On the Delta set of a singular arithmetical congruence monoid

Paul Baginski; Scott T. Chapman; George J. Schaeffer

doi:10.5802/jtnb.615

Paul Baginski ¹ ; Scott T. Chapman ² ; George J. Schaeffer ³

¹ University of California at Berkeley Department of Mathematics Berkeley, California 94720
² Trinity University Department of Mathematics One Trinity Place San Antonio, TX. 78212-7200
³ Carnegie Mellon University Department of Mathematical Sciences Pittsburgh, PA 15213

Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 1, pp. 45-59.

Résumé
Abstract

Si $a$ et $b$ sont des entiers positifs, avec $a \leq b$ et $a^{2} \equiv a \mod b$ , l’ensemble

M_{a, b} = {x \in ℕ : x \equiv a \mod b ou x = 1}

est un monoïde multiplicatif, appelé monoïde de congruence arithmétique (ACM). Pour chaque monoïde avec ses unités $M^{\times}$ et pour chaque $x \in M ∖ M^{\times}$ , nous dirons que $t \in ℕ$ est une longueur de décomposition en facteurs de $x$ si et seulement s’il existe des éléments irréductibles $y_{1}, ..., y_{t} \in M$ tels que $x = y_{1} \dots y_{t}$ . Soit $ℒ (x) = {t_{1}, ..., t_{j}}$ l’ensemble des longueurs (avec $t_{i} < t_{i + 1}$ pour $i < j$ ). Le Delta-ensemble d’un élément $x$ est $Δ (x) = {t_{i + 1} - t_{i} : 1 \leq i < j}$ et le Delta-ensemble du monoïde $M$ est $Δ (M) = ⋃_{x \in M ∖ M^{\times}} Δ (x)$ . Nous examinons $Δ (M)$ quand $M = M_{a, b}$ est un ACM avec $pgcd (a, b) > 1$ . Cet ensemble est complètement caractérisé quand $pgcd (a, b) = p^{α}$ , $p$ un nombre premier et $α > 0$ . Quand $pgcd (a, b)$ a plus d’un facteur premier, nous donnons des bornes pour $Δ (M)$ .

If $a$ and $b$ are positive integers with $a \leq b$ and $a^{2} \equiv a \mod b$ , then the set

M_{a, b} = {x \in ℕ : x \equiv a \mod b or x = 1}

is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid $M$ with units $M^{\times}$ and any $x \in M ∖ M^{\times}$ we say that $t \in ℕ$ is a factorization length of $x$ if and only if there exist irreducible elements $y_{1}, ..., y_{t}$ of $M$ and $x = y_{1} \dots y_{t}$ . Let $ℒ (x) = {t_{1}, ..., t_{j}}$ be the set of all such lengths (where $t_{i} < t_{i + 1}$ whenever $i < j$ ). The Delta-set of the element $x$ is defined as the set of gaps in $ℒ (x)$ : $Δ (x) = {t_{i + 1} - t_{i} : 1 \leq i < k}$ and the Delta-set of the monoid $M$ is given by $⋃_{x \in M ∖ M^{\times}} Δ (x)$ . We consider the $Δ (M)$ when $M = M_{a, b}$ is an ACM with $gcd (a, b) > 1$ . This set is fully characterized when $gcd (a, b) = p^{α}$ for $p$ prime and $α > 0$ . Bounds on $Δ (M_{a, b})$ are given when $gcd (a, b)$ has two or more distinct prime factors

MR

DOI : 10.5802/jtnb.615

Affiliations des auteurs :

Paul Baginski ¹ ; Scott T. Chapman ² ; George J. Schaeffer ³

¹ University of California at Berkeley Department of Mathematics Berkeley, California 94720
² Trinity University Department of Mathematics One Trinity Place San Antonio, TX. 78212-7200
³ Carnegie Mellon University Department of Mathematical Sciences Pittsburgh, PA 15213

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Paul Baginski; Scott T. Chapman; George J. Schaeffer. On the Delta set of a singular arithmetical congruence monoid. Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 1, pp. 45-59. doi : 10.5802/jtnb.615. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.615/

