Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-06-06T00:18:27.741Z Has data issue: false hasContentIssue false
  • English
  • Français

On The Asymptotic Values of Length Functions In Krull And Finitely Generated commutative Monoids

Part of: Computational number theory Arithmetic rings and other special rings Semigroups

Published online by Cambridge University Press:  09 April 2009

S. T. Chapman  and
J. C. Rosales
S. T. Chapman
Affiliation:
Trinity UniversityDepartment of Mathematics 715 Stadium Drive San Antonio, Texas 78212-7200 USA e-mail: schapman@trinity.edu
J. C. Rosales
Affiliation:
Departamento de Álgebra Universidad de Granada E-18071 Granada Spain e-mail: jrosales@ugr.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let M be a commutative cancellative atomic monoid. We consider the behaviour of the asymptotic length functions and on M. If M is finitely generated and reduced, then we present an algorithm for the computation of both and where x is a nonidentity element of M. We also explore the values that the functions and can attain when M is a Krull monoid with torsion divisor class group, and extend a well-known result of Zaks and Skula by showing how these values can be used to characterize when M is half-factorial.

MSC classification

Secondary: 20M14: Commutative semigroups 20M25: Semigroup rings, multiplicative semigroups of rings 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations 11Y05: Factorization
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 74 , Issue 3 , June 2003 , pp. 421 - 436
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Anderson, D. D., Anderson, D. F., Chapman, S. T. and Smith, W. W., ‘Rational elasticity of factorizations in Krull domains’, Proc. Amer. Math. Soc. 117 (1993), 37–43.CrossRefGoogle Scholar
[2]Anderson, D. F. and Pruis, P., ‘Length functions on integral domains’, Proc. Amer. Math. Soc. 113 (1991), 933–937.CrossRefGoogle Scholar
[3]Carlitz, L., ‘A characterization of algebraic number fields with class number two’, Proc. Amer. Math. Soc. 11 (1960), 391–392.Google Scholar
[4]Chapman, S. T., On the Davenport constant, the cross number and their application in factorization theory, Lecture Notes in Pure and Appl. Math. 171 (Marcel Dekker, New York, 1995) pp. 167–190.Google Scholar
[5]Chapman, S. T., Krause, U. and Oeljeklaus, E., ‘Monoids determined by a homogeneous linear Diophantine equation and the half-factorial property’, J. Pure Appl. Algebra 151 (2000), 107–133.CrossRefGoogle Scholar
[6]Chapman, S. T. and Smith, W. W., ‘An analysis using the Zaks-Skula constant of element factorizations in Dedekind domains’, J. Algebra 159 (1993), 176–190.CrossRefGoogle Scholar
[7]Geroldinger, A., ‘Über nicht-eindeutige Zerlegunen in irreduzible Elements’, Math. Z. 197 (1988), 505–529.CrossRefGoogle Scholar
[8]Geroldinger, A. and Gao, W., ‘Half-factorial domains and half-factorial subsets of abelian groups’, Houston J. Math. 24 (1998), 593–611.Google Scholar
[9]Geroldinger, A. and Halter-Koch, F., ‘Non-unique factorizations in block semigroups and arithmetical applications’, Math. Slovaca 42 (1992), 641–661.Google Scholar
[10]Geroldinger, A. and Halter-Koch, F., ‘On the asymptotic behaviour of lengths of factorizations’, J. Pure Appl. Algebra 77 (1992), 239–252.CrossRefGoogle Scholar
[11]Halter-Koch, F., ‘Halbgruppen mit Divisorentheorie’, Exposition Math. 8 (1990), 27–66.Google Scholar
[12]Krause, U., ‘A characterization of algebraic number fields with cyclic class group of prime power order’, Math. Z. 186 (1984), 143–148.CrossRefGoogle Scholar
[13]Rosales, J. C., García-Sánchez, P. A. and García-García, J. I., ‘Atomic commutative monoids and their elasticity’, preprint.Google Scholar
[14]Rosales, J. C., García-Sánchez, P. A. and García-García, J. I., ‘Finitely generated commutative monoids (Nova Science Publishers, New York, 1999).Google Scholar
[15]Skula, L., ‘On c-semigroups’, ACTA Arith. 31 (1976), 247–257.CrossRefGoogle Scholar
[16]Zaks, A., ‘Half-factorial domains’, Bull. Amer. Math. Soc. 82 (1976), 721–723.CrossRefGoogle Scholar
[17]Zaks, A., ‘Half-factorial domains’, Israel J. Math. 37 (1980), 281–302.CrossRefGoogle Scholar