An algorithm to compute ω-primality in a numerical monoid

Anderson, David F.; Chapman, Scott T.; Kaplan, Nathan O.; Torkornoo, Desmond

doi:10.1007/s00233-010-9259-5

Cited by 26 publications

(63 citation statements)

References 7 publications

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“…By (1) we have that w n is a factorization of minimum length and, therefore, z = w n by (2). Hence α i ≥ n(r) for some i ∈ 0, N , as desired.…”

Section: Catenary Degreementioning

confidence: 75%

“…We have also proved that there exists a unique factorization of x of minimum length, which is (2). For the direct implication of (3), it suffices to observe that if α i ≥ d(r) for some i ∈ 1, N , then we can use the identity…”

Section: Catenary Degreementioning

confidence: 96%

“…Proof. To prove that (1) implies (2), suppose that r ∈ N. In this case, S r ∼ = N 0 . Since N 0 is a factorial monoid, ρ(S r ) = ρ(N 0 ) = 1.…”

Section: Catenary Degreementioning

confidence: 99%

“…To argue (2), it suffices to observe that r ∈ N implies that S r = (N 0 , +) is a factorial monoid and, therefore, ρ(S r ) = {1}.…”

Section: Catenary Degreementioning

confidence: 99%

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Factorization invariants of Puiseux monoids generated by geometric sequences

Chapman

Gotti

2019

Communications in Algebra

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We study some of the factorization invariants of the class of Puiseux monoids generated by geometric sequences, and we compare and contrast them with the known results for numerical monoids generated by arithmetic sequences. The class we study here consists of all atomic monoids of the form S r := r n | n ∈ N 0 , where r is a positive rational. As the atomic monoids S r are nicely generated, we are able to give detailed descriptions of many of their factorization invariants. One distinguishing characteristic of S r is that all its sets of lengths are arithmetic sequences of the same distance, namely |a − b|, where a, b ∈ N are such that r = a/b and gcd(a, b) = 1. We prove this, and then use it to study the elasticity and tameness of S r .

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“…By (1) we have that w n is a factorization of minimum length and, therefore, z = w n by (2). Hence α i ≥ n(r) for some i ∈ 0, N , as desired.…”

Section: Catenary Degreementioning

confidence: 75%

Section: Catenary Degreementioning

confidence: 96%

“…Proof. To prove that (1) implies (2), suppose that r ∈ N. In this case, S r ∼ = N 0 . Since N 0 is a factorial monoid, ρ(S r ) = ρ(N 0 ) = 1.…”

Section: Catenary Degreementioning

confidence: 99%

“…To argue (2), it suffices to observe that r ∈ N implies that S r = (N 0 , +) is a factorial monoid and, therefore, ρ(S r ) = {1}.…”

Section: Catenary Degreementioning

confidence: 99%

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Factorization invariants of Puiseux monoids generated by geometric sequences

Chapman

Gotti

2019

Communications in Algebra

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“…The ω(H, ·)-invariants (introduced in [17]) and the tame degrees are well-established invariants in the theory of nonunique factorizations which found much interest in recent literature (for example, see [4] for investigations in the context of integral domains, or [5] for investigations in numerical monoids). Whereas in v-noetherian monoids (these are monoids satisfying the ascending chain condition for v-ideals) we 3.…”

Section: Tame Monoids: Examples and First Propertiesmentioning

confidence: 99%

On the arithmetic of tame monoids with applications to Krull monoids and Mori domains

Geroldinger

Kainrath

2010

Journal of Pure and Applied Algebra

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Communicated by E.M. Friedlander MSC: 13A05; 13F05; 20M13 a b s t r a c tLet H be an atomic monoid (e.g., the multiplicative monoid of a noetherian domain). For an element b ∈ H, let ω(H, b) be the smallest N ∈ N 0 ∪ {∞} having the following property: if n ∈ N and a 1 , . . . , a n ∈ H are such that b divides a 1 · . . . · a n , then b already divides a subproduct of a 1 · . . . · a n consisting of at most N factors. The monoid H is called tame if sup{ω(H, u) | u is an atom of H} < ∞. This is a well-studied property in factorization theory, and for various classes of domains there are explicit criteria for being tame. In the present paper, we show that, for a large class of Krull monoids (including all Krull domains), the monoid is tame if and only if the associated Davenport constant is finite. Furthermore, we show that tame monoids satisfy the Structure Theorem for Sets of Lengths. That is, we prove that in a tame monoid there is a constant M such that the set of lengths of any element is an almost arithmetical multiprogression with bound M.

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$$\omega $$-Primality in arithmetic Leamer monoids

Chapman

Tripp

2019

Semigroup Forum

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An algorithm to compute ω-primality in a numerical monoid

Cited by 26 publications

References 7 publications

Factorization invariants of Puiseux monoids generated by geometric sequences

Factorization invariants of Puiseux monoids generated by geometric sequences

On the arithmetic of tame monoids with applications to Krull monoids and Mori domains

$$\omega $$-Primality in arithmetic Leamer monoids

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