Congruences on direct products of transformation and matrix monoids

Mal′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$'$$\end{document}cev described the congruences of the monoid Tn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}_n$$\end{document} of all full transformations on a finite set Xn={1,⋯,n}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_n=\{1, \dots ,n\}$$\end{document}. Since then, congruences have been characterized in various other monoids of (partial) transformations on Xn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_n$$\end{document}, such as the symmetric inverse monoid In\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {I}_n$$\end{document} of all injective partial transformations, or the monoid PTn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {PT}_n$$\end{document} of all partial transformations. The first aim of this paper is to describe the congruences of the direct products Qm×Pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_m\times P_n$$\end{document}, where Q and P belong to {T,PT,I}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\mathcal {T}, \mathcal {PT},\mathcal {I}\}$$\end{document}. Mal′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$'$$\end{document}cev also provided a similar description of the congruences on the multiplicative monoid Fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_n$$\end{document} of all n×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\times n$$\end{document} matrices with entries in a field F; our second aim is to provide a description of the principal congruences of Fm×Fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_m \times F_n$$\end{document}. 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.


Introduction
Let PT n denote the monoid of all partial transformations on the set X n = {1, . . . , n}. Let S n denote the symmetric group on X n . Let T n be full transformation monoid, that is, the semigroup of all transformations in PT n with domain X n ; and let I n be the symmetric inverse monoid, that is, the semigroup of all 1-1 maps contained in PT n . The congruences of these semigroups were described in the past: Mal cev [33] for T n , Šutov [38] for PT n and Liber [32] for I n .
In this paper we provide a description of the principal congruences of Q n × Q m (Theorem 3.11), where Q ∈ {PT , T , I}, and then use this result to provide the full description of all congruences of these semigroups (Theorem 4.11).
Similarly, for a field F, denote by F n the monoid of all n ×n matrices with entries in F. The congruences of F n have been described by Mal cev [34] (see also [27]). Here we provide a description of the principal congruences of F n × F m (Theorems 7. 8, 7.10, 7.12, and 7.13).
It is worth pointing out that the descriptions of the congruences of the semigroups S := i∈M Q i and T := where F is a field, M is a finite multiset of natural numbers, and Q ∈ {PT , T , I}, are in fact yielded by the results of this paper, modulo the use of heavy notation and very long, but not very informative, statements of theorems. (For more details we refer the reader to Sect. 5.) It is well known that the description of the congruence classes of a semigroup, contrary to what happens in a group or in a ring, poses special problems and usually requires very delicate considerations (see [26,Section 5.3]). Therefore it is no wonder that the study of congruences is among the topics attracting more attention when researching semigroups, something well illustrated by the fact that the few years of this century already witnessed the publication of more than 250 research papers on the topic.
Given the ubiquitous nature of the direct product construction, it comes quite as a surprise to realize that almost nothing is known about congruences on direct products of semigroups, even when the congruences on each factor of the product are known. Here we start that study trusting that this will be the first contribution in a long sequence of papers describing the congruences of direct products of transformation semigroups. Before closing this introduction it is also worth to add that we have been led to this problem, not just by the inner appeal of a natural idea (describing the congruences of direct products of very important classes of semigroups whose congruences were already known), but by considerations on the centralizer in T n of idempotent transformations. More about this will be said on the problems section at the end of the paper.
In order to outline the structure of the paper, we now introduce some notation. Let S be a finite monoid. We say that a, b ∈ S are H-related if aS = bS and Sa = Sb.
The elements a, b ∈ S are said to be D-related if SaS = SbS. In Sect. 2, we recall the description of the congruences on Q i , which we use in Sect. 3 to fully describe the principal congruences on Q m × Q n , for Q ∈ {PT , T , I}.
In Sect. 4, we show that a congruence θ on Q m ×Q n is determined by those unions of its classes that are also unions of D-classes, which we will call θ -dlocks. After presenting the possible types of θ -dlocks, whose properties are related to their H-classes, we shall describe θ within each such block, making use of the results obtained in Sect. 3. In Sect. 5 we give an idea of how the congruences look like on a semigroup of the form Q m × Q n × Q r . Section 6 is devoted to the description of the congruences on F n following [27], and we dedicate Sect. 7 to characterizing the principal congruences on F m × F n . We do so following the pattern of Sect. 3, doing the necessary adaptations. Describing the general congruences of F m × F n is notationally heavy but we trust the reader will be convinced that to do so nothing but straightforward adaptions of Sect. 4 are needed. The paper finishes with a set of problems.

Preliminaries
For clarity, we start by recalling some well known facts on the Green relations as well as the description of the congruences on an arbitrary Q m . The lattice of congruences of a semigroup S will be denoted by Con(S). For further details see [24].
Given f ∈ Q n we denote its domain by dom( f ), its image by im( f ), its kernel by ker( f ) and its rank (the size of the image of f ) by | f |.

Lemma 2.1 Let f, g ∈ Q n . Then
(1) f Dg iff | f | = |g|; (2) f Lg iff f and g have the same image; (3) f Rg iff f and g have the same domain and kernel; (4) f Hg iff f and g have the same domain, kernel, and image.
Let S be a monoid. A set I ⊆ S is said to be an ideal of S if S I S ⊆ I and an ideal I is said to be principal if there exists an element a ∈ S such that I = SaS. It is well known that all ideals in Q n are principal; in fact, given any ideal I ≤ Q n , then I = Q n a Q n , for all transformation a ∈ I of maximum rank. In addition we have Q n a Q n = {b ∈ Q n | |b| ≤ |a|}.
The Green relation J is defined on a monoid S as follows: for a, b ∈ S, Thus two elements are J -related if and only if they generate the same principal ideal. In a finite semigroup we have J = D (and this explains the definition of D used in Sect. 1). It is easy to see that if | f | < |g|, then S f S ⊆ SgS and in particular f ∈ SgS.
If f Hg then f and g have the same image, so we may speak about the set image im H of an H-class H . Given an H-class H of Q n , we can fix an arbitrary linear order on im H , say that im f = {a 1 < · · · < a | f | } for all f ∈ H . We define a right action · of the group S i with i = | f | on all elements in Q n of rank i: let ω ∈ S i and x ∈ dom( f ), then (x) f · ω = a jω where x f = a j with regard to the fixed ordering associated with the H-class of f . Note that the action · preserves H-classes. Hence for each ω ∈ S i and H-class H with | im H | = i, we may defineω H ∈ S im H , where S im H is the symmetric group over im H , such that (x) f · ω = ((x) f )ω H , for all f ∈ H , x ∈ im H .
The description of the congruences of Q n can be found in [24, sec. 6.3.15], and goes as follows.
Theorem 2.2 A non-universal congruence of Q n is associated with a pair (k, N ), where 1 ≤ k ≤ n, and N is a normal subgroup of S k ; and it is of the form θ(k, N ) defined as follows: for all f, g ∈ Q n , We write θ = θ(k, N ) or just θ if there is no ambiguity. It follows from the normality of N that the definition of θ(k, N ) is independent of the ordering associated with each of the H-classes of Q n . A similar independence result will hold for a corresponding construction in our main result.
The following will be applied later without further reference. Let g, g ∈ Q n . It follows from Theorem 2.2, that if (g, g ) ∈ H, then the principal congruence θ generated by (g, g ) is θ(|g|, N ), where N is the normal subgroup of S |g| generated by σ ∈ S |g| with g = g · σ , with respect to a fixed ordering of the image of g. If where k = max{|g|, |g |}, or the universal congruence if k = n. In either case, this is the Rees congruence defined by the ideal I k of all transformations of rank less or equal to k, i.e. θ I k .
From this description we see that if θ = θ(k, N ) ∈ Con(Q n ) and there exist f, g ∈ Q n , with | f | < |g| and ( f, g) ∈ θ , then |g| < k and the ideal generated by g is contained in a single θ -class.
For each n > 1 the congruences on each semigroup Q n form a chain [24, sec. 6.5.1]. Let ι S and ω S be, respectively, the trivial and the universal congruences on S. For k ∈ {1, . . . , n}, denote by ≡ ε k , ≡ A k and ≡ S k , the congruence associated to k and to the trivial, the alternating and the symmetric subgroup of S k , respectively. Finally, for k = 4, let ≡ V 4 be the congruence associated with the Klein 4-group. We have Let 0 * stand for 1 if Q = T and for 0 in the other cases. For 0 * ≤ i ≤ m, let I (m) i stand for the ideal of Q m consisting of all functions f with | f | ≤ i. We will usually just write I i if m is deducible from the context. Let θ I i stand for the Rees congruence on Q m defined by I i .

