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 $a\le b$ and ${a}^{2}\equiv a\phantom{\rule{0.277778em}{0ex}}\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}b$, then the set

 ${M}_{a,b}=\left\{x\in ℕ:x\equiv a\phantom{\rule{0.277778em}{0ex}}\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}b\phantom{\rule{4pt}{0ex}}\text{or}\phantom{\rule{4pt}{0ex}}x=1\right\}$

is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid $M$ with units ${M}^{×}$ and any $x\in M\setminus {M}^{×}$ we say that $t\in ℕ$ is a factorization length of $x$ if and only if there exist irreducible elements ${y}_{1},...,{y}_{t}$ of $M$ and $x={y}_{1}\cdots {y}_{t}$. Let $ℒ\left(x\right)=\left\{{t}_{1},...,{t}_{j}\right\}$ be the set of all such lengths (where ${t}_{i}<{t}_{i+1}$ whenever $i). The Delta-set of the element $x$ is defined as the set of gaps in $ℒ\left(x\right)$: $\Delta \left(x\right)=\left\{{t}_{i+1}-{t}_{i}:1\le i and the Delta-set of the monoid $M$ is given by ${\bigcup }_{x\in M\setminus {M}^{×}}\Delta \left(x\right)$. We consider the $\Delta \left(M\right)$ when $M={M}_{a,b}$ is an ACM with $gcd\left(a,b\right)>1$. This set is fully characterized when $gcd\left(a,b\right)={p}^{\alpha }$ for $p$ prime and $\alpha >0$. Bounds on $\Delta \left({M}_{a,b}\right)$ are given when $gcd\left(a,b\right)$ has two or more distinct prime factors

Si $a$ et $b$ sont des entiers positifs, avec $a\le b$ et ${a}^{2}\equiv a\phantom{\rule{0.277778em}{0ex}}\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}b$, l’ensemble

 ${M}_{a,b}=\left\{x\in ℕ:x\equiv a\phantom{\rule{0.277778em}{0ex}}\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}b\phantom{\rule{4pt}{0ex}}\text{ou}\phantom{\rule{4pt}{0ex}}x=1\right\}$

est un monoïde multiplicatif, appelé monoïde de congruence arithmétique (ACM). Pour chaque monoïde avec ses unités ${M}^{×}$ et pour chaque $x\in M\setminus {M}^{×}$, nous dirons que $t\in ℕ$ est une longueur de décomposition en facteurs de $x$ si et seulement s’il existe des éléments irréductibles ${y}_{1},...,{y}_{t}\in M$ tels que $x={y}_{1}\cdots {y}_{t}$. Soit $ℒ\left(x\right)=\left\{{t}_{1},...,{t}_{j}\right\}$ l’ensemble des longueurs (avec ${t}_{i}<{t}_{i+1}$ pour $i). Le Delta-ensemble d’un élément $x$ est $\Delta \left(x\right)=\left\{{t}_{i+1}-{t}_{i}:1\le i et le Delta-ensemble du monoïde $M$ est $\Delta \left(M\right)={\bigcup }_{x\in M\setminus {M}^{×}}\Delta \left(x\right)$. Nous examinons $\Delta \left(M\right)$ quand $M={M}_{a,b}$ est un ACM avec $pgcd\left(a,b\right)>1$. Cet ensemble est complètement caractérisé quand $pgcd\left(a,b\right)={p}^{\alpha }$, $p$ un nombre premier et $\alpha >0$. Quand $pgcd\left(a,b\right)$ a plus d’un facteur premier, nous donnons des bornes pour $\Delta \left(M\right)$.

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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