On the arithmetic of arithmetical congruence monoids

Abstract

Let N represent the positive integers and N 0 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 = {a} consists of a single element. If H Γ is an ACM, then we represent it with the notation M (a, b) = (a + bN 0) ∪ {1}, where a, b ∈ N and a 2 ≡ 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.

On the Delta set of a singular arithmetical congruence monoid

“…If d = 1 then we call M a,b regular and if d > 1, we call it singular. An elementary argument (see [3,Lemma 2.1]) shows that regular ACMs are precisely the Hilbert monoids described above. Singular ACMs are further characterized as being either local or global depending on whether d has one or more distinct prime factors.…”

“…Arithmetical congruence monoids have been addressed recently in the literature in [2] and [3] where the factorization properties of these monoids are explored. In particular, in [3,Theorem 2.4] it is shown that the elasticity of factorization (see [1] or [10] for a definition) of M a,b is finite if and only if M a,b is regular or local.…”

“…In particular, in [3,Theorem 2.4] it is shown that the elasticity of factorization (see [1] or [10] for a definition) of M a,b is finite if and only if M a,b is regular or local. A more in-depth general analysis of congruence monoids can be found in [8].…”

“…A more in-depth general analysis of congruence monoids can be found in [8]. Our purpose is to extend the work of [3] and On the Delta set of a singular ACM 47 provide a more precise examination of the factorization properties of the elements of M a,b into products of irreducible elements.…”

“…Many important factorization properties of ACMs have already been investigated; in particular, questions of factoriality, half-factoriality, and elasticity are covered extensively in [3]. Most relevant to our discussion will be the following results from [3] (Theorem 2.4 and Theorem 2.7).…”