Arithmetic of Congruence Monoids

Abstract

Let N represent the positive integers. Let n ∈ N and ⊆ N. Set n = x ∈ N ∃y ∈ x ≡ y mod n ∪ 1 . If n is closed under multiplication, it is known as a congruence monoid or CM. A classical result of James and Niven [15] is that for each n, exactly one CM admits unique factorization into products of irreducibles, namely n = x ∈ N gcd x n = 1 . In this article, we examine additional factorization properties of CMs. We characterize CMs that contain primes, and we determine elasticity for several classes of CMs and bound it for several others. Also, for several classes, we characterize half-factoriality and determine whether the elasticity is accepted and whether it is full.