On the local k-elasticities of Puiseux monoids

If $M$ is an atomic monoid and $x$ is a nonzero non-unit element of $M$, then the set of lengths $\mathsf{L}(x)$ of $x$ is the set of all possible lengths of factorizations of $x$, where the length of a factorization is the number of irreducible factors (counting repetitions). In a recent paper, F. Gotti and C. O'Neil studied the sets of elasticities $\mathcal{R}(P) := \{\sup \mathsf{L}(x)/\inf \mathsf{L}(x) : x \in P\}$ of Puiseux monoids $P$. Here we take this study a step forward and explore the local $k$-elasticities of the same class of monoids. We find conditions under which Puiseux monoids have all their local elasticities finite as well as conditions under which they have infinite local $k$-elasticities for sufficiently large $k$. Finally, we focus our study of the $k$-elasticities on the class of primary Puiseux monoids, proving that they have finite local $k$-elasticities if either they are boundedly generated and do not have any stable atoms or if they do not contain $0$ as a limit point

Factorization invariants of Puiseux monoids generated by geometric sequences

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 consists of all atomic monoids of the form $S_r := \langle r^n \mid n \in \mathbb{N}_0 \rangle,$ 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 $a,b \in \mathbb{N}$ are such that $r = a/b$ and $\text{gcd}(a,b) = 1$. We prove this, and then use it to study the elasticity and tameness of $S_r$.Comment: 23 pages, 3 table

Atomicity and Factorization of Puiseux Monoids

A Puiseux monoid is an additive submonoid of the nonnegative cone of rational numbers. Although Puiseux monoids are torsion-free rank-one monoids, their atomic structure is rich and highly complex. For this reason, they have been important objects to construct crucial examples in commutative algebra and factorization theory. In 1974 Anne Grams used a Puiseux monoid to construct the first example of an atomic domain not satisfying the ACCP, disproving Cohn's conjecture that every atomic domain satisfies the ACCP. Even recently, Jim Coykendall and Felix Gotti have used Puiseux monoids to construct the first atomic monoids with monoid algebras (over a field) that are not atomic, answering a question posed by Robert Gilmer back in the 1980s. This dissertation is focused on the investigation of the atomic structure and factorization theory of Puiseux monoids. Here we established various sufficient conditions for a Puiseux monoid to be atomic (or satisfy the ACCP). We do the same for two of the most important atomic properties: the finite-factorization property and the bounded-factorization property. Then we compare these four atomic properties in the context of Puiseux monoids. This leads us to construct and study several classes of Puiseux monoids with distinct atomic structure. Our investigation provides sufficient evidence to believe that the class of Puiseux monoids is the simplest class with enough complexity to find monoids satisfying almost every fundamental atomic behavior.Comment: 108 pages. arXiv admin note: text overlap with arXiv:1908.0922

On the Atomic Structure of Puiseux Monoids

In this paper, we study the atomic structure of the family of Puiseux monoids. Puiseux monoids are a natural generalization of numerical semigroups, which have been actively studied since mid-nineteenth century. Unlike numerical semigroups, the family of Puiseux monoids contains non-finitely generated representatives. Even more interesting is that there are many Puiseux monoids which are not even atomic. We delve into these situations, describing, in particular, a vast collection of commutative cancellative monoids containing no atoms. On the other hand, we find several characterization criteria which force Puiseux monoids to be atomic. Finally, we classify the atomic subfamily of strongly bounded Puiseux monoids over a finite set of primes.Comment: 21 pages. Some typos have been corrected and the exposition has been improve

On the factorization invariants of the additive structure of exponential Puiseux semirings

Exponential Puiseux semirings are additive submonoids of \qq_{\geq 0} generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we investigate some of the factorization invariants of exponential Puiseux semirings and briefly explore the connections of these properties with semigroup-theoretical invariants. Specifically, we prove that sets of lengths of atomic exponential Puiseux semirings are almost arithmetic progressions with a common bound, while unions of sets of lengths are arithmetic progressions. Additionally, we provide exact formulas to compute the catenary degrees of these monoids and show that minima and maxima of their sets of distances are always attained at Betti elements. We conclude by providing various characterizations of the atomic exponential Puiseux semirings with finite omega functions; in particular, we completely describe them in terms of their presentations.Comment: 20 pages, no figures. This version will appear in Journal of Algebra and Its Application

