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Gotti Marly - One of the best experts on this subject based on the ideXlab platform.
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Factorizations in reciprocal Puiseux Monoids
2021Co-Authors: Aguilera Cecilia, Gotti Marly, Hamelberg, Andre F.Abstract: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-invertible element $x$ can be written as a sum of irreducibles. The number of irreducibles in this sum is called a length of $x$. In this paper, we identify and investigate generalized classes of reciprocal Puiseux Monoids that are atomic. Moreover, for the atomic Monoids in the identified classes, we study the ascending chain condition on principal ideals and also the sets of lengths of their elements.Comment: 16 page
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Atomicity and Density of Puiseux Monoids
2020Co-Authors: Bras-amorós Maria, Gotti MarlyAbstract: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, as well as its set of irreducibles, spread through $\mathbb{R}_{\ge 0}$. First, we separate Puiseux Monoids according to their density in $\mathbb{R}_{\ge 0}$, and we characterize Monoids in each of these classes in terms of generating sets and sets of irreducibles. Then we study the density of the difference group, the root closure, and the conductor semigroup of a Puiseux Monoid. Finally, we prove that every Puiseux Monoid generated by a strictly increasing sequence of rationals is nowhere dense in $\mathbb{R}_{\ge 0}$ and has empty conductor.Comment: 14 page
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When is a Puiseux Monoid atomic?
2020Co-Authors: Chapman, Scott T., Gotti Felix, Gotti MarlyAbstract:A Puiseux Monoid is an additive subMonoid of the nonnegative rational numbers. If $M$ is a Puiseux Monoid, then the question of whether each non-invertible element of $M$ can be written as a sum of irreducible elements (that is, $M$ is atomic) is surprisingly difficult. Although various techniques have been developed over the past few years to identify subclasses of Puiseux Monoids that are atomic, no general characterization of such Monoids is known. Here we survey some of the most relevant aspects related to the atomicity of Puiseux Monoids. We provide characterizations of when $M$ is finitely generated, factorial, half-factorial, other-half-factorial, Pr\"ufer, seminormal, root-closed, and completely integrally closed. In addition to the atomicity, characterizations are also not known for when $M$ satisfies the ACCP, the bounded factorization property, or the finite factorization property. In each of these cases, we construct an infinite class of Puiseux Monoids satisfying the corresponding property.Comment: 24 pages; the previous version has been rewritten in a more friendly way. This version will appear in the American Mathematical Monthl
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On the set of molecules of numerical and Puiseux Monoids
2020Co-Authors: Gotti Marly, Tirador, Marcos M.Abstract: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 elements molecules. Molecules were first investigated by W. Narkiewicz in the context of algebraic number theory. More recently, F. Gotti and the first author studied molecules in the context of Puiseux Monoids. Here we address some aspects related to the size of the sets of molecules of various subclasses of Puiseux Monoids with different atomic behaviors. In particular, we positively answer the following recent realization conjecture: for each $m \in \mathbb{N}_{\ge 2}$ there exists a numerical Monoid whose set of molecules that are not atoms has cardinality $m$.Comment: 15 page
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Atomicity and Factorization of Puiseux Monoids
2020Co-Authors: Gotti MarlyAbstract: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
M Sularia - One of the best experts on this subject based on the ideXlab platform.
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on a class of residuated semilattice Monoids
Fuzzy Sets and Systems, 2003Co-Authors: P Flondor, M SulariaAbstract:In this paper, we consider the structure of residuated join semilattice Monoid and we study its first properties. The class of residuated join semilattice Monoids is a variety of algebras. A representation of this structure as residuated mappings is given. Then we present the notion of normal filter together with a characterization theorem of homomorphic images based on this notion. Different classes of residuated join semilattice Monoids are introduced.
Nyberg-brodda Carl-fredrik - One of the best experts on this subject based on the ideXlab platform.
