Numerical Semigroups: IMNS 2018

Preface
Contents
Counting Numerical Semigroups by Genus and Even Gaps via Kunz-Coordinate Vectors
	1 Introduction
	2 Apéry Set and Kunz-Coordinate Vector
	3 The Main Result and an Application to a Counting Problem
	References
Patterns on the Numerical Duplication by Their Admissibility Degree
	1 Introduction
	2 Preliminaries
	3 Patterns and Their Admissibility Degree
	4 Patterns Equivalent to the Arf Pattern
	5 Patterns on the Numerical Duplication
	6 Patterns on Rings
	References
Primality in Semigroup Rings
	1 Introduction
	2 Primal Elements in Monoids
	3 Primal Elements in a Graded Domain
	4 Primal Elements in Semigroup Rings
	References
Conjecture of Wilf: A Survey
	1 Introduction
		1.1 Terminology and Notation
		1.2 A Convenient Way to Visualize Numerical Semigroups
	2 Two Problems Posed by Wilf
		2.1 Wilf\'s Paper
		2.2 Problem (a.i): Wilf\'s Conjecture
		2.3 Problem (a.ii): Another Open Problem
		2.4 Problem (b): Counting Numerical Semigroups
	3 Some Classes of Wilf Semigroups
		3.1 The Type as an Important Ingredient
		3.2 Semigroups Given by Sets of Generators
		3.3 Semigroups with Nonnegative Eliahou Numbers
		3.4 Natural Constructions
		3.5 Semigroups with Small Multiplicity
		3.6 Semigroups with Large Embedding Dimension (Compared to the Multiplicity)
			3.6.1 Some Comments
		3.7 Semigroups with Big Multiplicity (and Possibly Small Embedding Dimension)
		3.8 Semigroups with Large Multiplicity (Compared to the Conductor)
			3.8.1 Some Comments
		3.9 Considering Unusual Invariants
		3.10 Families Described Through One Invariant
	4 Quasi-Generalization
	References
Gapsets of Small Multiplicity
	1 Introduction
	2 Gapset Filtrations
		2.1 The Canonical Partition
		2.2 Gapset Filtrations
	3 The Case m=3
	4 Some More General Tools
		4.1 On m-Extensions and m-Filtrations
		4.2 Gapset Filtrations Revisited
		4.3 A Compact Representation
		4.4 Complementing an m-Extension
		4.5 The Insertion Maps fi
	5 The Case m=4
		5.1 Concluding Remark
	References
Generic Toric Ideals and Row-Factorization Matrices in Numerical Semigroups
	1 Preliminaries
		1.1 Numerical Semigroups
		1.2 The Fibers of Elements in Numerical Semigroups
		1.3 Semigroup Rings
	2 Generic Toric Ideals
		2.1 Main Results
		2.2 Basic Fibers
	References
Symmetric (Not Complete Intersection) Semigroups Generated by Six Elements
	1 Introduction
	2 Symmetric (Not CI) Semigroups Generated by Six Integers and Polynomial Identities
	3 The Lower Bound for the Frobenius Numbers of Semigroups S6
	4 Are There Any Constraints on Betti\'s Numbers of Symmetric (Not CI) Semigroups S6?
	5 Symmetric (Not CI) Semigroups S6 with the W and W2 Properties
	References
Syzygies of Numerical Semigroup Rings, a Survey Through Examples
	1 Introduction
	2 Principal Matrices and Gorenstein Sequences of Length 4
	3 Arithmetic Sequences
	4 Gluing
	References
Irreducibility and Factorizations in Monoid Rings
	1 Introduction
	2 Notation and Background
		2.1 General Notation
		2.2 Monoids
		2.3 Factorizations
		2.4 Monoid Rings
	3 Irreducibility Criteria for Monoid Rings
		3.1 Extended Gauss\'s Lemma
		3.2 Extended Eisenstein\'s Criterion
	4 Factorizations in Monoid Algebras
	References
On the Molecules of Numerical Semigroups, Puiseux Monoids, and Puiseux Algebras
	1 Introduction
	2 Monoids, Atoms, and Molecules
		2.1 General Notation
		2.2 Monoids
		2.3 Atoms and Molecules
	3 Molecules of Numerical Semigroups
