Abstract: 
\def\N{{\sym N}}Let $\N$ represent the positive integers and $\N_0$ the
non-negative integers. If $b\in \N$ and ${{\mit\Gamma}}$ is a
multiplicatively closed subset of
$\mathbb{Z}_b=\mathbb{Z}/b\mathbb{Z}$, then the set $H_{{\mit\Gamma}}
=\{x\in \N \mid x+b\mathbb{Z}\in {{\mit\Gamma}}\}\cup\{1\}$ is a 
multiplicative submonoid of $\N$ known as a 
{\it congruence monoid.} An
{\it arithmetical congruence monoid} (or {\it 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
$M(a,b) =(a+b\N_0)\cup \{1\}$, where $a, b\in \N$
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.

1. Department of Mathematics
   Harvey Mudd College
   1250 N. Dartmouth Ave.
   Claremont, CA 91711, U.S.A.
   and
   Department of Mathematics
   University of California at Santa Barbara
   Santa Barbara, CA 93106, U.S.A.
   
2. Department of Mathematics
   The University of Iowa
   14 MacLean Hall
   Iowa City, IA 52242, U.S.A.
   and
   Mathematics Department, MS 136
   Rice University
   6100 S. Main St.
   Houston, TX 77005-1892, U.S.A.
   
3. Department of Mathematics
   Trinity University
   One Trinity Place
   San Antonio, TX 78212-7200, U.S.A.
   
4. Department of Mathematics
   Harvard University
   One Oxford Street
   Cambridge, MA 02138, U.S.A.
   and
   Mathematics Department
   University of California at Los Angeles
   Box 951555
   Los Angeles, CA 90095-1555, U.S.A