The Catenary and Tame Degree in Finitely Generated Commutative Cancellative Monoids (original) (raw)

On the set of catenary degrees of finitely generated cancellative commutative monoids

International Journal of Algebra and Computation, 2016

The catenary degree of an element [Formula: see text] of a cancellative commutative monoid [Formula: see text] is a nonnegative integer measuring the distance between the irreducible factorizations of [Formula: see text]. The catenary degree of the monoid [Formula: see text], defined as the supremum over all catenary degrees occurring in [Formula: see text], has been studied as an invariant of nonunique factorization. In this paper, we investigate the set [Formula: see text] of catenary degrees achieved by elements of [Formula: see text], focusing on the case where [Formula: see text] is finitely generated (where [Formula: see text] is known to be finite). Answering an open question posed by García-Sánchez, we provide a method to compute the smallest nonzero element of [Formula: see text] that parallels a well-known method of computing the maximum value. We also give several examples demonstrating certain extremal behavior for [Formula: see text].

The catenary and tame degree of numerical monoids

Forum Mathematicum

Studying ceratin combinatorial properties of non-unique factorizations have been a subject of recent literatures. Little is known about two combinatorial invariants, namely the catenary degree and the tame degree, even in the case of numerical monoids. In this paper we compute these invariants for a certain class of numerical monoids generated by generalized arithmetic sequences. We also show that the difference between the tame degree and the catenary degree can be arbitrary large even if the number of minimal generators is fixed.

Length-factoriality in commutative monoids and integral domains

Journal of Algebra, 2021

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 the setting of integral domains, lengthfactoriality can be taken as an alternative definition of a unique factorization domain. However, being a length-factorial monoid is in general weaker than being a factorial monoid (i.e., a unique factorization monoid). Here we further investigate length-factoriality. First, we offer two characterizations of a length-factorial monoid M , and we use such characterizations to describe the set of Betti elements and obtain a formula for the catenary degree of M. Then we study the connection between lengthfactoriality and purely long (resp., purely short) irreducibles, which are irreducible elements that appear in the longer (resp., shorter) part of any unbalanced factorization relation. Finally, we prove that an integral domain cannot contain purely short and a purely long irreducibles simultaneously, and we construct a Dedekind domain containing purely long (resp., purely short) irreducibles but not purely short (resp., purely long) irreducibles.

Factorization invariants in

2016

Let NA be the monoid generated by A = {a 1 ,. .. , an} ⊆ Z d. We introduce the homogeneous catenary degree of NA as the smallest N ∈ N with the following property: for each a ∈ NA and any two factorizations u, v of a, there exists factorizations u = w 1 ,. .. , wt = v of a such that, for every k, d(w k , w k+1) ≤ N, where d is the usual distance between factorizations, and the length of w k , |w k |, is less than or equal to max{|u|, |v|}. We prove that the homogeneous catenary degree of NA improves the monotone catenary degree as upper bound for the ordinary catenary degree, and we show that it can be effectively computed. We also prove that for half-factorial monoids, the tame degree and the ω-primality coincide, and that all possible catenary degrees of the elements of an affine semigroup of this kind occur as the catenary degree of one of its Betti elements.

The catenary and tame degrees on a numerical monoid are eventually periodic

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, including Krull monoids and numerical monoids. In this paper, we show for a given numerical monoid S that the sequences {c(s)} s∈S and {t(s)} s∈S are both eventually periodic. We show similar behavior for several functions related to the catenary degree which have recently appeared in the literature. These results nicely complement the known result that the sequence {∆(s)} s∈S of delta sets of S also satisfies a similar periodicity condition.

On the atomicity of monoid algebras

Journal of Algebra

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 = N 0 : he constructed an atomic integral domain R such that the polynomial ring R[x] is not atomic. However, the question of whether a monoid algebra F [x; M ] over a field F is atomic provided that M is atomic has been open since then. Here we offer a negative answer to this question. First, we find for any infinite cardinal κ a torsion-free atomic monoid M of rank κ satisfying that the monoid domain R[x; M ] is not atomic for any integral domain R. Then for every n ≥ 2 and for each field F of finite characteristic we exhibit a torsion-free atomic monoid of rank n such that F [x; M ] is not atomic. Finally, we construct a torsion-free atomic monoid M of rank 1 such that Z 2 [x; M ] is not atomic.

An Abstract Factorization Theorem and Some Applications

2021

We combine the language of monoids with the language of preorders to formulate an abstract factorization theorem with several applications. In particular, this leads to (i) a generalization of P.M. Cohn’s classical theorem on “atomic factorizations” from cancellative to Dedekind-finite monoids (and, hence, to a variety of rings that are not domains); (ii) a monoid-theoretic proof that every module of finite uniform dimension over a (commutative or non-commutative) ring R is isomorphic to a direct sum of finitely many indecomposable R-modules (in fact, we obtain the result as a special case of a general decomposition theorem for the objects of certain categories with finite products, where the indecomposable R-modules are characterized as the atoms of a certain “monoid of modules”). Also, we recover and extend an existence theorem of D.D. Anderson and S. Valdes-Leon on “irreducible factorizations” in commutative rings [RMJM 1996]; a refinement of Cohn’s theorem to “nearly cancellativ...

