On the Delta set of a singular arithmetical congruence monoid
Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 1, p. 45-59
Si a et b sont des entiers positifs, avec ab et a 2 a mod b , l’ensembleMa,b={x:xa mod boux=1}est un monoïde multiplicatif, appelé monoïde de congruence arithmétique (ACM). Pour chaque monoïde avec ses unités M × et pour chaque xMM × , nous dirons que t est une longueur de décomposition en facteurs de x si et seulement s’il existe des éléments irréductibles y 1 ,...,y t M tels que x=y 1 y t . Soit (x)={t 1 ,...,t j } l’ensemble des longueurs (avec t i <t i+1 pour i<j). Le Delta-ensemble d’un élément x est Δ(x)={t i+1 -t i :1i<j} et le Delta-ensemble du monoïde M est Δ(M)= xMM × Δ(x). Nous examinons Δ(M) quand M=M a,b est un ACM avec pgcd(a,b)>1. Cet ensemble est complètement caractérisé quand pgcd(a,b)=p α , p un nombre premier et α>0. Quand pgcd(a,b) a plus d’un facteur premier, nous donnons des bornes pour Δ(M).
If a and b are positive integers with ab and a 2 a mod b , then the setMa,b={x:xa mod borx=1}is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid M with units M × and any xMM × we say that t is a factorization length of x if and only if there exist irreducible elements y 1 ,...,y t of M and x=y 1 y t . Let (x)={t 1 ,...,t j } be the set of all such lengths (where t i <t i+1 whenever i<j). The Delta-set of the element x is defined as the set of gaps in (x): Δ(x)={t i+1 -t i :1i<k} and the Delta-set of the monoid M is given by xMM × Δ(x). We consider the Δ(M) when M=M a,b is an ACM with gcd(a,b)>1. This set is fully characterized when gcd(a,b)=p α for p prime and α>0. Bounds on Δ(M a,b ) are given when gcd(a,b) has two or more distinct prime factors
@article{JTNB_2008__20_1_45_0,
     author = {Baginski, Paul and Chapman, Scott T. and Schaeffer, George J.},
     title = {On the Delta set of a singular arithmetical congruence monoid},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {20},
     number = {1},
     year = {2008},
     pages = {45-59},
     doi = {10.5802/jtnb.615},
     mrnumber = {2434157},
     zbl = {pre05543190},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2008__20_1_45_0}
}
Baginski, Paul; Chapman, Scott T.; Schaeffer, George J. On the Delta set of a singular arithmetical congruence monoid. Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 1, pp. 45-59. doi : 10.5802/jtnb.615. http://www.numdam.org/item/JTNB_2008__20_1_45_0/

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