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Products of idempotent endomorphisms of an independence algebra of finite rank

Published online by Cambridge University Press:  20 January 2009

John Fountain  and
Andrew Lewin
John Fountain
Affiliation:
Department of MathematicsUniversity of YorkHeslingtonYork Y01 5DD
Andrew Lewin
Affiliation:
Department of MathematicsUniversity of YorkHeslingtonYork Y01 5DD
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Abstract

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Products of idempotents are investigated in the endomorphism monoid of an algebra belonging to a class of algebras which includes finite sets and finite dimensional vector spaces as special cases. It is shown that every endomorphism which is not an automorphism is a product of idempotent endomorphisms. This provides a common generalisation of earlier results of Howie and Erdos for the cases when the algebra is a set or vector space respectively.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 35 , Issue 3 , October 1992 , pp. 493 - 500
Copyright
Copyright © Edinburgh Mathematical Society 1992

References

REFERENCES

1.Dawlings, R. J. H., Semigroups of singular endomorphisms of vector spaces (PhD thesis, St Andrews, 1980).Google Scholar
2.Dawlings, R. J. H., The semigroup of singular endomorphisms of a finite dimensional vector space, in Semigroups (Eds. Hall, T. E., Jones, P. R., Preston, G. B., Academic Press, Sydney, 1980), 121–131.CrossRefGoogle Scholar
3.Erdos, J. A., On products of idempotent matrices, Glasgow Math. J. 8 (1967), 118–122.CrossRefGoogle Scholar
4.Gould, V. A. R., Endomorphism monoids of independence algebras, preprint.Google Scholar
5.Grätzer, G., Universal algebra (Van Nostrand, Princeton, 1968).Google Scholar
6.Hall, T. E., On regular semigroups, J. Algebra 24 (1973), 1–24.CrossRefGoogle Scholar
7.Howie, J. M., The subsemigroup generated by the idempotents of a full transformation semigroup, J. London Math. Soc. 41 (1966), 707–716.CrossRefGoogle Scholar
8.Howie, J. M., An introduction to semigroup theory (Academic Press, London, 1976).Google Scholar
9.McKenzie, R. N., McNulty, G. F. and Taylor, W. F., Algebra, lattices, varieties, Vol. I (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. Roy. Soc. Edinburgh A, 100 (1985), 123–138.CrossRefGoogle Scholar