# Felix GottiMassachusetts Institute of Technology | MIT · Department of Mathematics

Felix Gotti

Ph.D. in Mathematics

## About

73

Publications

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730

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Introduction

Felix Gotti currently works at the Department of Mathematics of MIT. Felix does research in commutative algebra, algebraic combinatorics, and arithmetic of semigroups.

**Skills and Expertise**

Additional affiliations

September 2020 - present

July 2020 - present

Education

August 2014 - May 2019

## Publications

Publications (73)

Let M be an atomic monoid and let x be a non-unit element of M. The elasticity of x, denoted by r(x), is the ratio of its largest factorization length to its shortest factorization length, and it measures how far is x from having a unique factorization. The elasticity r(M) of M is the supremum of the elasticities of all non-unit elements of M. The...

It is well known that the number of non-isomorphic unit interval orders on [n] equals the n-th Catalan number. Using work of Skandera and Reed and work of Postnikov, we show that each unit interval order on [n] naturally induces a rank n positroid on [2n]. We call the positroids produced in this fashion unit interval positroids. We characterize the...

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...

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...

Let M be a commutative cancellative monoid. The set Δ(M), which consists of all positive integers which are distances between consecutive factorization lengths of elements in M, is a widely studied object in the theory of nonunique factorizations. If M is a Krull monoid with cyclic class group of order n ≥ 3, then it is well-known that Δ (M) ⊆ {1,…...

An integral domain is called atomic if every nonzero nonunit element factors into irreducibles. On the other hand, an integral domain is said to satisfy the ascending chain condition on principal ideals (ACCP) if every ascending chain of principal ideals stabilizes. It was asserted by P. Cohn back in the sixties that every atomic domain satisfies t...

A cancellative and commutative monoid $M$ is atomic if every non-invertible element of $M$ factors into irreducibles (also called atoms), and $M$ is hereditarily atomic if every submonoid of $M$ is atomic. In addition, $M$ is hereditary ACCP if every submonoid of $M$ satisfies the ascending chain condition on principal ideals (ACCP). Our primary pu...

An integral domain $R$ is atomic if each nonzero nonunit of $R$ factors into irreducibles. In addition, an integral domain $R$ satisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. Although it is not hard to verify that every integral...

If every subring of an integral domain is atomic, we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind domains in terms of atomic overrings. Then we use this characterization to determine the fields that are hereditarily...

An integral domain D is called an irreducible-divisor-finite domain (IDF-domain) if every nonzero element of D has finitely many irreducible divisors up to associates. The study of IDF-domains dates back to the seventies. In this paper, we investigate various aspects of the IDF property. In 2009, P. Malcolmson and F. Okoh proved that the IDF proper...

A subset $S$ of an integral domain $R$ is called a semidomain provided that the pairs $(S,+)$ and $(S, \cdot)$ are semigroups with identities. The study of factorizations in integral domains was initiated by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990, and this area has been systematically investigated since then. In this presentation,...

In this paper, we study factorizations in the additive monoids of positive algebraic valuations N0[α] of the semiring of polynomials N0[X] using a methodology introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. We begin by dete...

A subset S of an integral domain R is called a semidomain provided that the pairs (S, +) and (S, ·) are semigroups with identities. The study of factorizations in integral domains was initiated in 1990, and this area has been systematically investigated since then. In this paper, we study the divisibility and arithmetic of factorizations in the mor...

An integral domain $D$ is called an irreducible-divisor-finite domain (IDF-domain) if every nonzero element of $D$ has finitely many irreducible divisors up to associates. The study of IDF-domains dates back to the seventies. In this paper, we investigate various aspects of the IDF property. In 2009, P. Malcolmson and F. Okoh proved that the IDF pr...

If every subring of an integral domain is atomic, then we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind domains in terms of atomic overrings. Then we use this characterization to determine the fields that are heredit...

An integral domain is called atomic if every nonzero nonunit element factors into irreducibles. On the other hand, an integral domain is said to satisfy the ascending chain condition on principal ideals (ACCP) if every ascending chain of principal ideals terminates. It was asserted by Cohn back in the sixties that every atomic domain satisfies the...

A subsemiring S of \(\mathbb {R}\) is called a positive semiring provided that S consists of nonnegative numbers and \(1 \in S\). Here we study factorizations in both the additive monoid \((S,+)\) and the multiplicative monoid \((S\backslash \{0\}, \cdot )\). In particular, we investigate when, for a positive semiring S, both \((S,+)\) and \((S\bac...

In this paper, we address various aspects of divisibility by irreducibles in rings consisting of integer-valued polynomials. An integral domain is called atomic if every nonzero nonunit factors into irreducibles. Atomic domains that do not satisfy the ascending chain condition on principal ideals (ACCP) have proved to be elusive, and not many of th...

In this paper, a semigroup algebra consisting of polynomial expressions with coefficients in a field F and exponents in an additive submonoid M of Q≥0 is called a Puiseux algebra and denoted by F[M]. Here we study the atomic structure of Puiseux algebras. To begin with, we answer the isomorphism problem for the class of Puiseux algebras, that is, w...

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...

Let $S$ be a nonnegative semiring of the real line, called here a positive semiring. We study factorizations in both the additive monoid $(S,+)$ and the multiplicative monoid $(S\setminus\{0\}, \cdot)$. In particular, we investigate when, for a positive semiring $S$, both $(S,+)$ and $(S\setminus\{0\}, \cdot)$ have the following properties: atomici...

An atomic monoid M is called a length-factorial monoid (or an other-half-factorial monoid) if for each non-invertible element x∈M no two distinct factorizations of x have the same length. The notion of length-factoriality was introduced by Coykendall and Smith in 2011 as a dual of the well-studied notion of half-factoriality. They proved that in th...

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...

An atomic monoid $M$ is called a length-factorial monoid (or an other-half-factorial monoid) if for each non-invertible element $x \in M$ no two distinct factorizations of $x$ have the same length. The notion of length-factoriality was introduced by Coykendall and Smith in 2011 as a dual of the well-studied notion of half-factoriality. They proved...

An integral domain is atomic if every nonzero nonunit factors into irreducibles. Let R be an integral domain. We say that R is a bounded factorization domain if it is atomic and for every nonzero nonunit \(x \in R\), there is a positive integer N such that for any factorization \(x = a_1 \cdots a_n\) of x into irreducibles \(a_1, \dots , a_n\) in R...

An integral domain is atomic if every nonzero nonunit factors into irreducibles. Let $R$ be an integral domain. We say that $R$ is a bounded factorization domain if it is atomic and for every nonzero nonunit $x \in R$, there is a positive integer $N$ such that for any factorization $x = a_1 \cdots a_n$ of $x$ into irreducibles $a_1, \dots, a_n$ in...

In this paper, we study factorizations in the additive monoids of positive algebraic valuations $\mathbb{N}_0[\alpha]$ of the semiring of polynomials $\mathbb{N}_0[X]$ using a methodology introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. A cancellative commutative monoid is atomic if every non-invertible element factors into i...

An integral domain (or a commutative cancellative monoid) is atomic if every nonzero nonunit element factors into atoms, and it satisfies the ACCP if every ascending chain of principal ideals eventually stabilizes. The interplay between these two properties has been investigated since the 1970s. An atomic domain (or monoid) satisfies the finite fac...

Every torsion-free atomic monoid $M$ can be embedded into a real vector space via the inclusion $M \hookrightarrow \text{gp}(M) \hookrightarrow \mathbb{R} \otimes_{\mathbb{Z}} \text{gp}(M)$, where $\text{gp}(M)$ is the Grothendieck group of $M$. Let $\mathcal{C}$ be the class consisting of all submonoids (up to isomorphism) that can be embedded in...

For an integral domain R and a commutative cancellative monoid M, the ring consisting of all polynomial expressions with coefficients in R and exponents in M is called the monoid ring of M over R. An integral domain R is called atomic if every nonzero nonunit element can be written as a product of irreducibles. In the study of the atomicity of inte...

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...

An integral domain (or a commutative cancellative monoid) is atomic if every nonzero nonunit element is the product of irreducibles, and it satisfies the ACCP if every ascending chain of principal ideals eventually stabilizes. The interplay between these two properties has been investigated since the 1970s. An atomic domain (or monoid) satisfies th...

