Maximal and minimal congruences on some semigroups

In 2006, Sanwong and Sullivan described the maximal congruences on the semigroup N consisting of all non-negative integers under standard multiplication, and on the semigroup T(X) consisting of all total transformations of an infinite set X under composition. Here, we determine all maximal congruences on the semigroup ℤ _n under multiplication modulo n. And, when Y ⊆ X, we do the same for the semigroup T(X, Y) consisting of all elements of T(X) whose range is contained in Y. We also characterise the minimal congruences on T(X, Y). The third author thanks the Faculty of Science, Chiang Mai University, Thailand, for its financial assistance during his visit in February 2006