The computation of factorization invariants for affine semigroups

García-Sánchez, Pedro A.; O’Neill, Christopher; Webb, Gautam

doi:10.1142/s0219498819500191

Cited by 13 publications

(11 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This point of view prepared the ground for the development of algorithms computing arithmetical invariants in finitely generated monoids (we refer to [37] for a survey, and to [36,78] for a sample of further work in this direction). In particular, for numerical monoids there is a wealth of papers providing algorithms for determining arithmetical invariants and in some cases there are even precise values (formulas) for arithmetical invariants (in terms of the atoms or of other algebraic invariants; [25,38]). A further class of objects, for which precise formulas for arithmetical invariants are available, will be discussed in Section 6.…”

Section: Finitely Generated Monoidsmentioning

confidence: 99%

Factorization theory in commutative monoids

Geroldinger

Zhong

2020

Semigroup Forum

View full text Add to dashboard Cite

This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.

show abstract

Section: Finitely Generated Monoidsmentioning

confidence: 99%

Factorization theory in commutative monoids

Geroldinger

Zhong

2020

Semigroup Forum

View full text Add to dashboard Cite

show abstract

“…Given how central the functions that compute Z(n) and L(n) are, these functions have undergone numerous improvements since the early days of the numericalsgps package, and now run surprisingly fast even for reasonably large input. McN); [ 3,7,8,9,10 ] gap> LengthsOfFactorizationsElementWRTNumericalSemigroup(150, McN); [ 10,11,13,14,15,16,17,18,19,20,21,22,23,24,25 ] The numericalsgps package can also compute delta sets, both of numerical semigroups and of their elements. The original implementation of the latter function used Theorem 5 to compute the delta set of every element up to N, and only more recently was a more direct algorithm developed [14].…”

Section: Using Software To Guide Mathematical Inquisitionmentioning

confidence: 99%

“….60], n -> (n in S)); [ 7,10,12,14,17,19,20 Well-chosen plots can be an incredibly effective tool for visualizing such data. Figure 4 depicts the values output above; the repeating pattern in the right half of the plot is undeniable.…”

Section: Using Software To Guide Mathematical Inquisitionmentioning

confidence: 99%

Beyond Coins, Stamps, and Chicken McNuggets: An Invitation to Numerical Semigroups

Chapman

Garcia

O’Neill

2020

Foundations for Undergraduate Research in Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Clearly, the class C contains all affine monoids. Computational aspects of affine monoids and factorization invariants of halffactorial affine monoids have been studied in [18] and [17], respectively. Diophantine monoids form a special subclass of that one consisting of affine monoids and has been studied in [9].…”

Section: Monoids In Cmentioning

confidence: 99%

The system of sets of lengths and the elasticity of submonoids of a finite-rank free commutative monoid

Gotti

2019

J. Algebra Appl.

View full text Add to dashboard Cite

Every torsion-free atomic monoid M can be embedded into a real vector space via the inclusion M → gp(M ) → R⊗ Z gp(M ), where gp(M ) is the Grothendieck group of M . Let C be the class consisting of all submonoids (up to isomorphism) that can be embedded in a finite-rank free commutative monoid. Here we investigate how the atomic structure and factorization properties of members of C reflect in the combinatorics and geometry of their conic hulls cone(M ) ⊆ R ⊗ Z gp(M ). First, we establish geometric characterizations in terms of cone(M ) for a monoid M in C to be factorial, half-factorial, and other-half-factorial. Then we show that the submonoids of M determined by the faces of cone(M ) amount for all divisor-closed submonoids of M . Finally, we investigate the cones of finitary, primary, finitely primary, and strongly primary monoids in C (monoids in these classes have been relevant in the development of factorization theory). Along the way, we study the cones that can be realized by monoids in C and by finitary monoids in C.An atomic monoid M is half-factorial (or an HFM) provided that for all noninvertible x ∈ M , any two factorizations of x have the same number of irreducibles (counting repetitions). In addition, an integral domain is half-factorial (or an HFD) if its multiplicative monoid is an HFM. The concept of half-factoriality was first investigated by L. Carlitz in the context of algebraic number fields; he proved that an algebraic number field is an HFD if and only if its class group has size at most two [6]. However, the term "half-factorial domain" is due to A. Zaks [45]. In [46], Zaks studied Krull domains that are HFDs in terms of their divisor class groups. Parallel to this, L. Skula [42] and J.Śliwa [43], motivated by some questions of W. Narkiewicz on algebraic number theory [39, Chapter 9], carried out systematic studies of HFDs. Since then HFMs and HFDs have been actively studied (see [7] and references therein).Other-half-factoriality, on the other hand, is a dual version of half-factoriality, and it was introduced by J. Coykendall and W. Smith [12]. An atomic monoid M is called other-half-factorial (or an OHFM) provided that for all non-invertible x ∈ M no two distinct factorizations of x in M contain the same number of irreducibles (counting repetitions). Although an integral domain is a UFD if and only if its multiplicative monoid is an OHFM [12, Corollary 2.11], OHFMs are not always factorial or halffactorial, even in the class C. In the second part of this paper, we offer geometric and combinatorial characterizations for the HFMs in C and for the OHFMs in C.The study of primary monoids was initiated by T. Tamura [44] and M. Petrich [40] in the 1970s and has received a great deal of attention since then [34,35,20]. Primary monoids naturally appear in commutative algebra: an integral domain is 1-dimensional and local if and only if its multiplicative monoid is primary. One of the most useful subclasses of primary monoids in factorization theory is that one consisting of finitely primary ...

show abstract

The computation of factorization invariants for affine semigroups

Cited by 13 publications

References 26 publications

Factorization theory in commutative monoids

Factorization theory in commutative monoids

Beyond Coins, Stamps, and Chicken McNuggets: An Invitation to Numerical Semigroups

The system of sets of lengths and the elasticity of submonoids of a finite-rank free commutative monoid

Contact Info

Product

Resources

About