Minimal presentations for monoids with the ascending chain condition on principal ideals

Semigroup Forum, 2012

We show that the natural way to extend several key results concerning minimal presentations for finitely generated commutative cancellative reduced monoids, is to replace the finitely generated condition by the ascending chain condition on principal ideals. Keywords Monoid • Ascending condition on principal ideals • Minimal presentations • Betti elements • Catenary degree • Sets of distances All monoids in this paper are commutative, cancellative and reduced, so we will omit these adjectives in the sequel. By having a closer look at [8, Chap. 9] and [7] one realizes that if the finitely generated condition is replaced by the ascending chain condition on principal ideals most of the results remain true. This is due to the fact that in the monoids studied on those papers there are no infinite descending chains of elements with respect to the order induced by the monoid, and so induction can be performed. We show that this replacement allows us to easily generalize, on the one hand, the construction of minimal presentations, and on the other the definition and characterization of gluings for finitely generated monoids given in [8, Chap. 9] and [7], respectively. The motivation to do this, apart from the seek of a more general setting to work in, is twofold, as we explain next. Communicated by Jorge Almeida.

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Affine Semigroups Having a Unique Betti Element

Journal of Algebra and Its Applications, 2013

We characterize affine semigroups having one Betti element and we compute some relevant non-unique factorization invariants for these semigroups. As an example, we particularize our description to numerical semigroups.

Minimal presentations for monoids with the ascending chain condition on principal ideals

Semigroup Forum, 2012

Homogenization of a nonsymmetric embedding-dimension-three numerical semigroup

Involve, a Journal of Mathematics, 2014

Let n 1 , n 2 , n 3 be positive integers with gcd(n 1 , n 2 , n 3) = 1. For S = n 1 , n 2 , n 3 nonsymmetric, we give an alternative description, using elementary techniques, of a minimal presentation of its homogenizationS = (1, 0), (1, n 1), (1, n 2), (1, n 3). As a consequence, we show that this minimal presentation is unique. We recover Bresinsky's characterization of the Cohen-Macaulay property ofS and present a procedure to compute all possible catenary degrees of the elements ofS.

Measuring primality in numerical semigroups with embedding dimension three

Journal of Algebra and Its Applications, 2016

In this paper, we find the ω-value of the generators of any numerical semigroup with embedding dimension three. This allows us to determine all possible orderings of the ω-values of the generators. In addition, we relate the ω-value of the numerical semigroup to its catenary degree.

numericalsgps, a GAP package for numerical semigroups

ACM Communications in Computer Algebra, 2016

The package numericalsgps performs computations with and for numerical semigroups. Recently also affine semigroups are admitted as objects for calculations. This manuscript is a survey of what the package does, and at the same time of the trending topics on numerical semigroups.

The catenary and tame degree of numerical monoids

Forum Mathematicum

The computation of factorization invariants for affine semigroups

Journal of Algebra and Its Applications

We present several new algorithms for computing factorization invariant values over affine semigroups. In particular, we give (i) the first known algorithm to compute the delta set of any affine semigroup, (ii) an improved method of computing the tame degree of an affine semigroup, and (iii) a dynamic algorithm to compute catenary degrees of affine semigroup elements. Our algorithms rely on theoretical results from combinatorial commutative algebra involving Gröbner bases, Hilbert bases, and other standard techniques. Implementation in the computer algebra system GAP is discussed.

On the set of catenary degrees of finitely generated cancellative commutative monoids

International Journal of Algebra and Computation, 2016

Numerical Semigroups on Compound Sequences

Communications in Algebra, 2016

We generalize the geometric sequence {a p , a p-1 b, a p-2 b 2 , . . . , b p } to allow the p copies of a (resp. b) to all be different. We call the se- We consider numerical semigroups whose minimal set of generators form a compound sequence, and compute various semigroup and arithmetical invariants, including the Frobenius number, Apéry sets, Betti elements, and catenary degree. We compute bounds on the delta set and the tame degree.

Inside factorial monoids and integral domains

Journal of Algebra, 2002

We investigate two classes of monoids and integral domains, called inside and outside factorial, whose definitions are closely related in a divisor-theoretic manner to the concept of unique factorization. We prove that a monoid is outside factorial if and only if it is a Krull monoid with torsion class group, and that it is inside factorial if and only if its root-closure is a rational generalized Krull monoid with torsion class group. We determine the structure of Cale bases of inside factorial monoids and characterize inside factorial monoids among weakly Krull monoids. These characterizations carry over to integral domains. Inside factorial orders in algebraic number fields are charcterized by several other factorization properties.

