Computation of the ω-primality and asymptotic ω-primality with applications to numerical semigroups

García-García, J. I.; Moreno-Frías, M. A.; Vigneron-Tenorio, A.

doi:10.1007/s11856-014-1144-6

Cited by 19 publications

(23 citation statements)

References 16 publications

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“…Example 20. Let H be the affine semigroup generated by G = {(2, 14, 2), (5, 6, 1), (7,4,4), (9,3,5), (5,5,15), (6,9,12), (3,9,7), (10,1,3), (3,6,8)}.…”

Section: Lemma 14mentioning

confidence: 99%

“…So, the set of divisor-closed submonoids of H is {{(0, 0, 0}, (5,5,15) , (2, 14, 2) , (5, 6, 1) , (10, 1, 3) , (2, 14, 2), (5, 6, 1) , (5, 6, 1), (10, 1, 3) , (5, 5, 15), (10, 1, 3) , (2, 14, 2), (5,5,15), (3,9,7), (3,6,8) , H}.…”

Section: The Equations Of the 2-faces Arementioning

confidence: 99%

“…Return {d S | S ∈ D(H)}.We now illustrate the above algorithm with two examples. 9, 0),(10,11, 0),(15,5, 0), (0, 0, 1), (10, 0, 1)}. The monoid H is isomorphic to N 5 / ∼ M with M the subgroup of Z 5 with defining equations…”

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confidence: 99%

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On divisor-closed submonoids and minimal distances in finitely generated monoids

García-García

Marín-Aragón

Moreno-Frías

2019

Journal of Symbolic Computation

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We study the lattice of divisor-closed submonoids of finitely generated cancellative commutative monoids. In case the monoid is an affine semigroup, we give a geometrical characterization of such submonoids in terms of its cone. Finally, we use our results to give an algorithm for computing ∆ * (H), the set of minimal distances of H.All semigroups, monoids and groups appearing in this paper are commutative and thus in the sequel we omit this adjective when we refer to one of these objects. We denote by Z, N, and Q the set of integers, nonnegative integers, and rationals, respectively.Non-unique Factorization Theory was initiated in 1990s, it has its origins in algebraic number theory (see [23]), and its most important goal is to classify non-uniqueness of factorization occurring for different commutative monoids and domains. Since then, the study of factorizations in integral domains and monoids has become an active area of interest, and, in particularly, a class that plays a central role in factorization theory are Krull monoids (see [21]). In [18], we can find a large amount of results concerning properties of factorizations, and a detailed study of Krull monoids. In that work, parameters and properties are defined and studied mainly from a theoretical perspective. One paper that applies algorithmic methods, such as the found in [25], for the study of finitely generated commutative monoids to compute parameters of Non-unique Factorization Theory is [27]; it is remarkable that the algorithms given are illustrated with examples of monoids where these parameters are computed. Not only [27] is in this line of study, for instance, in [6], it is given a algorithm to compute the elasticity of a monoid, the catenary degree is computed in [24], the tame degree in [7], the ω-primality in [2] and [15], and the ∆-set of a monoid in [11].This work follows the above-mentioned line, and it is motivated by the study of the set of minimal distances of a Krull monoid H with finite class group G. This parameter, denoted by ∆ * (H), was first investigated in [17], and, among other works, it is also studied in [5], [8], [9], [19], and [20]. A result of [5] allows us to determine the minimal distance of certain Krull monoids with cyclic class group. In [8], the set of minimal distances of H is studied for Krull monoids with infinite class group. It is well-known (see [18]) that if a Krull monoid H has class group G with G finite, then there is a constant M ∈ N such that all the sets of lengths are almost arithmetical multiprogressions with bound M and difference d ∈ ∆ * (H). Recently,in [20], it is proved that max ∆ * (H) ≤ {exp(G) − 2, r(G) − 1} with exp(G) and r(G) as defined in [18, Appendix A] and the equality holds if every class of G contains a prime divisor. Usually, this parameter is usually defined for Krull monoids, but in [22] the monoid of non-negative solutions of systems of linear Diophantine equations was characterized as being the same as Krull monoids with finitely generated (torsion free) quotient group. Also, usin...

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“…Example 20. Let H be the affine semigroup generated by G = {(2, 14, 2), (5, 6, 1), (7,4,4), (9,3,5), (5,5,15), (6,9,12), (3,9,7), (10,1,3), (3,6,8)}.…”

Section: Lemma 14mentioning

confidence: 99%

Section: The Equations Of the 2-faces Arementioning

confidence: 99%

See 1 more Smart Citation

On divisor-closed submonoids and minimal distances in finitely generated monoids

García-García

Marín-Aragón

Moreno-Frías

2019

Journal of Symbolic Computation

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“…Figure 3 depicts the factorizations of 450 ∈ S along with all pairwise distances. There exists a 16-chain between any two factorizations of 450; one such 16-chain between (6,2,8) and (24,3,2) is depicted with bold red edges. Since every 16-chain between these factorizations contains the edge labeled 16, we have c(450) = 16.…”

Section: The Catenary Degreementioning

confidence: 99%

Factorization invariants in numerical monoids

O’Neill¹,

Pelayo²

2017

Algebraic and Geometric Methods in Discrete Mathematics

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Nonunique factorization in commutative monoids is often studied using factorization invariants, which assign to each monoid element a quantity determined by the factorization structure. For numerical monoids (co-finite, additive submonoids of N), several factorization invariants have received much attention in the recent literature. In this survey article, we give an overview of the length set, elasticity, delta set, ω-primality, and catenary degree invariants in the setting of numerical monoids. For each invariant, we present current major results in the literature and identify the primary open questions that remain.

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“…Bounding the eventual quasilinearity of ω-primality. For numerical monoids, the ω-function admits a predictable behavior for sufficiently large values, a result that appeared as [15,Theorem 3.6] and independently as [11,Corollary 20]. Before stating this result here as Theorem 6.8, we give a definition.…”

mentioning

confidence: 99%

On dynamic algorithms for factorization invariants in numerical monoids

2016

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Abstract. Studying the factorization theory of numerical monoids relies on understanding several important factorization invariants, including length sets, delta sets, and ω-primality. While progress in this field has been accelerated by the use of computer algebra systems, many existing algorithms are computationally infeasible for numerical monoids with several irreducible elements. In this paper, we present dynamic algorithms for the factorization set, length set, delta set, and ω-primality in numerical monoids and demonstrate that these algorithms give significant improvements in runtime and memory usage. In describing our dynamic approach to computing ω-primality, we extend the usual definition of this invariant to the quotient group of the monoid and show that several useful results naturally extend to this broader setting.

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Computation of the ω-primality and asymptotic ω-primality with applications to numerical semigroups

Cited by 19 publications

References 16 publications

On divisor-closed submonoids and minimal distances in finitely generated monoids

On divisor-closed submonoids and minimal distances in finitely generated monoids

Factorization invariants in numerical monoids

On dynamic algorithms for factorization invariants in numerical monoids

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