[John wrote](https://forum.azimuthproject.org/discussion/comment/18880/#Comment_18880):

> And finally, if you make \\(\mathbb{Z}/2\\) into a _ring_ with both addition and multiplication as its operations, mathematicians often call it \\(\mathbb{F}_2\\), because it's the [field with two elements](https://en.wikipedia.org/wiki/GF(2)).

Incidentally, fields -- along with rings, vector spaces, modules, etc -- are examples of objects which, like monoidal preorders, have multiple structures interacting in some way. Fields and rings have both a "multiplication" and an "addition", and impose constraints on how they may interact. Modules and vector spaces are even more fun, since they're basically just commutative groups (for vector addition) that are _acted upon_ by a ring or field (for scalar multiplication).

Each piece of these objects is interesting on its own: you can fit all of the objects above together using little more than [groups](https://en.wikipedia.org/wiki/Group_(mathematics)), [group actions](https://en.wikipedia.org/wiki/Group_action), and some coherence conditions between the structures. But you can often get more interesting structures from their composition than by studying the individual pieces.

We've already seen interesting structures arise this way in this very course. Preorders are another fundamental building block, and when we combine monoids and preorders in a certain way, we get monoidal preorders. \\(\mathcal{V}\\)-enriched categories combine a monoidal preorder (or a monoidal category more generally) with a binary function out of a set, satisfying certain coherence properties between these two pieces.

It wouldn't surprise me if there was a categorical way to frame this composition of structures. Like, a ring would be a special kind of object in the product category over rings.

> And finally, if you make \\(\mathbb{Z}/2\\) into a _ring_ with both addition and multiplication as its operations, mathematicians often call it \\(\mathbb{F}_2\\), because it's the [field with two elements](https://en.wikipedia.org/wiki/GF(2)).

Incidentally, fields -- along with rings, vector spaces, modules, etc -- are examples of objects which, like monoidal preorders, have multiple structures interacting in some way. Fields and rings have both a "multiplication" and an "addition", and impose constraints on how they may interact. Modules and vector spaces are even more fun, since they're basically just commutative groups (for vector addition) that are _acted upon_ by a ring or field (for scalar multiplication).

Each piece of these objects is interesting on its own: you can fit all of the objects above together using little more than [groups](https://en.wikipedia.org/wiki/Group_(mathematics)), [group actions](https://en.wikipedia.org/wiki/Group_action), and some coherence conditions between the structures. But you can often get more interesting structures from their composition than by studying the individual pieces.

We've already seen interesting structures arise this way in this very course. Preorders are another fundamental building block, and when we combine monoids and preorders in a certain way, we get monoidal preorders. \\(\mathcal{V}\\)-enriched categories combine a monoidal preorder (or a monoidal category more generally) with a binary function out of a set, satisfying certain coherence properties between these two pieces.

It wouldn't surprise me if there was a categorical way to frame this composition of structures. Like, a ring would be a special kind of object in the product category over rings.