Congruences on direct products of transformation and matrix monoids

Semigroup Forum, 97 (2018), 384-416
https://doi.org/10.1007/s00233-018-9931-8

Mal?cev described the congruences of the monoid ????? of all full transformations on a finite set ????????={1,…,????}. Since then, congruences have been characterized in various other monoids of (partial) transformations on ????????, such as the symmetric inverse monoid ????? of all injective partial transformations, or the monoid ?????? of all partial transformations. The first aim of this paper is to describe the congruences of the direct products ????????×????????, where Q and P belong to {?,??,?}. Mal?cev also provided a similar description of the congruences on the multiplicative monoid ???????? of all ????×???? matrices with entries in a field F; our second aim is to provide a description of the principal congruences of ????????×????????. 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.