Principal congruences on Q m × Q n
The aim of this section is to describe the principal congruences of Q m × Q n , when Q ∈ {I, T , PT }. We will start by transferring our notations to the setting of this product semigroup.
For functions f ∈ Q m ∪ Q n , let | f | once again stand for the size of the image of f , and for ( f, g) ∈ Q m × Q n let |( f, g)| = (| f |, |g|), where we order these pairs according to the partial order ≤ × ≤. Throughout, π 1 and π 2 denote the projections to the first and second factor.
We will start with some general lemmas about congruences on Q m × Q n . In case there is no danger of ambiguity, we will use the shorthand P for Q m × Q n , to simplify the writing. Lemma 3.1 Let θ be a congruence of Q m × Q n and fix f ∈ Q m ; let

Then
(1) θ f is a congruence on Q n ; Proof (1) That θ f is an equivalence on Q n is clear. The compatibility follows from the fact that Q m has an identity; indeed Similarly we prove the left compatibility.
In a similar way, given a congruence θ on Q m × Q n and fixed g ∈ Q n , we define The next result describes the ideals of Q n × Q m .
j is an ideal of P is obvious. Conversely, let I be an ideal of P, and ( f, g) ∈ I . It is self-evident that ( f, g) ∈ I | f | × I |g| . Then, by the definition of an ideal of P, we have P( f, g)P ⊆ I , for every ( f, g) ∈ I . Let ( f , g ) ∈ I | f | × I |g| . Then f ∈ Q m f Q m and g ∈ Q n gQ n so that ( f , g ) ∈ P( f, g)P ⊆ I . It follows that ∪ ( f,g)∈I I | f | × I |g| ⊆ I . Regarding the reverse inclusion, let ( f, g) ∈ I ; then ∪ ( f,g)∈I I | f | × I |g| ⊇ I . The result follows. (1) If Q ∈ {PT , I}, then θ contains a class I θ which is an ideal; (2) If Q = T and both π 1 (θ ) and π 2 (θ ) are non-trivial, then θ contains a class I θ which is an ideal; (3) θ contains at most one ideal class.
Proof (1) If Q is PT or I, then P has a zero, whose congruence class is easily seen to be a unique ideal of P.
(2) If Q is T , let c a denote the constant map whose image is {a}, for some a ∈ {1, . . . , m}. Since π 2 (θ ) is non-trivial, it follows that there exist f, f ∈ Q m and distinct g, g ∈ Q n such that We claim that I (m) × {c e } lies in a θ -class, and similarly Conversely, given any ( f, g) in the θ -class of (c a , c b ) and any ( f , g ) ∈ P, we have (3) The last assertion holds in all semigroups, as any ideal class is a (necessarily unique) zero element of the quotient semigroup P/θ .
For the remainder of this section we fix the following notations. Let Q ∈ {T , PT , I}, f, f ∈ Q m and g, g ∈ Q n . Let θ be a principal congruence on Q m × Q n generated by (( f, g), ( f , g )). Let θ 1 be the principal congruence generated by ( f, f ) in Q m and θ 2 be the principal congruence generated by (g, g ) in Q n .
Proof Let θ be the binary relation defined by the statement of the lemma. If |a| ≤ | f |, then a = u f v, for some u, v ∈ Q m and hence ((a, g), (a, g )) = Conversely, it is straightforward to check that θ is a congruence containing (( f, g), ( f, g )), thus θ ⊆ θ . The result follows.
The following corollary gives a more direct description of the congruences covered by Lemma 3.4 by incorporating the structure of the congruence θ 2 on Q n . By applying Theorem 2.2, we obtain (2) a = c and |a| = |c| ≤ | f |, |b|, |d| ≤ k.
If (g, g ) ∈ H and g = g then θ 2 = θ(k, N ), for k = |g| and N is the normal subgroup of S k generated by σ , where g = g · σ . Moreover, (a, b)θ (c, d) if and only if one of the following holds: (2) a = c, |b| = |d| = k, bHd and d = b · ω for some ω ∈ N ; (3) a = c, |a| = |c| ≤ | f |, |b|, |d| < |g|. Proof We will show that for j = | f |, θ I j ⊆ θ g . An analogous result for j = | f | follows symmetrically. So let us assume that | f | ≤ | f | = j. As ( f, f ) / ∈ H, f and f must differ in either image or kernel. We consider two cases.
First case: im f = im f .
∈ H. Now, the congruence θ generated by ( f h, f ) is contained in θ g and by Theorem 2.2, we have θ = θ I j . We get θ I j ⊆ θ g .
Second case: ker f = ker f .
implies that ker f ker f . As above, the regularity of Q m implies that there exists an idempotent h that is R-related to f ; thus h and f have the same kernel. Hence Theorem 3.7 Let θ be the congruence on Q m × Q n generated by (( f, g), ( f , g )), and assume that Then θ is the Rees congruence on Q m × Q n defined by the ideal I = I i × I k ∪ I j × I l .
Proof If Q = T , then pick two arbitrary constants z = c a and z = c b in Q m and Q n , respectively. If Q ∈ {PT , I}, let z, z be the empty maps in Q m and Q n .
As f = f and g = g , by Lemma 3.3, the congruence θ contains an ideal class K . As (z, z ) lies in the smallest ideal I 0 * × I 0 * of P, (z, z ) ∈ K . We claim that ( f, g) ∈ K .
To show this, note that by Lemma 3.6, either θ g or θ g contains the Rees congruence θ I max{i, j} . The dual of Lemma 3.6 guarantees that either θ I max{k,l} ⊆ θ f or θ I max{k,l} ⊆ θ f . Up to symmetry, there are two cases.
First case: θ I max{k,l} ⊆ θ f , θ I max{i, j} ⊆ θ g . We have that g, z ∈ I max{k,l} , so (g, z ) ∈ θ I max{k,l} ⊆ θ f , that is, ( f, g)θ ( f, z ). As f, z ∈ I max{i, j} , an analogous argument shows that ( f, z) ∈ θ g . By Lemma 3.1, we have θ g ⊆ θ z , and so ( f, z) ∈ θ z . Thus ( f, z )θ (z, z ) and hence Second case: θ I max{k,l} ⊆ θ f , θ I max{i, j} ⊆ θ g . We have g, z ∈ I max{k,l} so (g, z ) ∈ θ f and similarly ( f , z) ∈ θ g . Thus Let h ∈ S m be such that f h f = f (such h clearly exists). We then have Hence in both cases ( f, g) ∈ K . As K is a class of θ , then ( f , g ) ∈ K as well. Proof We start with some considerations having in mind the initial conditions. By the dual of Lemma 3.1, we have θ g = θ g ⊆ θ h for all h ∈ Q n with |h| ≤ k. By Lemma 3.6, we get θ I j ⊆ θ g = θ g , hence that θ I j ⊆ θ h for each such h. Assume w.l.o.g. that | f | = j. Now f, f ∈ I j and θ I j ⊆ θ g , so f θ g f . Thus ( f , g )θ ( f, g)θ ( f , g). Then (g, g ) ∈ θ f and therefore θ 2 ⊆ θ f , as θ 2 is the congruence generated by (g, g ). By Assume that |a|, |c| ≤ j, |b|, |d| ≤ k and bθ 2 d. Taking u = a we obtain θ 2 ⊆ θ a . As bθ 2 d, we get (a, b)θ (a, d). Now |d| ≤ k which, as mentioned at the beginning of the proof, implies that For the reverse inclusion, it suffices to check that θ is a congruence containing (( f, g), ( f , g )). We leave this straightforward verification to the reader.
Notice that we can once again give a more explicit description of θ by incorporating the classification of θ 2 given by Theorem 2.2. Corollary 3.10 Let ( f, g), ( f , g ) ∈ Q m × Q n , such that g = g , (g, g ) ∈ H and ( f, f ) / ∈ H. Let j = max{| f |, | f |} and k = |g| = |g |. Let g = g ·σ for σ ∈ S k with regard to some ordering associated with the H-class of g, and let N be the normal subgroup of S k generated by σ . If θ is the congruence on Q m × Q n generated by if and only if one of the following holds: (2) |a|, |c| ≤ j, |b| = k, bHd and d = b · ω for some ω ∈ N , and with regard to some ordering associated with the H-class of d; (3) |a|, |c| ≤ j and |b|, |d| < k.
We remark that there are obvious dual versions of Lemma 3.4 and Theorem 3.9 obtained by switching the roles of the coordinates. Apart from the trivial case that We will first extend the actions · of S i on Q m and of S j on Q n to a partial action of S i × S j on Q m × Q n . We define the action · of S i × S j on the set where in the first component · is applied with respect to the ordering of the H-class of f within Q m , and correspondingly in the second component.
As H-classes of Q m × Q n are products of H-classes of Q m and of Q n , it follows that the action · preserves H-classes. In addition, the action · is transitive on each H-class. If H f and H g stand for the H-classes of f in Q m and of g in Q n , we have ∈ S im g are as defined before Theorem 2.2. In this context, we will always consider S im f × S im g to be a subgroup of S m × S n in the natural way.

