Arithmetic Congruence Monoids: A Survey

We consider multiplicative monoids of the positive integers defined by a single congruence. If a and b are positive integers such that a≤ b and a2≡ a mod b, then such a monoid (known as an arithmetic congruence monoid or an ACM) can be described as Ma,b = (a+ bN0)∪{1}. In lectures on elementary number theory, Hilbert demonstrated to students the utility of the proof of the Fundamental Theorem of Arithmetic for Z by considering the arithmetic congruence monoid with a = 1 and b = 4. In M1,4, the element 441 has a nonunique factorization into irreducible elements as 9 · 49 = 212. ACMs have appeared frequently in the mathematical literature over the last decade. While their structures can be understood merely with rational number theory, their multiplicative behavior can become quite complex. We show that all ACMs fall into one of three mutually exclusive classes: regular (relating to a = 1), local (relating to gcd(a,b) = pk for some rational prime p), and global (gcd(a,b) is not a power of a prime). In each case, we examine the behavior of various invariants widely studied in the theory of nonunique factorizations. Our principal tool will be the construction of transfer homomorphisms from the Ma,b to monoids with simpler multiplicative structure.