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1 Colloquium Mathematicum

1 2007

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On the arithmetic of arithmetical congruence monoids

100%

Banister M. , Chaika J. , Chapman S. , Meyerson W.

nr 1

105-118

Let ℕ represent the positive integers and ℕ₀ the non-negative integers. If b ∈ ℕ and Γ is a multiplicatively closed subset of $ℤ_b = ℤ/bℤ$, then the set $H_Γ = {x ∈ ℕ | x + bℤ ∈ Γ} ∪ {1}$ is a multiplicative submonoid of ℕ known as a congruence monoid. An arithmetical congruence monoid (or ACM) is a congruence monoid where Γ = ā consists of a single element. If $H_Γ$ is an ACM, then we represent it with the notation M(a,b) = (a + bℕ₀) ∪ {1}, where a, b ∈ ℕ and a² ≡ a (mod b). A classical 1954 result of James and Niven implies that the only ACM which admits unique factorization of elements into products of irreducibles is M(1,2) = M(3,2). In this paper, we examine further factorization properties of ACMs. We find necessary and sufficient conditions for an ACM M(a,b) to be half-factorial (i.e., lengths of irreducible factorizations of an element remain constant) and further determine conditions for M(a,b) to have finite elasticity. When the elasticity of M(a,b) is finite, we produce a formula to compute it. Among our remaining results, we show that the elasticity of an ACM M(a,b) may not be accepted and show that if an ACM M(a,b) has infinite elasticity, then it is not fully elastic.