Systems of sets of lengths of Puiseux monoids

Gotti, Felix

doi:10.1016/j.jpaa.2018.08.004

Cited by 29 publications

(16 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These two papers generated a flood of work in this area, from both the pure [1,8,9,10,12,13,14,41] and the computational [5,2,16,17] points of view. Over the past three years, similar studies have emerged for additive submonoids of the nonnegative rational numbers, also known as Puiseux monoids [26,28,30,31,32,33].…”

Section: Prologuementioning

confidence: 99%

See 1 more Smart Citation

Factorization invariants of Puiseux monoids generated by geometric sequences

Chapman

Gotti

2019

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

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 .

show abstract

Section: Prologuementioning

confidence: 99%

“…The system of sets of lengths of numerical monoids has been studied in [1] and [23], while the system of sets of lengths of Puiseux monoids was first studied in [31]. In addition, a friendly introduction to sets of lengths and the role they play in factorization theory is surveyed in [19].…”

Section: Factorization Invariantsmentioning

confidence: 99%

Factorization invariants of Puiseux monoids generated by geometric sequences

Chapman

Gotti

2019

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…Hence [26,Proposition 3.2] guarantees that M is in C if and only if M is finitely generated. As a result, non-finitely generated submonoids of (Q ≥0 , +) such as 1/p | p is prime are finite-rank torsion-free monoids that do not belong to the class C. The atomic and factorization structures of submonoids of (Q ≥0 , +) have been fairly considered lately; see, for instance, [28,29,30]. Clearly, the Grothendieck group of a non-finitely generated submonoid of (Q ≥0 , +) cannot be free.…”

Section: The Cones Of 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.

Self Cite

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

“…Systems of sets of lengths have been studied in the context of Krull monoids [14], C-monoids [8], affine monoids [10], and submonoids of N d [19] (see also [12,7] for some recent work). Given that Puiseux monoids started receiving attention just a few years ago, little is known about its system of sets of lengths (for recent progress see [16,21]). In [3,Theorem 3.3], Chapman et al prove that the system of sets of lengths of a rational cyclic semiring is an arithmetic sequence.…”

Section: Introductionmentioning

confidence: 99%

On the sets of lengths of Puiseux monoids generated by multiple geometric sequences

Polo

2020

Preprint

View full text Add to dashboard Cite

It is well known that a Puiseux monoid M is atomic if and only if M contains a minimal set of generators. This characterization, whose verification involves a great deal of effort given that Puiseux monoids are not, in general, finitely generated, motivated a series of papers studying the atomicity of Puiseux monoids generated by structured sets such as monotone and geometric sequences. In this paper, we focus on the study of rational multicyclic monoids, that is, Puiseux monoids generated by multiple geometric sequences. In particular, we provide a complete description of the rational multicyclic monoids that are hereditarily atomic. Additionally, we show that the sets of lengths of certain family of atomic rational multicyclic monoids are finite unions of infinite multidimensional arithmetical progressions, a result we use to realize infinite arithmetic progressions as delta sets of some Puiseux monoids.

show abstract

Systems of sets of lengths of Puiseux monoids

Cited by 29 publications

References 24 publications

Factorization invariants of Puiseux monoids generated by geometric sequences

Factorization invariants of Puiseux monoids generated by geometric sequences

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

On the sets of lengths of Puiseux monoids generated by multiple geometric sequences

Contact Info

Product

Resources

About