Theorem 3.11
Let θ be the principal congruence on Q m × Q n generated by and σ 2 ∈ S k be such that f · σ 1 = f and g · σ 2 = g . Let N be the normal subgroup of S i × S k generated by the pair (σ 1 , σ 2 ). Then or one of the following hold: As f H f , we have that f and f have the same kernel and image. Together Let h ∈ T n be such that yh = x and it is identical otherwise. Then Let β be the congruence generated by the pair if and only if |a|, |c| ≤ i − 1, |b|, |d| ≤ k, and bθ 2,β d, where θ 2,β is the Q ncongruence generated by (g, g ) and hence is equal to θ 2 . By Theorem 2.2, the relation θ 2 restricted to I k−1 is the universal relation. Therefore the pairs ((a, b), (c, d)) that satisfy condition (1) or (3) are in β, but β's generating pair is in θ , and so they are in θ , as well.
∈ H, and by Theorem 3.9, as before, we conclude that θ must contain all pairs satisfying conditions (1) or (3).
Next suppose that Q = I. We have i ≥ 2. In this case, there exists Once again applying Theorem 3.9, we conclude that θ must contain all pairs satisfying conditions (1) or (3).
By symmetrically applying the above considerations to the second argument, we also show that θ contains the pairs that satisfy condition (2). The next step is to prove that, for any Q ∈ {T , PT , I} the pairs that satisfy condition (4) are also in θ .
Note that the group of units of Q m × Q n is S m × S n . We can choose (u, v) ∈ S m × S n such that both u f and vg are idempotent transformations.
Hence we may assume w.l.o.g. that f , g are idempotents.
Let H be the H-class of ( f, g). Then ( f , g ) ∈ H . As H contains an idempotent, H is a group. Moreover, it is easy to see that φ given by ( Let θ be the restriction of θ to H , then θ is a congruence on a group. Let K be the normal subgroup of H corresponding to θ , and K =K φ −1 . As an idempotent, As |a| = | f | and |b| = |g|, we have aJ f , and bJ g, so there exists h 1 , h 2 ∈ Q m and h 3 , h 4 ∈ Q n such that a = h 1 f h 2 , b = h 3 gh 4 . Once again as |a| = | f | and |b| = |g|, h 2 | im f is an injection and so is h 4 | im f . Hence w.l.o.g. we may assume that = h · τ s for all h ∈ π 1 (H ). We will writeτ 1 for the extension ofτ π 1 (H ) s to S m that is the identity on {1, . . . , m}\ im H , and use corresponding notation if τ 1 is replaced by other elements of S i or S k . Consider 2 , as these elements agree on im f and are the identity otherwise. Henceω 1 andτ 1 are conjugate in S m and so have the same cycle structure. The cycle structure ofω 1 is obtained from ω 1 by the addition of m − i trivial cycles. The same holds forτ 1 and τ 1 . It follows that τ 1 and ω 1 are conjugates in S i . Analogously,ω 2 = h 4τ2 h −1 4 , and τ 2 and ω 2 are conjugates in S k . Therefore, We obtain that as required.
It follows that all pairs that satisfy one of the conditions (1) to (4) are in θ .
We can get a more direct description of the congruence classes by using the following folklore result. Its proof is an easy exercise.
Under the conditions of Theorem 3.11, if both σ 1 , σ 2 are odd permutations, then d) if and only if one of the following holds:

