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Products of three idempotent transformations

Published online by Cambridge University Press:  17 April 2009

R. P. Sullivan  and
Rachel Thomas
R. P. Sullivan
Affiliation:
Department of Mathematics and Statistics, University of Western Australia, Nedlands WA 6009, Australia
Rachel Thomas
Affiliation:
Department of Mathematics and Statistics, University of Western Australia, Nedlands WA 6009, Australia
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Abstract

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In 1988 Howie, Robertson and Schein characterised the transformations of a finite set X that can be written as a product of two or of three idempotent transformations of X; and in 1989 Saito did the same for products of four idempotents. In 1998 Thomas extended the characterisation of two idempotents to arbitrary sets, and here we characterise products of three idempotents in general. We also define a notion of complexity for transformations of any set and use it to provide a different solution to the three-idempotent problem.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 68 , Issue 1 , August 2003 , pp. 57 - 71
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Clifford, A.H. and Preston, G.B., The algebraic theory of semigroups, Mathematical Surveys (Volume 1 and 2) 7 (American Mathematical Society, Providence, RI, 1961 and 1967).Google Scholar
[2]Howie, J.M., ‘The subsemigroup generated by the idempotents of a full transformation semigroup’, J. London Math. Soc. 41 (1966), 707–716.CrossRefGoogle Scholar
[3]Howie, J.M., An introduction to semigroup theory (Academic Press, London, 1976.).Google Scholar
[4]Howie, J.M., ‘Some subsemigroups of infinite full transformation semigroups’, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 159–167.CrossRefGoogle Scholar
[5]Howie, J.M. and Marques-Smith, M.P.O., ‘A nilpotent-generated semigroup associated with a semigroup of full transformations’, Proc. Royal Soc. Edinburgh Sect. A 108 (1988), 181–187.CrossRefGoogle Scholar
[6]Howie, J.M., Robertson, E.F. and Schein, B.M., ‘A combinatorial property of finite full transformation semigroups’, Proc. Roy. Soc. Edinburgh Sect. A 109 (1988), 319–328.CrossRefGoogle Scholar
[7]Marques, M.P.O., ‘A congruence-free semigroup associated with an infinite cardinal number’, Proc. Royal Soc. Edinburgh Sect. A 93 (1983), 245–257.CrossRefGoogle Scholar
[8]Marques-Smith, M.P.O. and Sullivan, R.P., ‘Nilpotents and congruences on semigroups of transformations with fixed rank’, Proc Royal Soc. Edinburgh Sect. A 125 (1995), 399–412.CrossRefGoogle Scholar
[9]Saito, T., ‘Products of four idempotents in finite full transformation semigroups’, Semigroup Forum 39 (1989), 179–193.CrossRefGoogle Scholar
[10]Thomas, R., ‘Products of two idempotent transformations over arbitrary sets and vector spaces’, Bull. Austral. Math. Soc. 57 (1998), 59–71.CrossRefGoogle Scholar
[11]Thomas, R., Products of idempotent transformations, MSc thesis (University of Western Australia, Nedlands, W.A. 1999).Google Scholar