The Congruences on the Semigroup of Balanced Transformations of an Infinite Set

In 1966, J. M. Howie (J. London Math. Soc.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. Soc. Edinburgh Sect. A88, 159–167, 169–184) and Marques (1983, Proc. Roy. Soc. Edinburgh Sect. A93, 245–257) showed that certain Rees quotient semigroups associated with H are congruence-free. Here, we describe all congruences on H.