Bibliographie
Cité par

[1] D.F. Anderson, Elasticity of factorizations in integral domains: a survey. Lecture Notes in Pure and Appl. Math. 189 (1997), 1–29. | MR | Zbl

[2] M. Banister, S. T. Chapman, J. Chaika, W. Meyerson, On a result of James and Niven concerning unique factorization in congruence semigroups. To appear in Elemente der Mathematik. | MR

[3] M. Banister, S. T. Chapman, J. Chaika, W. Meyerson, On the arithmetic of arithmetical congruence monoids. Colloq. Math. 108 (2007), 105–118. | MR

[4] C. Bowles, S.T. Chapman, N. Kaplan, D. Reiser, On $Δ$ -sets of numerical monoids. J. Pure Appl. Algebra 5 (2006), 1–24. | MR | Zbl

[5] S. T. Chapman, A. Geroldinger, Krull domains and their monoids, their sets of lengths, and associated combinatorial problems. Lecture Notes in Pure and Applied Mathematics 189 (1997), 73–112. | Zbl

[6] A. Geroldinger, Uber nicht-eindeutige Zerlegungen in irreduzible Elemente. Math. Z. 197 (1988), 505–529. | MR | Zbl

[7] A. Geroldinger, F. Halter-Koch, Non-unique factorizations. Algebraic, Combinatorial and Analytic Theory, Chapman & Hall/CRC, 2006. | MR | Zbl

[8] A. Geroldinger, F. Halter-Koch, Congruence monoids. Acta Arith. 112 (2004), 263–296. | MR | Zbl

[9] F. Halter-Koch, Arithmetical semigroups defined by congruences. Semigroup Forum 42 (1991), 59–62. | MR | Zbl

[10] F. Halter-Koch, Elasticity of factorizations in atomic monoids and integral domains. J. Théor. Nombres Bordeaux 7 (1995), 367–385. | Numdam | MR | Zbl

[11] F. Halter-Koch, C-monoids and congruence monoids in Krull domains. Lect. Notes Pure Appl. Math. 241 (2005), 71–98. | MR | Zbl

[12] R. D. James, I. Niven, Unique factorization in multiplicative systems. Proc. Amer. Math. Soc. 5 (1954), 834–838. | MR | Zbl

