On the arithmetic of arithmetical congruence monoids

Volume 108 / 2007

M. Banister, J. Chaika, S. T. Chapman, W. Meyerson Colloquium Mathematicum 108 (2007), 105-118 MSC: 20M14, 20D60, 13F05. DOI: 10.4064/cm108-1-9

Abstract

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.

Authors

  • M. BanisterDepartment 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.
    e-mail
  • J. ChaikaDepartment 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.
    e-mail
  • S. T. ChapmanDepartment of Mathematics
    Trinity University
    One Trinity Place
    San Antonio, TX 78212-7200, U.S.A.
    e-mail
  • W. MeyersonDepartment 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
    e-mail

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