## 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.