Factorizations in reciprocal Puiseux monoids

2021

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.

Notes on Grading Monoids

2006

Throughout this paper, a semigroup S will denote a torsion free grading monoid, and it is a non-zero semigroup with 0. The operation is written additively. The aim of this paper is to study semigroup version of an integral domain ([1],[3],[4] and [티).

On arithmetical numerical monoids with some generators omitted

Semigroup Forum, 2018

Numerical monoids (cofinite, additive submonoids of the non-negative integers) arise frequently in additive combinatorics, and have recently been studied in the context of factorization theory. Arithmetical numerical monoids, which are minimally generated by arithmetic sequences, are particularly well-behaved, admitting closed forms for many invariants that are difficult to compute in the general case. In this paper, we answer the question "when does omitting generators from an arithmetical numerical monoid S preserve its (well-understood) set of length sets and/or Frobenius number?" in two extremal cases: (i) we characterize which individual generators can be omitted from S without changing the set of length sets or Frobenius number; and (ii) we prove that under certain conditions, nearly every generator of S can be omitted without changing its set of length sets or Frobenius number.

Factorization in monoids by stratification of atoms and the Elliott Problem

2020

In an additive factorial monoid each element can be represented as a linear combination of irreducible elements (atoms) with uniquely determined coefficients running over all natural numbers. In this paper we develop for a wide class of non-factorial monoids a concept of stratification for atoms which allows to represent each element as a linear combination of atoms where the coefficients are uniquely determined when restricted in a particular way. This wide class includes inside factorial monoids and in particular simplicial affine semigroups. In the latter case the question of uniqueness is related to a problem studied by E. B. Elliott in a paper from 1903. For the monoid of all nonnegative solutions of a certain linear Diophantine equation in three variables, Elliott considers"simple sets of solutions"(atoms of the monoid) and looks for a method that gives"every set once only". We show that for simplicial affine semigroups in two dimensions a stratification is...

Uniquely presented finitely generated commutative monoids

Pacific Journal of Mathematics, 2010

A finitely generated commutative monoid is uniquely presented if it has a unique minimal presentation. We give necessary and sufficient conditions for finitely generated, combinatorially finite, cancellative, commutative monoids to be uniquely presented. We use the concept of gluing to construct commutative monoids with this property. Finally for some relevant families of numerical semigroups we describe the elements that are uniquely presented.

Factorization Properties of Congruence Monoids

2012

Let n ∈ N, Γ ⊆ N and define Γn = {x ∈ Zn | x ∈ Γ} the set of residues of elements of Γ modulo n. If Γn is multiplicatively closed we may define the following submonoid of the naturals: HΓn = {x ∈ N | x = γ, γ ∈ Γn}∪{1} known as a congruence monoid (CM). Unlike the naturals, many CMs enjoy the property of non-unique factorization into irreducibles. This opens the door to the study of arithmetic invariants associated with nonunique factorization theory; most important to us will be the concept of elasticity. In particular we give a complete characterization of when a given CM has finite elasticity. Throughout we explore the arithmetic properties of HΓn in terms of the arithmetic and algebraic properties of Γn.

Divisibility theory in commutative rings: Bezout monoids

Journal of Mathematical Sciences, 2012

A ubiquitous class of lattice ordered semigroups introduced by Bosbach in 1991, which we will call Bezout monoids, seems to be the appropriate structure for the study of divisibility in various classical rings like GCD domains (including UFD's), rings of low dimension (including semihereditary rings), as well as certain subdirect products of such rings and certain factors of such subdirect products. A Bezout monoid is a commutative monoid S with 0 such that under the natural partial order (for a, b ∈ S, a ≤ b ∈ S ⇐⇒ bS ⊆ aS), S is a distributive lattice, multiplication is distributive over both meets and joins, and for any x, y ∈ S, if d = x ∧ y and dx 1 = x, then there is a y 1 ∈ S with dy 1 = y and x 1 ∧ y 1 = 1. In the present paper, Bezout monoids are investigated by using filters and m-prime filters. We also prove analogues of the Pierce and the Grothendieck sheaf representations of rings for Bezout monoids. The question as to whether Bezout monoids describe divisibility in Bezout rings (rings whose finitely generated ideals are principal) is still open.

Presentations of finitely generated cancellative commutative monoids and nonnegative solutions of systems of linear equations

Discrete Applied Mathematics, 2006

Varying methods exist for computing a presentation of a finitely generated commutative cancellative monoid. We use an algorithm of Contejean and Devie [An efficient incremental algorithm for solving systems of linear diophantine equations, Inform. and Comput. 113 (1994) 143-172] to show how these presentations can be obtained from the nonnegative integer solutions to a linear system of equations. We later introduce an alternate algorithm to show how such a presentation can be efficiently computed from an integer basis.

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The Catenary and Tame Degree in Finitely Generated Commutative Cancellative Monoids (original) (raw)

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