Displaying similar documents to “Factorization properties of Krull monoids with infinite class group”

Elasticity of factorizations in atomic monoids and integral domains

Franz Halter-Koch (1995)

Journal de théorie des nombres de Bordeaux

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For an atomic domain R , its elasticity ρ ( R ) is defined by : ρ ( R ) = sup { m / n u 1 u m = v 1 v n for irreducible u j , v i R } . We study the elasticity of one-dimensional noetherian domains by means of the more subtle invariants μ m ( R ) defined by : μ m ( R ) = sup { n u 1 u m = u 1 v n for irreducible u j , v i R } . As a main result we characterize all orders in algebraic number fields having finite elasticity. On the way, we obtain a series of results concerning the invariants μ m and ρ for monoids and integral domains which are of independent interest.

The set of minimal distances in Krull monoids

Alfred Geroldinger, Qinghai Zhong (2016)

Acta Arithmetica

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Let H be a Krull monoid with class group G. Then every nonunit a ∈ H can be written as a finite product of atoms, say a = u 1 · . . . · u k . The set (a) of all possible factorization lengths k is called the set of lengths of a. If G is finite, then there is a constant M ∈ ℕ such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference d ∈ Δ*(H), where Δ*(H) denotes the set of minimal distances of H. We show that max Δ*(H) ≤ maxexp(G)-2,(G)-1 and that equality holds...

On the arithmetic of arithmetical congruence monoids

M. Banister, J. Chaika, S. T. Chapman, W. Meyerson (2007)

Colloquium Mathematicae

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Let ℕ represent the positive integers and ℕ₀ the non-negative integers. If b ∈ ℕ and Γ is a multiplicatively closed subset of b = / b , then the set H Γ = x | x + b 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...

Arithmetic of non-principal orders in algebraic number fields

Andreas Philipp (2010)

Actes des rencontres du CIRM

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Let R be an order in an algebraic number field. If R is a principal order, then many explicit results on its arithmetic are available. Among others, R is half-factorial if and only if the class group of R has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.

On delta sets and their realizable subsets in Krull monoids with cyclic class groups

Scott T. Chapman, Felix Gotti, Roberto Pelayo (2014)

Colloquium Mathematicae

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Let M be a commutative cancellative monoid. The set Δ(M), which consists of all positive integers which are distances between consecutive factorization lengths of elements in M, is a widely studied object in the theory of nonunique factorizations. If M is a Krull monoid with cyclic class group of order n ≥ 3, then it is well-known that Δ(M) ⊆ {1,..., n-2}. Moreover, equality holds for this containment when each class contains a prime divisor from M. In this note, we consider the question...

Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids

Víctor Blanco, Pedro A. García-Sánchez, Alfred Geroldinger (2010)

Actes des rencontres du CIRM

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Arithmetical invariants—such as sets of lengths, catenary and tame degrees—describe the non-uniqueness of factorizations in atomic monoids.We study these arithmetical invariants by the monoid of relations and by presentations of the involved monoids. The abstract results will be applied to numerical monoids and to Krull monoids.