On the Delta set of a singular arithmetical congruence monoid

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

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

Si $a$ et $b$ sont des entiers positifs, avec $a\le b$ et $a^2\equiv a\mathrm{mod}b$, l’ensemble

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

est un monoïde multiplicatif, appelé monoïde de congruence arithmétique (ACM). Pour chaque monoïde avec ses unités $M^{\times }$ et pour chaque $x\in M\setminus M^{\times }$, nous dirons que $t\in \mathbb{N}$ 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 $ℒ\left(x\right)=\{t_1,...,t_j\}$ l’ensemble des longueurs (avec $t_i<t_{i+1}$ pour $i<j$). Le Delta-ensemble d’un élément $x$ est $\Delta \left(x\right)=\{t_{i+1}-t_i:1\le i<j\}$ et le Delta-ensemble du monoïde $M$ est $\Delta \left(M\right)={\bigcup }_{x\in M\setminus M^{\times }}\Delta \left(x\right)$. Nous examinons $\Delta \left(M\right)$ quand $M=M_{a,b}$ est un ACM avec $pgcd(a,b)>1$. Cet ensemble est complètement caractérisé quand $pgcd(a,b)=p^{\alpha }$, $p$ un nombre premier et $\alpha >0$. Quand $pgcd(a,b)$ a plus d’un facteur premier, nous donnons des bornes pour $\Delta \left(M\right)$.

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

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

is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid $M$ with units $M^{\times }$ and any $x\in M\setminus M^{\times }$ we say that $t\in \mathbb{N}$ 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 $ℒ\left(x\right)=\{t_1,...,t_j\}$ 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 $ℒ\left(x\right)$: $\Delta \left(x\right)=\{t_{i+1}-t_i:1\le i<k\}$ and the Delta-set of the monoid $M$ is given by ${\bigcup }_{x\in M\setminus M^{\times }}\Delta \left(x\right)$. We consider the $\Delta \left(M\right)$ when $M=M_{a,b}$ is an ACM with $gcd(a,b)>1$. This set is fully characterized when $gcd(a,b)=p^{\alpha }$ for $p$ prime and $\alpha >0$. Bounds on $\Delta \left(M_{a,b}\right)$ are given when $gcd(a,b)$ has two or more distinct prime factors

[1] D.F. Anderson, Elasticity of factorizations in integral domains: a survey. Lecture Notes in Pure and Appl. Math. 189 (1997), 1–29. | MR | Zbl

[2] M. Banister, S. T. Chapman, J. Chaika, W. Meyerson, On a result of James and Niven concerning unique factorization in congruence semigroups. To appear in Elemente der Mathematik. | MR

[3] M. Banister, S. T. Chapman, J. Chaika, W. Meyerson, On the arithmetic of arithmetical congruence monoids. Colloq. Math. 108 (2007), 105–118. | MR

[4] C. Bowles, S.T. Chapman, N. Kaplan, D. Reiser, On $\Delta$-sets of numerical monoids. J. Pure Appl. Algebra 5 (2006), 1–24. | MR | Zbl

[5] S. T. Chapman, A. Geroldinger, Krull domains and their monoids, their sets of lengths, and associated combinatorial problems. Lecture Notes in Pure and Applied Mathematics 189 (1997), 73–112. | Zbl

[6] A. Geroldinger, Uber nicht-eindeutige Zerlegungen in irreduzible Elemente. Math. Z. 197 (1988), 505–529. | MR | Zbl

[7] A. Geroldinger, F. Halter-Koch, Non-unique factorizations. Algebraic, Combinatorial and Analytic Theory, Chapman & Hall/CRC, 2006. | MR | Zbl

[8] A. Geroldinger, F. Halter-Koch, Congruence monoids. Acta Arith. 112 (2004), 263–296. | MR | Zbl

[9] F. Halter-Koch, Arithmetical semigroups defined by congruences. Semigroup Forum 42 (1991), 59–62. | MR | Zbl

[10] F. Halter-Koch, Elasticity of factorizations in atomic monoids and integral domains. J. Théor. Nombres Bordeaux 7 (1995), 367–385. | Numdam | MR | Zbl

[11] F. Halter-Koch, C-monoids and congruence monoids in Krull domains. Lect. Notes Pure Appl. Math. 241 (2005), 71–98. | MR | Zbl

[12] R. D. James, I. Niven, Unique factorization in multiplicative systems. Proc. Amer. Math. Soc. 5 (1954), 834–838. | MR | Zbl