On the Elasticity of Generalized Arithmetical Congruence Monoids

Let a and b be positive integers with a <= b and a(2) a (mod b). The set M(a, b) = {n is an element of N vertical bar n = a(mod b)} boolean OR {1} is multiplicatively closed and known as an arithmetical congruence monoid (or ACM). It is well known that unique factorization need not occur in ACMs. In this paper, we investigate factorization results when we consider only elements of M(a, b) of sufficiently large size. More specifically, if M(a, b) is an ACM, we offer results concerning the elasticity of generalized ACMs (or GACMs) of the form M(r)(a, b) = {a+ kb is an element of M(a, b)vertical bar k >= r} boolean OR {1} where r is a nonnegative integer. 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 derive a formula for finite elasticity in the case a not equal 1.