On the arithmetic of arithmetical congruence monoids

M. Banister¹ ; J. Chaika² ; S. T. Chapman³ ; W. Meyerson⁴

¹ Department of Mathematics Harvey Mudd College 1250 N. Dartmouth Ave. Claremont, CA 91711, U.S.A. and Department of Mathematics University of California at Santa Barbara Santa Barbara, CA 93106, U.S.A.
² Department of Mathematics The University of Iowa 14 MacLean Hall Iowa City, IA 52242, U.S.A. and Mathematics Department, MS 136 Rice University 6100 S. Main St. Houston, TX 77005-1892, U.S.A.
³ Department of Mathematics Trinity University One Trinity Place San Antonio, TX 78212-7200, U.S.A.
⁴ Department of Mathematics Harvard University One Oxford Street Cambridge, MA 02138, U.S.A. and Mathematics Department University of California at Los Angeles Box 951555 Los Angeles, CA 90095-1555, U.S.A

Colloquium Mathematicum, Tome 108 (2007) no. 1, pp. 105-118

Résumé

Let ${\mathbb N}$ represent the positive integers and ${\mathbb N}_0$ the non-negative integers. If $b\in {\mathbb N}$ and ${{\mit\Gamma}}$ is a multiplicatively closed subset of $\mathbb{Z}_b=\mathbb{Z}/b\mathbb{Z}$, then the set $H_{{\mit\Gamma}} =\{x\in {\mathbb N} \mid x+b\mathbb{Z}\in {{\mit\Gamma}}\}\cup\{1\}$ is a multiplicative submonoid of ${\mathbb N}$ known as a congruence monoid. An arithmetical congruence monoid (or ACM) is a congruence monoid where ${{\mit\Gamma}}=\{\overline{a}\}$ consists of a single element. If $H_{{\mit\Gamma}}$ is an ACM, then we represent it with the notation $M(a,b) =(a+b{\mathbb N}_0)\cup \{1\}$, where $a, b\in {\mathbb N}$ and $a^2\equiv a \pmod{b}$. A classical 1954 result of James and Niven implies that the only ACM which admits unique factorization of elements into products of irreducibles is $M(1,2)=M(3,2)$. In this paper, we examine further factorization properties of ACMs. We find necessary and sufficient conditions for an ACM $M(a,b)$ to be half-factorial (i.e., lengths of irreducible factorizations of an element remain constant) and further determine conditions for $M(a,b)$ to have finite elasticity. When the elasticity of $M(a,b)$ is finite, we produce a formula to compute it. Among our remaining results, we show that the elasticity of an ACM $M(a,b)$ may not be accepted and show that if an ACM $M(a,b)$ has infinite elasticity, then it is not fully elastic.

Détail

DOI : 10.4064/cm108-1-9

Keywords: mathbb represent positive integers mathbb non negative integers mathbb mit gamma multiplicatively closed subset mathbb mathbb mathbb set mit gamma mathbb mid mathbb mit gamma cup multiplicative submonoid mathbb known congruence monoid arithmetical congruence monoid acm congruence monoid where mit gamma overline consists single element mit gamma acm represent notation mathbb cup where mathbb equiv pmod classical result james niven implies only acm which admits unique factorization elements products irreducibles paper examine further factorization properties acms necessary sufficient conditions acm half factorial lengths irreducible factorizations element remain constant further determine conditions have finite elasticity elasticity finite produce formula compute among remaining results elasticity acm may accepted acm has infinite elasticity fully elastic