On the Delta Set of a Singular Arithmetical Congruence Monoid

If a and b are positive integers with a ≤ b and a ≡ a mod b, then the set Ma,b = {x ∈ N : x ≡ a mod b or x = 1} is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid M with units M× and any x ∈ M \M× we say that t ∈ N is a factorization length of x if and only if there exist irreducible elements y1, . . . , yt of M and x = y1 · · · yt. Let L(x) = {t1, . . . , tj} be the set of all such lengths (where ti < ti+1 whenever i < j). The Delta-set of the element x is defined as the set of gaps in L(x): ∆(x) = {ti+1 − ti : 1 ≤ i < k} and the Delta-set of the monoid M is given by ⋃ x∈M\M× ∆(x). We consider the ∆(M) when M = Ma,b is an ACM with gcd(a, b) > 1. This set is fully characterized when gcd(a, b) = p for p prime and α > 0. Bounds on ∆(Ma,b) are given when gcd(a, b) has two or more distinct prime factors. The first author was supported by a Dept. of Homeland Security Graduate Fellowship. The third author received support from the National Science Foundation, Grant #DMS0353488. 2 Paul Baginski, S. T. Chapman, George J. Schaeffer