Hostname: page-component-7479d7b7d-t6hkb Total loading time: 0 Render date: 2024-07-12T22:50:31.034Z Has data issue: false hasContentIssue false
  • English
  • Français

Products of idempotent endomorphisms of an independence algebra of infinite rank

Published online by Cambridge University Press:  24 October 2008

John Fountain  and
Andrew Lewin
John Fountain
Affiliation:
Department of Mathematics, University of York, Heslington, York YO1 5DD
Andrew Lewin
Affiliation:
Department of Mathematics, University of York, Heslington, York YO1 5DD
Get access Rights & Permissions [Opens in a new window]

Abstract

In 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (under composition) of idempotent self-maps of the same set. In 1967, J. A. Erdos considered the analogous question for linear maps of a finite dimensional vector space and in 1985, Reynolds and Sullivan solved the problem for linear maps of an infinite dimensional vector space. Using the concept of independence algebra, the authors gave a common generalization of the results of Howie and Erdos for the cases of finite sets and finite dimensional vector spaces. In the present paper we introduce strong independence algebras and provide a common generalization of the results of Howie and Reynolds and Sullivan for the cases of infinite sets and infinite dimensional vector spaces.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 2 , September 1993 , pp. 303 - 319
Copyright
Copyright © Cambridge Philosophical Society 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Fountain, J. and Lewin, A.. Products of idempotent endomorphisms of an independence of finite rank. Proc. Edinburgh Math. Soc. 35 (1992), 493–500.CrossRefGoogle Scholar
[2]Gould, V. A. R.. Endomorphism monoids of independence algebras. Preprint, University of York.Google Scholar
[3]Grätzer, G.. Universal algebra (Van Nostrand, Princeton, 1968).Google Scholar
[4]Hall, T. E.. On regular semigroups. J. Algebra 24 (1973), 1–24.CrossRefGoogle Scholar
[5]Howie, J. M.. The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc. 41 (1966), 707–716.CrossRefGoogle Scholar
[6]Howie, J. M.. An introduction to semigroup theory. Academic Press, London, 1976.Google Scholar
[7]Howie, J. M., Some subsemigroups of infinite full transformation semigroups. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 159–167.Google Scholar
[8]Jones, P. R.. Exchange properties and basis properties for closure operators. Colloq. Math. 57 (1989), 29–33.CrossRefGoogle Scholar
[9]McKenzie, R. N., McNulty, G. F. and Taylor, W. F.. Algebra, lattices, varieties. Vol. 1. (Wadsworth, Monterey, 1983).Google Scholar
[10]Narkiewicz, W.. Independence in a certain class of abstract algebras. Fund. Math. 50 (1961/1962), 333–340.CrossRefGoogle Scholar
[11]Reynolds, M. A. and Sullivan, R. P.. Products of idempotent linear transformations. Proc. Royal Soc. Edinburgh Sect. A 100 (1985), 123–138.CrossRefGoogle Scholar