# On the Delta set of a singular arithmetical congruence monoid

• [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
• Volume: 20, Issue: 1, page 45-59
• ISSN: 1246-7405

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## Abstract

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If $a$ and $b$ are positive integers with $a\le b$ and ${a}^{2}\equiv a\phantom{\rule{0.277778em}{0ex}}\mathrm{mod}b$, then the set${M}_{a,b}=\left\{x\in ℕ:x\equiv a\phantom{\rule{0.277778em}{0ex}}\mathrm{mod}b\phantom{\rule{4pt}{0ex}}\text{or}\phantom{\rule{4pt}{0ex}}x=1\right\}$is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid $M$ with units ${M}^{×}$ and any $x\in M\setminus {M}^{×}$ 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 $x={y}_{1}\cdots {y}_{t}$. Let $ℒ\left(x\right)=\left\{{t}_{1},...,{t}_{j}\right\}$ 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 $ℒ\left(x\right)$: $\Delta \left(x\right)=\left\{{t}_{i+1}-{t}_{i}:1\le i<k\right\}$ and the Delta-set of the monoid $M$ is given by ${\bigcup }_{x\in M\setminus {M}^{×}}\Delta \left(x\right)$. We consider the $\Delta \left(M\right)$ when $M={M}_{a,b}$ is an ACM with $gcd\left(a,b\right)>1$. This set is fully characterized when $gcd\left(a,b\right)={p}^{\alpha }$ for $p$ prime and $\alpha >0$. Bounds on $\Delta \left({M}_{a,b}\right)$ are given when $gcd\left(a,b\right)$ has two or more distinct prime factors

## How to cite

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Baginski, Paul, Chapman, Scott T., and Schaeffer, George J.. "On the Delta set of a singular arithmetical congruence monoid." Journal de Théorie des Nombres de Bordeaux 20.1 (2008): 45-59. <http://eudml.org/doc/10832>.

@article{Baginski2008,
abstract = {If $a$ and $b$ are positive integers with $a\le b$ and $a^2\equiv a\;\mathrm\{mod\}b$, then the set$M\_\{a,b\}=\lbrace x\in \mathbb\{N\}:\text\{x\equiv a\;\mathrm\{mod\}b or x=1\}\rbrace$is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid $M$ with units $M^\times$ and any $x\in M\setminus M^\times$ we say that $t\in \mathbb\{N\}$ is a factorization length of $x$ if and only if there exist irreducible elements $y_1,\ldots ,y_t$ of $M$ and $x=y_1\cdots y_t$. Let $\mathcal\{L\}(x)=\lbrace t_1,\ldots ,t_j\rbrace$ be the set of all such lengths (where $t_i&lt;t_\{i+1\}$ whenever $i&lt;j$). The Delta-set of the element $x$ is defined as the set of gaps in $\mathcal\{L\}(x)$: $\Delta (x)=\lbrace t_\{i+1\}-t_i:1\le i&lt;k\rbrace$ and the Delta-set of the monoid $M$ is given by $\bigcup _\{x\in M\setminus M^\times \}\Delta (x)$. We consider the $\Delta (M)$ when $M=M_\{a,b\}$ is an ACM with $\gcd (a,b)&gt;1$. This set is fully characterized when $\gcd (a,b)=p^\alpha$ for $p$ prime and $\alpha &gt;0$. Bounds on $\Delta (M_\{a,b\})$ are given when $\gcd (a,b)$ has two or more distinct prime factors},
affiliation = {University of California at Berkeley Department of Mathematics Berkeley, California 94720; Trinity University Department of Mathematics One Trinity Place San Antonio, TX. 78212-7200; Carnegie Mellon University Department of Mathematical Sciences Pittsburgh, PA 15213},
author = {Baginski, Paul, Chapman, Scott T., Schaeffer, George J.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {non-unique factorizations; arithmetical congruence monoids; half-factorial monoids; elasticities of factorizations; delta-sets; sets of gaps; factorizations into irreducibles; lengths of factorizations},
language = {eng},
number = {1},
pages = {45-59},
publisher = {Université Bordeaux 1},
title = {On the Delta set of a singular arithmetical congruence monoid},
url = {http://eudml.org/doc/10832},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Baginski, Paul
AU - Chapman, Scott T.
AU - Schaeffer, George J.
TI - On the Delta set of a singular arithmetical congruence monoid
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 1
SP - 45
EP - 59
AB - If $a$ and $b$ are positive integers with $a\le b$ and $a^2\equiv a\;\mathrm{mod}b$, then the set$M_{a,b}=\lbrace x\in \mathbb{N}:\text{x\equiv a\;\mathrm{mod}b or x=1}\rbrace$is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid $M$ with units $M^\times$ and any $x\in M\setminus M^\times$ we say that $t\in \mathbb{N}$ is a factorization length of $x$ if and only if there exist irreducible elements $y_1,\ldots ,y_t$ of $M$ and $x=y_1\cdots y_t$. Let $\mathcal{L}(x)=\lbrace t_1,\ldots ,t_j\rbrace$ be the set of all such lengths (where $t_i&lt;t_{i+1}$ whenever $i&lt;j$). The Delta-set of the element $x$ is defined as the set of gaps in $\mathcal{L}(x)$: $\Delta (x)=\lbrace t_{i+1}-t_i:1\le i&lt;k\rbrace$ and the Delta-set of the monoid $M$ is given by $\bigcup _{x\in M\setminus M^\times }\Delta (x)$. We consider the $\Delta (M)$ when $M=M_{a,b}$ is an ACM with $\gcd (a,b)&gt;1$. This set is fully characterized when $\gcd (a,b)=p^\alpha$ for $p$ prime and $\alpha &gt;0$. Bounds on $\Delta (M_{a,b})$ are given when $\gcd (a,b)$ has two or more distinct prime factors
LA - eng
KW - non-unique factorizations; arithmetical congruence monoids; half-factorial monoids; elasticities of factorizations; delta-sets; sets of gaps; factorizations into irreducibles; lengths of factorizations
UR - http://eudml.org/doc/10832
ER -

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