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Elements in a Numerical Semigroup with Factorizations of the Same Length

  Published:2010-07-26
 Printed: Mar 2011
  • S. T. Chapman,
    Sam Houston State University, Department of Mathematics and Statistics, Huntsville, TX, U.S.A.
  • P. A. García-Sánchez,
    Departamento de Álgebra, Universidad de Granada, Granada, España
  • D. Llena,
    Departamento de Geometría, Topología y Química Orgánica, Universidad de Almería, Almería, España
  • J. Marshall,
    Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM, U.S.A.
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Abstract

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: numerical monoid, numerical semigroup, non-unique factorization numerical monoid, numerical semigroup, non-unique factorization
MSC Classifications: 20M14, 20D60, 11B75 show english descriptions Commutative semigroups
Arithmetic and combinatorial problems
Other combinatorial number theory
20M14 - Commutative semigroups
20D60 - Arithmetic and combinatorial problems
11B75 - Other combinatorial number theory
 

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