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Here's my very basic attempt to visualize some of the key structures: https://github.com/mavam/abstract-algebra-cheatsheet.

Is anyone aware of something similar, but much more comprehensive?

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One slight correction: for Field, the commutative group structure of multiplication should be on S\{0}, not on all of S.

I've almost-exclusively seen "Commutative Groups" referred to as "Abelian Groups". Is there a reason you use "Commutative Groups"? I'll admit my algebra is quite limited and I may be missing a subtlety.

There is no subtlety here. That groups which are commutative are referred to as "abelian" is purely historical. Refering to them as commutative helps understanding for people who are not fammillar with them, and helps highlight the structural simmilarity that the diagram is attempting to show.

Sadly, I find it unlikely that we will get mathamaticians to agree to stop calling them Abelian groups, so learners will have to learn that name eventually.

That's why I always call it partition sort instead of quicksort.

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