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On the Word Problem for Special Monoids
2021Co-Authors: Nyberg-brodda Carl-fredrikAbstract:A Monoid is special if it admits a presentation in which all the defining relations are of the form $w = 1$. In this article, we study the word problem, in the sense of Duncan & Gilman, for special Monoids, and relate the language-theoretic properties of this set to the word problem for the group of units of the Monoid, by developing the theory of the words representing invertible words. We show that a special Monoid has regular word problem if and only if it is a finite group. When $\mathcal{C}$ is a class of languages satisfying some restrictions modelled on the context-free languages, we show that a special Monoid has word problem in $\mathcal{C}$ if and only if its group of units has word problem in $\mathcal{C}$. As a corollary, we generalise the Muller-Schupp theorem to special Monoids: a finitely presented special Monoid has context-free word problem if and only if its group of units is virtually free. This completely answers a question from 1993 of Book & Otto, and for the class of special Monoids answers a question from 2004 of Duncan & Gilman. We show that it is decidable whether a one-relator special Monoid has context-free word problem. Finally, by proving that any context-free Monoid has decidable rational subset membership problem, we also obtain a large class of special Monoids for which this problem is decidable.Comment: 31 page
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On the Word Problem for Special Monoids
2021Co-Authors: Nyberg-brodda Carl-fredrikAbstract:A Monoid is called special if it admits a presentation in which all defining relations are of the form $w = 1$. Every group is special, but not every Monoid is special. In this article, we describe the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, for special Monoids in terms of their group of units. We prove that a special Monoid has context-free word problem if and only if its group of units is virtually free, giving a full generalisation of the Muller-Schupp theorem. This fully answers, for the class of special Monoids, a question posed by Duncan & Gilman in 2004. We describe the congruence classes of words in a special Monoid, and prove that these have the same language-theoretic properties as the word problem. This answers a question first posed by Zhang in 1992. As a corollary, we prove that it is decidable (in polynomial time) whether a special one-relation Monoid has context-free word problem. This completely answers another question from 1992, also posed by Zhang.Comment: 29 pages, 79 references. Significant revision from previous versio
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Non-finitely Generated Maximal Subgroups of Context-free Monoids
2021Co-Authors: Nyberg-brodda Carl-fredrikAbstract:A finitely generated group or Monoid is said to be context-free if it has context-free word problem. In this note, we give an example of a context-free Monoid, none of whose maximal subgroups are finitely generated. This answers a question of Brough, Cain & Pfeiffer on whether the group of units of a context-free Monoid is always finitely generated, and highlights some of the contrasts between context-free Monoids and context-free groups.Comment: 10 pages, 51 references. Final submitted versio
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On one-relator groups and units of special one-relation inverse Monoids
2021Co-Authors: Nyberg-brodda Carl-fredrikAbstract:This note investigates and clarifies some connections between the theory of one-relator groups and special one-relation inverse Monoids, i.e. those inverse Monoids with a presentation of the form $\operatorname{Inv}\langle A \mid w=1 \rangle$. We show that every one-relator group admits a special one-relation inverse Monoid presentation. We subsequently consider the classes ${\rm {\small ANY}}, {\rm {\small RED}}, {\rm {\small CRED}},$ and ${\rm {\small POS}}$ of one-relator groups which can be defined by special one-relation inverse Monoid presentations in which the defining word is arbitrary; reduced; cyclically reduced; or positive, respectively. We show that the inclusions ${\rm {\small ANY}} \supset {\rm {\small CRED}} \supset {\rm {\small POS}}$ are all strict. Conditional on a natural conjecture, we prove ${\rm {\small ANY}} \supset {\rm {\small RED}}$. Following this, we use the Benois algorithm recently devised by Gray & Ruskuc to produce an infinite family of special one-relation inverse Monoids which exhibit similar pathological behaviour (which we term O'Haresque) to the O'Hare Monoid with respect to computing the minimal invertible pieces of the defining word. Finally, we provide a counterexample to a conjecture by Gray & Ruskuc that the Benois algorithm always correctly computes the minimal invertible pieces of a special one-relation inverse Monoid.Comment: 22 pages, 74 reference
