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Nilpotents and congruences on semigroups of transformations with fixed rank

Published online by Cambridge University Press:  14 November 2011

M. Paula O. Marques-Smith  and
R. P. Sullivan
M. Paula O. Marques-Smith
Affiliation:
Departamento de Matematica, Universidade do Minho, 4700 Braga, Portugal
R. P. Sullivan
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia
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Abstract

In 1988, Howie and Marques-Smith studied Pm, a Rees quotient semigroup of transformations associated with a regular cardinal m, and described the elements which can be written as a product of nilpotents in Pm. In 1981, Marques proved that if Δm denotes the Malcev congruence on Pm, then Pm/Δm is congruence-free for any infinite m. In this paper, we describe the products of nilpotents in Pm when m is nonregular, and determine all the congruences on Pm when m is an arbitrary infinite cardinal. We also investigate when a nilpotent is a product of idempotents.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 2 , 1995 , pp. 399 - 412
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Clifford, A. H. and Preston, G. B.. The Algebraic Theory of Semigroups (Providence, RI: American Mathematical Society, Mathematical Surveys, No. 7, vols 1 and 2, 1961 and 1967).Google Scholar
2Howie, J. M.. The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc. 41 (1966), 707–16.CrossRefGoogle Scholar
3Howie, J. M.. An Introduction to Semigroup Theory (London: Academic Press, 1976).Google Scholar
4Howie, J. M.. Some subsemigroups of infinite full transformation semigroups. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 159–67.CrossRefGoogle Scholar
5Howie, J. M. and Paula, M.Marques-Smith, O.. A nilpotent-generated semigroup associated with a semigroup of full transformations. Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), 181–7.CrossRefGoogle Scholar
6Thomas, Jech. Set Theory (New York: Academic Press, 1978).Google Scholar
7Malcev, A. I.. Symmetric groupoids. Mat Sb. 31 (1952), 136–51 (in Russian); English translation by Heidi Semla and Bob Sullivan: Amer. Math. Soc. Trans. 113 (1979), 235–50.Google Scholar
8Paula, M.Marques, O.. A congruence-free semigroup associated with an infinite cardinal number. Proc. Roy. Soc. Edinburgh Sect. A 93 (1983), 245–57.Google Scholar
9Monk, J. D.. Introduction to Set Theory (New York: McGraw-Hill, 1969).Google Scholar
10Sullivan, R. P.. Semigroups generated by nilpotent transformations. J. Algebra 110 (1987), 324–43.CrossRefGoogle Scholar