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ON FACTORIZATION IN BLOCK MONOIDS FORMED BY $\{\bar{1},\bar{a}\}$ IN $\mathbb{Z}_{n}$

Published online by Cambridge University Press:  04 July 2003

Scott T. Chapman  and
William W. Smith
Scott T. Chapman
Affiliation:
Trinity University, Department of Mathematics, 715 Stadium Drive, San Antonio, TX 78212-7200, USA (schapman@trinity.edu)
William W. Smith
Affiliation:
The University of North Carolina at Chapel Hill, Department of Mathematics, Chapel Hill, NC 27599-3250, USA (wwsmith@email.unc.edu)
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Abstract

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We consider the factorization properties of block monoids on $\mathbb{Z}_n$ determined by subsets of the form $S_a=\{\bar{1},\bar{a}\}$. We denote such a block monoid by $\mathcal{B}_a(n)$. In §2, we provide a method based on the division algorithm for determining the irreducible elements of $\mathcal{B}_a(n)$. Section 3 offers a method to determine the elasticity of $\mathcal{B}_a(n)$ based solely on the cross number. Section 4 applies the results of §3 to investigate the complete set of elasticities of Krull monoids with divisor class group $\mathbb{Z}_n$.

AMS 2000 Mathematics subject classification: Primary 20M14; 20D60; 13F05

Keywords

block monoidelasticity of factorizationKrull monoid
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 46 , Issue 2 , June 2003 , pp. 257 - 267
Copyright
Copyright © Edinburgh Mathematical Society 2003