Here we discuss a variety of algebraic property that can be transfer from both a monoid and an integral domain to the monoid algebra they determine.

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...

This thesis is a compendium of three studies on which matroids and convex geometry play a central role and show their connections to Catalan combinatorics, tiling theory, and factorization theory. First, we study positroids in connection with rational Dyck paths. Then, we study certain matroids on the lattice points of a regular triangle in connect...

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...

Let M be a commutative cancellative monoid, and let R be an integral domain. The question of whether the monoid ring R[x;M] is atomic provided that both M and R are atomic dates back to the 1980s. In 1993, Roitman gave a negative answer to the question for M=N0: he constructed an atomic integral domain R such that the polynomial ring R[x] is not at...

Let $M$ be a commutative cancellative monoid, and let $R$ be an integral domain. The question of whether the monoid ring $R[x;M]$ is atomic provided that both $M$ and $R$ are atomic dates back to the 1980s. In 1993, Roitman gave a negative answer to the question for $M = \mathbb{N}_0$: he constructed an atomic integral domain $R$ such that the poly...

Let [Formula: see text] be an atomic monoid. For [Formula: see text], let [Formula: see text] denote the set of all possible lengths of factorizations of [Formula: see text] into irreducibles. The system of sets of lengths of [Formula: see text] is the set [Formula: see text]. On the other hand, the elasticity of [Formula: see text], denoted by [Fo...

For an integral domain $R$ and a commutative cancellative monoid $M$, the ring consisting of all polynomial expressions with coefficients in $R$ and exponents in $M$ is called the monoid ring of $M$ over $R$. An integral domain is called atomic if every nonzero nonunit element can be written as a product of irreducibles. In the investigation of the...

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...

Here we present two classes of non-atomic monoid algebras with atomic exponent monoids, answering a question that Robert Gilmer asked on the 1980's.

We discuss a question by R. Gilmer on the atomicity of monoid domains. Then we present two classes of monoid algebras, both giving negative answers to the Gilmer's question.

We exhibit some algebraic and factorization properties of monoid algebras of commutative cancellative torsion-free monoids. Given a field F, we focus, in particular, on properties that can be transferred from a monoid M to its corresponding monoid algebra F[M]. Special attention is given to monoid algebras of rank-1 monoids.

For a field F and a monoid M, the algebra of all polynomial expressions with coefficients in F and exponents in M is called the monoid algebra of M over F and is denoted by F[X; M]. The problem of determining conditions on M and F such that the algebra F[X;M] satisfies certain specified condition was first studied by Robert Gilmer and then by many...

Positroids are representable matroids inside the totally nonnegative Grassmannian; they have received significant attention since they were introduced by Postnikov in 2006. Even though positroids were brought to light only recently, they have found connections to cluster algebras, mirror symmetry, and free probability. On the other hand, unit inter...

An overview on rational Dyck positroids and their related combinatorial objects.

Let $H$ be an atomic monoid. For $x \in H$, let $\mathsf{L}(x)$ denote the set of all possible lengths of factorizations of $x$ into irreducibles. The system of sets of lengths of $H$ is the set $\mathcal{L}(H) = \{\mathsf{L}(x) \mid x \in H\}$. On the other hand, the elasticity of $x$, denoted by $\rho(x)$, is the quotient $\sup \mathsf{L}(x)/\inf...

Here we present a connection between tilings and certain matroids whose ground sets consist of the lattice points of a regular simplex. We provide characterizations for the independent sets, the circuits, and the flats in terms of tilings. We also use tilings to study the rank function and the connectedness of the presented matroids.

Here we present a few results on the elasticity and the sets of lengths of submonoids of a finite-rank free abelian monoid (i.e., generalized affine semigroups). A generalized affine semigroup with full system of sets of lengths is constructed. The Characterization Problem for the family of generalized affine semigroups is answered negatively. Then...

Here we talk about some recent results on the system of sets of lengths of Puiseux monoids.

