On the Elasticity of Generalized Arithmetical Congruence Monoids

An arithmetical congruence monoid (or ACM ) is a multiplicative monoid, which consists of an arithmetic sequence and the element 1. As they are traditionally defined, it is required that a ≤ b and a ≡ a (mod b) must hold to ensure closure. It is well known that unique factorization need not occur in ACMs. In this paper, we investigate factorization results when the requirement a ≤ b is dropped. More specifically, if M(a, b) is an ACM, we offer results concerning the elasticity of generalized ACMs (or GACMs) of the form Mr(a, b) = {a + kb ∈ M(a, b)|k ≥ r} ∪ {1}. We characterize when a generalized ACM is half-factorial (i.e. lengths of irreducible factorizations are constant). Moreover, we offer conditions, which force the elasticity to be infinite and offer a formula in the case a 6= 1 for when it is finite.