The set of minimal distances in Krull monoids

Alfred Geroldinger; Qinghai Zhong

Displaying similar documents to “The set of minimal distances in Krull monoids”

Factorization properties of Krull monoids with infinite class group

Wolfgang Hassler (2002)

Colloquium Mathematicae

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For a non-unit a of an atomic monoid H we call $L_{H} (a) = k \in ℕ | a = u ₁ . . . u_{k} w i t h i r r e d u c i b l e u_{i} \in H$ the set of lengths of a. Let H be a Krull monoid with infinite divisor class group such that each divisor class is the sum of a bounded number of prime divisor classes of H. We investigate factorization properties of H and show that H has sets of lengths containing large gaps. Finally we apply this result to finitely generated algebras over perfect fields with infinite divisor class group.

On the construction and the realization of wild monoids

Pavel Růžička (2018)

Archivum Mathematicum

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We develop elementary methods of computing the monoid $𝒱 (R)$ for a directly-finite regular ring $R$ . We construct a class of directly finite non-cancellative refinement monoids and realize them by regular algebras over an arbitrary field.

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 $ℤ_{b} = ℤ / b ℤ$ , 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...

К описанию моноидов, над которыми все полигоны имеют $ω$ -стабильную теорию

Т.Г. Мустафин (1990)

Algebra i Logika

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Unified-like product of monoids and its regularity property

Esra Kırmızı Çetinalp (2024)

Czechoslovak Mathematical Journal

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We first define a new monoid construction (called unified-like product $O ⋄_{Ω} J$ ) under a unified product $O ⋈ J$ and the Schützenberger product $O ⋄ J$ . We investigate whether this algebraic construction defined with operations of the unified and Schützenberger product specifies a monoid or not. Then, we obtain a presentation of this new product for any two monoids. Finally, we define the necessary and sufficient conditions for $O ⋄_{Ω} J$ to be regular.

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 : $ρ (R) = sup {m / n |u_{1} \dots u_{m} = v_{1} \dots v_{n}$ for irreducible $u_{j}, v_{i} \in R}$ . We study the elasticity of one-dimensional noetherian domains by means of the more subtle invariants $μ_{m} (R)$ defined by : $μ_{m} (R) = sup {n |u_{1} \dots u_{m} = u_{1} \dots v_{n}$ for irreducible $u_{j}, v_{i} \in R}$ . 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 minimal closed monoids for the Galois connection $End$ - $Con$

Danica Jakubíková-Studenovská, Reinhard Pöschel, Sándor Radelecki (2024)

Mathematica Bohemica

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The minimal nontrivial endomorphism monoids $M = End Con (A, F)$ of congruence lattices of algebras $(A, F)$ defined on a finite set $A$ are described. They correspond (via the Galois connection $End$ - $Con$ ) to the maximal nontrivial congruence lattices $Con (A, F)$ investigated and characterized by the authors in previous papers. Analogous results are provided for endomorphism monoids of quasiorder lattices $Quord (A, F)$ .

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.

The monoid of generalized hypersubstitutions of type τ = (n)

Wattapong Puninagool, Sorasak Leeratanavalee (2010)

Discussiones Mathematicae - General Algebra and Applications

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A (usual) hypersubstitution of type τ is a function which takes each operation symbol of the type to a term of the type, of the same arity. The set of all hypersubstitutions of a fixed type τ forms a monoid under composition, and semigroup properties of this monoid have been studied by a number of authors. In particular, idempotent and regular elements, and the Green’s relations, have been studied for type (n) by S.L. Wismath. A generalized hypersubstitution of type τ=(n) is a mapping...

Lifts for semigroups of endomorphisms of an independence algebra

João Araújo (2006)

Colloquium Mathematicae

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For a universal algebra , let End() and Aut() denote, respectively, the endomorphism monoid and the automorphism group of . Let S be a semigroup and let T be a characteristic subsemigroup of S. We say that ϕ ∈ Aut(S) is a lift for ψ ∈ Aut(T) if ϕ|T = ψ. For ψ ∈ Aut(T) we denote by L(ψ) the set of lifts of ψ, that is, $L (ψ) = {ϕ \in A u t (S) | ϕ |}_{T} = ψ$ $.$ Let be an independence algebra of infinite rank and let S be a monoid of monomorphisms such that G = Aut() ≤ S ≤ End(). It is obvious that G is characteristic...

Every braid admits a short sigma-definite expression

Jean Fromentin (2011)

Journal of the European Mathematical Society

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A result by Dehornoy (1992) says that every nontrivial braid admits a $σ$ -definite expression, defined as a braid word in which the generator $σ_{i}$ with maximal index $i$ appears with exponents that are all positive, or all negative. This is the ground result for ordering braids. In this paper, we enhance this result and prove that every braid admits a $σ$ -definite word expression that, in addition, is quasi-geodesic. This establishes a longstanding conjecture. Our proof uses the dual braid monoid...

A note on minimal zero-sum sequences over ℤ

Papa A. Sissokho (2014)

Acta Arithmetica

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A zero-sum sequence over ℤ is a sequence with terms in ℤ that sum to 0. It is called minimal if it does not contain a proper zero-sum subsequence. Consider a minimal zero-sum sequence over ℤ with positive terms $a ₁, . . ., a_{h}$ and negative terms $b ₁, . . ., b_{k}$ . We prove that h ≤ ⌊σ⁺/k⌋ and k ≤ ⌊σ⁺/h⌋, where $σ ⁺ = \sum_{i = 1}^{h} a_{i} = - \sum_{j = 1}^{k} b_{j}$ . These bounds are tight and improve upon previous results. We also show a natural partial order structure on the collection of all minimal zero-sum sequences over the set i∈ ℤ : -n ≤ i ≤ n for any positive...