The structure of all congruences on Q m × Q n
We now look at our main aim: to determine the structure of all congruences on Q m × Q n . When studying a congruence θ on Q m × Q n , as in the case of Q n (Theorem 2.2), we realize that the θ -classes are intrinsically related to the D-classes of Q m × Q n . We shall show that θ is determined by some minimal blocks of θ -classes, called here θ -dlocks, which are also unions of D-classes. The strategy will be to determine the possible types of θ -dlocks and to describe θ within such blocks. Throughout this section, θ denotes a congruence on Q m × Q n . To avoid some minor technicalities, we assume that Q m and Q n are non-trivial, i.e. we exclude the factors T 0 , T 1 , PT 0 , I 0 .
(1) X is a union of θ -classes as well as a union of D-classes; (2) No proper non-empty subset of X satisfies (1).
In other words, the θ -dlocks are the classes of the equivalence relation generated by D ∪ θ . We will just write dlock if θ is understood by context, and will describe dlocks by listing their D-classes. Concretely, let D i, j be the D-class of all pairs ( f, g) such that | f | = i, |g| = j. Recall that 0 * refers to 0 for Q ∈ {I, PT } and to 1 for First we describe the various configurations of dlocks with respect to the H-classes they contain. To this end, we will divide the dlocks into 9 different types. Definition 4.2 Let X be a θ -dlock. We say that X has first component type We define the second component type of X dually. Finally we say that X has type V W for V, W ∈ {ε, H, F} if it has first component type V and second component type W .
Clearly, we obtain all possible types of dlocks. Next, we will describe the congruence θ by means of its restriction to each of its dlocks.
Proof Let X be a θ -dlock of type F F and assume that its first component type F is witnessed by , all sets of the form I i × {b}, where |b| ≤ k, are contained in congruence classes of θ . By a dual argument {a} × I k is contained in a θ class for |a| ≤ i. By choosing |a| = 0 * = |b|, we see that these sets intersect. It follows that θ and hence θ have a class that contains D 0 * ,0 * . It is straightforward to check that such a θ -congruence class Y is an ideal of Q m × Q n . By Lemma 3.2, all ideals are unions of D-classes. Therefore Y is a dlock that contains X . As dlocks are disjoint X = Y , and so X is a single θ -class that is an ideal.
Conversely, suppose that I is an ideal class of θ , and that P indexes the D-classes intersecting I . By Lemma 3.2, the ideal I is a union of D-classes, so that D P = I , and I is a dlock. It is a dlock of type F F unless either π 1 (I ) or π 2 (I ) consists of a single H-class. The listed exceptions are the only way this can happen, as we assumed that Q n , Q m are non-trivial.
In particular, by Lemma 3.3, the congruence θ has at most one dlock of type F F. If it exists, the unique dlock of type F F is the θ -class that contains D 0 * ,0 * . By Lemma 3.2, we can visualize this dlock as a "landscape" (see Fig. 1).

