Congruences on direct products of transformation and matrix monoids

Abstract

Mal $$'$$ cev described the congruences of the monoid $$\mathcal {T}_n$$ of all full transformations on a finite set $$X_n=\{1, \dots ,n\}$$ . Since then, congruences have been characterized in various other monoids of (partial) transformations on $$X_n$$ , such as the symmetric inverse monoid $$\mathcal {I}_n$$ of all injective partial transformations, or the monoid $$\mathcal {PT}_n$$ of all partial transformations. The first aim of this paper is to describe the congruences of the direct products $$Q_m\times P_n$$ , where Q and P belong to $$\{\mathcal {T}, \mathcal {PT},\mathcal {I}\}$$ . Mal $$'$$ cev also provided a similar description of the congruences on the multiplicative monoid $$F_n$$ of all $$n\times n$$ matrices with entries in a field F; our second aim is to provide a description of the principal congruences of $$F_m \times F_n$$ . 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.