In this paper we investigate a family of matroids introduced by Ardila and Billey to study one-dimensional intersections of complete flag arrangements of C n. The set of lattice points T n,d inside the regular simplex obtained by intersecting the nonnegative cone of R d with the affine hyperplane x 1 + · · · + x d = n − 1 is the ground set of a mat...

In this paper we investigate a family of matroids introduced by Ardila and Billey to study one-dimensional intersections of complete flag arrangements of $\mathbb{C}^n$. The set of lattice points $T_{n,d}$ inside the regular simplex obtained by intersecting the nonnegative cone of $\mathbb{R}^d$ with the affine hyperplane $x_1 + \dots + x_d = n-1$...

In this paper, we construct three families of semigroup algebras using Puiseux monoids, recently-studied additive submonoids of $\mathbb{Q}$. The first family consists of semigroup algebras of certain atomic Puiseux monoids called primary. Properties of primary Puiseux monoids will allow us to show that the semigroup algebras they determine are ato...

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...

In this paper we study the system of sets of lengths of non-finitely generated atomic Puiseux monoids (a Puiseux monoid is an additive submonoid of $\mathbb{Q}_{\ge 0}$). We begin by presenting a BF-monoid $M$ with full system of sets of lengths, which means that for each subset $S$ of $\mathbb{Z}_{\ge 2}$ there exists an element $x \in M$ whose se...

A rational Dyck path of type (m,d) is an increasing unit-step lattice path from (0,0) to (m,d) that never goes above the diagonal line y = (d/m)x. On the other hand, a positroid of rank d on the ground set [d+m] is a special type of matroid coming from the totally nonnegative Grassmannian. In this paper we describe how to naturally assign a rank d...

There are several families of atomic monoids whose arithmetical invariants have received a great deal of attention during the last two decades. In particular, the factorization theory of finitely generated monoids, Krull monoids, and C-monoids has been systematically studied. Puiseux monoids, which are additive submonoids consisting of nonnegative...

Here we present a class of positroids that can be naturally produced using the class of unit interval orders. We call such positroids unit interval positroids. We show how the canonical interval representation of a unit interval order P and the canonical antiadjacency matrix of P encode the same Dyck path, which we call the Dyck path associated to...

It is well known that the number of non-isomorphic unit interval orders on [n] equals the n-th Catalan number. Using work of Skandera and Reed and work of Postnikov, we show that each unit interval order on [n] naturally induces a rank n positroid on [2n]. We call the positroids produced in this fashion unit interval positroids. We characterize the...

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...

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid (Formula presented.), consider the family of “shifted” monoids (Formula presented.) obtained by adding (Formula presented.) to each generator of (Formula presented.). In this paper, we examine minimal relations among the generators of (Formula present...

It is well known that the number of non-isomorphic unit interval orders on [n] equals the n-th Catalan number. Using work of Skandera and Reed and work of Postnikov, we show that each unit interval order on [n] naturally induces a rank n positroid on [2n]. We call the positroids produced in this fashion unit interval positroids. We characterize the...

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 rational Dyck path of type (m,d) is an increasing unit-step lattice path from (0,0) to (m,d) in Z^2 that never goes above the diagonal line y = (d/m)x. On the other hand, a positroid of rank d on the ground set [d+m] is a special type of matroid coming from the totally nonnegative Grassmannian. In this paper we describe how to naturally assign a...

Given an ambient ordered field K, a positive monoid is a countably generated additive submonoid of the nonnegative cone of K. In this paper, we first generalize a few atomic features exhibited by Puiseux monoids of the field of rational numbers to the more general setting of positive monoids of Archimedean fields, accordingly arguing that such gene...

We give an overview of some properties of the atomic configuration of certain subfamilies of Puiseux monoids that can be characterized by order and the topology of the same monoids.

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...

Here we offer a brief introduction to numerical semigroups and some of their non-unique factorization properties. The purpose of this presentation is completely motivational, and it is mainly addressed to undergraduate students having little or no previous exposure to factorization theory.

## Questions

Question (1)

Definitions of atomic monoid, factorization, and lengths can be found in https://arxiv.org/pdf/1711.06961.pdf.

## Projects

Projects (3)

Understand the combinatorial structure of positroids and matroids, as well as find connections between them and other families of combinatorial objects.