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The Geometry of Special Monoids
2021Co-Authors: Nyberg-brodda Carl-fredrikAbstract:A Monoid is said to be special if it admits a presentation in which all defining relations are of the form $w = 1$. Groups are familiar examples of special Monoids. This article studies the geometric and structural properties of the Cayley graphs of finitely presented special Monoids, building on work by Zhang and Gray-Steinberg. It is shown that the right Cayley graph $\Gamma$ of a special Monoid $M$ is a context-free graph, in the sense of Muller & Schupp, if and only if the group of units of $M$ is virtually free. This generalises the geometric aspect of the well-known Muller-Schupp Theorem from groups to special Monoids. Furthermore, we completely characterise when the monadic second order theory of $\Gamma$ is decidable: this is precisely when the group of units is virtually free. This completely answers for the class of special Monoids a question of Kuske & Lohrey from 2006. As a corollary, we obtain that the rational subset membership problem for $M$ is decidable when the group of units of $M$ is virtually free, extending results of Kambites & Render. We also show that the class of special Monoids with virtually free group of units is the same as the class of special Monoids with right Cayley graph quasi-isometric to a tree as undirected graphs. The above results are proven by developing two general constructions for graphs which preserve context-freeness, of independent interest. The first takes a context-free graph and constructs a tree of copies of this graph. The second is a bounded determinisation of the resulting tree of copies.Comment: 56 pages, 16 figure
Felix Gotti - One of the best experts on this subject based on the ideXlab platform.
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factorizations in upper triangular matrices over information semialgebras
Journal of Algebra, 2020Co-Authors: Nicholas R Baeth, Felix GottiAbstract:Abstract 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 factorization property (FFP) if every element has only finitely many factorizations, and it satisfies the bounded factorization property (BFP) if for each element there is a bound for the number of atoms in each of its factorizations. These two properties have been systematically studied since they were introduced by Anderson, Anderson, and Zafrullah in 1990. Noetherian domains satisfy the BFP, while Dedekind domains satisfy the FFP. It is well known that, for commutative cancellative Monoids (and, in particular, for multiplicative Monoids of integral domains), FFP ⇒ BFP ⇒ ACCP ⇒ atomic. For n ≥ 2 , we show that each of these four properties transfers back and forth between an information semialgebra S (certain commutative cancellative semiring) and the multiplicative Monoid T n ( S ) • consisting of n × n upper triangular matrices over S. We also show that a similar transfer behavior takes place if one replaces T n ( S ) • by its subMonoid U n ( S ) consisting of upper triangular matrices with units along their main diagonals. As a consequence, we find that the atomic chain FFP ⇒ BFP ⇒ ACCP ⇒ atomic also holds for the two classes comprising the noncommutative Monoids T n ( S ) • and U n ( S ) . Finally, we construct various rational information semialgebras to verify that, in general, none of the established implications is reversible.
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puiseux Monoids and transfer homomorphisms
Journal of Algebra, 2018Co-Authors: Felix GottiAbstract:Abstract There are several families of atomic Monoids whose arithmetical invariants have received a great deal of attention during the last two decades. The factorization theory of finitely generated Monoids, strongly primary Monoids, Krull Monoids, and C-Monoids are among the most systematically studied. Puiseux Monoids, which are additive subMonoids of Q ≥ 0 consisting of nonnegative rational numbers, have only been studied recently. In this paper, we provide evidence that this family comprises plenty of Monoids with a basically unexplored atomic structure. We do this by showing that the arithmetical invariants of the well-studied atomic Monoids mentioned earlier cannot be transferred to most Puiseux Monoids via homomorphisms that preserve atomic configurations, i.e., transfer homomorphisms. Specifically, we show that transfer homomorphisms from a non-finitely generated atomic Puiseux Monoid to a finitely generated Monoid do not exist. We also find a large family of Puiseux Monoids that fail to be strongly primary. In addition, we prove that the only nontrivial Puiseux Monoid that accepts a transfer homomorphism to a Krull Monoid is N 0 . Finally, we classify the Puiseux Monoids that happen to be C-Monoids.
