Results in Mathematics

, Volume 66, Issue 1–2, pp 227–245 | Cite as

Accepted Elasticity in Local Arithmetic Congruence Monoids

  • Lorin Crawford
  • Vadim Ponomarenko
  • Jason Steinberg
  • Marla Williams
Article

Abstract

For certain \({a,b \in \mathbb{N}}\), an Arithmetic Congruence MonoidM(a, b) is a multiplicatively closed subset of \({\mathbb{N}}\) given by \({\{x\in\mathbb{N}:x \equiv a \pmod{b}\} \cup\{1\}}\). An irreducible in this monoid is any element that cannot be factored into two elements, each greater than 1. Each monoid element (apart from 1) may be factored into irreducibles in at least one way. The elasticity of a monoid element (apart from 1) is the longest length of a factorization into irreducibles, divided by the shortest length of a factorization into irreducibles. The elasticity of the monoid is the supremum of the elasticities of the monoid elements. A monoid has accepted elasticity if there is some monoid element that has the same elasticity as the monoid. An Arithmetic Congruence Monoid is local if gcd(a, b) is a prime power (apart from 1). It has already been determined whether Arithmetic Congruence Monoids have accepted elasticity in the non-local case; we make make significant progress in the local case, i.e. for many values of a, b.

Mathematics Subject Classification (2010)

20M14 20D60 13F05 

Keywords

Non-unique factorization arithmetical congruence monoid accepted elasticity elasticity of factorization 

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References

  1. 1.
    Anderson D.D., Anderson D.F., Chapman S.T., Smith W.W.: Rational elasticity of factorizations in Krull domains. Proc. Amer. Math. Soc 117(1), 37–43 (1993)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Anderson DD, Preisser J.: Factorization in integral domains without identity. Results Math. 55(3-4), 249–264 (2009)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Anderson, D.F.: Elasticity of factorizations in integral domains: a survey. In: Factorization in integral domains (Iowa City, IA, 1996), vol. 189 of Lecture Notes in Pure and Appl. Math., pp. 1–29. Dekker, New York (1997)Google Scholar
  4. 4.
    Baginski, P., Chapman, S.T.: Arithmetic congruence monoids: a survey. In: Combinatorial and additive number theory: contributions from CANT 2011. Springer, forthcomingGoogle Scholar
  5. 5.
    Baginski P., Chapman S.T., Crutchfield C., Grace Kennedy K., Wright M.: Elastic properties and prime elements. Results Math. 49(3-4), 187–200 (2006)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Baginski P., Chapman S.T., Schaeffer G.J.: On the delta set of a singular arithmetical congruence monoid. J. Théor. Nombres Bordeaux 20(1), 45–59 (2008CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Banister M., Chaika J., Chapman S.T., Meyerson W.: On a result of James and Niven concerning unique factorization in congruence semigroups. Elem. Math. 62(2), 68–72 (2007)MATHMathSciNetGoogle Scholar
  8. 8.
    Banister M., Chaika J., Chapman S.T., Meyerson W.: On the arithmetic of arithmetical congruence monoids. Colloq. Math. 108(1), 105–118 (2007)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Banister M., Chaika J., Chapman S.T., Meyerson W.: A theorem on accepted elasticity in certain local arithmetical congruence monoids. Abh. Math. Semin. Univ. Hambg. 79(1), 79–86 (2009)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Banister, M., Chaika, J., Meyerson, W.: Technical report, Trinity University REU (2003)Google Scholar
  11. 11.
    Chapman S.T., Steinberg D.: On the elasticity of generalized arithmetical congruence monoids. Results Math. 58(3–4), 221–231 (2010)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Fontana, M., Houston, E., Lucas, T.: Factoring ideals in integral domains, volume~14 of Lecture Notes of the unione matematica Italiana. Springer, Heidelberg (2013)Google Scholar
  13. 13.
    Geroldinger, A.: Additive group theory and non-unique factorizations. In: Combinatorial number theory and additive group theory, Adv. Courses Math. CRM Barcelona, pp. 1–86. Birkhäuser, Basel (2009)Google Scholar
  14. 14.
    Geroldinger A., Halter-Koch F.: Congruence monoids. Acta Arith. 112(3), 263–296 (2004)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Geroldinger, A., Halter-Koch, F.: Non-unique factorizations, vol. 278 of Pure and Applied Mathematics (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL (2006) Algebraic, combinatorial and analytic theoryGoogle Scholar
  16. 16.
    Halter-Koch, F.: C-monoids and congruence monoids in Krull domains. In: Arithmetical properties of commutative rings and monoids, vol. 241 of Lect. Notes Pure Appl. Math., pp. 71–98. Chapman & Hall/CRC, Boca Raton, FL (2005)Google Scholar
  17. 17.
    Hardy G.H.: Ramanujan: twelve lectures on subjects suggested by his life and work. Chelsea Publishing Company, New York (1959)Google Scholar
  18. 18.
    Hungerford, T.W.: Algebra. Holt, Rinehart and Winston, Inc., New York (1974)Google Scholar
  19. 19.
    Jenssen M., Montealegre D., Ponomarenko V.: Irreducible factorization lengths and the elasticity problem within \({\mathbb{N}}\). Amer. Math. Monthly 120(4), 322–328 (2013)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Niven, I., Zuckerman, H.S.: An introduction to the theory of numbers. John Wiley & Sons, New York-Chichester-Brisbane, fourth edition (1980)Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Lorin Crawford
    • 1
  • Vadim Ponomarenko
    • 2
  • Jason Steinberg
    • 3
  • Marla Williams
    • 4
  1. 1.Clark Atlanta UniversityAtlantaUSA
  2. 2.San Diego State UniversitySan DiegoUSA
  3. 3.Princeton UniversityPrincetonUSA
  4. 4.Willamette UniversitySalemUSA

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