# NumericalSgps

Last news since Aachen 2nd GAP days

• Documentation

• Code review

• New contributions

• Good semigroups

• New functions

• Future plans

# Documentation

• Revised
• Contribution section contents merged in the corresponding sections
• Over 130 pages

# Code Review

1. Cleaning and improvements
2. Bug fixes
3. Attributes, properties and methods instead of global functions
4. Inference rules (thanks Sebastian!)
5. Corrected and simplified usage of GAP type objects and declarations of GAP representations (thanks Max!)
6. NumSgpsTests() modified (Thanks Alexander!)

# New contributions

• Alessio Sammartano
• Chris O'Neill
• Klara Stokes
• Ignacio Ojeda and C. J. Ávila
• Alfredo Sánchez-R. Navarro
• Giuseppe Zito
• Andrés Herrera Poyatos
• Benjamín Alarcón Heredia

## Alessio Sammartano

He already had several contributions to graded ring of a numerical semigroup ring (see contributions section in the manual)

His new contributions are

## Klara Stokes

We implemented several functions related to patterns for ideals of numerical semigroups

• K. Stokes Patterns of ideals of numerical semigroups. Semigroup Forum 93 (2016), no. 1, 180–200.
AsIdealOfNumericalSemigroup
BoundForConductorOfImageOfPattern
ApplyPatternToIdeal

## Chris O'Neill

We were able to find a way to compute the  Delta set of an affine semigroup (via Gröbner bases)

DeltaSetOfAffineSemigroup​

He already contributed with the implementation of Delta sets in numerical semigroups (see contributions section in the manual)

## I. Ojeda and C. J. Moreno Ávila

Calculation of the Frobenius number and Apéry set of a numerical semigroup using Gröbner basis

Needs 4ti2Interface or 4ti2gap

A faster version for Apéry sets is available with Singular and 4ti2Interface

## Alfredo Sánchez-R. Navarro

He helped in the implementation of tame degrees for affine and numerical semigroups

Also several types of catenary degrees

Pointed some useless code in a couple of functions

## Giuseppe Zito

He improved the implementations of the following ​

ArfNumericalSemigroupsWithFrobeniusNumber ArfNumericalSemigroupsWithFrobeniusNumberUpTo ArfNumericalSemigroupsWithGenus ArfNumericalSemigroupsWithGenusUpTo ArfCharactersOfArfNumericalSemigroup

He is currently working on Arf good semigroups

## Andrés Herrera-Poyatos

He improved the implementation of

IsSelfReciprocalUnivariatePolynomial

IsKroneckerPolynomial

## Benjamín Alarcón Heredia

Helped with the implementation of Feng-Rao distances and numbers

# Good semigroups

A submonoid $$S$$ of $$(\mathbb{N}^n,+)$$ is a good semigroup if

(G1) for all $$a,b\in S$$, $$a\wedge b\in S$$
(G2) if $$a,b\in S$$ and $$a_i=b_i$$ for some $$i\in \{1,\ldots,n\}$$, then there exists $$c\in S$$ such that $$c_i>a_i=b_i$$, $$c_j\ge\min\{a_j,b_j \}$$ for $$j\in\{1,\ldots,n\}\setminus\{i\}$$ and $$c_j=\min\{a_j,b_j \}$$ if $$a_j\ne b_j$$
(G3) there exists $$C\in S$$ such that $$C+\mathbb N^n\subseteq S$$

This picture was produced with GAP and tikz

# Good semigroups

## Semigroups of values

$$\mathrm{v}\left(\mathbb{K}[\![x,y]\!]/(y^4-2x^3y^2-4x^5y+x^6-x^7)(y^2-x^3)\right)$$

# Good semigroups

## Semigroup duplication

A numerical semigroup $$S$$ and a relative ideal $$E$$ of $$S$$ with $$E\subseteq S$$,
$$S\bowtie E= D\cup (E\times E)\cup\{ a\wedge b\mid a\in D, b\in E\times E\},$$
where $$D=\{(s,s) \mid s\in S\}$$

# Good semigroups

## Semigroup amalgamation

For $$S$$ and $$T$$ numerical semigroups, $$g:S\to T$$ a monoid morphism (and thus multiplication by an integer) and $$E$$ a relative ideal of $$T$$ with $$E\subseteq T$$
(here $$D=\{(s,ks) \mid s\in S\}$$)

S\bowtie^g E=D\cup(g^{-1}(E)\times E)\cup \{a\wedge b\mid a\in D, b\in g^{-1}(E)\times E\}
$S\bowtie^g E=D\cup(g^{-1}(E)\times E)\cup \{a\wedge b\mid a\in D, b\in g^{-1}(E)\times E\}$

# New functions

Apart from the functions already mentioned in the contributions section, we added the following

GeneratorsKhalerDifferentials
ModuleGenerators_Global
RatliffRushNumberOfIdealOfNumericalSemigroup
AsymptoticRatliffRushNumberOfIdealOfNumericalSemigroup
RatliffRushClosureOfIdealOfNumericalSemigroup
DenumerantOfElementInNumericalSemigroup MaximumDegreeOfElementWRTNumericalSemigroup MaximalDenumerantOfElementInNumericalSemigroup
IsSuperSymmetricNumericalSemigroup

# New functions (II)

​IsOrdinaryNumericalSemigroup (synonym IsOrdinary)
​IsAcuteNumericalSemigroup (synonym IsAcute)
DesertsOfNumericalSemigroup
GraverBasis
CanonicalBasisOfKernelCongruence
MultiplicitySequenceOfNumericalSemigroup
LipmanSemigroup
HolesOfNumericalSemigroup
KunzCoordinatesOfNumericalSemigroup
KunzPolytope
InductiveSemigroup
MultipleOfNumericalSemigroup
NumericalSemigroupByAffineMap
SemigroupOfValuesOfPlaneCurve (needs singular)
WilfNumber
EliahouNumber
ProfileOfNumericalSemigroup
EliahouSlicesOfNumericalSemigroup​
LatticePathAssociatedToNumericalSemigroup​
MoebiusFunctionAssociatedToNumericalSemigroup​

# Future plans

• Semigroup of values of a curve
• Arf good semigroups via multiplicity trees
• Add functions that produce dot and tikz (we already have some that produce dot; use viz.js or to be used with jupyter gap kernel)

An example with the jupyter-gap kernel