Congruences on Direct Products of Transformation and Matrix Monoids

TL;DR

This paper characterizes the congruences on direct products of transformation and matrix monoids, extending Malcev's foundational work to new classes and multi-factor products, and discusses open problems in the area.

Contribution

It provides a comprehensive description of congruences on direct products of various transformation monoids and matrix monoids, expanding the understanding of their algebraic structure.

Findings

Congruences on $Q_m imes P_n$ for $Q, P ext{ in }\\{T, PT, In\}\

Principal congruences on $F_m imes F_n$ for matrix monoids

Discussion of congruences in products of more than two semigroups

Abstract

Malcev described the congruences of the monoid $T_n$ 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 $In_n$ 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 $Q_m\times P_n$, where $Q$ and $P$ belong to $\{T, PT,In\}$. Malcev 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 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 a fairly large number of open problems.