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.