### Abstract

In 1966, J. M. Howie (J. London Math. Sec. 41, 707-716) showed that the semigroup generated by all non-identity idempotent transformations of an infinite set X is the disjoint union of two semigroups, one of which is denoted by H and consists of all balanced transformations of X (that is, all transformations whose defect, shift, and collapse are equal and infinite). Subsequently, J. M. Howie (1981, Proc. Roy Sec. Edinburgh Sect. A 88, 159-167, 169-184) and Marques (1983, Proc. Roy. Sec. Edinburgh Sect. A 93, 245-257) showed that certain Rees quotient semigroups associated with H are congruence-free. Here, we describe all congruences on H. (C) 2000 Academic Press.

Original language | English |
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Pages (from-to) | 1-30 |

Journal | Journal of Algebra |

Volume | 234 |

DOIs | |

Publication status | Published - 2000 |

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## Cite this

Marques-Smith, M. P. O., & Sullivan, R. (2000). The congruences on the semigroup of balanced transformations of an infinite set.

*Journal of Algebra*,*234*, 1-30. https://doi.org/10.1006/jabr.2000.8391