Arithmetic of Congruence Monoids
Arielle Fujiwara, J. K. Gibson, Matthew Jenssen, Daniel Montealegre, Vadim Ponomarenko, Ari Tenzer · Communications in Algebra · 2016
Let ℕ represent the positive integers. Let n ∈ ℕ and Γ ⊆ ℕ. Set Γn = {x ∈ ℕ: ∃ y ∈ Γ, x ≡ ymodn} ∪ {1}. If Γn is closed under multiplication, it is known as a congruence monoid or CM. A classical result of James and Niven [15 James, R. D., Niven, I. (1954). Unique factorization in multiplicative systems. Proc. Amer. Math. Soc. 5:834–838.[Crossref], [Web of Science ®] , [Google Scholar]] is that for each n, exactly one CM admits unique factorization into products of irreducibles, namely Γn = {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.