Marly Gotti

Apple Inc.

A Puiseux monoid is an additive submonoid of the real line consisting of rationals. We say that a Puiseux monoid is reciprocal if it can be generated by the reciprocals of the terms of a strictly increasing sequence of pairwise relatively primes positive integers. We say that a commutative and cancellative (additive) monoid is atomic if every non-i...

An additive submonoid of the nonnegative cone of the real line is called a positive monoid. Positive monoids consisting of rational numbers (also known as Puiseux monoids) have been the subject of several recent papers. Moreover, those generated by a geometric sequence have also received a great deal of recent attention. Our purpose is to survey ma...

Additive submonoids of the nonnegative rationals, also known as Puiseux monoids, are not unique factorization monoids (UFMs) in general. Indeed, the only unique factorization Puiseux monoids are those generated by one element. However, even if a Puiseux monoid is not a UFM, it may contain nonzero elements having exactly one factorization. We call s...

A Puiseux monoid is an additive submonoid of the nonnegative rational numbers. If M is a Puiseux monoid, then the question of whether each nonunit element of M can be written as a sum of irreducible elements (that is, M is atomic) is surprisingly difficult. For instance, although various techniques have been developed over the past few years to ide...

A Puiseux monoid is an additive submonoid consisting of non-negative rationals. Although the operation of addition is continuous with respect to the standard topology, the set of irreducibles of a Puiseux monoid is, in general, difficult to describe. Here, we use topological density to understand how much a Puiseux monoid, as well as its set of irr...

Additive submonoids of $\mathbb{Q}_{\ge 0}$, also known as Puiseux monoids, are not unique factorization monoids (UFMs) in general. Indeed, the only unique factorization Puiseux monoids are those generated by one element. However, even if a Puiseux monoid is not a UFM, it may contain nonzero elements having exactly one factorization. We call such e...

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 molecule is a nonzero non-unit element of an integral domain (resp., commutative cancellative monoid) having a unique factorization into irreducibles (resp., atoms). Here we study the molecules of Puiseux monoids as well as the molecules of their corresponding semigroup algebras, which we call Puiseux algebras. We begin by presenting, in the cont...

A Puiseux monoid is a submonoid of $(\mathbb{Q},+)$ consisting of nonnegative rational numbers. Although the operation of addition is continuous with respect to the standard topology, the set of irreducibles of a Puiseux monoid is, in general, difficult to describe. In this paper, we use topological density to understand how much a Puiseux monoid,...

The atomic structure and some of the factorization invariants of Puiseux monoids (i.e., additive submonoids consisting of nonnegative rationals) have been investigated during the last three years. Here we offer a tour through the atomicity and factorizations of Puiseux monoids.

A Puiseux monoid is an additive submonoid of the nonnegative cone of $\mathbb{Q}$. Puiseux monoids exhibit, in general, a complex atomic structure. For instance, although various techniques have been developed in the past few years to identify subclasses of atomic Puiseux monoids, no characterization of atomic Puiseux monoids has been found so far....

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 Sr:=〈rn|n∈N0〉, where r is a positive rational. As the atom...

Most undergraduate level abstract algebra texts use \(\mathbb {Z}[\sqrt{-5}]\) as an example of an integral domain which is not a unique factorization domain (or UFD) by exhibiting two distinct irreducible factorizations of a nonzero element. But such a brief example, which requires merely an understanding of basic norms, only scratches the surface...

We study here 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 the atomic monoids of the form $S_r := \langle r^n \mid n \in \mathbb{N}_0 \rang...

In this presentation, we shall go over various monoidal and arithmetic properties of subsemirings of the field of rationals.

Introduction to the Factorization Theory of Puiseux Monoids

An element of an additive commutative cancellative monoid is a molecule if it has a unique factorization. Here we present some results on the sets of molecules of Puiseux monoids. In particular, we talk about results on the possible cardinalities of the sets of molecules and the sets of those molecules failing to be irreducibles for numerical semig...

In this paper, we study the atomic structure of Puiseux monoids generated by monotone sequences. To understand this atomicity, it is often useful to know whether the monoid is bounded, in the sense that it has a bounded generating set. We provide necessary and sufficient conditions for atomicity and boundedness to be transferred from a monotone Pui...

I present some results on the atomicity of monotone Puiseux monoids

If $P$ is an atomic monoid and $x$ is a nonzero non-unit element of $P$, 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 elasticiti...

Most undergraduate level abstract algebra texts use $\mathbb{Z}[\sqrt{-5}]$ as an example of an integral domain which is not a unique factorization domain (or UFD) by exhibiting two distinct irreducible factorizations of a nonzero element. But such a brief example, which requires merely an understanding of basic norms, only scratches the surface of...

We compute the catenary degree of elements contained in numerical monoids generated by arithmetic sequences. We find that this can be done by describing each element in terms of the cardinality of its length set and of its set of factorizations. As a corollary, we find for such monoids that the catenary degree becomes fixed on large elements. This...

We say that an element of an additive commutative cancellative monoid is a molecule if it has a unique factorization, i.e., if it can be expressed in a unique way as a sum of irreducibles. In this paper, we study the sets of molecules of Puiseux monoids (additive submonoids of the nonnegative rational numbers). In particular, we present results on...

For every labeled forest F with set of vertices [n] we can consider the subgroup G of the symmetric group S_n that is generated by all the cycles determined by all maximal paths of F. We say that G is the chain group of the forest F. In this paper we study the relation between a forest and its chain group. In particular, we find the chain groups of...

A dense Puiseux monoid is an additive submonoid of the nonnegative rationals whose topological closure is the set of nonnegative reals. It follows immediately that every Puiseux monoid failing to be dense is atomic. However, the atomic structure of dense Puiseux monoids is significantly complex. Dense Puiseux monoids can be antimatter, atomic, or a...

Let M be a commutative cancellative monoid. For m a nonunit in M, the catenary degree of m, denoted c(m), and the tame degree of m, denoted t(m), are combinatorial constants that describe the relationships between differing irreducible factorizations of m. These constants have been studied carefully in the literature for various kinds of monoids, i...