Arithmetical invariants of local quaternion orders

Baeth, Nicholas R.; Smertnig, Daniel

doi:10.4064/aa170601-13-8

Cited by 12 publications

(61 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Wishing to have a full picture of the sets of elasticities of cyclic rational semirings, we propose the following conjecture. [4,7,15,43]. In particular, the unions of sets of lengths and the local elasticities of Puiseux monoids have been considered in [34].…”

Section: Catenary Degreementioning

confidence: 99%

Factorization invariants of Puiseux monoids generated by geometric sequences

Chapman

Gotti

2019

Communications in Algebra

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: Catenary Degreementioning

confidence: 99%

Factorization invariants of Puiseux monoids generated by geometric sequences

Chapman

Gotti

2019

Communications in Algebra

View full text Add to dashboard Cite

show abstract

“…4. In a forthcoming paper [7], Baeth and Smertnig verify the Structure Theorem for Unions for local quaternion orders. Their result and the present Theorem 3.6 reveal the first non-commutative monoids H for which it could be shown that L(H) satisfies the Structure Theorem for Unions without showing that L(H) = L(B) for some commutative monoid B (see also Corollary 4.4).…”

Section: Remark 37mentioning

confidence: 94%

Sets of lengths in atomic unit-cancellative finitely presented monoids

Geroldinger

Schwab

2018

Colloq. Math.

View full text Add to dashboard Cite

For an element a of a monoid H, its set of lengths L(a) ⊂ N is the set of all positive integers k for which there is a factorization a = u 1 ·. . .·u k into k atoms. We study the system L(H) = {L(a) | a ∈ H} with a focus on the unions U k (H) ⊂ N which are the unions of all sets of lengths containing a given k ∈ N. The Structure Theorem for Unions -stating that for all sufficiently large k, the sets U k (H) are almost arithmetical progressions with the same difference and global bound -has found much attention for commutative monoids and domains. We show that it holds true for the not necessarily commutative monoids in the title satisfying suitable algebraic finiteness conditions. Furthermore, we give an explicit description of the system of sets of lengths of monoids Bn = a, b | ba = b n for n ∈ N ≥2 . Based on this description, we show that the monoids Bn are not transfer Krull, which implies that their systems L(Bn) are distinct from systems of sets of lengths of commutative Krull monoids and others.2010 Mathematics Subject Classification. 20M13, 20M05, 13A05. Key words and phrases. sets of lengths, unions of sets of lengths, finitely presented monoids, Möbius monoid.

show abstract

“…The structure of unions of sets of lengths has been studied for a wide range of monoids and domains (see [14,4,15,52] for recent progress).…”

Section: Note That Inf{αmentioning

confidence: 99%

On strongly primary monoids, with a focus on Puiseux monoids

Geroldinger,

Gotti,

Tringali

2019

Preprint

View full text Add to dashboard Cite

Primary and strongly primary monoids and domains play a central role in the ideal and factorization theory of commutative monoids and domains. Among others, it is known that primary monoids satisfying the ascending chain condition on divisorial ideals (e.g., numerical monoids) are strongly primary; and the multiplicative monoid of non-zero elements of a one-dimensional local domain is primary and it is strongly primary if the domain is noetherian. In the present paper, we focus on the study of additive submonoids of the non-negative rationals, called Puiseux monoids. It is easy to see that Puiseux monoids are primary monoids, and we provide conditions ensuring that they are strongly primary. Then we study local and global tameness of strongly primary Puiseux monoids; most notably, we establish an algebraic characterization of when a Puiseux monoid is globally tame. Moreover, we obtain a result on the structure of sets of lengths of all locally tame strongly primary monoids.

show abstract

Arithmetical invariants of local quaternion orders

Cited by 12 publications

References 20 publications

Factorization invariants of Puiseux monoids generated by geometric sequences

Factorization invariants of Puiseux monoids generated by geometric sequences

Sets of lengths in atomic unit-cancellative finitely presented monoids

On strongly primary monoids, with a focus on Puiseux monoids

Contact Info

Product

Resources

About