Lemma 4.4
Let X be a θ -dlock. Then X is of type εF or HF if and only if there exist 0 * ≤ i ≤ m, 1 ≤ j ≤ n, and N S i , such that X = D P with P = {i} × {0 * , . . . , j}, and for every ( f, g) ∈ X, If X satisfies these requirements, then X is of type HF exactly when N = ε i .
Proof Let X = D P be a θ -dlock. Suppose X is of type εF or HF. Then for every θ -class C contained in X , we have π 1 (C) contained in an H-class of Q m by the definitions of first component types ε and H. Hence π 1 (C) is contained in a D-class of Q m for all such C. Since the elements of π 1 (X ) that are θ -related are pairwise either equal or H-related, π 1 (X ) must be a single D-class of Q m , say D i , by the definition of θ -dlock. Therefore P = {i} × K for some non-empty K ⊆ {0 * , . . . , n}. Let j be the largest element in K . We claim that there exists (( f, g), ( f , g )) ∈ θ with (g, g ) / ∈ H and ( f, g) ∈ D i, j . For otherwise [(f ,ḡ)] θ would be contained in an H-class for every (f ,ḡ) ∈ D i, j , and then {(i, j)} would index a dlock contained in X , and X being minimal would imply D i, j = X . However, X is a dlock of type F or HF, and we have a contradiction.
So there exists Let θ be the principal congruence generated by (( f, g), ( f , g )). Either Corollary 3.5 or the dual of Theorem 3.9 is applicable to θ -the first one if f = f , and hence N = ε i , and the second one otherwise.
We claim that the sets from (2) or (3) are also congruence classes of θ . As θ ⊆ θ the sets in (2) Now let E be a congruence class of θ that is contained in X . Then E must be a union of sets from (2) or (3), and in particular, must intersect D i, j , say (f ,ĝ) ∈ E ∩ D i, j . Let (f ,ḡ)θ (f ,ĝ). Then |ḡ| ≤ j, since E ⊆ X = D P . Moreover (f ,f ) ∈ H as X is a dlock of type εF or HF. Now, if β is given byf =f · β, then β ∈ N , by the maximality of N . It follows that E is one of the sets from (2), (3), and thus contained in Notice that j cannot be 0. In fact, if j = 0, then π 2 (X ) only contains one element, which contradicts the definition of type F or HF. We have concluded the proof of the "if" direction of the first statement of the lemma.
The "only if" direction now follows directly from the description (1), provided that there exists (g, g ) / ∈ H, (g, g ) ∈ π 2 (X ). This holds as Q n is non-trivial and j ≥ 1. Finally, the last statement follows directly from (1) and the definitions of type εF or HF.
If X is a dlock of type HF or εF, we call the group N S i from Lemma 4.4 the normal subgroup associated with X . Clearly, a dual version of Lemma 4.4 holds for dlocks X of type Fε or FH. Moreover, every congruence θ has at most one dlock of type HF.
Proof By Lemma 4.4, we have ε i = N S i , which implies that i ≥ 2. By the description (1), we may find (( f, g), ( f , g ) Let θ be the principal congruence generated by (( f, g), ( f , g )). Then θ is described in the dual of Corollary 3.10. By this corollary, if C := {0 * , . . . , i − 1} × {0 * , . . . , j} then D C is one equivalence class of θ , and hence contained in an equivalence class of θ . As j ≥ 1, i − 1 ≥ 1, and we excluded the case that Q n is trivial, π 1 (D C ) and π 2 (D C ) both contain more then one H-class. Hence C is contained in the index set of a θ -dlockX of type F F. Now assume that θ has a potentially different dlock X = D P of type HF, where P = {i } × {0 * , . . . , j }. By applying our previous results to X , we get that D i −1,0 * must also lie in a θ -dlockX of type F F. As noted after Lemma 4.3, dlocks of type F F are unique, and so D i −1,0 * ⊆X . So both D i−1,0 * ⊆X and D i −1,0 * ⊆X , but D i,0 * X and D i ,0 * X , as these D-classes lie in the HF-dlocks X and X . By Lemma 4.3, the index set ofX is downwards closed, hence there is a uniqueī such that D¯i ,0 * ⊆X , D¯i +1,0 * X . It follows that i =ī + 1 = i . Thus both X and X contain D i,0 * and therefore X = X .
We may visualize the statement of Lemma 4.5 by saying that a θ -dlock of type HF must lie on the "most eastern slope" of the dlock of type F F. Once again, a dual result holds for dlocks of type FH. Figure 2 shows the possible positions for dlocks of type HF and FH in relation to a dlock of type F F. Proof If i = 0 * , the statement quantifies over the empty set, so assume this is not the case. It suffices to show the statement for the case that k = i − 1, the remaining values By Lemma 4.4 we may find (( f, g), ( f, g )) ∈ θ ∩ X 2 , such that ( f, g) ∈ D i, j , (g, g ) / ∈ H. Note that this implies that |g | ≤ j. Let θ be the principal congruence generated by (( f, g), ( f, g )). Then θ is described in the dual of Corollary 3.10. By this corollary, if |f | = i − 1, then It follows that {i − 1} × {0 * , . . . , j} is the index set of a θ -dlock and hence contained in the index set of a θ -dlock X . As ((f , g), (f , g )) ∈ θ ⊆ θ , and (g, g ) / ∈ H, it follows that X is of type F F, HF, or εF, as these three options cover all cases where π 2 (D P ) H. The result follows. Proof Assume that i 1 < i 2 < i 3 with i 1 , i 3 ∈ π 1 (J ). We want to show that i 2 ∈ π 1 (J ), as well. As i 1 , i 3 ∈ π 1 (J ), X 1 = D {i 1 }×{0 * ,..., j i 1 } and X 3 = D {i 3 }×{0 * ,..., j i 3 } are θ -dlocks of type εF, for some j i 1 , j i 3 . Applying Lemma 4.6 to X 3 , we get that D {i 2 }×{0 * ,..., j i 3 } is contained in θ -dlock X of type F F, HF, or εF.
If X were of type HF or F F, then by Lemmas 4.5 or 4.3, respectively, D {i 1 }×{0 * ,..., j i 3 } would be contained in a dlock of type F F, which in the latter case would be the dlock X itself. In particular, D i 1 ,0 * would be contained in a dlock of type F F. However, as i 1 ∈ J , D i 1 ,0 * is contained in the dlock X 1 of type εF, a contradiction. Hence X is of type εF, and i 2 ∈ π 1 (J ). It follows that π 1 (J ), if not empty, is a set of consecutive integers. Now let (i, j i ), (i − 1, j i−1 ) ∈ J . As above, we have that D {i−1}×{0 * ,..., j i } is contained in a θ -dlock X of type F F, HF, or εF. As (i − 1, j i−1 ) ∈ J , this must Moreover, in this situation, Proof Let X be of type HH, εH, Hε, or εε. Then π 1 (θ ∩ X 2 ) ⊆ H and π 2 (θ ∩ X 2 ) ⊆ H. It follows that each H-class in X is a union of θ -classes. Therefore every D-class in X is a union of θ -classes as well, and by the minimality property of a dlock, there is only one D-class in X . It follows that X = D i, j for some (i, j). Now as θ ∩ X 2 ⊆ H × H, having (( f, g), ( f , g )) ∈ θ ∩ X 2 implies that f = f · σ, g = g · τ for some σ ∈ S i and τ ∈ S j . Let N ⊆ S i × S j be the set of all (σ, τ ) that correspond to some (( f, g), ( f , g )) ∈ θ ∩ X 2 .
If N = {(id S i , id S j )} then θ is the identity on X , and θ ∩ X 2 is given by (4) with the choice N = ε i × ε j .
It remains to show that N is a normal subgroup of S i × S j . Let (σ, τ ), (σ , τ ) ∈ N , and ( f, g) ∈ X arbitrary, then where the first implication follows as (4) describes θ on X , and the second implication as · is a group action. Thus N is a subgroup of S i × S j . Now if (σ, τ ) ∈ N , then by either Theorem 3.11, Lemma 3.4, or the dual of Lemma 3.4 (applied to any (( f, g), ( f · σ, g ·τ )) ∈ θ ∩ X 2 ), N contains the normal subgroup generated by (σ, τ ). Therefore N is a subgroup generated by a union of normal subgroups, and hence it is itself normal. The converse statement is immediate, and the characterization of the various types follows directly from the definition of the types and from (4).
If X = D i, j is a dlock of type HH, εH, Hε, or εε, we will call N S i × S j from Lemma 4.8 the normal subgroup associated with X . Proof By Lemma 4.8, we have π 1 (N ) = ε i , π 2 (N ) = ε j , and so i, j ≥ 2. Also by Lemma 4.8, there are (( f, g), ( f , g )) ∈ θ ∩ X 2 with f = f , g = g . Let us fix such a pair, and let θ be the principal congruence generated by it. Then θ ⊆ θ and θ is described in Theorem 3.11. Letḡ,ḡ ∈ Q m be transformations of rank j − 1 that are in different H-classes. Such elements clearly exist. By Theorem 3.11(2), we get (( f,ḡ), ( f ,ḡ )) ∈ θ ⊆ θ .
As f = f , (ḡ,ḡ ) / ∈ H, the θ -dlock X containing D i, j−1 is either of type F F or HF, depending on the existence or not of a pair ((f ,ĝ), (f ,ĝ ) In the first case, we are done, so assume that X is of type HF. Then the restriction of θ to X 2 is given in Lemma 4.4. Let N S i be the normal subgroup of X . We now wish to prove that π 1 (N ) ⊆ N .
The last statement follows dually. The result means that the dlocks of type HH can only occupy the "valleys" in the landscape formed by the dlocks of F F, HF, and FH (see Fig. 4).

