Congruences on direct products of transformation and matrix monoids

Malcev described the congruences of the monoid Tn of all full transformations on a finite set Xn = {1,..., n}. Since then, congruences have been characterized in various other monoids of ( partial) transformations on Xn, such as the symmetric inverse monoid In of all injective partial transformations, or the monoid PT n of all partial transformations. The first aim of this paper is to describe the congruences of the direct products Qm x Pn, where Q and P belong to {T, PT, I}. Malcev also provided a similar description of the congruences on the multiplicative monoid Fn of all n xn matrices with entries in a field F; our second aim is to provide a description of the principal congruences of Fm x Fn. The paper finishes with some comments on the congruences of products of more than two transformation semigroups, and on a number of related open problems.