scholarly journals Irreducible decomposition of binomial ideals

2016 ◽  
Vol 152 (6) ◽  
pp. 1319-1332 ◽  
Author(s):  
Thomas Kahle ◽  
Ezra Miller ◽  
Christopher O’Neill
Keyword(s):  
Number Theory ◽  
Duke Math ◽  
Commutative Monoid ◽  
Commutative Monoids ◽  
Irreducible Decomposition ◽  
Binomial Ideal ◽  
Binomial Ideals ◽  
Mesoprimary Decomposition

Building on coprincipal mesoprimary decomposition [Kahle and Miller, Decompositions of commutative monoid congruences and binomial ideals, Algebra and Number Theory 8 (2014), 1297–1364], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of irreducible binomial ideals, thus answering a question of Eisenbud and Sturmfels [Binomial ideals, Duke Math. J. 84 (1996), 1–45].

PRESENTATIONS OF FINITELY GENERATED SUBMONOIDS OF FINITELY GENERATED COMMUTATIVE MONOIDS

2002 ◽  
Vol 12 (05) ◽  
pp. 659-670 ◽  
Author(s):  
J. C. ROSALES ◽  
P. A. GARCÍA-SÁNCHEZ ◽  
J. I. GARCÍA-GARCÍA
Keyword(s):  
Commutative Monoid ◽  
Finitely Generated ◽  
Commutative Monoids ◽  
Algorithmic Method ◽  
Simple Equations ◽  
Torsion Free

We give an algorithmic method for computing a presentation of any finitely generated submonoid of a finitely generated commutative monoid. We use this method also for calculating the intersection of two congruences on ℕp and for deciding whether or not a given finitely generated commutative monoid is t-torsion free and/or separative. The last section is devoted to the resolution of some simple equations on a finitely generated commutative monoid.

scholarly journals ON PRESENTATIONS OF COMMUTATIVE MONOIDS

1999 ◽  
Vol 09 (05) ◽  
pp. 539-553 ◽  
Author(s):  
J. C. ROSALES ◽  
P. A. GARCÍA-SÁNCHEZ ◽  
J. M. URBANO-BLANCO
Keyword(s):  
Commutative Monoid ◽  
Commutative Monoids ◽  
Minimal Presentations

In this paper, we introduce the concept of a strongly reduced monoid and we characterize the minimal presentations for such monoids. As a consequence, we give a method to obtain a presentation for any commutative monoid.

scholarly journals Fractional parts of Dedekind sums

2016 ◽  
Vol 12 (05) ◽  
pp. 1137-1147
Author(s):  
William D. Banks ◽  
Igor E. Shparlinski
Keyword(s):  
Number Theory ◽  
Uniform Distribution ◽  
Exponential Sums ◽  
Kloosterman Sums ◽  
Duke Math ◽  
Dedekind Sums ◽  
Bilinear Forms ◽  
Recent Improvement ◽  
Double Exponential ◽  
Invent Math

Using a recent improvement by Bettin and Chandee to a bound of Duke, Friedlander and Iwaniec [Bilinear forms with Kloosterman fractions, Invent. Math. 128 (1997) 23–43] on double exponential sums with Kloosterman fractions, we establish a uniformity of distribution result for the fractional parts of Dedekind sums [Formula: see text] with [Formula: see text] and [Formula: see text] running over rather general sets. Our result extends earlier work of Myerson [Dedekind sums and uniform distribution, J. Number Theory 28 (1988) 233–239] and Vardi [A relation between Dedekind sums and Kloosterman sums, Duke Math. J. 55 (1987) 189–197]. Using different techniques, we also study the least denominator of the collection of Dedekind sums [Formula: see text] on average for [Formula: see text].

scholarly journals The system of sets of lengths and the elasticity of submonoids of a finite-rank free commutative monoid

2019 ◽  
Vol 19 (07) ◽  
pp. 2050137 ◽  
Author(s):  
Felix Gotti
Keyword(s):  
Finite Rank ◽  
Convex Cone ◽  
The Other ◽  
Extremal System ◽  
Commutative Monoid ◽  
Commutative Monoids ◽  
Other Hand ◽  
Sets Of Lengths

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 [Formula: see text], is the quotient [Formula: see text] and the elasticity of [Formula: see text] is the supremum of the set [Formula: see text]. The system of sets of lengths and the elasticity of [Formula: see text] both measure how far [Formula: see text] is from being half-factorial, i.e. [Formula: see text] for each [Formula: see text]. Let [Formula: see text] denote the collection comprising all submonoids of finite-rank free commutative monoids, and let [Formula: see text]. In this paper, we study the system of sets of lengths and the elasticity of monoids in [Formula: see text]. First, we construct for each [Formula: see text] a monoid in [Formula: see text] having extremal system of sets of lengths. It has been proved before that the system of sets of lengths does not characterize (up to isomorphism) monoids in [Formula: see text]. Here we use our construction to extend this result to [Formula: see text] for any [Formula: see text]. On the other hand, it has been recently conjectured that the elasticity of any monoid in [Formula: see text] is either rational or infinite. We conclude this paper by proving that this is indeed the case for monoids in [Formula: see text] and for any monoid in [Formula: see text] whose corresponding convex cone is polyhedral.

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

2016 ◽  
Vol 26 (03) ◽  
pp. 565-576 ◽  
Author(s):  
Christopher O’Neill ◽  
Vadim Ponomarenko ◽  
Reuben Tate ◽  
Gautam Webb
Keyword(s):  
Nonnegative Integer ◽  
Nonzero Element ◽  
Commutative Monoid ◽  
Finitely Generated ◽  
Commutative Monoids ◽  
Maximum Value ◽  
Extremal Behavior ◽  
Catenary Degree ◽  
Open Question

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].

scholarly journals Decompositions of commutative monoid congruences and binomial ideals

2014 ◽  
Vol 8 (6) ◽  
pp. 1297-1364 ◽  
Author(s):  
Thomas Kahle ◽  
Ezra Miller
Keyword(s):  
Commutative Monoid ◽  
Binomial Ideals

scholarly journals Groups and monoids of Pythagorean triples connected to conics

2017 ◽  
Vol 15 (1) ◽  
pp. 1323-1331
Author(s):  
Nadir Murru ◽  
Marco Abrate ◽  
Stefano Barbero ◽  
Umberto Cerruti
Keyword(s):  
Commutative Monoid ◽  
Commutative Monoids ◽  
Pythagorean Triples ◽  
Pythagorean Triple

Abstract We define operations that give the set of all Pythagorean triples a structure of commutative monoid. In particular, we define these operations by using injections between integer triples and 3 × 3 matrices. Firstly, we completely characterize these injections that yield commutative monoids of integer triples. Secondly, we determine commutative monoids of Pythagorean triples characterizing some Pythagorean triple preserving matrices. Moreover, this study offers unexpectedly an original connection with groups over conics. Using this connection, we determine groups composed by Pythagorean triples with the studied operations.

Multiplicative Number Theory I

2006 ◽  
Author(s):  
Hugh L. Montgomery ◽  
Robert C. Vaughan
Keyword(s):  
Number Theory
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