Lemma 4.10
Proof As N is non-trivial, j ≥ 2. Let σ generate N as a normal subgroup in S j . Let ( f, g) ∈ X and set g = g · σ . Then (( f, g), ( f, g )) ∈ θ . Let θ be the principal congruence generated by this pair. Then θ ⊆ θ and θ is described in Lemma 3.4. Letḡ,ḡ ∈ Q m both be transformations of rank j − 1 that are in different Hclasses. Such elements clearly exist. By Lemma 3.4, (( f,ḡ), ( f,ḡ )) ∈ θ ⊆ θ . As ∈ H orf =f . Now assume that i > 0 * , and letf ∈ Q n be a transformation of rank i − 1. Once again by Lemma 3.4, it follows that θ , and therefore θ , contains ((f , g), (f , g )). Let X be the dlock containing D i−1, j , then ((f , g), (f , g )) ∈ θ ∩ X 2 . It follows that X must be of a type for which π 2 (θ ∩ X 2 ) is not the identity. The six listed types in the statement of the theorem are exactly those for which this condition is satisfied.
Now ((f , g), (f , g )) = ((f , g), (f ·id S i−1 , g·σ )) ∈ θ . In the fourth case, i.e. when X is a dlock of type FH with normal subgroupN , we have that σ ∈N . Similarly in the fifth and sixth cases, we get that (id S i−1 , σ ) ∈N . As σ generates N as a normal subgroup, the statements in the last three cases follow.

Fig. 5 A possible configuration for dlocks of all types other than εε
We conclude that the dlocks of type εH can be placed onto the "west-facing" slopes of the landscape made up of the dlocks of type F F, HF, or εF. For any such slope the dlocks of type εH, must be "staked on the top of each other", with the initial εH-dlock being placed on either a "step" of the F F-HF-εF-landscape or on a dlock of type HH or FH (see Fig. 5). Symmetric statements hold for dlocks of type Hε.
We will not derive any additional conditions for dlocks of type εε, so we may use them to fill out the remaining "spaces" in our landscape without violating any conclusion achieved so far.
The results of this section give us tight constraint about the structure of any congruence θ on Q m × Q n . In our next theorem we will state that all the conditions we have derived so far are in fact sufficient to define a congruence.   B is a part of type HH, εH, Hε, or εε then  Suppose that on each P-part B we define a binary relation θ B as follows: Fε, let ( f, g)θ B ( f , g ) if and only if gHg and g = g · σ for some σ ∈ N B ; (iv) If B has type HH, εH, Hε, or εε, let ( f, g) Let θ = ∪ B∈P θ B . Then θ is a congruence on Q m × Q n . Conversely, every congruence on Q m × Q n can be obtained in this way.
Proof The "converse" part of this last theorem follows from Lemmas 4.3 to 4.10 and, where applicable, their dual versions. To show that θ is a congruence involves checking for each (( f, g), ( f , g )) ∈ θ , that the principal congruence generated by (( f, g), ( f , g )) is contained in θ , using our results on principal congruences from Sect. 3. The proof is straightforward, but it requires the verification of many different cases. We have opted not to write it here to limit the length of the article.

Observation 4.12
We remark that Theorem 4.11 also holds for the more general case of semigroups of the form Q m × P n , where Q, P ∈ {PT , T , I}, provided that the expression 0 * is interpreted in the context of the relevant factor and the exceptional cases (1)(a) and (1)(b) are are conditional on individual factor types. In fact, nearly all our results and proofs carry over to the case of Q m × P n without any other adjustments. The exceptions are Lemma 3.3, Theorem 3.7, and Theorem 3.11, which require simple and straightforward modifications.

Products of three transformation semigroups
As said above, the results of this paper essentially solve the problem of describing the congruences of Q n 1 × Q n 2 × Q n 3 . . . × Q n k , the product of finitely many transformation semigroups of the types considered, although the resulting description of the congruences would require heavy statements and notation, but not much added value.
To illustrate our point, we have included a series of figures that give an idea of how the dlock-structure of a triple product looks like. In Figs. 6,7,8,9,10,11,12,13, and 14, each D-class is represented by a cube, and D-classes belonging to the same dlock are combined into a colour-coded polytope. The figure is orientated so that the cube representing the D-class of D 0 * ,0 * ,0 * is furthest away from the observer and obstructed from view.
To reduce the number of required colors, types that are obtained by a permutation of the coordinates have the same colour. Each figure adds the dlocks from one such colour group to the previous figure. For example, Fig. 6 contains one grey dlock of type F F F, while Fig. 7 adds three red dlocks of types F FH, FHF and HF F.
The following pairs of figures, from the dual and triple product case, can be considered to be in correspondence with each other: Figs To obtain a final configuration from Fig. 14, one needs to fill out all remaining spaces with cubes that represent dlock-type εεε. Put together, the figures demonstrate a large number, but not all, of the possible configuration of dlock-types. Fig. 6 F, F, F   Fig. 7 F, F, H  Fig. 8 F, H, H   Fig. 9 F, F, ε   Fig. 10 F, H, ε   Fig. 11 F, ε, ε   Fig. 12 H, H, H   Fig. 13 H, H, ε  Fig. 14 H, ε, ε

Matrix monoids
Let F be a field with multiplicative unit group F * . Consider the multiplicative monoid F n of all n × n-matrices over F. We will, throughout, identify matrices with their induced (left) linear transformation on the vector space F n . The rank, kernel, and image of a matrix are now defined with regard to their usual meanings from linear algebra. Note that in particular the definition of kernel is now different from the notation of kernel used in the section on transformation semigroups. In addition, matrix multiplication corresponds to a composition of linear transformations that is left-to right, and hence inverse from the situation for transformation monoids.
In this section, we will determine the principal congruences on the monoid F m × F n . As it turns out, this case closely mirrors the situation of the semigroup PT m ×PT n . In many cases, transferring the proofs of the previous sections to our new setting requires only an adaptation of notation. In those cases, we will leave it to the reader to make the relevant changes.
Other than notional changes, the main difference from the situation on PT m ×PT n corresponds to the description of the congruence generated by a pair of the form (( f, g), ( f, g )). For matrix monoids, this congruence properly relates to the congruence generated by some ((A, B), (λA, B )), where λ ∈ F * . Hence our description needs to be adapted to take care of the extra parameter λ.
We will start by recalling several facts about the monoids F n . Recall that two matrices are R-related if they have the same image, L-related if they have the same kernel, H-related if they have the same image and kernel, and D-related if they have the same rank.
We let e i, j , for 1 ≤ i, j ≤ n, be the elements of the standard linear basis of F n and set E i = e 1,1 + e 2,2 + · · · + e i,i . We identify the linear group GL(i, F) with the maximal subgroup of F n that contains E i . Denote the identity matrix on F n by 1 and the zero matrix by 0. We have 1 = E n , and we set E 0 = 0.
The description of the congruences of F n can be found in [34]. We will however use the following slightly different description from [27]. While this is an unpublished source, the two characterization only differ on condition (b) of the following description, and it is an easy exercise to check that they are indeed equivalent. (2) if μ ≤ n − 1 there exist subgroups G μ+1 , G μ+2 , . . . , G n of F * such that G n ⊆ G n−1 ⊆ · · · ⊆ G μ+1 and G μ+1 E μ ⊆Ḡ μ ; (3) two matrices A and B are in R if and only if one of the following conditions holds: (a) rank A < μ and rank B < μ; (b) rank A = rank B = μ, and there exist s 1 , s 2 ∈ GL(n, F) such that s 1 As 2 and s 1 Bs 2 are both in GL(μ, F), and belong to the same coset of GL(μ, F) moduloḠ μ ; (c) rank A = rank B = i, for some μ < i ≤ n, and A = λB for some λ ∈ G i .

