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.
The theory of non-unique factorizations in integral domains and monoids is a very active area of current research (see both [1] and [4] to view recent trends in this work). To demonstrate the phenomena of non-unique factorizations, we consider a result from the classical setting on uniqueness of factorizations by James and Niven [11]. We proceed as follows: Let N represent the natural numbers and suppose that M ⊆ N is a multiplicative semigroup. M is called a congruence semigroup if there exists a natural number n such . Im Hilbertschen Monoid 1+4N 0 = {1, 5, 9, 13, . . .} (N 0 = natürliche Zahlen inklusive Null) ist die Zerlegung in irreduzible Faktoren nicht eindeutig: Es gilt zum Beispiel 441 = 9 · 49 = 21 · 21. Hilberts Monoid ist ein Beispiel einer Kongruenz-Halbgruppe. Ein klassisches Resultat von James und Niven besagt, dass in einer KongruenzHalbgruppe M genau dann der Fundamentalsatz der Arithmetik gilt, wenn M aus allen Zahlen besteht, die relativ prim zu einer festen Zahl n ∈ N sind. Die Autoren der vorliegenden Arbeit untersuchen das andere Extrem, nämlich den Fall, wo M aus allen Zahlen besteht, die nicht relativ prim zu einer festen Zahl n ∈ N sind. Sie zeigen, dass in diesem Fall wenigstens die Anzahl der Primfaktoren bei der Zerlegung einer Zahl eindeutig ist.
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