On the Delta set of a singular arithmetical congruence monoid

Paul Baginski; Scott T. Chapman; George J. Schaeffer

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

Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 1, pp. 45-59.

Abstract
Résumé

If $a$ and $b$ are positive integers with $a \leq b$ and $a^{2} \equiv a \mod b$ , then the set

M_{a, b} = {x \in ℕ : x \equiv a \mod b or x = 1}

is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid $M$ with units $M^{\times}$ and any $x \in M ∖ M^{\times}$ 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 $x = y_{1} \dots y_{t}$ . Let $ℒ (x) = {t_{1}, ..., t_{j}}$ 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 $ℒ (x)$ : $Δ (x) = {t_{i + 1} - t_{i} : 1 \leq i < k}$ and the Delta-set of the monoid $M$ is given by $⋃_{x \in M ∖ M^{\times}} Δ (x)$ . 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 $a \leq b$ et $a^{2} \equiv a \mod b$ , l’ensemble

M_{a, b} = {x \in ℕ : x \equiv a \mod b ou x = 1}

est un monoïde multiplicatif, appelé monoïde de congruence arithmétique (ACM). Pour chaque monoïde avec ses unités $M^{\times}$ et pour chaque $x \in M ∖ M^{\times}$ , 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 $x = y_{1} \dots y_{t}$ . Soit $ℒ (x) = {t_{1}, ..., t_{j}}$ l’ensemble des longueurs (avec $t_{i} < t_{i + 1}$ pour $i < j$ ). Le Delta-ensemble d’un élément $x$ est $Δ (x) = {t_{i + 1} - t_{i} : 1 \leq i < j}$ et le Delta-ensemble du monoïde $M$ est $Δ (M) = ⋃_{x \in M ∖ M^{\times}} Δ (x)$ . 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 $Δ (M)$ .

MR: 2434157

DOI: 10.5802/jtnb.615

Author's affiliations:

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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