I have never taken a formal course in Abstract Algebra (yet), but I am interested in learning more about the subject, as I know it is extremely important in Modern Mathematics and a powerful tool beyond (like in, say, Physics).
However, in my limited exposure to the subject, I have found the taxonomy of the various Algebraic Structures very difficult to follow. For example, in basic Group Theory, there are groups, but also semigroups, monoids, semilattices, quasigroups, loops, and Abelian groups. I have no difficulty in understanding the definitions of each, but understanding (or maybe more accurately, remembering) the interrelationships between them is confusing and difficult. I suppose this is largely because many of the names of these structures (like many names in math, unfortunately) are very non-descriptive.
I have seen some authors create a flowchart-like graphic to help illustrate these relationships (e.g. a monoid is a semigroup with identity, or equivalently a group without inverses, etc.), and while helpful, they are always rather limited in scope and always deal with a linear progression of the structure hierarchy and do not include structures that "branch out" so to speak from the main hierarchy (e.g. quasigroups and semilattices which are special cases of magmas and semigroups respectively, but have no direct inclusive relationship to, say, monoids)
So my question is: Are there any robust graphics illustrating the interrelationships between the various algebraic structures?
Personally, I'm mainly interested in such graphics for Groups and Rings (especially Rings), but I leave this question open to answers for more advanced structures too (like modules).
Best Answer
Here is my own attempt at organizing the interrelationships for Groups and Rings. But as I said, I am an amateur in the subject at best, and so this may be incorrect in some places. If so, I would be really grateful to any corrections anyone can give me.
The way I've chosen to organize the structures is as a downwardly flowing web, whose nodes are the various algebraic structures, and whose directional edges denote the axioms that need to be added to the upstream structure in order to produce the downstream structure. This is done in a symmetric way, so that the axiom together with its upstream structure is both necessary and sufficient for producing the next downstream structure.
Groups
Rings
Note: I know there are different methods to define Rings (e.g. whether it includes a multiplicative identity or not). In the following graphic I adopt the convention where a ring does contain $1$, and those that don't are called "rngs".
Also, because some of the ring-like structures (e.g. near-rings) branch out early in the hierarchy before reaching an "officially" named structure like semirings, I had to invent some of my own names for various primitive ringoids to provide the nodes for the branching. Such homemade names I have enclosed in "quotes" to distinguish them from the more standard names.
These graphics were generated using draw.io