Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T09:29:03.780Z Has data issue: false hasContentIssue false
  • English
  • Français

Relatively free algebras with weak exchange properties

Part of: Algebraic structures Varieties

Published online by Cambridge University Press:  09 April 2009

John Fountain  and
Victoria Gould
John Fountain
Affiliation:
Department of Mathematics, University of York Heslington York, YO10 5DD, UK, e-mail: jbfl@york.ac.uk, varg1@york.ac.uk
Victoria Gould
Affiliation:
Department of Mathematics, University of York Heslington York, YO10 5DD, UK, e-mail: jbfl@york.ac.uk, varg1@york.ac.uk
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.

We consider algebras for which the operation PC of pure closure of subsets satisfies the exchange property. Subsets that are independent with respect to PC are directly independent. We investigate algebras in which PC satisfies the exchange property and which are relatively free on a directly independent generating subset. Examples of such algebras include independence algebras and dinitely generated free modules over principal ideal domains.

Keywords

free generators(direct) independenceexchange property

MSC classification

Secondary: 08B20: Free algebras 08A05: Structure theory 08A60: Unary algebras 08A62: Finitary algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 75 , Issue 3 , December 2003 , pp. 355 - 384
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Bergman, G. M., ‘Constructing division rings as module-theoretic direct limits’, Trans. Amer. Math. Soc. 354 (2002), 2079–2114.CrossRefGoogle Scholar
[2]Cameron, P. J. and Szabó, C., ‘Independence algebras’, J. London Math. Soc. 61 (2000), 321–334.CrossRefGoogle Scholar
[3]Cohn, P. M., Universal algebra (Harper & Row, New York, 1965).Google Scholar
[4]Cohn, P. M., Free rings and their relations (Academic Press, London, 1971).Google Scholar
[5]Cohn, P. M., Free rings and their relations, 2nd edition (Academic Press, London, 1985).Google Scholar
[6]Glazek, K., ‘Some old and new problems in indenpendence theory’, Colloq. Math. 42 (1979), 127–189.CrossRefGoogle Scholar
[7]Gould, V., ‘Indenpendence algebras’, Algebra Universalis 33 (1995), 294–318.CrossRefGoogle Scholar
[8]Grätzer, G., Universal algebra (Van Nostrand, Princeton, N.J., 1968).Google Scholar
[9]Howie, J. M., Fundamentals of semigroup theory (Oxford University Press, Oxford, 1995).CrossRefGoogle Scholar
[10]Kaplansky, I., ‘Elementary divisors and modules’, Trans. Amer. Math. Soc. 66 (1949), 464–491.CrossRefGoogle Scholar
[11]Kilp, M., Knauer, U. and Mikhalev, A. V., Monoids, acts and categories (Walter de Gruyter, Berlin, 2000).CrossRefGoogle Scholar
[12]McKenzie, R. N., McNulty, G. F. and Taylor, W. T., Algebra, lattices, vaieties (Wadsworth, Monterey, CA, 1983).Google Scholar
[13]Narkiewicz, W., ‘Independence in a certain class of abstract algebras’, Fund. Math. 50 (1961/1962), 333–340.CrossRefGoogle Scholar
[14]Schmidt, J., ‘Mehrstufige Austauschstrukturen’, Z. Math. Logik Grundlagen Math. 2 (1956), 233–249.CrossRefGoogle Scholar