Nils Olsson; Christopher O'Neill; Derek Rawling Atomic density of arithmetical congruence monoids, Semigroup Forum, Volume 108 (2024) no. 2, pp. 432-437 | DOI:10.1007/s00233-024-10426-w | Zbl:1548.20094
Scott T. Chapman; Christopher O'Neill; Vadim Ponomarenko On length densities, Forum Mathematicum, Volume 34 (2022) no. 2, pp. 293-306 | DOI:10.1515/forum-2021-0149 | Zbl:1491.13027
Felix Gotti The system of sets of lengths and the elasticity of submonoids of a finite-rank free commutative monoid, Journal of Algebra and its Applications, Volume 19 (2020) no. 7, p. 18 (Id/No 2050137) | DOI:10.1142/s0219498820501376 | Zbl:1471.20039
Jacob Hartzer; Christopher O'Neill On the periodicity of irreducible elements in arithmetical congruence monoids, Integers, Volume 17 (2017), p. paper | Zbl:1433.20015
Stefan Colton; Nathan Kaplan The realization problem for delta sets of numerical semigroups, Journal of Commutative Algebra, Volume 9 (2017) no. 3 | DOI:10.1216/jca-2017-9-3-313
Arielle Fujiwara; Joseph Gibson; Matthew O. Jenssen; Daniel Montealegre; Vadim Ponomarenko; Ari Tenzer Arithmetic of congruence monoids., Communications in Algebra, Volume 44 (2016) no. 8, pp. 3407-3421 | DOI:10.1080/00927872.2015.1065874 | Zbl:1350.20042
Alfred Geroldinger; Florian Kainrath; Andreas Reinhart Arithmetic of seminormal weakly Krull monoids and domains, Journal of Algebra, Volume 444 (2015), pp. 201-245 | DOI:10.1016/j.jalgebra.2015.07.026 | Zbl:1327.13006
J. I. García-García; M. A. Moreno-Frías; A. Vigneron-Tenorio Computation of delta sets of numerical monoids., Monatshefte für Mathematik, Volume 178 (2015) no. 3, pp. 457-472 | DOI:10.1007/s00605-015-0785-9 | Zbl:1343.20061
Christopher O'Neill and Roberto Pelayo; Roberto Pelayo How Do You Measure Primality?, The American Mathematical Monthly, Volume 122 (2015) no. 2, p. 121 | DOI:10.4169/amer.math.monthly.122.02.121
Paul Baginski; Scott Chapman Arithmetic congruence monoids: a survey, Combinatorial and additive number theory. Selected papers based on the presentations at the conferences CANT 2011 and 2012, New York, NY, USA, May 2011 and May 2012, New York, NY: Springer, 2014, pp. 15-38 | DOI:10.1007/978-1-4939-1601-6_2 | Zbl:1371.11002
Lorin Crawford; Vadim Ponomarenko; Jason Steinberg; Marla Williams Accepted elasticity in local arithmetic congruence monoids., Results in Mathematics, Volume 66 (2014) no. 1-2, pp. 227-245 | DOI:10.1007/s00025-014-0374-6 | Zbl:1308.20052
Jason Haarmann; Ashlee Kalauli; Aleesha Moran; Christopher O'Neill; Roberto Pelayo Factorization properties of Leamer monoids., Semigroup Forum, Volume 89 (2014) no. 2, pp. 409-421 | DOI:10.1007/s00233-014-9578-z | Zbl:1317.20055
Matthew Jenssen; Daniel Montealegre; Vadim Ponomarenko Irreducible Factorization Lengths and the Elasticity Problem within ℕ, The American Mathematical Monthly, Volume 120 (2013) no. 4, p. 322 | DOI:10.4169/amer.math.monthly.120.04.322
S. T. Chapman; P. A. García-Sánchez; D. Llena; A. Malyshev; D. Steinberg On the delta set and the Betti elements of a BF-monoid., Arabian Journal of Mathematics, Volume 1 (2012) no. 1, pp. 53-61 | DOI:10.1007/s40065-012-0019-0 | Zbl:1277.20071
Lance Bryant; James Hamblin; Lenny Jones A Variation on the Money-Changing Problem, The American Mathematical Monthly, Volume 119 (2012) no. 5, p. 406 | DOI:10.4169/amer.math.monthly.119.05.406
S. T. Chapman; Jay Daigle; Rolf Hoyer; Nathan Kaplan Delta Sets of Numerical Monoids Using Nonminimal Sets of Generators, Communications in Algebra, Volume 38 (2010) no. 7, p. 2622 | DOI:10.1080/00927870903045165
Paul Baginski; S. T. Chapman; Ryan Rodriguez; George J. Schaeffer; Yiwei She On the Delta set and catenary degree of Krull monoids with infinite cyclic divisor class group., Journal of Pure and Applied Algebra, Volume 214 (2010) no. 8, pp. 1334-1339 | DOI:10.1016/j.jpaa.2009.10.015 | Zbl:1193.20071
S. T. Chapman; David Steinberg On the elasticity of generalized arithmetical congruence monoids., Results in Mathematics, Volume 58 (2010) no. 3-4, pp. 221-231 | DOI:10.1007/s00025-010-0062-0 | Zbl:1203.20058
M. Banister; J. Chaika; S. T. Chapman; W. Meyerson A theorem on accepted elasticity in certain local arithmetical congruence monoids., Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Volume 79 (2009) no. 1, pp. 79-86 | DOI:10.1007/s12188-008-0012-x | Zbl:1194.20057
Donald Adams; Rene Ardila; David Hannasch; Audra Kosh; Hanah McCarthy; Vadim Ponomarenko; Ryan Rosenbaum Bifurcus semigroups and rings, Involve, a Journal of Mathematics, Volume 2 (2009) no. 3, p. 351 | DOI:10.2140/involve.2009.2.351

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