On morphisms of commutative monoids

García, Juan Ignacio García; Moreno-Frías, M. A.

doi:10.1007/s00233-011-9349-z

Cited by 2 publications

(4 citation statements)

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“…A recent paper by García-García and Moreno [5] treat problems on morphisms of commutative monoids which at first sight could be thought as similar to the ones treated in this subsection, but we have not discovered any strong connection.…”

Section: Isomorphisms Of Rqns'smentioning

confidence: 65%

“…gap> fn:=NumericalSemigroupsWithFrobeniusNumber(n);; gap> nirrn:=Filtered(fn,s->not IsIrreducibleNumericalSemigroup(s));; gap> List(nirrn,s->SmallElementsOfNumericalSemigroup(s)); [ [ 0, 5, 7, 9 ], [ 0, 6,7,9 ], [ 0, 7,9 ], [ 0, 5, 6, 9 ], [ 0, 3, 6, 9 ], [ 0, 5,9 ], [ 0, 6,9 ], [ 0, 9 ] ]…”

Section: A Table Of Semigroups With Small Frobenius Numbersmentioning

confidence: 99%

“…gap> N := NumericalSemigroup(4,11,13,18);; gap> SmallElementsOfNumericalSemigroup(N); [ 0, 4,8,11,12,13,15 ] From basic results on irreducible numerical semigroups (see [6]) it follows that an irreducible numerical semigroup with 6 small elements must have Frobenius number 11 or 12. These can be computed as follows: Example 6.4. gap> n := 11;; gap> irrn := IrreducibleNumericalSemigroupsWithFrobeniusNumber(n);; gap> List(irrn,s->SmallElementsOfNumericalSemigroup(s)); [ [ 0, 5, 7, 8, 9, 10, 12 ], [ 0, 4,5,8,9,10,12 ], [ 0, 3, 6, 7, 9, 10, 12 ], [ 0, 6, 7, 8, 9, 10, 12 ], [ 0, 2, 4, 6, 8, 10, 12 ], [ 0, 4, 6, 8, 9, 10, 12 ] ] gap> n := 12;; gap> irrn := IrreducibleNumericalSemigroupsWithFrobeniusNumber(n);; gap> List(irrn,s->SmallElementsOfNumericalSemigroup(s)); [ [ 0, 7, 8, 9, 10, 11, 13 ], [ 0, 5, 8, 9, 10, 11, 13 ] ]…”

Section: A Table Of Semigroups With Small Frobenius Numbersmentioning

confidence: 99%

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Rees quotients of numerical semigroups

Delgado

Fernandes

2013

Port. Math.

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We introduce a class of finite semigroups obtained by considering Rees quotients of numerical semigroups. Several natural questions concerning this class, as well as particular subclasses obtained by considering some special ideals, are answered while others remain open. We exhibit nice presentations for these semigroups and prove that the Rees quotients by ideals of N, the positive integers under addition, constitute a set of generators for the pseudovariety of commutative and nilpotent semigroups.

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Section: Isomorphisms Of Rqns'smentioning

confidence: 65%

Section: A Table Of Semigroups With Small Frobenius Numbersmentioning

confidence: 99%

Section: A Table Of Semigroups With Small Frobenius Numbersmentioning

confidence: 99%

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Rees quotients of numerical semigroups

Delgado

Fernandes

2013

Port. Math.

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“…Since N r+k is free, the map π : N r+k → Z d1 × · · · × Z dr × N k is a monoid morphism (see [14]) verifying that π(a * j ) = a * j for all 1 ≤ j ≤ r + k. Therefore π(H) = H and π |H : H → H is a monoid morphism. Note that for every subset J ⊂ H we have π −1 |H (J) = π −1 (J)∩H.…”

mentioning

confidence: 99%

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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On morphisms of commutative monoids

Cited by 2 publications

References 3 publications

Rees quotients of numerical semigroups

Rees quotients of numerical semigroups

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

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