On strongly primary monoids, with a focus on Puiseux monoids

Primary and strongly primary monoids and domains play a central role in the ideal and factorization theory of commutative monoids and domains. It is well-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.Comment: 25 pages. It will appear in Journal of Algebr

Structural properties of subadditive families with applications to factorization theory

Let $H$ be a multiplicatively written monoid. Given $k\in{\bf N}^+$, we denote by $\mathscr U_k$ the set of all $\ell\in{\bf N}^+$ such that $a_1\cdots a_k=b_1\cdots b_\ell$ for some atoms $a_1,\ldots,a_k,b_1,\ldots,b_\ell\in H$. The sets $\mathscr U_k$ are one of the most fundamental invariants studied in the theory of non-unique factorization, and understanding their structure is a basic problem in the field: In particular, it is known that, in many cases of interest, these sets are almost arithmetic progressions with the same difference and bound for all large $k$, namely, $H$ satisfies the Structure Theorem for Unions. The present paper improves the current state of the art on this problem. More precisely, we show that, under mild assumptions on $H$, not only does the Structure Theorem for Unions hold, but there also exists $\mu\in{\bf N}^+$ such that, for every $M\in{\bf N}$, the sequences $$\bigl((\mathscr U_k-\inf\mathscr U_k)\cap[\![0,M]\!]\bigr)_{k\ge 1} \quad\text{and}\quad \bigl((\sup\mathscr U_k-\mathscr U_k)\cap[\![0,M]\!]\bigr)_{k\ge 1}$$ are $\mu$-periodic from some point on. The result applies, e.g., to (the multiplicative monoid of) all commutative Krull domains (e.g., Dedekind domains) with finite class group; a variety of weakly Krull commutative domains (including all orders in number fields with finite elasticity); some maximal orders in central simple algebras over global fields; and all numerical monoids. Large parts of the proofs are worked out in a "purely additive model", by inquiring into the properties of what we call a subadditive family, i.e., a collection $\mathscr L$ of subsets of $\bf N$ such that, for all $L_1,L_2\in\mathscr L$, there is $L\in\mathscr L$ with $L_1+L_2\subseteq L$.Comment: 22 pp., no figures. Fixed a few typos and updated statements and definitions after realizing that what is proved in the paper is slighly stronger than what claimed in the previous version. To appear in Israel Journal of Mathematic

On length densities

For a commutative cancellative monoid $M$, we introduce the notion of the length density of both a nonunit $x\in M$, denoted $\mathrm{LD}(x)$, and the entire monoid $M$, denoted $\mathrm{LD}(M)$. This invariant is related to three widely studied invariants in the theory of non-unit factorizations, $L(x)$, $\ell(x)$, and $\rho(x)$. We consider some general properties of $\mathrm{LD}(x)$ and $\mathrm{LD}(M)$ and give a wide variety of examples using numerical semigroups, Puiseux monoids, and Krull monoids. While we give an example of a monoid $M$ with irrational length density, we show that if $M$ is finitely generated, then $\mathrm{LD}(M)$ is rational and there is a nonunit element $x\in M$ with $\mathrm{LD}(M)=\mathrm{LD}(x)$ (such a monoid is said to have accepted length density). While it is well-known that the much studied asymptotic versions of $L(x)$, $\ell (x)$ and $\rho (x)$ (denoted $\overline{L}(x)$, $\overline{\ell}(x)$, and $\overline{\rho} (x)$) always exist, we show the somewhat surprising result that $\overline{\mathrm{LD}}(x) = \lim_{n\rightarrow \infty} \mathrm{LD}(x^n)$ may not exist. We also give some finiteness conditions on $M$ that force the existence of $\overline{\mathrm{LD}}(x)$

Factorization Theory in Commutative Monoids

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.Comment: Semigroup Forum, to appea

On the arithmetic of monoids of ideals

We study the algebraic and arithmetic structure of monoids of invertible ideals (more precisely, of $r$-invertible $r$-ideals for certain ideal systems $r$) of Krull and weakly Krull Mori domains. We also investigate monoids of all nonzero ideals of polynomial rings with at least two indeterminates over noetherian domains. Among others, we show that they are not transfer Krull but they share several arithmetical phenomena with Krull monoids having infinite class group and prime divisors in all classes.Comment: Prop. 5.13 replaced Example 5.13; accepted for publication in Arkiv for Matemati