		3.1 Betty Elements
		3.2 On the Sizes of the Sets of Molecules
	4 Molecules of Puiseux Monoids
		4.1 Molecules of Generic Puiseux Monoids
		4.2 Molecules of Prime Reciprocal Monoids
	5 Molecules of Puiseux Algebras
	References
Arf Numerical Semigroups with Multiplicity 9 and 10
	1 Introduction
	2 Arf Numerical Semigroups
	3 Arf Numerical Semigroups with Multiplicity 9
	4 Arf Numerical Semigroups with Multiplicity 10
	References
Numerical Semigroup Rings of Maximal Embedding Dimension with Determinantal Defining Ideals
	1 Introduction
	2 Proof of Theorem 2
		2.1 Proof of (3) (2) and the Assertion (a)
		2.2 Proof of (2) (4)
		2.3 Proof of (b) and (c)
	3 Examples
	References
Embedding Dimension of a Good Semigroup
	1 Introduction
	2 Semiring Associated to a Good Semigroup and Irreducible Absolutes
		2.1 Semiring ΓS and Basic Properties
		2.2 A System of Generators of ΓS as a Semiring
	3 Embedding Dimension of a Good Semigroup
		3.1 An Inferior Bound for the Embedding Dimension
		3.2 A Superior Bound for the Embedding Dimension
		3.3 An Algorithm for the Computation of the Embedding Dimension of a Semigroup SN2
	4 Properties of Embedding Dimension
		4.1 Relationship Between Embedding Dimension of a Ring and Embedding Dimension of Its Value Semigroup
		4.2 Relationship Between Embedding Dimension and Multiplicity
	References
On Multi-Index Filtrations Associated to Weierstraß Semigroups
	1 Introduction
	2 Terminology and Notation
		2.1 Branches and Parametrizations
		2.2 Divisors on Smooth Curves
	3 Brill-Noether Theory for Curves
	4 The Weierstraß Semigroup at Several Points
		4.1 Dimension of the Riemann-Roch Quotients with Respect to Pi and Associated Functions
		4.2 Computing the Weierstraß Semigroup at Two Points
	5 Computational Aspects Using Singular
		5.1 Hints of Usage of brnoeth.lib
		5.2 Procedures Generalizing to Several Points
	References
On the Hilbert Function of Four-Generated Numerical SemigroupRings
	1 Introduction
	2 Preliminaries
	3 The Case Embedding Dimension 4
	4 Cases with Cardinality of A2≤ 4
	5 Case D2 = {n3, n4}
	6 The Hilbert Function Does Not Decrease at Level 3
	7 Conclusions
	References
Lattice Ideals, Semigroups and Toric Codes
	1 Introduction
	2 The Algebraic Approach
	3 Lattice Ideals and Subgroups of the Torus TX
		3.1 Degenerate Tori
	4 Vanishing Ideals of Subsgroups of TX
		4.1 The Length of the Code Cα,YQ
	References
The Number of Star Operations on Numerical Semigroups and on Related Integral Domains
	1 Introduction
	2 Notation
	3 Star Operations
	4 Estimates Through the Genus
	5 Estimates Through the Multiplicity
	6 Multiplicity 3
	7 Prime Multiplicity
	8 Linear Families
	9 Algorithms and Explicit Data
	10 The Ring Version
	References
Torsion in Tensor Products over One-Dimensional Domains
	1 Introduction
	2 Some Evidence
	References
Almost Symmetric Numerical Semigroups with Odd Generators
	1 Introduction
	2 Basic Concepts
	3 Symmetric Semigroups
	4 Pseudo-Symmetric Semigroups
	5 Almost Symmetric Semigroups with Type Three
	References
Poincaré Series on Good Semigroup Ideals
	1 Introduction
	2 Preliminaries
		2.1 Good Semigroups and Their Ideals
		2.2 Distance of Semigroup Ideals
		2.3 Canonical Semigroup Ideals
		2.4 Value Semigroups
	3 Distance and Duality
	4 Symmetry of the Poincaré Series
	References
A Short Proof of Bresinski\'s Theorem on Gorenstein Semigroup Rings Generated by Four Elements
	1 Basic Concepts
	2 The Proof
	References