Taking that one step further, a monoid homomorphism is a function that transforms one monoid into another while preserving that essential structure (homo-morph: same-shape), so that map-then-reduce is the same as reduce-then-map. Being able to swap the two might be a useful optimization if one is more expensive than the other, for example. e.g. `e^(a+b) = e^a*e^b` turns addition into multiplication. Fourier transforms turn convolution (expensive, complicated) into multiplication (cheap, simple), if you know what that is.

In some other contexts, it's useful to talk about transforming your problem while preserving its essential structure. e.g. in engineering a Fourier transform is a common isomorphism (invertible homomorphism) which lets you transform your problem into an easier one in the frequency domain, solve it, and then pull that solution back into the normal domain. But to understand what's going on with preserving structures, you need to first understand what structures are even present in your problems in the first place, and what it means to preserve them.

This stuff isn't strictly necessary to understand to get real work done, but without it, you get lots of engineers that feel like the techniques they learn in e.g. a differential equations class are essentially random magic tricks with no scaffold for them to organize the ideas.

Another useful purpose of these concepts is to have the vocabulary to ask questions: A semigroup is a monoid without a unit. Given a semigroup, can you somehow add a unit to make a monoid without breaking the existing multiplication? A group is a monoid where the multiplication has inverses/division (So if your unit is called 1, then for any x, there's a "1/x" where x/x = 1). Can you take a monoid and somehow add inverses to make it into a group? etc. In a programming context, these are generic questions about how to make better APIs (e.g. see [0]). It also turns out that groups exactly capture the notion of symmetry, so they're useful for things like geometry, physics, and chemistry. If the symmetries of the laws of physics include shifts, rotations, Lorentz boosts, and adding certain terms to the electromagnetic potential, can I somehow understand those things individually, and then somehow understand the "group of the universe" as being made out of those pieces (plus some others) put together? Can we catalogue all of the possible symmetries of molecules (which can tell you something about the the states they can be in and corresponding energy levels), ideally in terms of some comprehensible set of building blocks? etc.

[0] https://izbicki.me/blog/gausian-distributions-are-monoids