Arithmetic of Congruence Monoids

Abstract

Let ℕ represent the positive integers. Let n ∈ ℕ and Γ ⊆ ℕ. Set Γ_n = {x ∈ ℕ: ∃ y ∈ Γ, x ≡ ymodn} ∪ {1}. If Γ_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 Γ_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.

Key Words:

• Arithmetic congruence monoid
• Congruence monoid
• Elasticity
• Half-factorial
• Nonunique factorization

2010 Mathematics Subject Classification:

• 20M13
• 11A51
• 20M14

ACKNOWLEDGMENTS

The authors would like to acknowledge an anonymous referee, whose efforts improved the exposition of this article.