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Congruences on semigroups generated by injective nilpotent transformations

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).