Numerical semigroups with Apéry sets of unique expression
type: Item
instance of: scholarly article
title: Numerical semigroups with Apéry sets of unique expression
zbMATH Open document ID: 0955.20039
author: Yanyan Li
published in: Journal of Algebra
publication date: 2000-11-26T00:00:00Z
review text: A numerical semigroup is a subset \(S\) of the set of nonnegative integers \(\mathbb{N}\) closed under addition, containing the identity element of \(\mathbb{N}\) and generating the set of integers as a group. For \(n\in S\setminus\{0\}\), the set \(\{s\in S\mid s-n\not\in S\}\) is called the Apéry set of \(n\). Let \(\{n_0<\cdots<n_p\}\) be the unique minimal system of generators of \(S\). Denote by \(F\) the free monoid on the indeterminates \(\{X_0<\cdots<X_p\}\) and let \(\phi\colon F\to S\), \(\phi(X_i)=n_i\) be the monoid epimorphism. Obviously, \(S\cong F/\ker\phi\). A presentation of \(S\) is a finite subset \(\rho\) of \(F\times F\) such that the congruence of \(\phi\) is the least congruence on \(F\) containing \(\rho\). The knowledge of a minimal presentation of a semigroup has applications in ring theory by using the concept of semigroup ring. In this paper, the author gives a minimal presentation for a numerical semigroup such that all the elements of its Apéry set with respect to \(n_0\) have a unique expression. Interesting examples of this type of semigroups are supplied.
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 1441819
zbMATH Keywords: numerical semigroups
zbMATH Keywords: presentations of semigroups
zbMATH Keywords: symmetric semigroups
zbMATH Keywords: Apéry sets
zbMATH Keywords: generators
zbMATH Keywords: minimal presentations
reviewed by: Carlos Galindo
Wikidata QID: Q58843070
MaRDI profile type: Publication
full work available at URL: jabr.1999.8202
OpenAlex ID: W2012879953
cites work: Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains
cites work: Semi-groupes d'entiers et application aux branches
cites work: Generators and relations of abelian semigroups and semigroup rings
cites work: The Value-Semigroup of a One-Dimensional Gorenstein Ring
cites work: Q5343411
cites work: Q4951130
cites work: AN ALGORITHMIC METHOD TO COMPUTE A MINIMAL RELATION FOR ANY NUMERICAL SEMIGROUP
cites work: On numerical semigroups
cites work: On symmetric numerical semigroups
cites work: On numerical semigroups with high embedding dimension
cites work: On free affine semigroups
cites work: Cohen-Macaulay local rings of maximal embedding dimension
DOI: 10.1006/JABR.1999.8202
description: scientific article; zbMATH DE number 1441819