On the Delta set of a singular arithmetical congruence monoid

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 -