Theorem 6.1 A binary non-universal relation R on F n is a congruence if and only if
In addition, we need the following result from [27].

Lemma 6.2 A matrix A ∈ F n is a non-zero scalar multiple of the identity matrix if and only if A fixes all subspaces of F n of dimension n − 1.
The following will be applied later without further reference. Let A, B ∈ F n . From Theorem 6.1, the principal congruence of F n generated by (A, B) corresponds to the following parameters in the theorem: • If A = λB for some λ ∈ F * , then μ = 0, G n = · · · = G rank A+1 = {1}, G rank A = · · · = G 1 = λ , andḠ 0 = {0}; • If rank A = rank B, A = λB for all λ ∈ F * , and AHB, then μ = rank A, G n = · · · = G μ+1 = {1}, andḠ μ is the normal subgroup of GL(μ, F) that corresponds to the congruence generated by the pair (s 1 As 2 , s 1 Bs 2 ), where s 1 , s 2 ∈ GL(n, F) are such that s 1 As 2 , s 1 Bs 2 ∈ GL(rank A, F); For 0 ≤ i ≤ m, let I (n) i stand for the ideal of F n consisting of all matrices A with rank A ≤ i. We will usually just write I i if n is deducible from the context. Let θ I i stand for the Rees congruence on F n defined by the idealI i .

Principal congruences on F m × F n
For A ∈ F m ∪F n , let |A| = rank A, and for (A, B) ∈ F m ×F n let |(A, B)| = (|A|, |B|), where we order these pairs according to the partial order ≤ × ≤. Throughout, π 1 and π 2 denote the projections from F m × F n to the first and second factor, respectively.
We will start with some general lemmas concerning congruences on F m × F n .

Then
(1) θ A is a congruence on F n ; (2) if A ∈ F m and |A | ≤ |A|, then θ A ⊆ θ A ; Proof The proof of this lemma is virtual identical to the proof of Lemma 3.1, and is obtained from it by the syntactic substitutions f → A, g → B, Q → F.
In an analogous construction, given a congruence θ on F m × F n and fixed B ∈ F n , we define θ B := {(A, A ) ∈ F m × F m | (A, B)θ (A , B)}. The next result describes the ideals of F n × F m .

Lemma 7.3 Let θ be a congruence on F m × F n . Then
(1) θ contains a class I θ which is an ideal; (2) θ contains at most one ideal class.
For the remainder of this section we fix the following notation. Let K , K ∈ F m and L , L ∈ F n . Let θ be a principal congruence on F m × F n generated by ((K , L), (K , L )). Let θ 1 be the principal congruence generated by (K , K ) on F m and θ 2 be the principal congruence generated by (L , L ) on We claim that the definition of H is independent of our choice for (s 1 , s 2 ), (s 3 , s 4 ). For suppose that (t 1 , t 2 ), (t 3 , t 4 ) ∈ S m × S n are such that (t 1 , Let V, W be the subspaces of F m and F n generated by the columns of E i and E j , respectively. It is easy to check that left multiplication by (t 1 , t 2 )(s 1 , t 4 ), for all A ∈ GL(i, F)×GL( j, F). Now, (5) shows that in the group GL(i, F)× GL( j, F), we have t = s −1 . Thus, in GL(i, F) × GL( j, F), (t 1 , t 2 )(K , L )(t 3 , t 4 ) = s(s 1 , s 2 )(K , L )(s 3 , s 4 )s −1 is a conjugate of (s 1 , s 2 )(K , L )(s 3 , s 4 ) and thus generates the same normal subgroup.
If K = λK for some λ ∈ F * , it is easy to see that the normal subgroup H of GL(i, F) × GL( j, F) associated with ((K , L), (K , L )) is contained in F * E i × GL( j, F). We then associate a normal subgroupĤ of F * × GL( j, F) with the pair ((K , L), (K , L )), takingĤ as the image of H under the canonical map from F * E i × GL( j, F) to F * ×GL( j, F). If L = λL for some λ ∈ F * , we dually associate a normal subgroup of GL(i, F) × F * .
Letθ be the restriction of θ to the group GL(i, F) × GL( j, F), and R the normal subgroup of GL(i, F) × GL( j, F) corresponding toθ. By (6), we have Proof We have K HK and MHM so, by Lemma 7.4, we may assume that (K , It is straightforward to check that the normal subgroup of F * × GL(k, F) associated with ((E k , E j ), (λE k , L )) isĤ .
LetH be the normal subgroup of GL(k, F) × GL( j, F) associated with this pair, so thatĤ = φ(H ), where φ is the natural isomorphism from F * E k × GL( j, F) to F * × GL( j, F). LetH be the normal subgroup of GL(k, F) × GL( j, F) associated with the pair ((M, N ), (M , N )).
The result now follows with Lemma 7.4.
It is clear that a dual version of Lemma 7.5 holds as well, so let us look at the principal congruences on F m × F n . Proof This theorem can be shown by applying Lemma 7.5 followed by its dual. However, we will give a short direct proof.
Let θ be the binary relation defined by the statement of this theorem.
Conversely, it is straightforward to check that θ is a congruence containing ((K , L), (K , L )), therefore θ ⊆ θ . The result follows. Lemma 7.7 Let θ be the principal congruence on F m × F n generated by ((K , L), (K , L )), and let j = max{|K |, |K |}. If (K , K ) / ∈ H, then θ L or θ L contains the Rees congruence θ I j of F m .
We remark that the following proof is essentially equivalent to the proof of Lemma 3.6. As the technical adaptations required to transform one lemma into the other are more complex than in previous cases, we have decided to provide a complete proof.
Proof We will show that for j = |K |, θ I j ⊆ θ L . An anologous result for j = |K | follows symmetrically. So let us assume that |K | ≤ |K | = j. As (K , K ) / ∈ H, K and K must differ in either the image or the kernel. We consider two cases.
First case: im K = im K . As |K | = j ≥ |K |, then im K im K . As F m is regular, there exists an idempotent M ∈ F m such that MRK . Hence im M = im K , and as M is idempotent, We have that (M K , L ) = (M, E n )(K , L )θ (M, E n )(K , L) = (M K, L) = (K , L)θ (K , L ) and so (M K , K ) ∈ θ L on F m . As im K im K , and im M = im K , the transformations M K and K have different images, and it follows that (M K , K ) / ∈ H. Now, the congruence θ generated by (M K , K ) is contained in θ L and by Theorem 6.1 and the remarks following it, we have θ = θ I j . We get θ I j ⊆ θ L . Second case: ker K = ker K . Now |K | = j ≥ |K |, implies that ker K ker K . As above, the regularity of F m implies that there exists an idempotent M that is L-related to K ; thus M and K have the same kernel and K M = K . Hence (K M, L ) = (K , L )(M, E n )θ (K , L)(M, E n ) = (K M, L) = (K , L)θ (K , L ) and so (K M, K ) ∈ θ L . Now ker K = ker M ⊆ ker(K M). As ker K ker K , we get ker K = ker(K M) and so (K M, K ) / ∈ H. As above, by Theorem 6.1 and the remarks following it, we get θ I j ⊆ θ L . Theorem 7.8 Let θ be the congruence on F m × F n generated by ((K , L), (K , L )). Assume that (K , K ) / ∈ H, (L , L ) / ∈ H, |K | = i, |K | = j, |L| = k, and |L | = l. Then θ is the Rees congruence on F m × F n defined by the ideal J = I i × I k ∪ I j × I l .
The proof of this theorem is once again essentially the proof of Theorem 3.7.