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atomicity and boundedness of monotone puiseux Monoids
Semigroup Forum, 2018Co-Authors: Felix Gotti, Marly GottiAbstract:In this paper, we study the atomic structure of Puiseux Monoids generated by monotone sequences. To understand this atomic structure, it is often useful to know whether the Monoid has a bounded generating set. We provide necessary and sufficient conditions for the atomicity and boundedness to be transferred from a monotone Puiseux Monoid to all its subMonoids. Finally, we present two special subfamilies of monotone Puiseux Monoids and fully classify their atomic structure.
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puiseux Monoids and transfer homomorphisms
arXiv: Commutative Algebra, 2017Co-Authors: Felix GottiAbstract:There are several families of atomic Monoids whose arithmetical invariants have received a great deal of attention during the last two decades. The factorization theory of finitely generated Monoids, strongly primary Monoids, Krull Monoids, and C-Monoids are among the most systematically studied. Puiseux Monoids, which are additive subMonoids of $\mathbb{Q}_{\ge 0}$ consisting of nonnegative rational numbers, have only been studied recently. In this paper, we provide evidence that this family comprises plenty of Monoids with a basically unexplored atomic structure. We do this by showing that the arithmetical invariants of the well-studied atomic Monoids mentioned earlier cannot be transferred to most Puiseux Monoids via homomorphisms that preserve atomic configurations, i.e., transfer homomorphisms. Specifically, we show that transfer homomorphisms from a non-finitely generated atomic Puiseux Monoid to a finitely generated Monoid do not exist. We also find a large family of Puiseux Monoids that fail to be strongly primary. In addition, we prove that the only nontrivial Puiseux Monoid that accepts a transfer homomorphism to a Krull Monoid is $\mathbb{N}_0$. Finally, we classify the Puiseux Monoids that happen to be C-Monoids.
Polo Harold - One of the best experts on this subject based on the ideXlab platform.
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Approximating length-based invariants in atomic Puiseux Monoids
2021Co-Authors: Polo HaroldAbstract:A numerical Monoid is a cofinite additive subMonoid of the nonnegative integers, while a Puiseux Monoid is an additive subMonoid of the nonnegative cone of the rational numbers. Using that a Puiseux Monoid is an increasing union of copies of numerical Monoids, we prove that some of the factorization invariants of these two classes of Monoids are related through a limiting process. This allows us to extend results from numerical to Puiseux Monoids. We illustrate the versatility of this technique by recovering various known results about Puiseux Monoids.Comment: This version will appear in Algebra and Discrete Mathematic
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Approximating length-based invariants in atomic Puiseux Monoids
2020Co-Authors: Polo HaroldAbstract:A numerical Monoid is a cofinite additive subMonoid of the nonnegative integers, while a Puiseux Monoid is an additive subMonoid of the nonnegative cone of the rational numbers. Using that a Puiseux Monoid is an increasing union of copies of numerical Monoids, we prove that some of the factorization invariants of these two classes of Monoids are related through a limiting process. This allows us to extend results from numerical to Puiseux Monoids. We illustrate the versatility of this technique by recovering various known results about Puiseux Monoids
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On the atomic structure of exponential Puiseux Monoids and semirings
2020Co-Authors: Albizu-campos Sofía, Bringas Juliet, Polo HaroldAbstract:A Puiseux Monoid is an additive subMonoid of the nonnegative cone of the rational numbers. We say that a Puiseux Monoid $M$ is exponential provided that there exist a positive rational $r$ and a set $S$ consisting of nonnegative integers, which contains $0$, such that $M$ is generated by the set $\{r^s \mid s \in S\}$. If $M$ is multiplicatively closed then we say that $M$ is an exponential Puiseux semiring. Here we study the atomic properties of exponential Puiseux Monoids and semirings. First, we characterize atomic exponential Puiseux Monoids, and we prove that the finite factorization property, the bounded factorization property, and the ACCP coincide in this context. Then we proceed to offer a necessary condition and a sufficient condition for an exponential Puiseux Monoid to satisfy the ACCP. We conclude by describing the exponential Puiseux Monoids that are semirings.Comment: 17 pages. The exposition of the paper was improved. This version will appear in Communications in Algebr
