and A. de Mier, Semigroup Forum 75, No. 3, 677-682 (2007; Zbl 1128.20046)]).

In this manuscript the authors count admissible paths for special values of \(p\) and \(q\). It is also shown how some invariants of the numerical semigroup associated to a path are related with the coordinates of the integer points in the path. Also some special families of numerical semigroups have special shapes of paths.

The authors are also interested in determining when a path is admissible, or when some particular families of paths are admissible. For these families formulas for the Frobenius number and genus are given.

The paper contains a good amount of pictures and examples to ease the understanding of the proofs of the theoretical results presented.

Let \(c\) be the conductor of \(H_{pq}\), that is, the least nonnegative integer fulfilling \(c+\mathbb N\subseteq H_{pq}\). Every gap \(l\) in \(H_{pq}\) (positive integer not in \(H_{pq}\)) can be expressed uniquely as \(l=c-1-(ap+bq)\), for some \((a,b)\in\mathbb N^2\), defining in this way a bijection between \(\mathbb N\setminus H_{pq}\) and \(\Delta_0=\{(a,b)\in\mathbb N\mid aX+bY\leq c-1\}\).

Since \(H_{pq}\subseteq H\), we can obtain \(H\) from \(H_{pq}\) by filling some gaps. Also every gap of \(H\) is a gap of \(H_{pq}\), and thus corresponds to a point in \(\Delta_0\). The fact that \(s\in H\) implies that \(s+p,s+q\in H\), implies that the region corresponding to elements of \(H\) in \(\Delta_0\) is a union of rectangles. Consequently, the set of gaps of \(H\) is determined by the region of \(\mathbb N^2\) of elements between a path in \(\mathbb N^2\) (with the shape of a stair) and the line \(ax+by=c-1\).

Not all paths in \(\Delta_0\) correspond to a numerical semigroup, whence a path that corresponds to a numerical semigroup containing \(p\) and \(q\) is called admissible path. Hence counting paths allows counting numerical semigroups (this is not the first time that paths are used to count numerical semigroups [see M. Bras-Amorós and A. de Mier, Semigroup Forum 75, No. 3, 677-682 (2007; Zbl 1128.20046)]).

In this manuscript the authors count admissible paths for special values of \(p\) and \(q\). It is also shown how some invariants of the numerical semigroup associated to a path are related with the coordinates of the integer points in the path. Also some special families of numerical semigroups have special shapes of paths.

The authors are also interested in determining when a path is admissible, or when some particular families of paths are admissible. For these families formulas for the Frobenius number and genus are given.

The paper contains a good amount of pictures and examples to ease the understanding of the proofs of the theoretical results presented.

Reviewer: Pedro A. García Sánchez (Granada)

##### Keywords:

numerical semigroups; lattice paths; Apéry sets; Catalan numbers; genus; Frobenius numbers; pseudo-Frobenius numbers; types; symmetric semigroups; pseudo-symmetric semigroups
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\textit{E. Kunz} and \textit{R. Waldi}, Semigroup Forum 89, No. 3, 664--691 (2014; Zbl 1325.20053)

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