On the Delta set of a singular arithmetical congruence monoid
Journal de Théorie des Nombres de Bordeaux, Volume 20 (2008) no. 1, pp. 45-59.

If a and b are positive integers with ab and a 2 amodb, then the set

Ma,b={x:xamodborx=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

Si a et b sont des entiers positifs, avec ab et a 2 amodb, l’ensemble

Ma,b={x:xamodboux=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).

Received:
Published online:
DOI: 10.5802/jtnb.615
Paul Baginski 1; Scott T. Chapman 2; George J. Schaeffer 3

1 University of California at Berkeley Department of Mathematics Berkeley, California 94720
2 Trinity University Department of Mathematics One Trinity Place San Antonio, TX. 78212-7200
3 Carnegie Mellon University Department of Mathematical Sciences Pittsburgh, PA 15213
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Paul Baginski; Scott T. Chapman; George J. Schaeffer. On the Delta set of a singular arithmetical congruence monoid. Journal de Théorie des Nombres de Bordeaux, Volume 20 (2008) no. 1, pp. 45-59. doi : 10.5802/jtnb.615. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.615/

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