On the arithmetic of arithmetical congruence monoids
Volume 108 / 2007
Abstract
Let ${\mathbb N}$ represent the positive integers and ${\mathbb N}_0$ the non-negative integers. If $b\in {\mathbb N}$ and ${{\mit\Gamma}}$ is a multiplicatively closed subset of $\mathbb{Z}_b=\mathbb{Z}/b\mathbb{Z}$, then the set $H_{{\mit\Gamma}} =\{x\in {\mathbb N} \mid x+b\mathbb{Z}\in {{\mit\Gamma}}\}\cup\{1\}$ is a multiplicative submonoid of ${\mathbb N}$ known as a congruence monoid. An arithmetical congruence monoid (or ACM) is a congruence monoid where ${{\mit\Gamma}}=\{\overline{a}\}$ consists of a single element. If $H_{{\mit\Gamma}}$ is an ACM, then we represent it with the notation $M(a,b) =(a+b{\mathbb N}_0)\cup \{1\}$, where $a, b\in {\mathbb N}$ and $a^2\equiv a \pmod{b}$. A classical 1954 result of James and Niven implies that the only ACM which admits unique factorization of elements into products of irreducibles is $M(1,2)=M(3,2)$. In this paper, we examine further factorization properties of ACMs. We find necessary and sufficient conditions for an ACM $M(a,b)$ to be half-factorial (i.e., lengths of irreducible factorizations of an element remain constant) and further determine conditions for $M(a,b)$ to have finite elasticity. When the elasticity of $M(a,b)$ is finite, we produce a formula to compute it. Among our remaining results, we show that the elasticity of an ACM $M(a,b)$ may not be accepted and show that if an ACM $M(a,b)$ has infinite elasticity, then it is not fully elastic.
