What can we actually do with congruence relations, specifically?

Question

Let $T$ denote a Lawvere theory and $X$ denote a $T$-algebra.

Under my preferred definitions:

• A subalgebra of $X$ consists of a $T$-algebra $Y$ together with an injective homomorphism $Y \rightarrow X$.
• A quotient of $X$ consists of a $T$-algebra $Y$ together with a surjective homomorphism $X \rightarrow Y$.

Also:

• A congruence on $X$ is a subalgebra of $X^2$ that happens to be an equivalence relation.

Now it turns out that we can go back and forth between congruence relations and quotient objects. So "congruence relation" is basically a technique that allows us to include the poset of quotients of $X$ into the poset of subalgebras of $X^2$. Okay, but how does this help us? What can we actually do with this inclusion?

Question. What are some specific examples of things that can be done with congruence relations, which would be very hard or even impossible to do without them?

For example, one thing we can do is "pull back" any predicate defined on subalgebras of $X^2$ to obtain a predicate on quotients of $X$. However, a good answer should be more specific than this; I'd like specific examples of where this is done and why it is important.

Answer 1

Congruence relations are useful in universal algebra for many reasons. You mentioned the connection with specific quotients (or "homomorphic images") of the algebra and it's true that the whole lattice of congruence relations of a particular algebra reveals all the ways in which this algebra can be decomposed as a (subdirect) product of smaller (quotient) algebras.

You also suggested that "'congruence relation' is basically a technique that allows us to include the poset of quotients of $X$ into the poset of subalgebras of $X^2$. Okay, but how does this help us?"

It's true there is a one-to-one correspondence between the homomorphic images (or quotients) and the congruence relations, so the poset of quotients can be identified with the congruence lattice, which is a sublattice of $X^2$. How does this help us? One way is to consider the shape of the congruence lattice, which often provides valuable information about the algebra.

There is a deep theory of congruence lattices and what they tell us about the underlying algebras. Probably the best reference for this theory is The Shape of Congruence Lattices, by Kearnes and Kiss.

The real power of congruence relations is in characterizing whole varieties (equational classes) of algebras according to properties of the congruence lattices of algebras in the variety. Much is known about "congruence distributive" (CD) varieties, as well as congruence permutable (CP) and congruence modular (CM) varieties, to name a few classes of varieties that have been extensively studied.

A variety is CD (CM) if every algebra in the variety has a distributive (modular) congruence lattice. A variety is CP if every pair, $\theta, \phi$, of congruences in every algebra in the variety permutes, that is, $\theta \circ \phi = \phi \circ \theta$, where $\circ$ denotes the usual relation composition.

The monograph of Kearnes and Kiss mentioned above is fairly advanced. A good modern treatment of the elementary theory is Cliff Bergman's book. Alternatively, look here for background on basic universal algebra.

To take a simple example, suppose you have an algebra $A$ and you find that all of the congruences of $A$ permute with one another. (Already this can be useful for determining whether certain subdirect products are actually direct products.) You might then want to check whether the whole variety generated by $A$ is congruence permutable. If so, then you know that there is a ternary function $m(x,y,z)$, built using the operations of $A$ and possibly projections, that satisfies the equation $m(x,y,y) = x = m(y,y,x)$ for all $x, y \in A$. Such functions are called "Malcev terms." Having a Malcev term around can be very useful and can make it possible, or at least easier, to prove things about an algebra or variety of algebras.

Examples of proofs that exploit properties of congruence lattices abound in the literature. Besides the Kearnes and Kiss monograph mentioned above, some good examples are in the 2009 paper by Freese and Valeriote. That paper has important practical applications, and some of the algorithms we use to compute with, and determine properties of, finite algebras were not computationally feasible before the results in Freese and Valeriote.