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Elements in a Numerical Semigroup with Factorizations of the Same Length

Published online by Cambridge University Press:  20 November 2018

S. T. Chapman ,
P. A. García-Sánchez ,
D. Llena  and
J. Marshall
S. T. Chapman
Affiliation:
Sam Houston State University, Department of Mathematics and Statistics, Huntsville, TX, U.S.A.e-mail: scott.chapman@shsu.edu
P. A. García-Sánchez
Affiliation:
Departamento de Álgebra, Universidad de Granada, Granada, Españae-mail: pedro@ugr.es
D. Llena
Affiliation:
Departamento de Geometría, Topología y Química Orgánica, Universidad de Almería, Almería, Españae-mail: dllena@ual.es
J. Marshall
Affiliation:
Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM, U.S.A.e-mail: jormars@sandia.gov
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Abstract

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Questions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions.

Keywords

20M1420D6011B75numerical monoidnumerical semigroupnon-unique factorization
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 1 , 01 March 2011 , pp. 39 - 43
Copyright
Copyright © Canadian Mathematical Society 2011

References

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