Congruences on direct products of transformation and matrix monoids

Mal ' cev described the congruences of the monoid 𝒯_n of all full transformations on a finite set X_n={1, … ,n} . Since then, congruences have been characterized in various other monoids of (partial) transformations on X_n , such as the symmetric inverse monoid ℐ_n 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 Q_m× P_n , where Q and P belong to {𝒯, 𝒫𝒯,ℐ} . Mal ' cev also provided a similar description of the congruences on the multiplicative monoid F_n of all n× n matrices with entries in a field F ; our second aim is to provide a description of the principal congruences of F_m × F_n . 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.