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Congruences on semigroups generated by injective nilpotent transformations

  • M. Paula O. Marques-Smith (a1) and R.P. Sullivan (a2)

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In 1987, Sullivan characterised the elements of the semigroup NI(X) generated by the nilpotents in I(X), the symmetric inverse semigroup on an infinite set X; and, in the same year, Gomes and Howie did the same for finite X. In 1999, Marques-Smith and Sullivan determined all the ideals of NI(X) for arbitrary X. In this paper, we use that work to describe all the congruences on NI(X).

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References

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[1]Clifford, A.H. and Preston, G.B., The algebraic theory of semigroups, Mathematical Surveys, 7, (Vol. 1 and Vol. 2) (American Mathematical Society, Providence, RI, 1961 and 1967).
[2]Gomes, G.M.S. and Howie, J.M., ‘Nilpotents in finite symmetric inverse semigroups’, Proc. Edinburgh Math. Soc. 30 (1987), 383395.
[3]Howie, J.M., An introduction to semigroup theory (Academic Press, London, 1976).
[4]Howie, J.M. and Marques-Smith, M.P.O., ‘Inverse semigroups generated by nilpotent transformations’, Proc. Royal Soc. Edinburgh Sect. A 99 (1984), 153162.
[5]Hrbacek, K. and Jech, T., Introduction to set theory, (Second edition) (Marcel Dekker, New York, 1984).
[6]Marques-Smith, M.P.O. and Sullivan, R.P., ‘The ideal structure of nilpotent-generated transformation semigroups’, Bull. Austral. Math. Soc. 60 (1999), 303318.
[7]Scheiblich, H.E., ‘Concerning congruences on symmetric inverse semigroups’, Czechoslovak Math. J. 23 (1973), 110.
[8]Sullivan, R.P., ‘Semigroups generated by nilpotent transformations’, J. Algebra 110 (1987), 324343.
[9]Sullivan, R.P., ‘Congruences on transformation semigroups with fixed rank’, Proc. London Math. Soc. 70 (1995), 556580.
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Congruences on semigroups generated by injective nilpotent transformations

  • M. Paula O. Marques-Smith (a1) and R.P. Sullivan (a2)

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