Concrete Applications of Lattices to Algebra

Question

The importance of lattices to algebra (or any field of mathematics really) should be fairly obvious. Specifically, we always have a complete lattice of subobjects (and a lattice of strong subobjects etc.) and a complete lattice of congruences.

However, even though I like to claim this all the time I don't actually good way to illustrate the usefulness of this theory.

What are some concrete application of lattices to algebra?

This could be anything ranging from an alternate (better) proof of a classical theorem, to help in computing the subgroups of a finite group (if this is ever a thing, I'm only speculating here) or anything else, that you can hopefully motivate for a "classical" algebraist.

Answer 1

There are many answers to this question, but one application of lattices is described in this MSE post. Quoting from the answer, "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."

Another concrete application of lattices: determine properties of a group from the shape of its lattice of subgroups. This and many other such applications are described in Roland Schmidt's Subgroup lattices of groups.

Here's a simple example of this. Suppose $G$ is a group such that the lattice on the left in this drawing is isomorphic to the subgroup lattice of $G$, or isomorphic to an interval in the subgroup lattice of $G$. Then $G$ is not solvable.

(The same can be said of the lattice on the right in the drawing.)

It might be a worthwhile exercise to try to characterize lattices that cannot appear as intervals in subgroup lattices of finite solvable groups.

Incidentally, it is not known whether there exists a finite group that has the lattice on the right as an interval in its subgroup lattice. See this MO post. On the other hand, it can be shown that the lattice on the left is an interval in a subgroup lattice of a finite group.

A nice survey of this application of lattices is here.