Corollary 7.9
Under the conditions of Theorem 7.8, if i ≤ j and k ≤ l then θ = θ I j ×I l .
The proof of the following Theorem corresponds to the proof of Theorem 3.9. Notice that we can once again give a more explicit description of θ by incorporating the classification of θ 2 given by Theorem 6.1. Switching the roles of the coordinates we get an obvious dual version of Theorem 7.10.

Theorem 7.12
Let θ be the principal congruence on F m × F n generated by ((K , L), (K , L )), where K = λK for λ ∈ F * , and LHL , such that L is not a scalar multiple of L. Let |K | = i, |L| = j,Ĥ the the normal subgroup of F * × GL( j, F) associated with the pair ((K , L), (K , L )), and G the subgroup of F * generated by λ. Then, for (M, N ) Proof Let θ be the relation defined by the statement of the theorem. It is straightforward to check that θ is a congruence.
Conversely, θ contains the pairs from (1) trivially, and the ones from (2) by Lemma 7.5. It remains to show that θ contains the pairs from (3). By multiplying with suitable (E n , s 1 ), (E n , s 2 ) ∈ GL(m, F) × GL(n, F) on the left and right, we may assume that L = E j , L ∈ GL( j, F), and that L is not a scalar multiple of E j . For the following considerations, we will identify the ideal F n E j F n with F j . Now, as L is not a scalar multiple of E j , by Lemma 6.2, there exists a subspace V of F j with dimension j −1 that is not preserved by L . Let A ∈ F j be such that A has rank j − 1 and is the identity on V . Then (K , Considering those elements in F m × F n again, we see that V = im(E j A) = im(L A), and so (E j A, L A) / ∈ H. By applying Theorem 7.10 to the pair ((K , E j A), (K , L A)), we see that the pairs in (3) belong to θ .
In view of Theorems 7.6, 7.8, 7.10, 7.12, and, where applicable, their dual versions, it remains to determine the principal congruence θ when K HK , LHL , K = λK for all λ ∈ F * and L = λL for all λ ∈ F * . Proof Let θ be the relation defined by the statement of the theorem. It is straightforward to check that θ is a congruence.
It remains to show that θ contains the pairs from (3), (4), and (5). Using an analogous argument to the last part of the proof of Theorem 7.12, we can find an A ∈ F n such that (K , L A)θ (K , E j A), where L A and E j A have rank j − 1 but have different images and hence are not H-related. Now the congruence β generated by ((K , L A), (K , E j A)) is contained in θ and an application of Corollary 7.11 to β shows that β contains all pairs from (3) and (5), and then so does θ . Finally, θ contains the pairs in (4) by a symmetric argument.

Problems
In this final section we propose a number of problems motivated by the results above. Clearly, the most natural is the following.

Problem 8.1 Let S and T be any transformation semigroup whose congruences have been described. Find the congruences of S × T .
The description of the congruences of T n provided in [33] is used in [37] to describe the endomorphisms of T n . Regarding the monoid F n , its congruences are known since 1953 [34], but the description of End(F n ) is still to be done. Let G be a permutation group contained in the symmetric group S n and let t ∈ T n ; let G, t denote the submonoid of T n generated by G and t. These monoids proved to be a source of exciting new results involving different parts of Mathematics such as group and semigroup theories, combinatorics, number theory, linear algebra or computational algebra. (For an illustration see [2,[5][6][7][8][9][10][11][28][29][30][31]35,36,39] and the references therein.) In [30], the congruences of the monoids S n , t are described, and so in this context one may ask the following.

Problem 8.3
Let G ≤ S n be a permutation group (in particular a 2-transitive or imprimitive group) and let t, q ∈ T n \S n . Describe the congruences and the endomorphisms of G, t and of G, t × G, q .
The previous problems admit linear analogous.

Problem 8.4
Let V be a finite dimension vector space and let G ≤ Aut(V ); let t, q ∈ End(V )\ Aut(V ). Describe the congruences and the endomorphisms of G, t and of G, t × G, p . For some results on linear monoids of the form G, t we refer the reader to [15].
The next natural step is to ask for analogous results for the endomorphism monoid of an independence algebra, as sets and vector spaces are examples of such algebras. See [12,17,25].

Problem 8.5 Let A and B be finite dimension independence algebras. Describe the congruences on End(A) × End(B).
To tackle this problem one should rely on the classification of these algebras [17,40] as in [4].
Clearly, the same type of questions may be posed for other algebras.
A first step towards the solution of this last problem would be to solve it for an SC-ranked free M-act and for an SC-ranked free module over an ℵ 1 -Noetherian ring [16].
We recall that we were driven to the results in this paper by some considerations on centralizers of idempotents. Let e 2 = e ∈ T n , and denote by C T n (e) the centralizer of the idempotent e in T n , that is, C T n (e) = { f ∈ T n | f e = e f }. The monoid C T n (e) has very interesting features, in particular it generalizes both T n and PT n . See [3,13,14].

Problem 8.7
Describe the congruences of C T n (e), starting with the regular case. See [3].
The solution of this problem requires a complete description of the congruences of PT n × PT m , and that was what prompted us to write this paper.
Finally, we cannot avoid thinking about the partition monoid [1,[18][19][20][21]; this has a very rich structure and has been attracting increasing attention.