James East asked for some pictures of some congruences lattices of some famous semigroups, and I thought I’d post them here.

The pictures in this post were created using the Semigroups package for GAP (the development version). The particular code for finding congruence lattices was written by Michael Torpey and me. Given a finite semigroup S and a set P of pairs of elements of S, the Semigroups package finds a presentation <A|R> for S using a version of the Froidure-Pin algorithm algorithm, and then runs the Todd-Coxeter procedure on the presentation <A|R,P>.

Transformation semigroups

Perhaps the most famous transformation semigroups are the full transformation monoid, the partial transformation monoid, and the symmetric inverse monoid. The congruences lattices of these three monoids are all chains, described many years ago by Libor and Mal’cev.

In a recent paper of Araujo, Bentz, and Gomes, the congruences of direct products of the full transformation monoid \(T_n\), \(n\in\mathbb{N}\), were described. Here are the congruence lattices \(T_2 \times T_2\) and \(T_2 \times T_3\):

The Catalan monoid \(C_n\) consists of those order-preserving and decreasing transformations \(f\) of degree \(n\). The congruence lattice of \(C_4\) looks like this:

The transformations preserving a partition P of \({1, \ldots, n}\) form a monoid; let’s denote it by \(T(P)\). In a recent paper with Joao Araujo, Wolfram Bentz, and Csaba Schneider, we determined the minimum size of a generating set for \(T(P)\). Here are the congruence lattices of \(T(1 + 2)\) and \(T(2 + 2)\):

Semigroups of binary relations (or Boolean matrices)

Perhaps the most famous semigroup of Boolean matrices is the one containing all of the Boolean matrices, called the full Boolean matrix monoid and denote \(\mathbf{B}_n\) where \(n\in\mathbb{N}\) is the dimension of the matrices. The congruences of this semigroup are not known, which is perhaps not surprising since many more elementary things are not known about \(\mathbf{B}_n\) such as the minimum number of matrices required to generate it, the number of its Green’s \(\mathscr{J}\)-classes and so on.

Here is the lattice of congruences of \(\mathbf{B}_3\):

There are some more semigroups of Boolean matrices which we might consider famous because of their connects to famous problems in other areas.

The Hall monoids

The Hall monoid is the monoid consisting of all solutions to Hall’s Marriage Problem:

Hall’s Marriage Problem. Given two groups \(A\) and \(B\) of people:

  • for every person \(a\) in \(A\) there is a subset \(B_a\subseteq B\) such that \(a\) would be happy to marry anyone in \(B_a\); and

  • every person in \(B\) is happy to marry anyone who wants to marry them.

Is it possible to marry everyone so that everyone is happy?

Hall’s Marriage Theorem. [Hall 1935] Everyone is happy if and only if for any subset \(W\) of \(\{B_a : a\in A\}\),

For example, if \(A = {1,2,3}\) and \(B = {1,2,3,4,5}\), and \(B_1 = {1, 2, 3}\), \(B_2 = {1, 4, 5}\), \(B_3 = {3, 5}\), then possible “marriages” are \((1, 1), (2, 4), (3, 5)\) and \((1, 2), (2, 1), (3, 3)\).

But if \(B_1 = {2, 3, 4, 5}\), \(B_2 = {4, 5}\), \(B_3 = {5}, \quad B_4 = {4}\), then there are no possible “marriages” since if \(W = {B_2, B_3, B_4}\), then

Suppose that Then this can be described by a Boolean matrix where row \(i\) is the “characteristic” function of \(B_i\):

$$ \begin{pmatrix} 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 \end{pmatrix} $$

and “marriages’’ become permutation matrices contained this matrix.

We say that such a matrix is a Hall matrix if it contains a permutation matrix, and let \(\mathbf{H}_{n}\) denote the monoid of consisting of all Hall matrices of dimension \(n\).

The number of Hall matrices, or the size of the Hall monoid \(\mathbf{H}_n\) for some small values of \(n\) can be found on the OEIS.

Here is the lattice of congruences of \(\mathbf{H}_3\):

The Gossip monoids

The Gossip monoid is another monoid of Boolean matrices arising from a famous combinatorial problem. The Gossip monoid \(\mathbf{G}_n\) is the monoid of \(n\) dimensional Boolean matrices which is generated by the Boolean matrices corresponding to the equivalence relations.

Here is the lattice of congruences of \(\mathbf{G}_3\):

Some more Boolean matrix monoids

Here is the lattice of congruences of the monoid of all \(2\times 2\) unitriangular Boolean matrices: