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On the arithmetic of arithmetical congruence monoids

M. Banister; J. Chaika; S. T. Chapman; W. Meyerson — 2007

Colloquium Mathematicae

Let ℕ represent the positive integers and ℕ₀ the non-negative integers. If b ∈ ℕ and Γ is a multiplicatively closed subset of $ℤ_{b} = ℤ / b ℤ$ , then the set $H_{Γ} = x \in ℕ | x + b ℤ \in Γ \cup 1$ 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...

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